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University please visit us at - - -00:00:11.410 --> 00:00:15.540 align:start position:0% -Stanford University please visit us at -stanford.edu - -00:00:15.540 --> 00:00:15.550 align:start position:0% - - - -00:00:15.550 --> 00:00:17.520 align:start position:0% - -let's<00:00:16.270> come<00:00:16.420> back<00:00:16.450> to<00:00:16.720> general<00:00:16.960> relativity<00:00:17.050> or - -00:00:17.520 --> 00:00:17.530 align:start position:0% -let's come back to general relativity or - - -00:00:17.530 --> 00:00:22.380 align:start position:0% -let's come back to general relativity or -any<00:00:17.740> questions<00:00:20.010> about<00:00:21.010> the<00:00:21.280> contraction<00:00:21.640> yes - -00:00:22.380 --> 00:00:22.390 align:start position:0% -any questions about the contraction yes - - -00:00:22.390 --> 00:00:25.019 align:start position:0% -any questions about the contraction yes -yes<00:00:22.600> yes<00:00:22.780> yes<00:00:22.990> yes<00:00:23.140> yes<00:00:23.380> 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is simply defined as them as the -inverse<00:01:27.820> of<00:01:28.240> the<00:01:28.929> matrix<00:01:29.170> with<00:01:29.920> lower<00:01:30.159> indices - -00:01:30.539 --> 00:01:30.549 align:start position:0% -inverse of the matrix with lower indices - - -00:01:30.549 --> 00:01:33.120 align:start position:0% -inverse of the matrix with lower indices -that<00:01:31.060> means<00:01:31.840> if<00:01:31.960> you<00:01:32.079> multiply<00:01:32.289> them<00:01:32.920> together - -00:01:33.120 --> 00:01:33.130 align:start position:0% -that means if you multiply them together - - -00:01:33.130 --> 00:01:37.410 align:start position:0% -that means if you multiply them together -as<00:01:33.609> matrices<00:01:35.759> what's<00:01:36.759> the<00:01:36.880> result<00:01:37.179> of - -00:01:37.410 --> 00:01:37.420 align:start position:0% -as matrices what's the result of - - -00:01:37.420 --> 00:01:39.539 align:start position:0% -as matrices what's the result of -multiplying<00:01:37.990> 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will<00:01:51.759> bother<00:01:51.939> writing<00:01:52.179> the - -00:01:52.349 --> 00:01:52.359 align:start position:0% -everywhere else will bother writing the - - -00:01:52.359 --> 00:01:55.859 align:start position:0% -everywhere else will bother writing the -zeros<00:01:52.689> in<00:01:53.189> that's<00:01:54.299> statement<00:01:55.299> number<00:01:55.509> 1<00:01:55.630> so<00:01:55.780> we - -00:01:55.859 --> 00:01:55.869 align:start position:0% -zeros in that's statement number 1 so we - - -00:01:55.869 --> 00:01:57.989 align:start position:0% -zeros in that's statement number 1 so we -have<00:01:55.990> these<00:01:56.079> two<00:01:56.259> matrices<00:01:56.819> which<00:01:57.819> are - -00:01:57.989 --> 00:01:57.999 align:start position:0% -have these two matrices which are - - -00:01:57.999 --> 00:01:59.910 align:start position:0% -have these two matrices which are -related<00:01:58.299> by<00:01:58.600> the<00:01:58.659> fact<00:01:58.990> that<00:01:59.289> their<00:01:59.619> product - -00:01:59.910 --> 00:01:59.920 align:start position:0% -related by the fact that their product - - -00:01:59.920 --> 00:02:04.289 align:start position:0% -related by the fact that their product -is<00:02:00.459> the<00:02:01.240> unit<00:02:01.509> matrix<00:02:02.939> let's<00:02:03.939> just<00:02:04.119> talk<00:02:04.270> about - -00:02:04.289 --> 00:02:04.299 align:start position:0% -is the unit matrix let's just talk about - - -00:02:04.299 --> 00:02:06.959 align:start position:0% -is the unit matrix let's just talk about -multiplying<00:02:04.990> matrices<00:02:05.409> for<00:02:05.770> a<00:02:05.799> minute<00:02:06.069> if<00:02:06.369> I - -00:02:06.959 --> 00:02:06.969 align:start position:0% -multiplying matrices for a minute if I - - -00:02:06.969 --> 00:02:10.580 align:start position:0% -multiplying matrices for a minute if I -write<00:02:07.179> matrices<00:02:07.420> a<00:02:08.110> matrix<00:02:08.679> in<00:02:08.890> the<00:02:09.069> form<00:02:09.429> of - -00:02:10.580 --> 00:02:10.590 align:start position:0% -write matrices a matrix in the form of - - -00:02:10.590 --> 00:02:13.199 align:start position:0% -write matrices a matrix in the form of -let's<00:02:11.590> not<00:02:11.680> let's<00:02:12.190> not<00:02:12.250> worry<00:02:12.550> for<00:02:12.790> the<00:02:12.880> moment - -00:02:13.199 --> 00:02:13.209 align:start position:0% -let's not let's not worry for the moment - - -00:02:13.209 --> 00:02:17.040 align:start position:0% -let's not let's not worry for the moment -about<00:02:13.300> upper<00:02:13.720> and<00:02:13.870> lower<00:02:14.020> indices<00:02:15.930> upper<00:02:16.930> and - -00:02:17.040 --> 00:02:17.050 align:start position:0% -about upper and lower indices upper and - - -00:02:17.050 --> 00:02:18.690 align:start position:0% -about upper and lower indices upper and -lower<00:02:17.200> indices<00:02:17.530> are<00:02:17.890> very<00:02:18.069> important<00:02:18.580> when - -00:02:18.690 --> 00:02:18.700 align:start position:0% -lower indices are very important when - - -00:02:18.700 --> 00:02:20.130 align:start position:0% -lower indices are very important when -you're<00:02:18.790> doing<00:02:19.000> differential<00:02:19.480> geometry<00:02:19.660> and - -00:02:20.130 --> 00:02:20.140 align:start position:0% -you're doing differential geometry and - - -00:02:20.140 --> 00:02:22.920 align:start position:0% -you're doing differential geometry and -so<00:02:20.290> forth<00:02:20.590> but<00:02:20.800> just<00:02:20.980> matrices<00:02:21.580> and<00:02:21.930> matrix - -00:02:22.920 --> 00:02:22.930 align:start position:0% -so forth but just matrices and matrix - - -00:02:22.930 --> 00:02:25.210 align:start position:0% -so forth but just matrices and matrix -with<00:02:23.439> entries<00:02:23.680> a<00:02:24.099> B - -00:02:25.210 --> 00:02:25.220 align:start position:0% -with entries a B - - -00:02:25.220 --> 00:02:28.210 align:start position:0% -with entries a B -and<00:02:26.090> I<00:02:26.450> want<00:02:26.630> to<00:02:26.720> multiply<00:02:26.900> it<00:02:27.350> by<00:02:28.070> another - -00:02:28.210 --> 00:02:28.220 align:start position:0% -and I want to multiply it by another - - -00:02:28.220 --> 00:02:32.650 align:start position:0% -and I want to multiply it by another -matrix<00:02:29.290> with<00:02:30.290> whose<00:02:30.830> entries<00:02:31.400> are<00:02:31.430> or<00:02:32.270> whose - -00:02:32.650 --> 00:02:32.660 align:start position:0% -matrix with whose entries are or whose - - -00:02:32.660 --> 00:02:35.140 align:start position:0% -matrix with whose entries are or whose -indices<00:02:33.230> are<00:02:33.590> another<00:02:34.430> name<00:02:34.640> another<00:02:34.670> word - -00:02:35.140 --> 00:02:35.150 align:start position:0% -indices are another name another word - - -00:02:35.150 --> 00:02:43.380 align:start position:0% -indices are another name another word -another<00:02:35.510> letter<00:02:35.780> for<00:02:35.960> a<00:02:35.990> matrix<00:02:37.960> q<00:02:39.340> Q<00:02:40.340> okay<00:02:40.730> Q - -00:02:43.380 --> 00:02:43.390 align:start position:0% -another letter for a matrix q Q okay Q - - -00:02:43.390 --> 00:02:45.850 align:start position:0% -another letter for a matrix q Q okay Q -we're<00:02:44.390> multiplying<00:02:44.660> the<00:02:44.960> matrix<00:02:45.110> M<00:02:45.470> by<00:02:45.830> the - -00:02:45.850 --> 00:02:45.860 align:start position:0% -we're multiplying the matrix M by the - - -00:02:45.860 --> 00:02:48.850 align:start position:0% -we're multiplying the matrix M by the -matrix<00:02:46.280> Q<00:02:47.080> all<00:02:48.080> right<00:02:48.110> so<00:02:48.410> the<00:02:48.500> way<00:02:48.590> that<00:02:48.620> you - -00:02:48.850 --> 00:02:48.860 align:start position:0% -matrix Q all right so the way that you - - -00:02:48.860 --> 00:02:57.940 align:start position:0% -matrix Q all right so the way that you -multiply<00:02:49.370> it<00:02:49.400> is<00:02:50.170> just<00:02:54.490> you<00:02:55.490> put<00:02:56.240> M<00:02:56.480> a<00:02:56.930> B<00:02:57.380> next - -00:02:57.940 --> 00:02:57.950 align:start position:0% -multiply it is just you put M a B next - - -00:02:57.950 --> 00:03:03.040 align:start position:0% -multiply it is just you put M a B next -day<00:02:58.100> q<00:02:58.730> BC<00:02:59.540> and<00:03:00.850> then<00:03:01.850> you<00:03:02.060> multiply<00:03:02.990> the - -00:03:03.040 --> 00:03:03.050 align:start position:0% -day q BC and then you multiply the - - -00:03:03.050 --> 00:03:05.920 align:start position:0% -day q BC and then you multiply the -entropy<00:03:03.560> entries<00:03:04.130> and<00:03:04.370> sum<00:03:05.000> over<00:03:05.270> the - -00:03:05.920 --> 00:03:05.930 align:start position:0% -entropy entries and sum over the - - -00:03:05.930 --> 00:03:09.460 align:start position:0% -entropy entries and sum over the -repeated<00:03:06.440> index<00:03:07.210> that<00:03:08.210> gives<00:03:08.540> you<00:03:08.720> the<00:03:09.020> matrix - -00:03:09.460 --> 00:03:09.470 align:start position:0% -repeated index that gives you the matrix - - -00:03:09.470 --> 00:03:16.570 align:start position:0% -repeated index that gives you the matrix -product<00:03:10.930> mq<00:03:12.610> AC<00:03:14.980> all<00:03:15.980> right<00:03:16.160> so<00:03:16.310> what<00:03:16.460> does - -00:03:16.570 --> 00:03:16.580 align:start position:0% -product mq AC all right so what does - - -00:03:16.580 --> 00:03:22.650 align:start position:0% -product mq AC all right so what does -that<00:03:16.610> mean<00:03:16.970> about<00:03:17.780> G<00:03:18.650> mu<00:03:18.860> nu<00:03:18.890> and<00:03:20.860> G<00:03:21.860> mu<00:03:22.010> nu<00:03:22.040> here - -00:03:22.650 --> 00:03:22.660 align:start position:0% -that mean about G mu nu and G mu nu here - - -00:03:22.660 --> 00:03:25.420 align:start position:0% -that mean about G mu nu and G mu nu here -if<00:03:23.660> we<00:03:23.900> multiply<00:03:24.410> this<00:03:24.470> matrix<00:03:24.860> by<00:03:25.250> this - -00:03:25.420 --> 00:03:25.430 align:start position:0% -if we multiply this matrix by this - - -00:03:25.430 --> 00:03:27.610 align:start position:0% -if we multiply this matrix by this -matrix<00:03:25.970> we<00:03:26.120> should<00:03:26.330> get<00:03:26.420> the<00:03:26.570> unit<00:03:26.840> matrix<00:03:26.990> all - -00:03:27.610 --> 00:03:27.620 align:start position:0% -matrix we should get the unit matrix all - - -00:03:27.620 --> 00:03:29.170 align:start position:0% -matrix we should get the unit matrix all -right<00:03:27.740> so<00:03:27.860> let's<00:03:28.070> let's<00:03:28.340> use<00:03:28.550> the<00:03:28.700> rule<00:03:28.910> for - -00:03:29.170 --> 00:03:29.180 align:start position:0% -right so let's let's use the rule for - - -00:03:29.180 --> 00:03:32.020 align:start position:0% -right so let's let's use the rule for -matrix<00:03:29.510> multiplication<00:03:30.580> first<00:03:31.580> we<00:03:31.820> write - -00:03:32.020 --> 00:03:32.030 align:start position:0% -matrix multiplication first we write - - -00:03:32.030 --> 00:03:35.949 align:start position:0% -matrix multiplication first we write -down<00:03:32.270> the<00:03:32.630> matrix<00:03:33.010> G<00:03:34.270> now<00:03:35.270> let's<00:03:35.480> give<00:03:35.660> it<00:03:35.810> I - -00:03:35.949 --> 00:03:35.959 align:start position:0% -down the matrix G now let's give it I - - -00:03:35.959 --> 00:03:38.110 align:start position:0% -down the matrix G now let's give it I -don't<00:03:36.140> want<00:03:36.260> to<00:03:36.320> use<00:03:36.470> mu<00:03:37.160> nu<00:03:37.370> I'm<00:03:37.550> going<00:03:37.820> to<00:03:37.910> use - -00:03:38.110 --> 00:03:38.120 align:start position:0% -don't want to use mu nu I'm going to use - - -00:03:38.120 --> 00:03:44.190 align:start position:0% -don't want to use mu nu I'm going to use -some<00:03:38.600> arbitrary<00:03:39.500> indices<00:03:40.010> alpha<00:03:40.250> and<00:03:40.640> beta - -00:03:44.190 --> 00:03:44.200 align:start position:0% - - - -00:03:44.200 --> 00:03:47.140 align:start position:0% - -and<00:03:45.200> next<00:03:45.560> to<00:03:45.650> it<00:03:45.680> this<00:03:46.100> is<00:03:46.160> the<00:03:46.370> matrix<00:03:46.580> with - -00:03:47.140 --> 00:03:47.150 align:start position:0% -and next to it this is the matrix with - - -00:03:47.150 --> 00:03:51.699 align:start position:0% -and next to it this is the matrix with -lower<00:03:47.360> indices<00:03:47.750> next<00:03:48.320> to<00:03:48.410> it<00:03:48.560> we<00:03:48.709> put<00:03:50.709> the - -00:03:51.699 --> 00:03:51.709 align:start position:0% -lower indices next to it we put the - - -00:03:51.709 --> 00:03:53.620 align:start position:0% -lower indices next to it we put the -other<00:03:51.950> matrix<00:03:52.430> that<00:03:52.610> were<00:03:52.730> multiplying<00:03:53.300> by<00:03:53.450> it - -00:03:53.620 --> 00:03:53.630 align:start position:0% -other matrix that were multiplying by it - - -00:03:53.630 --> 00:04:03.449 align:start position:0% -other matrix that were multiplying by it -right<00:03:54.110> next<00:03:54.440> to<00:03:54.530> it<00:03:55.420> G<00:03:57.700> B<00:03:59.980> or<00:04:00.980> beta<00:04:01.370> gamma - -00:04:03.449 --> 00:04:03.459 align:start position:0% -right next to it G B or beta gamma - - -00:04:03.459 --> 00:04:06.790 align:start position:0% -right next to it G B or beta gamma -that's<00:04:04.459> the<00:04:04.610> same<00:04:04.880> combination<00:04:05.720> or<00:04:06.470> the<00:04:06.590> same - -00:04:06.790 --> 00:04:06.800 align:start position:0% -that's the same combination or the same - - -00:04:06.800 --> 00:04:11.410 align:start position:0% -that's the same combination or the same -logic<00:04:07.130> that<00:04:07.430> we<00:04:07.580> have<00:04:07.700> here<00:04:07.940> ma<00:04:08.180> b<00:04:08.720> QB<00:04:09.320> c<00:04:10.420> GG - -00:04:11.410 --> 00:04:11.420 align:start position:0% -logic that we have here ma b QB c GG - - -00:04:11.420 --> 00:04:14.410 align:start position:0% -logic that we have here ma b QB c GG -alpha<00:04:11.810> beta<00:04:12.050> G<00:04:12.620> beta<00:04:12.830> gamma<00:04:13.160> and<00:04:13.430> sum<00:04:14.239> over - -00:04:14.410 --> 00:04:14.420 align:start position:0% -alpha beta G beta gamma and sum over - - -00:04:14.420 --> 00:04:16.229 align:start position:0% -alpha beta G beta gamma and sum over -beta - -00:04:16.229 --> 00:04:16.239 align:start position:0% -beta - - -00:04:16.239 --> 00:04:20.320 align:start position:0% -beta -that<00:04:17.239> gives<00:04:17.660> us<00:04:17.840> a<00:04:18.080> matrix<00:04:18.799> with<00:04:19.730> an<00:04:19.850> alpha - -00:04:20.320 --> 00:04:20.330 align:start position:0% -that gives us a matrix with an alpha - - -00:04:20.330 --> 00:04:22.930 align:start position:0% -that gives us a matrix with an alpha -entry<00:04:20.810> or<00:04:20.989> alpha<00:04:21.350> index<00:04:21.799> and<00:04:22.070> a<00:04:22.160> gamma<00:04:22.400> index - -00:04:22.930 --> 00:04:22.940 align:start position:0% -entry or alpha index and a gamma index - - -00:04:22.940 --> 00:04:25.690 align:start position:0% -entry or alpha index and a gamma index -now<00:04:23.870> notice<00:04:24.260> over<00:04:24.470> here<00:04:24.919> we<00:04:25.130> didn't<00:04:25.370> pay<00:04:25.550> any - -00:04:25.690 --> 00:04:25.700 align:start position:0% -now notice over here we didn't pay any - - -00:04:25.700 --> 00:04:27.880 align:start position:0% -now notice over here we didn't pay any -attention<00:04:25.820> to<00:04:26.360> covariant<00:04:27.110> and<00:04:27.200> Convery<00:04:27.710> and - -00:04:27.880 --> 00:04:27.890 align:start position:0% -attention to covariant and Convery and - - -00:04:27.890 --> 00:04:30.510 align:start position:0% -attention to covariant and Convery and -covariant<00:04:28.760> and<00:04:28.850> contravariant<00:04:29.110> components - -00:04:30.510 --> 00:04:30.520 align:start position:0% -covariant and contravariant components - - -00:04:30.520 --> 00:04:33.070 align:start position:0% -covariant and contravariant components -we<00:04:31.520> just<00:04:31.760> thought<00:04:32.120> about<00:04:32.300> ordinary<00:04:32.540> matrix - -00:04:33.070 --> 00:04:33.080 align:start position:0% -we just thought about ordinary matrix - - -00:04:33.080 --> 00:04:37.659 align:start position:0% -we just thought about ordinary matrix -products<00:04:33.820> here<00:04:34.820> we<00:04:35.120> have<00:04:35.360> a<00:04:36.340> matrix<00:04:37.340> with - -00:04:37.659 --> 00:04:37.669 align:start position:0% -products here we have a matrix with - - -00:04:37.669 --> 00:04:38.740 align:start position:0% -products here we have a matrix with -covariant<00:04:38.360> and - -00:04:38.740 --> 00:04:38.750 align:start position:0% -covariant and - - -00:04:38.750 --> 00:04:40.810 align:start position:0% -covariant and -see<00:04:38.930> Zorka<00:04:39.590> tensor<00:04:39.920> with<00:04:40.070> covariant<00:04:40.580> indices - -00:04:40.810 --> 00:04:40.820 align:start position:0% -see Zorka tensor with covariant indices - - -00:04:40.820 --> 00:04:42.940 align:start position:0% -see Zorka tensor with covariant indices -multiplying<00:04:41.810> a<00:04:41.990> tensor<00:04:42.320> with<00:04:42.440> contravariant - -00:04:42.940 --> 00:04:42.950 align:start position:0% -multiplying a tensor with contravariant - - -00:04:42.950 --> 00:04:45.580 align:start position:0% -multiplying a tensor with contravariant -indices<00:04:43.460> and<00:04:43.940> summing<00:04:44.300> over<00:04:44.390> beta<00:04:44.750> but<00:04:45.410> that's - -00:04:45.580 --> 00:04:45.590 align:start position:0% -indices and summing over beta but that's - - -00:04:45.590 --> 00:04:47.940 align:start position:0% -indices and summing over beta but that's -the<00:04:45.800> matrix<00:04:46.010> product<00:04:46.730> of<00:04:46.910> these<00:04:47.060> two<00:04:47.300> matrices - -00:04:47.940 --> 00:04:47.950 align:start position:0% -the matrix product of these two matrices - - -00:04:47.950 --> 00:04:51.310 align:start position:0% -the matrix product of these two matrices -what<00:04:48.950> is<00:04:49.130> the<00:04:49.280> matrix<00:04:49.460> product<00:04:50.120> the<00:04:50.840> matrix - -00:04:51.310 --> 00:04:51.320 align:start position:0% -what is the matrix product the matrix - - -00:04:51.320 --> 00:04:55.810 align:start position:0% -what is the matrix product the matrix -product<00:04:51.830> is<00:04:52.700> the<00:04:53.540> matrix<00:04:54.020> with<00:04:54.950> ones<00:04:55.310> along - -00:04:55.810 --> 00:04:55.820 align:start position:0% -product is the matrix with ones along - - -00:04:55.820 --> 00:04:56.860 align:start position:0% -product is the matrix with ones along -the<00:04:55.880> diagonal - -00:04:56.860 --> 00:04:56.870 align:start position:0% -the diagonal - - -00:04:56.870 --> 00:05:00.880 align:start position:0% -the diagonal -that's<00:04:57.740> the<00:04:58.010> chronica<00:04:58.550> Delta<00:04:59.140> all<00:05:00.140> right<00:05:00.260> now - -00:05:00.880 --> 00:05:00.890 align:start position:0% -that's the chronica Delta all right now - - -00:05:00.890 --> 00:05:05.380 align:start position:0% -that's the chronica Delta all right now -it's<00:05:01.130> from<00:05:02.050> from<00:05:03.460> the<00:05:04.460> notation<00:05:05.180> and - -00:05:05.380 --> 00:05:05.390 align:start position:0% -it's from from the notation and - - -00:05:05.390 --> 00:05:08.590 align:start position:0% -it's from from the notation and -consistency<00:05:06.260> of<00:05:06.560> notation<00:05:07.190> I<00:05:07.460> could<00:05:08.300> write - -00:05:08.590 --> 00:05:08.600 align:start position:0% -consistency of notation I could write - - -00:05:08.600 --> 00:05:15.490 align:start position:0% -consistency of notation I could write -that<00:05:08.930> this<00:05:09.800> is<00:05:10.010> Delta<00:05:10.460> Alpha<00:05:10.940> Gamma<00:05:13.180> yeah<00:05:14.500> yeah - -00:05:15.490 --> 00:05:15.500 align:start position:0% -that this is Delta Alpha Gamma yeah yeah - - -00:05:15.500 --> 00:05:17.620 align:start position:0% -that this is Delta Alpha Gamma yeah yeah -we<00:05:16.250> don't<00:05:16.370> really<00:05:16.790> want<00:05:16.970> to<00:05:17.030> put<00:05:17.180> that<00:05:17.360> gamma - -00:05:17.620 --> 00:05:17.630 align:start position:0% -we don't really want to put that gamma - - -00:05:17.630 --> 00:05:19.390 align:start position:0% -we don't really want to put that gamma -upstairs<00:05:18.230> Downstairs - -00:05:19.390 --> 00:05:19.400 align:start position:0% -upstairs Downstairs - - -00:05:19.400 --> 00:05:22.300 align:start position:0% -upstairs Downstairs -why<00:05:20.330> not<00:05:20.630> well<00:05:21.080> just<00:05:21.410> a<00:05:21.500> consistency<00:05:22.280> of - -00:05:22.300 --> 00:05:22.310 align:start position:0% -why not well just a consistency of - - -00:05:22.310 --> 00:05:24.340 align:start position:0% -why not well just a consistency of -tensor<00:05:22.910> notation<00:05:23.290> covariant<00:05:24.290> and - -00:05:24.340 --> 00:05:24.350 align:start position:0% -tensor notation covariant and - - -00:05:24.350 --> 00:05:28.210 align:start position:0% -tensor notation covariant and -contravariant<00:05:25.810> sort<00:05:26.810> of<00:05:26.900> suggests<00:05:27.440> so<00:05:27.800> pretty - -00:05:28.210 --> 00:05:28.220 align:start position:0% -contravariant sort of suggests so pretty - - -00:05:28.220 --> 00:05:30.720 align:start position:0% -contravariant sort of suggests so pretty -much<00:05:28.400> insist<00:05:28.880> that<00:05:29.270> we<00:05:29.900> put<00:05:30.140> the<00:05:30.260> gamma - -00:05:30.720 --> 00:05:30.730 align:start position:0% -much insist that we put the gamma - - -00:05:30.730 --> 00:05:34.150 align:start position:0% -much insist that we put the gamma -upstairs<00:05:31.780> but<00:05:32.780> it<00:05:32.870> is<00:05:33.080> just<00:05:33.410> the<00:05:33.710> kronecker - -00:05:34.150 --> 00:05:34.160 align:start position:0% -upstairs but it is just the kronecker - - -00:05:34.160 --> 00:05:37.930 align:start position:0% -upstairs but it is just the kronecker -delta<00:05:34.550> 0<00:05:35.540> if<00:05:35.900> alpha<00:05:36.140> does<00:05:36.530> not<00:05:36.680> equal<00:05:36.980> gamma<00:05:37.190> 1 - -00:05:37.930 --> 00:05:37.940 align:start position:0% -delta 0 if alpha does not equal gamma 1 - - -00:05:37.940 --> 00:05:41.950 align:start position:0% -delta 0 if alpha does not equal gamma 1 -if<00:05:38.300> alpha<00:05:38.750> equals<00:05:39.170> gamma<00:05:40.330> so<00:05:41.330> intentional - -00:05:41.950 --> 00:05:41.960 align:start position:0% -if alpha equals gamma so intentional - - -00:05:41.960 --> 00:05:44.830 align:start position:0% -if alpha equals gamma so intentional -notation<00:05:42.530> the<00:05:43.130> Kronecker<00:05:43.700> Delta<00:05:43.730> is<00:05:44.300> always<00:05:44.810> a - -00:05:44.830 --> 00:05:44.840 align:start position:0% -notation the Kronecker Delta is always a - - -00:05:44.840 --> 00:05:46.990 align:start position:0% -notation the Kronecker Delta is always a -thing<00:05:45.200> with<00:05:45.590> an<00:05:45.740> upstairs<00:05:46.070> index<00:05:46.490> and<00:05:46.940> a - -00:05:46.990 --> 00:05:47.000 align:start position:0% -thing with an upstairs index and a - - -00:05:47.000 --> 00:05:49.480 align:start position:0% -thing with an upstairs index and a -downstairs<00:05:47.420> index<00:05:47.900> so<00:05:48.800> from<00:05:48.980> now<00:05:49.130> on<00:05:49.310> whenever - -00:05:49.480 --> 00:05:49.490 align:start position:0% -downstairs index so from now on whenever - - -00:05:49.490 --> 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that's number -one<00:05:56.270> now<00:05:57.050> number<00:05:57.980> two<00:05:58.190> let's<00:05:59.090> talk<00:05:59.480> about - -00:05:59.800 --> 00:05:59.810 align:start position:0% -one now number two let's talk about - - -00:05:59.810 --> 00:06:08.400 align:start position:0% -one now number two let's talk about -raising<00:06:00.500> and<00:06:01.070> lowering<00:06:03.880> the<00:06:04.880> a<00:06:06.520> tensor<00:06:07.520> index - -00:06:08.400 --> 00:06:08.410 align:start position:0% -raising and lowering the a tensor index - - -00:06:08.410 --> 00:06:14.710 align:start position:0% -raising and lowering the a tensor index -supposing<00:06:09.410> I<00:06:09.650> have<00:06:10.070> a<00:06:10.370> tensor<00:06:11.860> with<00:06:13.720> upstairs - -00:06:14.710 --> 00:06:14.720 align:start position:0% -supposing I have a tensor with upstairs - - -00:06:14.720 --> 00:06:19.150 align:start position:0% -supposing I have a tensor with upstairs -indices<00:06:15.380> and<00:06:15.940> I<00:06:16.940> want<00:06:17.210> to<00:06:17.540> lower<00:06:18.140> one<00:06:18.830> of<00:06:18.860> the - -00:06:19.150 --> 00:06:19.160 align:start position:0% -indices and I want to lower one of the - - -00:06:19.160 --> 00:06:21.580 align:start position:0% -indices and I want to lower one of the -indices<00:06:19.670> so<00:06:20.660> let's<00:06:20.930> take<00:06:21.170> some<00:06:21.200> general - -00:06:21.580 --> 00:06:21.590 align:start position:0% -indices so let's take some general - - -00:06:21.590 --> 00:06:27.760 align:start position:0% -indices so let's take some general -tensor<00:06:22.630> T<00:06:23.980> beta<00:06:24.980> gamma<00:06:25.370> and<00:06:25.840> I<00:06:26.840> want<00:06:27.110> to<00:06:27.410> lower - -00:06:27.760 --> 00:06:27.770 align:start position:0% -tensor T beta gamma and I want to lower - - -00:06:27.770 --> 00:06:32.080 align:start position:0% -tensor T beta gamma and I want to lower -the<00:06:28.250> beta<00:06:28.490> index<00:06:29.830> what<00:06:30.830> I<00:06:30.860> do<00:06:30.980> is<00:06:31.310> I<00:06:31.460> multiply - -00:06:32.080 --> 00:06:32.090 align:start position:0% -the beta index what I do is I multiply - - -00:06:32.090 --> 00:06:39.450 align:start position:0% -the beta index what I do is I multiply -it<00:06:32.470> by<00:06:33.820> G<00:06:34.820> alpha<00:06:35.210> beta<00:06:36.110> and<00:06:37.330> sum<00:06:38.330> over<00:06:38.360> beta - -00:06:39.450 --> 00:06:39.460 align:start position:0% -it by G alpha beta and sum over beta - - -00:06:39.460 --> 00:06:43.060 align:start position:0% -it by G alpha beta and sum over beta -that<00:06:40.460> lowers<00:06:40.940> the<00:06:41.120> index<00:06:41.590> and<00:06:42.590> it<00:06:42.740> gives<00:06:42.980> me - -00:06:43.060 --> 00:06:43.070 align:start position:0% -that lowers the index and it gives me - - -00:06:43.070 --> 00:06:48.490 align:start position:0% -that lowers the index and it gives me -the<00:06:43.490> tensor<00:06:43.970> T<00:06:45.310> alpha<00:06:46.310> gamma - -00:06:48.490 --> 00:06:48.500 align:start position:0% -the tensor T alpha gamma - - -00:06:48.500 --> 00:06:50.870 align:start position:0% -the tensor T alpha gamma -okay<00:06:49.500> let's<00:06:49.740> compare<00:06:50.190> that<00:06:50.250> with<00:06:50.669> what's - -00:06:50.870 --> 00:06:50.880 align:start position:0% -okay let's compare that with what's - - -00:06:50.880 --> 00:06:57.100 align:start position:0% -okay let's compare that with what's -going<00:06:51.030> on<00:06:51.300> here<00:06:54.199> here<00:06:55.199> I<00:06:55.229> have<00:06:55.530> the<00:06:55.650> matrix<00:06:55.830> G - -00:06:57.100 --> 00:06:57.110 align:start position:0% -going on here here I have the matrix G - - -00:06:57.110 --> 00:07:02.450 align:start position:0% -going on here here I have the matrix G -with<00:06:58.110> upper<00:06:58.530> indices<00:06:59.690> so<00:07:00.690> we<00:07:01.350> can<00:07:01.650> it's<00:07:02.340> the - -00:07:02.450 --> 00:07:02.460 align:start position:0% -with upper indices so we can it's the - - -00:07:02.460 --> 00:07:05.330 align:start position:0% -with upper indices so we can it's the -replacement<00:07:02.850> if<00:07:03.389> you<00:07:03.419> like<00:07:03.720> for<00:07:04.650> T<00:07:04.800> beta<00:07:05.010> gamma - -00:07:05.330 --> 00:07:05.340 align:start position:0% -replacement if you like for T beta gamma - - -00:07:05.340 --> 00:07:08.390 align:start position:0% -replacement if you like for T beta gamma -in<00:07:05.699> this<00:07:05.940> formula<00:07:06.800> next<00:07:07.800> to<00:07:07.889> it<00:07:08.040> I<00:07:08.070> have<00:07:08.280> the - -00:07:08.390 --> 00:07:08.400 align:start position:0% -in this formula next to it I have the - - -00:07:08.400 --> 00:07:11.360 align:start position:0% -in this formula next to it I have the -metric<00:07:08.820> G<00:07:09.540> alpha<00:07:09.720> beta<00:07:10.110> which<00:07:10.770> is<00:07:10.919> exactly - -00:07:11.360 --> 00:07:11.370 align:start position:0% -metric G alpha beta which is exactly - - -00:07:11.370 --> 00:07:14.210 align:start position:0% -metric G alpha beta which is exactly -what<00:07:11.580> I<00:07:11.610> have<00:07:11.699> over<00:07:11.880> here<00:07:12.919> when<00:07:13.919> I<00:07:13.949> combine - -00:07:14.210 --> 00:07:14.220 align:start position:0% -what I have over here when I combine - - -00:07:14.220 --> 00:07:16.700 align:start position:0% -what I have over here when I combine -these<00:07:14.610> two<00:07:14.850> together<00:07:15.080> what<00:07:16.080> does<00:07:16.260> it<00:07:16.410> give<00:07:16.590> me - -00:07:16.700 --> 00:07:16.710 align:start position:0% -these two together what does it give me - - -00:07:16.710 --> 00:07:19.550 align:start position:0% -these two together what does it give me -according<00:07:17.160> to<00:07:17.190> this<00:07:17.400> formula<00:07:18.260> according<00:07:19.260> to - -00:07:19.550 --> 00:07:19.560 align:start position:0% -according to this formula according to - - -00:07:19.560 --> 00:07:20.510 align:start position:0% -according to this formula according to -this<00:07:19.710> formula - -00:07:20.510 --> 00:07:20.520 align:start position:0% -this formula - - -00:07:20.520 --> 00:07:23.300 align:start position:0% -this formula -it<00:07:21.000> gives<00:07:21.330> me<00:07:21.479> t<00:07:21.840> with<00:07:22.350> one<00:07:22.620> lower<00:07:22.860> and<00:07:23.190> one - -00:07:23.300 --> 00:07:23.310 align:start position:0% -it gives me t with one lower and one - - -00:07:23.310 --> 00:07:27.230 align:start position:0% -it gives me t with one lower and one -upper<00:07:23.669> index<00:07:24.000> but<00:07:24.570> T<00:07:24.750> is<00:07:24.930> just<00:07:24.960> G<00:07:25.729> so<00:07:26.729> it<00:07:27.000> gives - -00:07:27.230 --> 00:07:27.240 align:start position:0% -upper index but T is just G so it gives - - -00:07:27.240 --> 00:07:30.379 align:start position:0% -upper index but T is just G so it gives -me<00:07:27.419> the<00:07:27.750> metric<00:07:28.139> tensor<00:07:28.590> with<00:07:29.370> one<00:07:29.669> out<00:07:29.910> with - -00:07:30.379 --> 00:07:30.389 align:start position:0% -me the metric tensor with one out with - - -00:07:30.389 --> 00:07:36.350 align:start position:0% -me the metric tensor with one out with -one<00:07:30.600> lower<00:07:30.840> index<00:07:30.990> and<00:07:31.680> one<00:07:31.949> upper<00:07:32.520> index<00:07:35.360> so - -00:07:36.350 --> 00:07:36.360 align:start position:0% -one lower index and one upper index so - - -00:07:36.360 --> 00:07:38.150 align:start position:0% -one lower index and one upper index so -the<00:07:36.510> metric<00:07:36.840> tensor<00:07:37.229> with<00:07:37.410> one<00:07:37.650> lower<00:07:37.979> index - -00:07:38.150 --> 00:07:38.160 align:start position:0% -the metric tensor with one lower index - - -00:07:38.160 --> 00:07:44.450 align:start position:0% -the metric tensor with one lower index -and<00:07:38.940> one<00:07:39.270> upper<00:07:39.690> index<00:07:39.720> is<00:07:41.000> always<00:07:42.000> just<00:07:43.460> Delta - -00:07:44.450 --> 00:07:44.460 align:start position:0% -and one upper index is always just Delta - - -00:07:44.460 --> 00:07:48.529 align:start position:0% -and one upper index is always just Delta -alpha<00:07:44.850> gamma<00:07:46.520> it<00:07:47.520> doesn't<00:07:47.970> depend<00:07:48.270> on - -00:07:48.529 --> 00:07:48.539 align:start position:0% -alpha gamma it doesn't depend on - - -00:07:48.539 --> 00:07:49.100 align:start position:0% -alpha gamma it doesn't depend on -position - -00:07:49.100 --> 00:07:49.110 align:start position:0% -position - - -00:07:49.110 --> 00:07:55.190 align:start position:0% -position -it's<00:07:50.010> just<00:07:50.310> Delta<00:07:50.669> alpha<00:07:50.970> gamma<00:07:53.300> it's<00:07:54.300> one<00:07:55.080> of - -00:07:55.190 --> 00:07:55.200 align:start position:0% -it's just Delta alpha gamma it's one of - - -00:07:55.200 --> 00:07:58.219 align:start position:0% -it's just Delta alpha gamma it's one of -the<00:07:55.289> few<00:07:55.500> tensors<00:07:56.599> that<00:07:57.599> will<00:07:57.750> meet<00:07:58.020> in - -00:07:58.219 --> 00:07:58.229 align:start position:0% -the few tensors that will meet in - - -00:07:58.229 --> 00:08:02.390 align:start position:0% -the few tensors that will meet in -geometry<00:07:59.010> which<00:07:59.940> doesn't<00:08:00.419> depend<00:08:00.960> on<00:08:01.370> one<00:08:02.370> of - -00:08:02.390 --> 00:08:02.400 align:start position:0% -geometry which doesn't depend on one of - - -00:08:02.400 --> 00:08:05.390 align:start position:0% -geometry which doesn't depend on one of -the<00:08:02.520> few<00:08:02.729> sets<00:08:03.030> of<00:08:03.090> components<00:08:03.870> which<00:08:04.800> is<00:08:05.099> just - -00:08:05.390 --> 00:08:05.400 align:start position:0% -the few sets of components which is just - - -00:08:05.400 --> 00:08:08.659 align:start position:0% -the few sets of components which is just -a<00:08:05.520> numerical<00:08:05.729> set<00:08:06.539> of<00:08:07.070> components<00:08:08.070> which<00:08:08.310> does - -00:08:08.659 --> 00:08:08.669 align:start position:0% -a numerical set of components which does - - -00:08:08.669 --> 00:08:12.950 align:start position:0% -a numerical set of components which does -not<00:08:08.940> depend<00:08:09.210> on<00:08:09.630> position<00:08:11.060> G<00:08:12.060> over<00:08:12.300> here<00:08:12.599> may - -00:08:12.950 --> 00:08:12.960 align:start position:0% -not depend on position G over here may - - -00:08:12.960 --> 00:08:16.490 align:start position:0% -not depend on position G over here may -depend<00:08:13.320> on<00:08:13.440> position<00:08:14.840> G<00:08:15.840> over<00:08:16.110> here<00:08:16.320> may - -00:08:16.490 --> 00:08:16.500 align:start position:0% -depend on position G over here may - - -00:08:16.500 --> 00:08:18.740 align:start position:0% -depend on position G over here may -depend<00:08:16.860> on<00:08:16.979> position<00:08:17.190> but<00:08:17.789> because<00:08:18.270> what's - -00:08:18.740 --> 00:08:18.750 align:start position:0% -depend on position but because what's - - -00:08:18.750 --> 00:08:21.850 align:start position:0% -depend on position but because what's -over<00:08:18.990> here<00:08:19.260> is<00:08:19.380> always<00:08:19.889> the<00:08:20.099> inverse<00:08:20.580> of<00:08:20.820> this - -00:08:21.850 --> 00:08:21.860 align:start position:0% -over here is always the inverse of this - - -00:08:21.860 --> 00:08:24.680 align:start position:0% -over here is always the inverse of this -Delta<00:08:22.860> Alpha<00:08:23.220> Gamma<00:08:23.400> does<00:08:23.910> not<00:08:24.120> depend<00:08:24.539> on - -00:08:24.680 --> 00:08:24.690 align:start position:0% -Delta Alpha Gamma does not depend on - - -00:08:24.690 --> 00:08:26.750 align:start position:0% -Delta Alpha Gamma does not depend on -position<00:08:24.840> it's<00:08:25.470> always<00:08:25.860> just<00:08:26.099> this<00:08:26.250> matrix<00:08:26.610> so - -00:08:26.750 --> 00:08:26.760 align:start position:0% -position it's always just this matrix so - - -00:08:26.760 --> 00:08:29.150 align:start position:0% -position it's always just this matrix so -that's<00:08:26.940> the<00:08:27.090> answer<00:08:27.360> to<00:08:27.479> the<00:08:27.570> question<00:08:28.160> what - -00:08:29.150 --> 00:08:29.160 align:start position:0% -that's the answer to the question what - - -00:08:29.160 --> 00:08:33.560 align:start position:0% -that's the answer to the question what -is<00:08:29.460> the<00:08:29.789> tensor<00:08:30.210> the<00:08:30.539> the<00:08:31.430> metric<00:08:32.430> tensor<00:08:32.849> with - -00:08:33.560 --> 00:08:33.570 align:start position:0% -is the tensor the the metric tensor with - - -00:08:33.570 --> 00:08:35.990 align:start position:0% -is the tensor the the metric tensor with -one<00:08:33.870> lower<00:08:34.169> index<00:08:34.289> and<00:08:34.919> one<00:08:35.039> upper<00:08:35.370> index<00:08:35.400> and - -00:08:35.990 --> 00:08:36.000 align:start position:0% -one lower index and one upper index and - - -00:08:36.000 --> 00:08:37.459 align:start position:0% -one lower index and one upper index and -the<00:08:36.150> answer<00:08:36.450> is<00:08:36.599> that<00:08:36.810> it's<00:08:36.959> the<00:08:37.050> chronica - -00:08:37.459 --> 00:08:37.469 align:start position:0% -the answer is that it's the chronica - - -00:08:37.469 --> 00:08:47.600 align:start position:0% -the answer is that it's the chronica -delta<00:08:41.330> set<00:08:42.330> again<00:08:43.250> here<00:08:45.260> no<00:08:46.260> from<00:08:47.190> this - -00:08:47.600 --> 00:08:47.610 align:start position:0% -delta set again here no from this - - -00:08:47.610 --> 00:08:49.210 align:start position:0% -delta set again here no from this -combination<00:08:47.940> what<00:08:48.600> is<00:08:48.720> this<00:08:48.839> combination - -00:08:49.210 --> 00:08:49.220 align:start position:0% -combination what is this combination - - -00:08:49.220 --> 00:08:54.470 align:start position:0% -combination what is this combination -alright<00:08:50.420> in<00:08:51.420> this<00:08:51.600> combination<00:08:51.870> here<00:08:53.029> this<00:08:54.029> g - -00:08:54.470 --> 00:08:54.480 align:start position:0% -alright in this combination here this g - - -00:08:54.480 --> 00:08:58.100 align:start position:0% -alright in this combination here this g -is<00:08:54.750> playing<00:08:55.200> the<00:08:55.350> role<00:08:55.560> of<00:08:55.589> this<00:08:55.950> T<00:08:56.250> right<00:08:57.110> okay - -00:08:58.100 --> 00:08:58.110 align:start position:0% -is playing the role of this T right okay - - -00:08:58.110 --> 00:09:00.560 align:start position:0% -is playing the role of this T right okay -so<00:08:58.950> let's<00:08:59.310> let's<00:08:59.760> call<00:09:00.180> it - -00:09:00.560 --> 00:09:00.570 align:start position:0% -so let's let's call it - - -00:09:00.570 --> 00:09:01.560 align:start position:0% -so let's let's call it -T - -00:09:01.560 --> 00:09:01.570 align:start position:0% -T - - -00:09:01.570 --> 00:09:02.819 align:start position:0% -T -I<00:09:01.600> don't<00:09:01.839> want<00:09:01.959> to<00:09:02.019> call<00:09:02.199> it<00:09:02.320> a<00:09:02.410> I'm<00:09:02.470> gonna<00:09:02.620> call - -00:09:02.819 --> 00:09:02.829 align:start position:0% -I don't want to call it a I'm gonna call - - -00:09:02.829 --> 00:09:06.569 align:start position:0% -I don't want to call it a I'm gonna call -it<00:09:02.920> G<00:09:03.360> okay<00:09:04.360> but<00:09:04.959> all<00:09:05.709> I'm<00:09:05.980> doing<00:09:06.009> is - -00:09:06.569 --> 00:09:06.579 align:start position:0% -it G okay but all I'm doing is - - -00:09:06.579 --> 00:09:11.699 align:start position:0% -it G okay but all I'm doing is -substituting<00:09:08.490> wherever<00:09:09.490> I<00:09:09.579> see<00:09:09.940> G<00:09:10.600> here<00:09:11.110> I'm - -00:09:11.699 --> 00:09:11.709 align:start position:0% -substituting wherever I see G here I'm - - -00:09:11.709 --> 00:09:15.050 align:start position:0% -substituting wherever I see G here I'm -sorry<00:09:11.980> wherever<00:09:12.339> I<00:09:12.399> see<00:09:12.670> T<00:09:13.540> over<00:09:13.839> here<00:09:13.870> I'm - -00:09:15.050 --> 00:09:15.060 align:start position:0% -sorry wherever I see T over here I'm - - -00:09:15.060 --> 00:09:19.680 align:start position:0% -sorry wherever I see T over here I'm -substituting<00:09:16.060> G<00:09:16.329> over<00:09:16.600> here<00:09:17.670> okay<00:09:18.670> the<00:09:19.449> rule - -00:09:19.680 --> 00:09:19.690 align:start position:0% -substituting G over here okay the rule - - -00:09:19.690 --> 00:09:22.740 align:start position:0% -substituting G over here okay the rule -is<00:09:19.959> that<00:09:20.319> if<00:09:21.190> you<00:09:21.310> take<00:09:21.550> tea<00:09:21.910> with<00:09:22.149> two<00:09:22.389> upper - -00:09:22.740 --> 00:09:22.750 align:start position:0% -is that if you take tea with two upper - - -00:09:22.750 --> 00:09:26.340 align:start position:0% -is that if you take tea with two upper -index<00:09:23.110> and<00:09:24.040> multiply<00:09:24.670> it<00:09:24.730> by<00:09:25.000> G<00:09:25.029> you<00:09:25.870> get<00:09:26.110> tea - -00:09:26.340 --> 00:09:26.350 align:start position:0% -index and multiply it by G you get tea - - -00:09:26.350 --> 00:09:28.019 align:start position:0% -index and multiply it by G you get tea -with<00:09:26.589> one<00:09:26.620> upper<00:09:27.100> index<00:09:27.130> and<00:09:27.579> one<00:09:27.699> lower<00:09:27.910> index - -00:09:28.019 --> 00:09:28.029 align:start position:0% -with one upper index and one lower index - - -00:09:28.029 --> 00:09:33.410 align:start position:0% -with one upper index and one lower index -well<00:09:28.630> what<00:09:28.779> is<00:09:28.899> T<00:09:29.170> tea<00:09:29.529> is<00:09:29.589> G<00:09:29.889> in<00:09:30.190> this<00:09:30.459> formula - -00:09:33.410 --> 00:09:33.420 align:start position:0% - - - -00:09:33.420 --> 00:09:34.819 align:start position:0% - -[Music] - -00:09:34.819 --> 00:09:34.829 align:start position:0% -[Music] - - -00:09:34.829 --> 00:09:40.079 align:start position:0% -[Music] -Alpha<00:09:35.829> Gamma<00:09:36.779> yeah<00:09:37.779> yeah<00:09:38.500> yeah<00:09:38.949> equals<00:09:39.370> yeah - -00:09:40.079 --> 00:09:40.089 align:start position:0% -Alpha Gamma yeah yeah yeah equals yeah - - -00:09:40.089 --> 00:09:44.610 align:start position:0% -Alpha Gamma yeah yeah yeah equals yeah -oh<00:09:40.769> I'm<00:09:41.769> sorry<00:09:41.949> okay<00:09:42.370> yeah<00:09:42.519> this<00:09:43.319> this<00:09:44.319> yes - -00:09:44.610 --> 00:09:44.620 align:start position:0% -oh I'm sorry okay yeah this this yes - - -00:09:44.620 --> 00:09:47.790 align:start position:0% -oh I'm sorry okay yeah this this yes -it's<00:09:44.860> just<00:09:44.980> this<00:09:45.160> whole<00:09:45.370> left-hand<00:09:46.000> side<00:09:46.800> is - -00:09:47.790 --> 00:09:47.800 align:start position:0% -it's just this whole left-hand side is - - -00:09:47.800 --> 00:09:51.480 align:start position:0% -it's just this whole left-hand side is -equal<00:09:48.009> to<00:09:48.220> this<00:09:49.380> by<00:09:50.380> definition<00:09:50.709> that's<00:09:51.339> a - -00:09:51.480 --> 00:09:51.490 align:start position:0% -equal to this by definition that's a - - -00:09:51.490 --> 00:09:56.340 align:start position:0% -equal to this by definition that's a -that's<00:09:51.730> essentially<00:09:52.329> a<00:09:52.360> definition<00:09:55.350> two - -00:09:56.340 --> 00:09:56.350 align:start position:0% -that's essentially a definition two - - -00:09:56.350 --> 00:09:58.680 align:start position:0% -that's essentially a definition two -definitions<00:09:56.800> going<00:09:57.250> on<00:09:57.430> one<00:09:58.389> is<00:09:58.569> the - -00:09:58.680 --> 00:09:58.690 align:start position:0% -definitions going on one is the - - -00:09:58.690 --> 00:10:00.480 align:start position:0% -definitions going on one is the -definition<00:09:59.139> of<00:09:59.380> how<00:09:59.529> you<00:09:59.589> raise<00:10:00.190> and<00:10:00.459> lower - -00:10:00.480 --> 00:10:00.490 align:start position:0% -definition of how you raise and lower - - -00:10:00.490 --> 00:10:03.329 align:start position:0% -definition of how you raise and lower -indices<00:10:01.029> and<00:10:01.600> the<00:10:02.139> other<00:10:02.319> is<00:10:02.649> the<00:10:02.829> definition - -00:10:03.329 --> 00:10:03.339 align:start position:0% -indices and the other is the definition - - -00:10:03.339 --> 00:10:06.000 align:start position:0% -indices and the other is the definition -of<00:10:03.550> G<00:10:03.850> with<00:10:04.060> upper<00:10:04.420> indices<00:10:04.750> that<00:10:05.350> it's<00:10:05.889> the - -00:10:06.000 --> 00:10:06.010 align:start position:0% -of G with upper indices that it's the - - -00:10:06.010 --> 00:10:07.889 align:start position:0% -of G with upper indices that it's the -inverse<00:10:06.459> of<00:10:06.790> the<00:10:06.940> G<00:10:07.149> with<00:10:07.360> lowering<00:10:07.660> these - -00:10:07.889 --> 00:10:07.899 align:start position:0% -inverse of the G with lowering these - - -00:10:07.899 --> 00:10:14.009 align:start position:0% -inverse of the G with lowering these -indices<00:10:08.620> right<00:10:09.579> so<00:10:11.880> whenever<00:10:12.880> you<00:10:13.209> find - -00:10:14.009 --> 00:10:14.019 align:start position:0% -indices right so whenever you find - - -00:10:14.019 --> 00:10:17.009 align:start position:0% -indices right so whenever you find -yourself<00:10:14.199> wanting<00:10:15.010> to<00:10:15.220> use<00:10:15.699> G<00:10:16.300> with<00:10:16.510> one<00:10:16.690> index - -00:10:17.009 --> 00:10:17.019 align:start position:0% -yourself wanting to use G with one index - - -00:10:17.019 --> 00:10:20.550 align:start position:0% -yourself wanting to use G with one index -upper<00:10:17.500> and<00:10:17.709> one<00:10:17.740> index<00:10:18.100> lower<00:10:18.870> it's<00:10:19.870> the<00:10:20.199> it's - -00:10:20.550 --> 00:10:20.560 align:start position:0% -upper and one index lower it's the it's - - -00:10:20.560 --> 00:10:24.210 align:start position:0% -upper and one index lower it's the it's -the<00:10:20.620> chronica<00:10:21.100> Delta<00:10:21.430> now<00:10:22.209> that<00:10:22.800> that<00:10:23.800> tells - -00:10:24.210 --> 00:10:24.220 align:start position:0% -the chronica Delta now that that tells - - -00:10:24.220 --> 00:10:26.009 align:start position:0% -the chronica Delta now that that tells -us<00:10:24.310> something<00:10:24.910> yeah<00:10:25.029> supposing<00:10:25.600> you<00:10:25.690> want<00:10:25.899> to - -00:10:26.009 --> 00:10:26.019 align:start position:0% -us something yeah supposing you want to - - -00:10:26.019 --> 00:10:28.319 align:start position:0% -us something yeah supposing you want to -contract<00:10:26.440> an<00:10:27.279> upper<00:10:27.610> index<00:10:27.880> and<00:10:28.149> a<00:10:28.180> lower - -00:10:28.319 --> 00:10:28.329 align:start position:0% -contract an upper index and a lower - - -00:10:28.329 --> 00:10:36.319 align:start position:0% -contract an upper index and a lower -index<00:10:30.870> supposing<00:10:31.870> you<00:10:31.930> have<00:10:32.079> some<00:10:32.230> tensor<00:10:34.949> T - -00:10:36.319 --> 00:10:36.329 align:start position:0% -index supposing you have some tensor T - - -00:10:36.329 --> 00:10:39.990 align:start position:0% -index supposing you have some tensor T -alpha<00:10:37.329> beta<00:10:37.569> and<00:10:38.139> you<00:10:38.279> want<00:10:39.279> to<00:10:39.399> form<00:10:39.610> a<00:10:39.699> scalar - -00:10:39.990 --> 00:10:40.000 align:start position:0% -alpha beta and you want to form a scalar - - -00:10:40.000 --> 00:10:43.439 align:start position:0% -alpha beta and you want to form a scalar -out<00:10:40.240> of<00:10:40.360> it<00:10:40.480> to<00:10:41.410> form<00:10:41.680> a<00:10:41.800> scalar<00:10:42.250> out<00:10:42.430> of<00:10:42.579> it<00:10:42.730> you - -00:10:43.439 --> 00:10:43.449 align:start position:0% -out of it to form a scalar out of it you - - -00:10:43.449 --> 00:10:47.220 align:start position:0% -out of it to form a scalar out of it you -can<00:10:43.690> multiply<00:10:44.290> it<00:10:44.880> by<00:10:45.880> the<00:10:46.389> metric<00:10:46.870> or<00:10:46.990> are<00:10:47.079> you - -00:10:47.220 --> 00:10:47.230 align:start position:0% -can multiply it by the metric or are you - - -00:10:47.230 --> 00:10:51.059 align:start position:0% -can multiply it by the metric or are you -first<00:10:47.439> of<00:10:47.589> all<00:10:47.970> you<00:10:48.970> can<00:10:49.180> just<00:10:49.209> identify<00:10:50.069> beta - -00:10:51.059 --> 00:10:51.069 align:start position:0% -first of all you can just identify beta - - -00:10:51.069 --> 00:10:53.100 align:start position:0% -first of all you can just identify beta -with<00:10:51.279> alpha<00:10:51.310> and<00:10:51.730> contract<00:10:52.389> the<00:10:52.689> beta<00:10:52.930> with - -00:10:53.100 --> 00:10:53.110 align:start position:0% -with alpha and contract the beta with - - -00:10:53.110 --> 00:10:55.620 align:start position:0% -with alpha and contract the beta with -alpha<00:10:53.430> or<00:10:54.430> you<00:10:54.639> can<00:10:54.819> just<00:10:54.970> think<00:10:55.149> of<00:10:55.300> it<00:10:55.420> as - -00:10:55.620 --> 00:10:55.630 align:start position:0% -alpha or you can just think of it as - - -00:10:55.630 --> 00:11:00.809 align:start position:0% -alpha or you can just think of it as -multiplying<00:10:56.439> it<00:10:56.769> by<00:10:57.750> G<00:10:58.750> beta<00:10:59.459> Delta<00:11:00.459> beta - -00:11:00.809 --> 00:11:00.819 align:start position:0% -multiplying it by G beta Delta beta - - -00:11:00.819 --> 00:11:04.439 align:start position:0% -multiplying it by G beta Delta beta -alpha<00:11:02.160> which<00:11:03.160> is<00:11:03.189> just<00:11:03.579> G<00:11:03.850> in<00:11:03.910> some<00:11:04.269> other - -00:11:04.439 --> 00:11:04.449 align:start position:0% -alpha which is just G in some other - - -00:11:04.449 --> 00:11:05.970 align:start position:0% -alpha which is just G in some other -notation<00:11:05.050> let's<00:11:05.199> write<00:11:05.380> several<00:11:05.620> different - -00:11:05.970 --> 00:11:05.980 align:start position:0% -notation let's write several different - - -00:11:05.980 --> 00:11:08.999 align:start position:0% -notation let's write several different -formulas<00:11:07.199> certain<00:11:08.199> formulas<00:11:08.680> all<00:11:08.889> right - -00:11:08.999 --> 00:11:09.009 align:start position:0% -formulas certain formulas all right - - -00:11:09.009 --> 00:11:14.069 align:start position:0% -formulas certain formulas all right -first<00:11:09.760> of<00:11:09.939> all<00:11:10.029> there's<00:11:10.660> T<00:11:11.910> alpha<00:11:12.910> beta<00:11:13.149> which - -00:11:14.069 --> 00:11:14.079 align:start position:0% -first of all there's T alpha beta which - - -00:11:14.079 --> 00:11:15.300 align:start position:0% -first of all there's T alpha beta which -if<00:11:14.230> I<00:11:14.319> want<00:11:14.529> to<00:11:14.620> make<00:11:14.769> a<00:11:14.800> scale - -00:11:15.300 --> 00:11:15.310 align:start position:0% -if I want to make a scale - - -00:11:15.310 --> 00:11:18.480 align:start position:0% -if I want to make a scale -out<00:11:15.490> of<00:11:15.640> it<00:11:15.820> I<00:11:16.080> multiply<00:11:17.080> it<00:11:17.110> by<00:11:17.470> G<00:11:17.500> alpha<00:11:18.279> beta - -00:11:18.480 --> 00:11:18.490 align:start position:0% -out of it I multiply it by G alpha beta - - -00:11:18.490 --> 00:11:22.290 align:start position:0% -out of it I multiply it by G alpha beta -and<00:11:18.820> sum<00:11:19.510> of<00:11:19.750> alpha<00:11:20.140> and<00:11:20.260> beta<00:11:20.440> that<00:11:21.180> makes<00:11:22.180> a - -00:11:22.290 --> 00:11:22.300 align:start position:0% -and sum of alpha and beta that makes a - - -00:11:22.300 --> 00:11:26.010 align:start position:0% -and sum of alpha and beta that makes a -scaler<00:11:23.970> alright<00:11:24.970> that's<00:11:25.330> one<00:11:25.540> way<00:11:25.720> to<00:11:25.750> make<00:11:25.960> a - -00:11:26.010 --> 00:11:26.020 align:start position:0% -scaler alright that's one way to make a - - -00:11:26.020 --> 00:11:28.890 align:start position:0% -scaler alright that's one way to make a -scalar<00:11:26.380> another<00:11:27.340> way<00:11:27.550> which<00:11:28.090> is<00:11:28.270> equal<00:11:28.720> and - -00:11:28.890 --> 00:11:28.900 align:start position:0% -scalar another way which is equal and - - -00:11:28.900 --> 00:11:32.190 align:start position:0% -scalar another way which is equal and -you<00:11:28.990> can<00:11:29.170> prove<00:11:29.440> this<00:11:29.680> I<00:11:30.690> leave<00:11:31.690> it<00:11:31.839> to<00:11:31.870> you<00:11:32.050> to - -00:11:32.190 --> 00:11:32.200 align:start position:0% -you can prove this I leave it to you to - - -00:11:32.200 --> 00:11:36.710 align:start position:0% -you can prove this I leave it to you to -prove<00:11:32.440> is<00:11:32.800> to<00:11:33.730> take<00:11:33.940> T<00:11:34.330> with<00:11:34.870> lower<00:11:35.170> components - -00:11:36.710 --> 00:11:36.720 align:start position:0% -prove is to take T with lower components - - -00:11:36.720 --> 00:11:40.829 align:start position:0% -prove is to take T with lower components -and<00:11:37.720> multiply<00:11:38.320> it<00:11:38.400> by<00:11:39.400> G<00:11:39.540> we<00:11:40.540> raised - -00:11:40.829 --> 00:11:40.839 align:start position:0% -and multiply it by G we raised - - -00:11:40.839 --> 00:11:44.220 align:start position:0% -and multiply it by G we raised -components<00:11:41.650> same<00:11:42.010> thing<00:11:42.779> that<00:11:43.779> you<00:11:43.960> need<00:11:44.110> to - -00:11:44.220 --> 00:11:44.230 align:start position:0% -components same thing that you need to - - -00:11:44.230 --> 00:11:46.310 align:start position:0% -components same thing that you need to -work<00:11:44.380> out<00:11:44.560> okay<00:11:45.400> to<00:11:45.700> see<00:11:45.880> that<00:11:46.029> that's<00:11:46.180> true - -00:11:46.310 --> 00:11:46.320 align:start position:0% -work out okay to see that that's true - - -00:11:46.320 --> 00:11:53.970 align:start position:0% -work out okay to see that that's true -but<00:11:47.320> you<00:11:47.650> can<00:11:47.830> also<00:11:49.740> take<00:11:50.740> it<00:11:50.800> to<00:11:51.190> be<00:11:51.930> T<00:11:52.980> with - -00:11:53.970 --> 00:11:53.980 align:start position:0% -but you can also take it to be T with - - -00:11:53.980 --> 00:11:57.140 align:start position:0% -but you can also take it to be T with -one<00:11:54.310> upper<00:11:54.850> index<00:11:55.180> one<00:11:55.930> lower<00:11:56.200> index - -00:11:57.140 --> 00:11:57.150 align:start position:0% -one upper index one lower index - - -00:11:57.150 --> 00:12:00.570 align:start position:0% -one upper index one lower index -multiplied<00:11:58.150> by<00:11:58.480> G<00:11:58.870> with<00:11:59.170> one<00:11:59.410> lower<00:11:59.710> index<00:11:59.860> and - -00:12:00.570 --> 00:12:00.580 align:start position:0% -multiplied by G with one lower index and - - -00:12:00.580 --> 00:12:03.600 align:start position:0% -multiplied by G with one lower index and -one<00:12:00.850> upper<00:12:01.240> index<00:12:01.529> they're<00:12:02.529> all<00:12:02.770> three<00:12:03.220> equal - -00:12:03.600 --> 00:12:03.610 align:start position:0% -one upper index they're all three equal - - -00:12:03.610 --> 00:12:07.380 align:start position:0% -one upper index they're all three equal -to<00:12:03.760> each<00:12:03.880> other<00:12:04.089> and<00:12:04.500> in<00:12:05.500> this<00:12:05.740> form<00:12:06.190> this<00:12:07.150> is - -00:12:07.380 --> 00:12:07.390 align:start position:0% -to each other and in this form this is - - -00:12:07.390 --> 00:12:15.500 align:start position:0% -to each other and in this form this is -just<00:12:07.750> a<00:12:08.970> chronica<00:12:09.970> delta<00:12:11.700> x<00:12:12.700> t<00:12:13.089> alpha<00:12:13.600> beta - -00:12:15.500 --> 00:12:15.510 align:start position:0% -just a chronica delta x t alpha beta - - -00:12:15.510 --> 00:12:19.170 align:start position:0% -just a chronica delta x t alpha beta -now<00:12:16.510> the<00:12:16.570> chronica<00:12:17.140> Delta<00:12:17.560> is<00:12:18.130> either<00:12:18.460> 0<00:12:18.850> or<00:12:18.970> 1 - -00:12:19.170 --> 00:12:19.180 align:start position:0% -now the chronica Delta is either 0 or 1 - - -00:12:19.180 --> 00:12:21.540 align:start position:0% -now the chronica Delta is either 0 or 1 -depending<00:12:19.780> on<00:12:20.410> whether<00:12:20.620> alpha<00:12:21.010> equals<00:12:21.339> beta - -00:12:21.540 --> 00:12:21.550 align:start position:0% -depending on whether alpha equals beta - - -00:12:21.550 --> 00:12:24.060 align:start position:0% -depending on whether alpha equals beta -so<00:12:22.300> it's<00:12:22.480> really<00:12:22.810> just<00:12:22.960> an<00:12:23.140> instruction<00:12:23.860> to - -00:12:24.060 --> 00:12:24.070 align:start position:0% -so it's really just an instruction to - - -00:12:24.070 --> 00:12:29.040 align:start position:0% -so it's really just an instruction to -add<00:12:24.610> T<00:12:25.120> 1<00:12:25.360> 1<00:12:25.600> plus<00:12:26.260> T<00:12:26.470> 2<00:12:26.680> 2<00:12:26.950> plus<00:12:27.610> T<00:12:27.820> 3<00:12:27.850> 3<00:12:28.390> plus<00:12:28.810> T<00:12:29.020> 4 - -00:12:29.040 --> 00:12:29.050 align:start position:0% -add T 1 1 plus T 2 2 plus T 3 3 plus T 4 - - -00:12:29.050 --> 00:12:32.480 align:start position:0% -add T 1 1 plus T 2 2 plus T 3 3 plus T 4 -4<00:12:29.560> and<00:12:29.980> so<00:12:30.700> it's<00:12:30.880> exactly<00:12:31.270> the<00:12:31.450> same<00:12:31.660> as<00:12:31.990> T - -00:12:32.480 --> 00:12:32.490 align:start position:0% -4 and so it's exactly the same as T - - -00:12:32.490 --> 00:12:37.650 align:start position:0% -4 and so it's exactly the same as T -alpha<00:12:33.490> alpha<00:12:34.410> T<00:12:35.410> 1<00:12:35.589> 1<00:12:35.800> plus<00:12:36.160> T<00:12:36.339> 2<00:12:36.490> 2<00:12:36.730> plus<00:12:37.030> T<00:12:37.210> 3<00:12:37.240> 3 - -00:12:37.650 --> 00:12:37.660 align:start position:0% -alpha alpha T 1 1 plus T 2 2 plus T 3 3 - - -00:12:37.660 --> 00:12:43.650 align:start position:0% -alpha alpha T 1 1 plus T 2 2 plus T 3 3 -and<00:12:37.870> so<00:12:37.990> forth<00:12:38.260> I<00:12:39.240> just<00:12:40.240> do<00:12:40.540> this<00:12:40.720> too<00:12:42.660> because - -00:12:43.650 --> 00:12:43.660 align:start position:0% -and so forth I just do this too because - - -00:12:43.660 --> 00:12:45.480 align:start position:0% -and so forth I just do this too because -there<00:12:43.960> was<00:12:44.080> a<00:12:44.110> hole<00:12:44.500> that<00:12:44.710> I<00:12:44.770> left<00:12:45.040> out<00:12:45.250> of<00:12:45.280> the - -00:12:45.480 --> 00:12:45.490 align:start position:0% -there was a hole that I left out of the - - -00:12:45.490 --> 00:12:47.370 align:start position:0% -there was a hole that I left out of the -discussion<00:12:45.670> what<00:12:46.360> the<00:12:46.540> metric<00:12:46.870> tensor<00:12:47.230> is - -00:12:47.370 --> 00:12:47.380 align:start position:0% -discussion what the metric tensor is - - -00:12:47.380 --> 00:12:55.990 align:start position:0% -discussion what the metric tensor is -with<00:12:47.560> one<00:12:47.740> upper<00:12:48.130> and<00:12:48.250> one<00:12:48.280> lower<00:12:48.400> index - -00:12:55.990 --> 00:12:56.000 align:start position:0% - - - -00:12:56.000 --> 00:13:01.840 align:start position:0% - -well<00:12:57.000> you<00:12:57.180> have<00:12:57.420> to<00:12:57.600> keep<00:12:57.779> track<00:12:58.279> in<00:12:59.279> general - -00:13:01.840 --> 00:13:01.850 align:start position:0% - - - -00:13:01.850 --> 00:13:03.889 align:start position:0% - -you<00:13:02.850> should<00:13:03.060> think<00:13:03.240> whenever<00:13:03.510> you<00:13:03.690> write<00:13:03.870> a - -00:13:03.889 --> 00:13:03.899 align:start position:0% -you should think whenever you write a - - -00:13:03.899 --> 00:13:10.240 align:start position:0% -you should think whenever you write a -tensor<00:13:05.480> think<00:13:06.480> of<00:13:06.660> ordering<00:13:08.060> the<00:13:09.060> indices - -00:13:10.240 --> 00:13:10.250 align:start position:0% -tensor think of ordering the indices - - -00:13:10.250 --> 00:13:16.310 align:start position:0% -tensor think of ordering the indices -alpha<00:13:11.250> beta<00:13:12.209> gamma<00:13:13.050> delta<00:13:14.240> sigma<00:13:15.240> now<00:13:16.019> unless - -00:13:16.310 --> 00:13:16.320 align:start position:0% -alpha beta gamma delta sigma now unless - - -00:13:16.320 --> 00:13:18.500 align:start position:0% -alpha beta gamma delta sigma now unless -the<00:13:16.589> tensor<00:13:17.070> happens<00:13:17.730> to<00:13:17.910> have<00:13:18.120> certain - -00:13:18.500 --> 00:13:18.510 align:start position:0% -the tensor happens to have certain - - -00:13:18.510 --> 00:13:19.430 align:start position:0% -the tensor happens to have certain -symmetries - -00:13:19.430 --> 00:13:19.440 align:start position:0% -symmetries - - -00:13:19.440 --> 00:13:22.250 align:start position:0% -symmetries -it<00:13:20.370> may<00:13:20.610> have<00:13:20.850> symmetries<00:13:21.570> which<00:13:21.779> tell<00:13:22.050> you - -00:13:22.250 --> 00:13:22.260 align:start position:0% -it may have symmetries which tell you - - -00:13:22.260 --> 00:13:23.449 align:start position:0% -it may have symmetries which tell you -that<00:13:22.290> it<00:13:22.620> doesn't<00:13:22.769> change<00:13:23.070> if<00:13:23.370> you - -00:13:23.449 --> 00:13:23.459 align:start position:0% -that it doesn't change if you - - -00:13:23.459 --> 00:13:24.740 align:start position:0% -that it doesn't change if you -interchange<00:13:23.940> the<00:13:24.029> indices<00:13:24.450> but<00:13:24.630> let's - -00:13:24.740 --> 00:13:24.750 align:start position:0% -interchange the indices but let's - - -00:13:24.750 --> 00:13:26.930 align:start position:0% -interchange the indices but let's -suppose<00:13:24.959> it<00:13:25.140> doesn't<00:13:25.320> then<00:13:26.310> then<00:13:26.610> you<00:13:26.700> have<00:13:26.820> to - -00:13:26.930 --> 00:13:26.940 align:start position:0% -suppose it doesn't then then you have to - - -00:13:26.940 --> 00:13:29.470 align:start position:0% -suppose it doesn't then then you have to -keep<00:13:27.180> track<00:13:27.510> of<00:13:27.810> the<00:13:27.990> order<00:13:28.260> of<00:13:28.380> these<00:13:28.470> indices - -00:13:29.470 --> 00:13:29.480 align:start position:0% -keep track of the order of these indices - - -00:13:29.480 --> 00:13:32.240 align:start position:0% -keep track of the order of these indices -and<00:13:30.480> if<00:13:30.810> you<00:13:30.899> decide<00:13:31.230> to<00:13:31.260> raise<00:13:31.649> the<00:13:32.010> beta - -00:13:32.240 --> 00:13:32.250 align:start position:0% -and if you decide to raise the beta - - -00:13:32.250 --> 00:13:49.240 align:start position:0% -and if you decide to raise the beta -index<00:13:35.930> put<00:13:36.930> it<00:13:37.019> over<00:13:37.140> here<00:13:38.029> okay<00:13:39.060> right<00:13:40.100> so - -00:13:49.240 --> 00:13:49.250 align:start position:0% - - - -00:13:49.250 --> 00:13:52.190 align:start position:0% - -what<00:13:50.250> I<00:13:50.279> said<00:13:50.519> is<00:13:50.579> if<00:13:51.240> it's<00:13:51.660> a<00:13:51.720> symmetric - -00:13:52.190 --> 00:13:52.200 align:start position:0% -what I said is if it's a symmetric - - -00:13:52.200 --> 00:13:56.389 align:start position:0% -what I said is if it's a symmetric -tensor<00:13:52.560> then<00:13:52.740> it<00:13:52.860> doesn't<00:13:53.130> matter<00:13:55.250> just<00:13:56.250> put - -00:13:56.389 --> 00:13:56.399 align:start position:0% -tensor then it doesn't matter just put - - -00:13:56.399 --> 00:13:59.019 align:start position:0% -tensor then it doesn't matter just put -it<00:13:56.490> directly<00:13:56.610> above<00:13:57.120> yeah<00:13:57.500> in<00:13:58.500> other<00:13:58.680> words - -00:13:59.019 --> 00:13:59.029 align:start position:0% -it directly above yeah in other words - - -00:13:59.029 --> 00:14:03.860 align:start position:0% -it directly above yeah in other words -right<00:14:00.029> may<00:14:00.750> I<00:14:00.810> look<00:14:01.020> at<00:14:01.140> directly<00:14:01.560> above<00:14:02.870> in - -00:14:03.860 --> 00:14:03.870 align:start position:0% -right may I look at directly above in - - -00:14:03.870 --> 00:14:09.980 align:start position:0% -right may I look at directly above in -other<00:14:04.020> words<00:14:04.290> think<00:14:04.800> mentally<00:14:08.329> of<00:14:09.329> putting - -00:14:09.980 --> 00:14:09.990 align:start position:0% -other words think mentally of putting - - -00:14:09.990 --> 00:14:15.800 align:start position:0% -other words think mentally of putting -some<00:14:11.899> partitions<00:14:12.899> in<00:14:13.850> and<00:14:14.850> you<00:14:15.180> can<00:14:15.329> slide<00:14:15.600> the - -00:14:15.800 --> 00:14:15.810 align:start position:0% -some partitions in and you can slide the - - -00:14:15.810 --> 00:14:18.440 align:start position:0% -some partitions in and you can slide the -indices<00:14:16.230> up<00:14:16.410> and<00:14:16.589> down<00:14:17.029> with<00:14:18.029> the<00:14:18.149> metric - -00:14:18.440 --> 00:14:18.450 align:start position:0% -indices up and down with the metric - - -00:14:18.450 --> 00:14:29.500 align:start position:0% -indices up and down with the metric -tensor - -00:14:29.500 --> 00:14:29.510 align:start position:0% - - - -00:14:29.510 --> 00:14:54.270 align:start position:0% - -[Music] - -00:14:54.270 --> 00:14:54.280 align:start position:0% - - - -00:14:54.280 --> 00:14:57.910 align:start position:0% - -[Music] - -00:14:57.910 --> 00:14:57.920 align:start position:0% - - - -00:14:57.920 --> 00:15:00.050 align:start position:0% - -the<00:14:58.920> original<00:14:59.340> question<00:14:59.550> of<00:14:59.790> what's<00:14:59.970> the - -00:15:00.050 --> 00:15:00.060 align:start position:0% -the original question of what's the - - -00:15:00.060 --> 00:15:02.930 align:start position:0% -the original question of what's the -relation<00:15:00.600> between<00:15:00.750> the<00:15:01.820> contravariant<00:15:02.820> and - -00:15:02.930 --> 00:15:02.940 align:start position:0% -relation between the contravariant and - - -00:15:02.940 --> 00:15:06.380 align:start position:0% -relation between the contravariant and -covariant<00:15:03.030> components<00:15:03.630> of<00:15:04.380> vectors<00:15:04.770> and<00:15:05.490> as<00:15:06.270> I - -00:15:06.380 --> 00:15:06.390 align:start position:0% -covariant components of vectors and as I - - -00:15:06.390 --> 00:15:09.140 align:start position:0% -covariant components of vectors and as I -told<00:15:06.600> you<00:15:06.750> they<00:15:07.080> are<00:15:07.160> connected<00:15:08.160> with - -00:15:09.140 --> 00:15:09.150 align:start position:0% -told you they are connected with - - -00:15:09.150 --> 00:15:11.660 align:start position:0% -told you they are connected with -expanding<00:15:09.900> the<00:15:10.200> vector<00:15:10.590> in<00:15:10.920> terms<00:15:11.130> of<00:15:11.310> basis - -00:15:11.660 --> 00:15:11.670 align:start position:0% -expanding the vector in terms of basis - - -00:15:11.670 --> 00:15:14.000 align:start position:0% -expanding the vector in terms of basis -vectors<00:15:12.060> or<00:15:12.350> projecting<00:15:13.350> the<00:15:13.500> vector<00:15:13.830> onto - -00:15:14.000 --> 00:15:14.010 align:start position:0% -vectors or projecting the vector onto - - -00:15:14.010 --> 00:15:17.710 align:start position:0% -vectors or projecting the vector onto -basis<00:15:14.520> vectors<00:15:15.260> the<00:15:16.260> same<00:15:16.500> is<00:15:16.680> true<00:15:16.830> of<00:15:16.920> them - -00:15:17.710 --> 00:15:17.720 align:start position:0% -basis vectors the same is true of them - - -00:15:17.720 --> 00:15:31.100 align:start position:0% -basis vectors the same is true of them -of<00:15:18.720> tensors<00:15:19.790> but<00:15:27.890> if<00:15:28.890> you<00:15:29.040> like<00:15:29.630> it's<00:15:30.630> just<00:15:30.960> a - -00:15:31.100 --> 00:15:31.110 align:start position:0% -of tensors but if you like it's just a - - -00:15:31.110 --> 00:15:36.610 align:start position:0% -of tensors but if you like it's just a -way<00:15:31.470> of<00:15:33.740> forming<00:15:34.740> tensors<00:15:35.280> out<00:15:35.550> of<00:15:35.760> tensors - -00:15:36.610 --> 00:15:36.620 align:start position:0% -way of forming tensors out of tensors - - -00:15:36.620 --> 00:15:39.170 align:start position:0% -way of forming tensors out of tensors -forming<00:15:37.620> tensors<00:15:38.040> with<00:15:38.220> upper<00:15:38.520> indices<00:15:38.820> out - -00:15:39.170 --> 00:15:39.180 align:start position:0% -forming tensors with upper indices out - - -00:15:39.180 --> 00:15:43.640 align:start position:0% -forming tensors with upper indices out -of<00:15:39.300> thought<00:15:39.510> enters<00:15:40.170> with<00:15:40.320> lower<00:15:40.530> indices<00:15:42.650> and - -00:15:43.640 --> 00:15:43.650 align:start position:0% -of thought enters with lower indices and - - -00:15:43.650 --> 00:15:46.100 align:start position:0% -of thought enters with lower indices and -you<00:15:43.770> give<00:15:43.980> them<00:15:44.100> the<00:15:44.220> same<00:15:44.430> name<00:15:44.810> if<00:15:45.810> they're - -00:15:46.100 --> 00:15:46.110 align:start position:0% -you give them the same name if they're - - -00:15:46.110 --> 00:15:52.910 align:start position:0% -you give them the same name if they're -related<00:15:46.680> by<00:15:49.520> by<00:15:50.520> this<00:15:50.580> operation<00:15:51.770> they<00:15:52.770> have - -00:15:52.910 --> 00:15:52.920 align:start position:0% -related by by this operation they have - - -00:15:52.920 --> 00:15:59.240 align:start position:0% -related by by this operation they have -the<00:15:53.010> same<00:15:53.190> information<00:15:53.520> in<00:15:54.000> them<00:15:58.010> geometry<00:15:59.010> is - -00:15:59.240 --> 00:15:59.250 align:start position:0% -the same information in them geometry is - - -00:15:59.250 --> 00:16:01.280 align:start position:0% -the same information in them geometry is -the<00:15:59.490> connection<00:16:00.000> between<00:16:00.050> expanding<00:16:01.050> things - -00:16:01.280 --> 00:16:01.290 align:start position:0% -the connection between expanding things - - -00:16:01.290 --> 00:16:03.170 align:start position:0% -the connection between expanding things -and<00:16:01.470> basis<00:16:01.800> vectors<00:16:02.190> and<00:16:02.460> projecting<00:16:03.030> onto - -00:16:03.170 --> 00:16:03.180 align:start position:0% -and basis vectors and projecting onto - - -00:16:03.180 --> 00:16:06.770 align:start position:0% -and basis vectors and projecting onto -basis<00:16:03.720> vectors<00:16:04.550> so<00:16:05.550> I'll<00:16:05.700> I'm<00:16:06.300> really<00:16:06.660> gonna - -00:16:06.770 --> 00:16:06.780 align:start position:0% -basis vectors so I'll I'm really gonna - - -00:16:06.780 --> 00:16:08.510 align:start position:0% -basis vectors so I'll I'm really gonna -leave<00:16:06.990> that<00:16:07.170> for<00:16:07.350> you<00:16:07.410> to<00:16:07.680> go<00:16:07.770> and<00:16:07.950> look<00:16:08.070> up<00:16:08.220> and - -00:16:08.510 --> 00:16:08.520 align:start position:0% -leave that for you to go and look up and - - -00:16:08.520 --> 00:16:12.320 align:start position:0% -leave that for you to go and look up and -find<00:16:08.880> some<00:16:09.300> good<00:16:10.610> good<00:16:11.610> discussion<00:16:12.180> of - -00:16:12.320 --> 00:16:12.330 align:start position:0% -find some good good discussion of - - -00:16:12.330 --> 00:16:16.310 align:start position:0% -find some good good discussion of -covariant<00:16:12.990> and<00:16:13.110> contravariant<00:16:14.930> indices<00:16:15.930> in - -00:16:16.310 --> 00:16:16.320 align:start position:0% -covariant and contravariant indices in - - -00:16:16.320 --> 00:16:19.460 align:start position:0% -covariant and contravariant indices in -the<00:16:16.350> geometric<00:16:16.680> meaning<00:16:18.170> there's<00:16:19.170> lots<00:16:19.350> of - -00:16:19.460 --> 00:16:19.470 align:start position:0% -the geometric meaning there's lots of - - -00:16:19.470 --> 00:16:22.920 align:start position:0% -the geometric meaning there's lots of -places<00:16:19.620> to<00:16:19.830> look<00:16:19.950> it<00:16:20.190> up<00:16:20.280> yeah - -00:16:22.920 --> 00:16:22.930 align:start position:0% - - - -00:16:22.930 --> 00:16:34.329 align:start position:0% - -G<00:16:23.930> has<00:16:24.139> to<00:16:24.199> be<00:16:24.440> invertible<00:16:27.610> yes<00:16:32.259> right<00:16:33.339> alright - -00:16:34.329 --> 00:16:34.339 align:start position:0% -G has to be invertible yes right alright - - -00:16:34.339 --> 00:16:44.670 align:start position:0% -G has to be invertible yes right alright -if<00:16:35.740> ya<00:16:41.529> in<00:16:42.529> ordinary<00:16:43.009> geometry<00:16:43.670> which<00:16:43.819> means - -00:16:44.670 --> 00:16:44.680 align:start position:0% -if ya in ordinary geometry which means - - -00:16:44.680 --> 00:16:47.110 align:start position:0% -if ya in ordinary geometry which means -geometry<00:16:45.680> of<00:16:45.860> spatial<00:16:46.310> surfaces<00:16:46.819> and<00:16:46.970> so - -00:16:47.110 --> 00:16:47.120 align:start position:0% -geometry of spatial surfaces and so - - -00:16:47.120 --> 00:16:49.179 align:start position:0% -geometry of spatial surfaces and so -forth<00:16:47.389> where<00:16:47.720> time<00:16:48.170> is<00:16:48.199> not<00:16:48.680> different<00:16:49.069> than - -00:16:49.179 --> 00:16:49.189 align:start position:0% -forth where time is not different than - - -00:16:49.189 --> 00:16:52.509 align:start position:0% -forth where time is not different than -space<00:16:49.519> and<00:16:49.819> you<00:16:49.879> don't<00:16:49.910> have<00:16:50.949> not<00:16:51.949> Minkowski - -00:16:52.509 --> 00:16:52.519 align:start position:0% -space and you don't have not Minkowski - - -00:16:52.519 --> 00:16:56.139 align:start position:0% -space and you don't have not Minkowski -geometry<00:16:53.120> then<00:16:53.930> all<00:16:54.379> lengths<00:16:55.250> are<00:16:55.550> always - -00:16:56.139 --> 00:16:56.149 align:start position:0% -geometry then all lengths are always - - -00:16:56.149 --> 00:17:00.850 align:start position:0% -geometry then all lengths are always -positive<00:16:56.860> obviously<00:16:58.420> not<00:16:59.420> so<00:16:59.720> in<00:17:00.139> relativity - -00:17:00.850 --> 00:17:00.860 align:start position:0% -positive obviously not so in relativity - - -00:17:00.860 --> 00:17:04.299 align:start position:0% -positive obviously not so in relativity -where<00:17:01.699> there<00:17:01.879> are<00:17:02.120> space<00:17:02.990> like<00:17:03.259> time<00:17:03.740> like<00:17:04.010> and - -00:17:04.299 --> 00:17:04.309 align:start position:0% -where there are space like time like and - - -00:17:04.309 --> 00:17:06.819 align:start position:0% -where there are space like time like and -light<00:17:04.490> like<00:17:04.730> directions<00:17:05.539> but<00:17:06.470> an<00:17:06.559> ordinary - -00:17:06.819 --> 00:17:06.829 align:start position:0% -light like directions but an ordinary - - -00:17:06.829 --> 00:17:08.919 align:start position:0% -light like directions but an ordinary -geometry<00:17:07.640> all<00:17:07.850> distances<00:17:08.750> are<00:17:08.900> always - -00:17:08.919 --> 00:17:08.929 align:start position:0% -geometry all distances are always - - -00:17:08.929 --> 00:17:13.770 align:start position:0% -geometry all distances are always -positive<00:17:09.350> so<00:17:10.069> that<00:17:10.819> means<00:17:11.059> if<00:17:11.240> you<00:17:11.360> take<00:17:11.569> a<00:17:11.630> DX - -00:17:13.770 --> 00:17:13.780 align:start position:0% -positive so that means if you take a DX - - -00:17:13.780 --> 00:17:18.149 align:start position:0% -positive so that means if you take a DX -let's<00:17:14.980> for<00:17:15.980> space<00:17:16.280> let's<00:17:16.669> just<00:17:16.909> use<00:17:17.150> M<00:17:17.449> and<00:17:17.720> n - -00:17:18.149 --> 00:17:18.159 align:start position:0% -let's for space let's just use M and n - - -00:17:18.159 --> 00:17:20.620 align:start position:0% -let's for space let's just use M and n -instead<00:17:19.159> of<00:17:19.309> MU<00:17:19.459> and<00:17:19.610> nu<00:17:19.669> mu<00:17:20.030> and<00:17:20.240> nu<00:17:20.270> will<00:17:20.510> be - -00:17:20.620 --> 00:17:20.630 align:start position:0% -instead of MU and nu mu and nu will be - - -00:17:20.630 --> 00:17:27.039 align:start position:0% -instead of MU and nu mu and nu will be -reserved<00:17:20.990> for<00:17:21.110> space-time<00:17:21.740> the<00:17:22.640> X<00:17:22.909> M<00:17:23.179> DX<00:17:24.110> n<00:17:26.049> G - -00:17:27.039 --> 00:17:27.049 align:start position:0% -reserved for space-time the X M DX n G - - -00:17:27.049 --> 00:17:37.330 align:start position:0% -reserved for space-time the X M DX n G -MN<00:17:29.470> m<00:17:31.090> n<00:17:32.090> of<00:17:33.039> X<00:17:34.570> this<00:17:35.570> must<00:17:35.929> always<00:17:36.200> be<00:17:36.679> greater - -00:17:37.330 --> 00:17:37.340 align:start position:0% -MN m n of X this must always be greater - - -00:17:37.340 --> 00:17:39.789 align:start position:0% -MN m n of X this must always be greater -than<00:17:37.610> zero<00:17:37.940> strictly<00:17:38.539> greater<00:17:38.720> than<00:17:38.990> zero<00:17:39.409> as - -00:17:39.789 --> 00:17:39.799 align:start position:0% -than zero strictly greater than zero as - - -00:17:39.799 --> 00:17:42.610 align:start position:0% -than zero strictly greater than zero as -long<00:17:40.549> as<00:17:40.730> the<00:17:40.909> X's<00:17:41.299> are<00:17:41.419> not<00:17:41.539> in<00:17:41.659> the<00:17:41.750> X's<00:17:42.409> could - -00:17:42.610 --> 00:17:42.620 align:start position:0% -long as the X's are not in the X's could - - -00:17:42.620 --> 00:17:44.220 align:start position:0% -long as the X's are not in the X's could -be<00:17:42.740> small<00:17:43.070> they<00:17:43.400> don't<00:17:43.700> have<00:17:43.789> to<00:17:43.880> be - -00:17:44.220 --> 00:17:44.230 align:start position:0% -be small they don't have to be - - -00:17:44.230 --> 00:17:47.470 align:start position:0% -be small they don't have to be -infinitesimal<00:17:45.230> just<00:17:45.559> small<00:17:46.330> think<00:17:47.330> of<00:17:47.390> them - -00:17:47.470 --> 00:17:47.480 align:start position:0% -infinitesimal just small think of them - - -00:17:47.480 --> 00:17:49.750 align:start position:0% -infinitesimal just small think of them -as<00:17:47.600> just<00:17:47.780> small<00:17:48.080> little<00:17:48.640> components<00:17:49.640> of - -00:17:49.750 --> 00:17:49.760 align:start position:0% -as just small little components of - - -00:17:49.760 --> 00:17:52.899 align:start position:0% -as just small little components of -vectors<00:17:50.289> every<00:17:51.289> vector<00:17:51.679> has<00:17:52.100> a<00:17:52.130> positive - -00:17:52.899 --> 00:17:52.909 align:start position:0% -vectors every vector has a positive - - -00:17:52.909 --> 00:17:56.260 align:start position:0% -vectors every vector has a positive -squared<00:17:53.390> length<00:17:54.669> square<00:17:55.669> root<00:17:55.789> of<00:17:55.880> any<00:17:56.030> power - -00:17:56.260 --> 00:17:56.270 align:start position:0% -squared length square root of any power - - -00:17:56.270 --> 00:18:00.010 align:start position:0% -squared length square root of any power -the<00:17:56.840> vector<00:17:57.350> is<00:17:57.500> always<00:17:57.799> positive<00:17:58.150> and<00:17:59.150> so - -00:18:00.010 --> 00:18:00.020 align:start position:0% -the vector is always positive and so - - -00:18:00.020 --> 00:18:02.649 align:start position:0% -the vector is always positive and so -this<00:18:00.230> has<00:18:00.409> to<00:18:00.559> be<00:18:00.650> greater<00:18:00.770> than<00:18:00.980> zero<00:18:01.340> now<00:18:02.330> if - -00:18:02.649 --> 00:18:02.659 align:start position:0% -this has to be greater than zero now if - - -00:18:02.659 --> 00:18:08.620 align:start position:0% -this has to be greater than zero now if -you<00:18:03.169> have<00:18:03.530> a<00:18:03.919> matrix<00:18:05.919> which<00:18:07.090> when<00:18:08.090> you - -00:18:08.620 --> 00:18:08.630 align:start position:0% -you have a matrix which when you - - -00:18:08.630 --> 00:18:11.590 align:start position:0% -you have a matrix which when you -multiply<00:18:09.350> it<00:18:09.760> on<00:18:10.760> the<00:18:11.120> left - -00:18:11.590 --> 00:18:11.600 align:start position:0% -multiply it on the left - - -00:18:11.600 --> 00:18:13.930 align:start position:0% -multiply it on the left -tons<00:18:12.230> of<00:18:12.350> vector<00:18:12.679> entire<00:18:13.070> on<00:18:13.250> the<00:18:13.340> right<00:18:13.520> times - -00:18:13.930 --> 00:18:13.940 align:start position:0% -tons of vector entire on the right times - - -00:18:13.940 --> 00:18:16.330 align:start position:0% -tons of vector entire on the right times -the<00:18:14.059> same<00:18:14.270> vector<00:18:14.750> and<00:18:14.960> it<00:18:15.440> must<00:18:15.620> be<00:18:15.770> positive - -00:18:16.330 --> 00:18:16.340 align:start position:0% -the same vector and it must be positive - - -00:18:16.340 --> 00:18:18.190 align:start position:0% -the same vector and it must be positive -then<00:18:17.210> there's<00:18:17.419> something<00:18:17.600> you<00:18:17.840> can<00:18:18.020> say<00:18:18.169> about - -00:18:18.190 --> 00:18:18.200 align:start position:0% -then there's something you can say about - - -00:18:18.200 --> 00:18:20.460 align:start position:0% -then there's something you can say about -that<00:18:18.470> matrix<00:18:18.860> did<00:18:19.250> anybody<00:18:19.549> know<00:18:19.640> what<00:18:19.760> it<00:18:19.789> is - -00:18:20.460 --> 00:18:20.470 align:start position:0% -that matrix did anybody know what it is - - -00:18:20.470 --> 00:18:25.210 align:start position:0% -that matrix did anybody know what it is -it's<00:18:21.470> positive<00:18:22.130> definite<00:18:22.730> and<00:18:23.320> that<00:18:24.320> has<00:18:24.740> and - -00:18:25.210 --> 00:18:25.220 align:start position:0% -it's positive definite and that has and - - -00:18:25.220 --> 00:18:28.730 align:start position:0% -it's positive definite and that has and -that<00:18:25.340> means<00:18:25.640> exactly<00:18:26.090> what - -00:18:28.730 --> 00:18:28.740 align:start position:0% - - - -00:18:28.740 --> 00:18:31.590 align:start position:0% - -positive-definite<00:18:29.740> i<00:18:29.950> ghen<00:18:30.159> values<00:18:30.580> so<00:18:31.480> it - -00:18:31.590 --> 00:18:31.600 align:start position:0% -positive-definite i ghen values so it - - -00:18:31.600 --> 00:18:34.619 align:start position:0% -positive-definite i ghen values so it -means<00:18:31.840> that<00:18:32.049> the<00:18:32.289> eigenvalues<00:18:33.220> of<00:18:33.249> GM<00:18:33.879> in<00:18:34.269> are - -00:18:34.619 --> 00:18:34.629 align:start position:0% -means that the eigenvalues of GM in are - - -00:18:34.629 --> 00:18:37.529 align:start position:0% -means that the eigenvalues of GM in are -positive<00:18:35.379> that's<00:18:35.529> about<00:18:35.830> all<00:18:36.299> that's<00:18:37.299> about - -00:18:37.529 --> 00:18:37.539 align:start position:0% -positive that's about all that's about - - -00:18:37.539 --> 00:18:43.470 align:start position:0% -positive that's about all that's about -the<00:18:37.749> only<00:18:37.990> thing<00:18:38.619> that<00:18:40.470> that<00:18:41.470> is<00:18:41.980> required<00:18:42.669> and - -00:18:43.470 --> 00:18:43.480 align:start position:0% -the only thing that that is required and - - -00:18:43.480 --> 00:18:45.539 align:start position:0% -the only thing that that is required and -of<00:18:43.960> course<00:18:44.200> if<00:18:44.320> all<00:18:44.529> the<00:18:44.710> eigenvalues<00:18:45.039> are - -00:18:45.539 --> 00:18:45.549 align:start position:0% -of course if all the eigenvalues are - - -00:18:45.549 --> 00:18:49.139 align:start position:0% -of course if all the eigenvalues are -positive<00:18:47.070> well<00:18:48.070> then<00:18:48.369> the<00:18:48.460> only<00:18:48.639> eigenvalues - -00:18:49.139 --> 00:18:49.149 align:start position:0% -positive well then the only eigenvalues - - -00:18:49.149 --> 00:18:50.749 align:start position:0% -positive well then the only eigenvalues -are<00:18:49.240> positive<00:18:49.269> yeah - -00:18:50.749 --> 00:18:50.759 align:start position:0% -are positive yeah - - -00:18:50.759 --> 00:18:54.029 align:start position:0% -are positive yeah -any<00:18:51.759> vector<00:18:52.090> will<00:18:52.720> have<00:18:52.929> a<00:18:52.960> positive<00:18:53.320> squared - -00:18:54.029 --> 00:18:54.039 align:start position:0% -any vector will have a positive squared - - -00:18:54.039 --> 00:18:55.529 align:start position:0% -any vector will have a positive squared -length<00:18:54.429> so<00:18:54.759> if<00:18:54.850> you<00:18:54.940> think<00:18:55.059> of<00:18:55.179> this<00:18:55.299> is<00:18:55.450> the - -00:18:55.529 --> 00:18:55.539 align:start position:0% -length so if you think of this is the - - -00:18:55.539 --> 00:18:58.919 align:start position:0% -length so if you think of this is the -squared<00:18:55.840> length<00:18:55.929> now<00:18:56.799> in<00:18:57.070> relativity<00:18:57.929> let's - -00:18:58.919 --> 00:18:58.929 align:start position:0% -squared length now in relativity let's - - -00:18:58.929 --> 00:19:00.539 align:start position:0% -squared length now in relativity let's -go<00:18:59.080> back<00:18:59.289> to<00:18:59.529> special<00:18:59.889> relativity<00:18:59.919> and - -00:19:00.539 --> 00:19:00.549 align:start position:0% -go back to special relativity and - - -00:19:00.549 --> 00:19:05.820 align:start position:0% -go back to special relativity and -special<00:19:01.330> relativity<00:19:02.279> the<00:19:03.279> tau<00:19:03.580> squared<00:19:04.830> is - -00:19:05.820 --> 00:19:05.830 align:start position:0% -special relativity the tau squared is - - -00:19:05.830 --> 00:19:11.389 align:start position:0% -special relativity the tau squared is -equal<00:19:06.279> to<00:19:06.460> DT<00:19:06.879> squared<00:19:08.159> minus<00:19:09.159> the<00:19:09.190> x<00:19:09.490> squared - -00:19:11.389 --> 00:19:11.399 align:start position:0% -equal to DT squared minus the x squared - - -00:19:11.399 --> 00:19:14.399 align:start position:0% -equal to DT squared minus the x squared -minus<00:19:12.399> dy<00:19:12.549> squared<00:19:13.059> minus<00:19:13.389> DZ<00:19:13.720> squared<00:19:14.049> or - -00:19:14.399 --> 00:19:14.409 align:start position:0% -minus dy squared minus DZ squared or - - -00:19:14.409 --> 00:19:16.320 align:start position:0% -minus dy squared minus DZ squared or -whatever<00:19:14.710> and<00:19:15.100> special<00:19:15.610> relativity<00:19:15.999> for - -00:19:16.320 --> 00:19:16.330 align:start position:0% -whatever and special relativity for - - -00:19:16.330 --> 00:19:18.930 align:start position:0% -whatever and special relativity for -example<00:19:17.129> now<00:19:18.129> if<00:19:18.220> I<00:19:18.279> want<00:19:18.490> to<00:19:18.549> put<00:19:18.730> in<00:19:18.850> the - -00:19:18.930 --> 00:19:18.940 align:start position:0% -example now if I want to put in the - - -00:19:18.940 --> 00:19:21.570 align:start position:0% -example now if I want to put in the -speed<00:19:19.179> of<00:19:19.330> light<00:19:19.570> I<00:19:19.980> can<00:19:20.980> put<00:19:21.129> in<00:19:21.220> the<00:19:21.309> speed<00:19:21.549> of - -00:19:21.570 --> 00:19:21.580 align:start position:0% -speed of light I can put in the speed of - - -00:19:21.580 --> 00:19:24.930 align:start position:0% -speed of light I can put in the speed of -light<00:19:21.850> in<00:19:22.090> to<00:19:22.509> one<00:19:22.809> of<00:19:22.899> two<00:19:23.080> ways<00:19:23.289> I<00:19:23.799> can<00:19:24.549> either - -00:19:24.930 --> 00:19:24.940 align:start position:0% -light in to one of two ways I can either - - -00:19:24.940 --> 00:19:27.899 align:start position:0% -light in to one of two ways I can either -multiply<00:19:25.980> this<00:19:26.980> one<00:19:27.220> by<00:19:27.249> the<00:19:27.460> square<00:19:27.789> of<00:19:27.820> the - -00:19:27.899 --> 00:19:27.909 align:start position:0% -multiply this one by the square of the - - -00:19:27.909 --> 00:19:29.399 align:start position:0% -multiply this one by the square of the -speed<00:19:27.999> I<00:19:28.240> need<00:19:28.509> to<00:19:28.749> give<00:19:28.899> these<00:19:29.049> the<00:19:29.200> same - -00:19:29.399 --> 00:19:29.409 align:start position:0% -speed I need to give these the same - - -00:19:29.409 --> 00:19:32.070 align:start position:0% -speed I need to give these the same -units<00:19:29.830> this<00:19:30.759> has<00:19:30.940> units<00:19:31.149> of<00:19:31.389> length<00:19:31.600> squared - -00:19:32.070 --> 00:19:32.080 align:start position:0% -units this has units of length squared - - -00:19:32.080 --> 00:19:33.840 align:start position:0% -units this has units of length squared -this<00:19:32.320> has<00:19:32.470> units<00:19:32.799> of<00:19:32.889> x<00:19:33.159> squared<00:19:33.580> they're<00:19:33.759> not - -00:19:33.840 --> 00:19:33.850 align:start position:0% -this has units of x squared they're not - - -00:19:33.850 --> 00:19:36.480 align:start position:0% -this has units of x squared they're not -compatible<00:19:34.240> to<00:19:34.919> make<00:19:35.919> them<00:19:36.039> have<00:19:36.190> the<00:19:36.279> same - -00:19:36.480 --> 00:19:36.490 align:start position:0% -compatible to make them have the same - - -00:19:36.490 --> 00:19:39.810 align:start position:0% -compatible to make them have the same -units<00:19:36.909> I<00:19:37.119> need<00:19:37.539> some<00:19:38.080> speeds<00:19:38.379> of<00:19:38.559> light<00:19:38.769> I<00:19:39.039> can - -00:19:39.810 --> 00:19:39.820 align:start position:0% -units I need some speeds of light I can - - -00:19:39.820 --> 00:19:41.039 align:start position:0% -units I need some speeds of light I can -either<00:19:39.940> put<00:19:40.149> the<00:19:40.269> speed<00:19:40.480> of<00:19:40.600> light<00:19:40.779> squared - -00:19:41.039 --> 00:19:41.049 align:start position:0% -either put the speed of light squared - - -00:19:41.049 --> 00:19:44.609 align:start position:0% -either put the speed of light squared -here<00:19:41.879> but<00:19:42.879> in<00:19:42.970> that<00:19:43.119> case<00:19:43.419> what<00:19:43.990> is<00:19:44.080> the<00:19:44.110> units - -00:19:44.609 --> 00:19:44.619 align:start position:0% -here but in that case what is the units - - -00:19:44.619 --> 00:19:49.230 align:start position:0% -here but in that case what is the units -of<00:19:44.980> this<00:19:45.490> quantity<00:19:47.759> well<00:19:48.759> let's<00:19:48.909> just<00:19:49.119> not - -00:19:49.230 --> 00:19:49.240 align:start position:0% -of this quantity well let's just not - - -00:19:49.240 --> 00:19:50.489 align:start position:0% -of this quantity well let's just not -worry<00:19:49.269> about<00:19:49.629> for<00:19:49.899> the<00:19:49.929> moment<00:19:50.110> let's<00:19:50.440> just - -00:19:50.489 --> 00:19:50.499 align:start position:0% -worry about for the moment let's just - - -00:19:50.499 --> 00:19:53.310 align:start position:0% -worry about for the moment let's just -define<00:19:51.299> yeah<00:19:52.299> it's<00:19:52.450> just<00:19:52.570> a<00:19:52.749> race<00:19:52.899> this<00:19:53.110> side - -00:19:53.310 --> 00:19:53.320 align:start position:0% -define yeah it's just a race this side - - -00:19:53.320 --> 00:19:56.190 align:start position:0% -define yeah it's just a race this side -for<00:19:53.529> a<00:19:53.559> minute<00:19:53.679> and<00:19:54.840> let's<00:19:55.840> just<00:19:55.960> consider - -00:19:56.190 --> 00:19:56.200 align:start position:0% -for a minute and let's just consider - - -00:19:56.200 --> 00:19:59.450 align:start position:0% -for a minute and let's just consider -this<00:19:56.559> expression<00:19:57.029> what<00:19:58.029> are<00:19:58.119> its<00:19:58.240> units - -00:19:59.450 --> 00:19:59.460 align:start position:0% -this expression what are its units - - -00:19:59.460 --> 00:20:02.460 align:start position:0% -this expression what are its units -length<00:20:00.460> squared<00:20:00.879> all<00:20:01.450> terms<00:20:01.990> in<00:20:02.200> it<00:20:02.320> have - -00:20:02.460 --> 00:20:02.470 align:start position:0% -length squared all terms in it have - - -00:20:02.470 --> 00:20:04.440 align:start position:0% -length squared all terms in it have -units<00:20:02.769> of<00:20:02.830> length<00:20:03.070> squared<00:20:03.460> that's<00:20:04.269> a - -00:20:04.440 --> 00:20:04.450 align:start position:0% -units of length squared that's a - - -00:20:04.450 --> 00:20:07.230 align:start position:0% -units of length squared that's a -perfectly<00:20:04.840> fine<00:20:05.139> definition<00:20:06.059> nothing<00:20:07.059> wrong - -00:20:07.230 --> 00:20:07.240 align:start position:0% -perfectly fine definition nothing wrong - - -00:20:07.240 --> 00:20:07.739 align:start position:0% -perfectly fine definition nothing wrong -with<00:20:07.450> it - -00:20:07.739 --> 00:20:07.749 align:start position:0% -with it - - -00:20:07.749 --> 00:20:11.070 align:start position:0% -with it -but<00:20:08.619> I<00:20:08.860> just<00:20:09.159> prefer<00:20:09.369> it's<00:20:09.879> a<00:20:09.999> it's<00:20:10.659> a<00:20:10.809> pure - -00:20:11.070 --> 00:20:11.080 align:start position:0% -but I just prefer it's a it's a pure - - -00:20:11.080 --> 00:20:13.889 align:start position:0% -but I just prefer it's a it's a pure -personal<00:20:11.710> preference<00:20:12.220> to<00:20:12.909> define<00:20:13.299> the<00:20:13.509> proper - -00:20:13.889 --> 00:20:13.899 align:start position:0% -personal preference to define the proper - - -00:20:13.899 --> 00:20:17.129 align:start position:0% -personal preference to define the proper -time<00:20:14.169> so<00:20:14.710> that<00:20:14.889> it<00:20:14.950> has<00:20:15.070> units<00:20:15.399> of<00:20:15.519> time<00:20:16.139> the - -00:20:17.129 --> 00:20:17.139 align:start position:0% -time so that it has units of time the - - -00:20:17.139 --> 00:20:19.080 align:start position:0% -time so that it has units of time the -given<00:20:17.440> units<00:20:17.769> of<00:20:17.889> time<00:20:18.159> we<00:20:18.490> don't<00:20:18.700> want<00:20:18.879> to<00:20:18.940> put - -00:20:19.080 --> 00:20:19.090 align:start position:0% -given units of time we don't want to put - - -00:20:19.090 --> 00:20:22.259 align:start position:0% -given units of time we don't want to put -a<00:20:19.119> ski<00:20:19.360> C<00:20:19.539> squared<00:20:19.869> here<00:20:20.730> we<00:20:21.730> want<00:20:21.940> to<00:20:21.999> put<00:20:22.179> the - -00:20:22.259 --> 00:20:22.269 align:start position:0% -a ski C squared here we want to put the - - -00:20:22.269 --> 00:20:25.169 align:start position:0% -a ski C squared here we want to put the -C<00:20:22.450> squared<00:20:22.659> in<00:20:22.929> the<00:20:23.019> denominator<00:20:23.169> here<00:20:24.179> it's - -00:20:25.169 --> 00:20:25.179 align:start position:0% -C squared in the denominator here it's - - -00:20:25.179 --> 00:20:27.419 align:start position:0% -C squared in the denominator here it's -just<00:20:25.269> the<00:20:25.509> same<00:20:25.749> quantity<00:20:26.409> except<00:20:26.830> divided<00:20:27.279> by - -00:20:27.419 --> 00:20:27.429 align:start position:0% -just the same quantity except divided by - - -00:20:27.429 --> 00:20:33.259 align:start position:0% -just the same quantity except divided by -C<00:20:27.669> squared<00:20:28.029> and<00:20:28.330> I<00:20:29.049> call<00:20:29.350> that<00:20:29.379> D<00:20:30.039> tau<00:20:30.220> squared - -00:20:33.259 --> 00:20:33.269 align:start position:0% - - - -00:20:33.269 --> 00:20:36.149 align:start position:0% - -now<00:20:34.269> obviously<00:20:34.450> I've<00:20:35.169> put<00:20:35.440> back<00:20:35.649> the<00:20:35.830> speed<00:20:36.039> of - -00:20:36.149 --> 00:20:36.159 align:start position:0% -now obviously I've put back the speed of - - -00:20:36.159 --> 00:20:41.070 align:start position:0% -now obviously I've put back the speed of -light<00:20:36.369> and<00:20:36.700> so<00:20:39.269> later<00:20:40.269> on<00:20:40.509> when<00:20:40.659> we<00:20:40.690> throw<00:20:40.990> away - -00:20:41.070 --> 00:20:41.080 align:start position:0% -light and so later on when we throw away - - -00:20:41.080 --> 00:20:42.509 align:start position:0% -light and so later on when we throw away -the<00:20:41.289> speed<00:20:41.529> of<00:20:41.649> light<00:20:41.830> we'll<00:20:42.039> just<00:20:42.220> call<00:20:42.399> this - -00:20:42.509 --> 00:20:42.519 align:start position:0% -the speed of light we'll just call this - - -00:20:42.519 --> 00:20:50.730 align:start position:0% -the speed of light we'll just call this -one<00:20:42.929> but<00:20:45.389> what's<00:20:46.389> the<00:20:46.480> question<00:20:46.840> anyway<00:20:49.740> yeah - -00:20:50.730 --> 00:20:50.740 align:start position:0% -one but what's the question anyway yeah - - -00:20:50.740 --> 00:20:52.529 align:start position:0% -one but what's the question anyway yeah -yeah<00:20:51.309> yeah<00:20:51.460> okay<00:20:51.759> all<00:20:51.909> right - -00:20:52.529 --> 00:20:52.539 align:start position:0% -yeah yeah okay all right - - -00:20:52.539 --> 00:20:54.419 align:start position:0% -yeah yeah okay all right -right<00:20:53.049> so<00:20:53.230> the<00:20:53.470> speed<00:20:53.740> of<00:20:53.830> light<00:20:53.980> is<00:20:54.129> not<00:20:54.159> the - -00:20:54.419 --> 00:20:54.429 align:start position:0% -right so the speed of light is not the - - -00:20:54.429 --> 00:20:58.100 align:start position:0% -right so the speed of light is not the -issue<00:20:54.610> here<00:20:55.240> but<00:20:55.690> quite<00:20:55.929> obviously<00:20:56.610> there<00:20:57.610> are - -00:20:58.100 --> 00:20:58.110 align:start position:0% -issue here but quite obviously there are - - -00:20:58.110 --> 00:21:00.989 align:start position:0% -issue here but quite obviously there are -directions<00:20:59.110> in<00:20:59.679> which<00:20:59.740> the<00:20:59.980> tau<00:21:00.399> squared<00:21:00.820> is - -00:21:00.989 --> 00:21:00.999 align:start position:0% -directions in which the tau squared is - - -00:21:00.999 --> 00:21:04.799 align:start position:0% -directions in which the tau squared is -positive<00:21:02.009> there<00:21:03.009> are<00:21:03.249> for<00:21:03.610> example<00:21:03.809> the - -00:21:04.799 --> 00:21:04.809 align:start position:0% -positive there are for example the - - -00:21:04.809 --> 00:21:06.600 align:start position:0% -positive there are for example the -directions<00:21:05.470> in<00:21:05.710> which<00:21:05.740> all<00:21:06.159> the<00:21:06.340> space - -00:21:06.600 --> 00:21:06.610 align:start position:0% -directions in which all the space - - -00:21:06.610 --> 00:21:08.580 align:start position:0% -directions in which all the space -components<00:21:07.360> are<00:21:07.480> zero<00:21:07.749> and<00:21:08.110> there's<00:21:08.350> only<00:21:08.470> a - -00:21:08.580 --> 00:21:08.590 align:start position:0% -components are zero and there's only a - - -00:21:08.590 --> 00:21:13.350 align:start position:0% -components are zero and there's only a -time<00:21:08.889> component<00:21:09.190> a<00:21:10.559> time<00:21:11.559> like<00:21:11.769> vector<00:21:12.360> there - -00:21:13.350 --> 00:21:13.360 align:start position:0% -time component a time like vector there - - -00:21:13.360 --> 00:21:16.109 align:start position:0% -time component a time like vector there -are<00:21:13.659> directions<00:21:14.529> in<00:21:14.710> which<00:21:14.860> D<00:21:15.100> tau<00:21:15.340> squared<00:21:15.820> is - -00:21:16.109 --> 00:21:16.119 align:start position:0% -are directions in which D tau squared is - - -00:21:16.119 --> 00:21:21.269 align:start position:0% -are directions in which D tau squared is -negative<00:21:16.330> or<00:21:17.139> pure<00:21:17.470> space<00:21:17.769> like<00:21:18.129> vector<00:21:20.279> so - -00:21:21.269 --> 00:21:21.279 align:start position:0% -negative or pure space like vector so - - -00:21:21.279 --> 00:21:24.330 align:start position:0% -negative or pure space like vector so -the<00:21:21.580> tau<00:21:21.759> squared<00:21:22.240> despite<00:21:22.840> the<00:21:23.259> notation<00:21:23.950> as - -00:21:24.330 --> 00:21:24.340 align:start position:0% -the tau squared despite the notation as - - -00:21:24.340 --> 00:21:27.749 align:start position:0% -the tau squared despite the notation as -the<00:21:25.210> square<00:21:25.509> of<00:21:25.539> something<00:21:25.990> is<00:21:26.590> not<00:21:27.129> really - -00:21:27.749 --> 00:21:27.759 align:start position:0% -the square of something is not really - - -00:21:27.759 --> 00:21:30.659 align:start position:0% -the square of something is not really -positive<00:21:27.970> definite<00:21:28.889> space<00:21:29.889> like<00:21:30.190> vectors - -00:21:30.659 --> 00:21:30.669 align:start position:0% -positive definite space like vectors - - -00:21:30.669 --> 00:21:32.730 align:start position:0% -positive definite space like vectors -have<00:21:30.909> negative<00:21:31.389> D<00:21:31.509> tau<00:21:31.690> squared<00:21:32.110> time<00:21:32.529> like - -00:21:32.730 --> 00:21:32.740 align:start position:0% -have negative D tau squared time like - - -00:21:32.740 --> 00:21:35.850 align:start position:0% -have negative D tau squared time like -vectors<00:21:33.220> have<00:21:33.519> positive<00:21:34.240> tau<00:21:34.419> squared<00:21:34.869> what - -00:21:35.850 --> 00:21:35.860 align:start position:0% -vectors have positive tau squared what - - -00:21:35.860 --> 00:21:37.619 align:start position:0% -vectors have positive tau squared what -about<00:21:36.070> light<00:21:36.309> like<00:21:36.549> vectors<00:21:37.059> light<00:21:37.360> like - -00:21:37.619 --> 00:21:37.629 align:start position:0% -about light like vectors light like - - -00:21:37.629 --> 00:21:41.009 align:start position:0% -about light like vectors light like -vectors<00:21:38.169> are<00:21:38.710> vectors<00:21:39.690> which<00:21:40.690> have<00:21:40.779> the - -00:21:41.009 --> 00:21:41.019 align:start position:0% -vectors are vectors which have the - - -00:21:41.019 --> 00:21:44.999 align:start position:0% -vectors are vectors which have the -property<00:21:41.559> that<00:21:41.740> the<00:21:42.700> tau<00:21:43.230> equals<00:21:44.230> DX<00:21:44.619> for - -00:21:44.999 --> 00:21:45.009 align:start position:0% -property that the tau equals DX for - - -00:21:45.009 --> 00:21:48.149 align:start position:0% -property that the tau equals DX for -example<00:21:45.460> divided<00:21:45.850> by<00:21:45.940> C<00:21:46.210> or<00:21:46.240> equivalently<00:21:47.159> the - -00:21:48.149 --> 00:21:48.159 align:start position:0% -example divided by C or equivalently the - - -00:21:48.159 --> 00:21:51.739 align:start position:0% -example divided by C or equivalently the -x<00:21:48.399> equals<00:21:48.639> C<00:21:48.999> D<00:21:49.179> tau<00:21:49.389> and<00:21:49.990> then<00:21:50.230> sorry<00:21:50.440> C<00:21:50.710> DT - -00:21:51.739 --> 00:21:51.749 align:start position:0% -x equals C D tau and then sorry C DT - - -00:21:51.749 --> 00:21:56.399 align:start position:0% -x equals C D tau and then sorry C DT -those<00:21:52.749> are<00:21:52.990> vectors<00:21:53.409> and<00:21:54.179> then<00:21:55.179> nonzero<00:21:55.840> the - -00:21:56.399 --> 00:21:56.409 align:start position:0% -those are vectors and then nonzero the - - -00:21:56.409 --> 00:21:59.700 align:start position:0% -those are vectors and then nonzero the -components<00:21:57.129> are<00:21:57.249> nonzero<00:21:57.460> the<00:21:58.240> X<00:21:58.389> and<00:21:58.629> DT<00:21:58.960> but - -00:21:59.700 --> 00:21:59.710 align:start position:0% -components are nonzero the X and DT but - - -00:21:59.710 --> 00:22:02.369 align:start position:0% -components are nonzero the X and DT but -they<00:21:59.860> correspond<00:22:00.490> to<00:22:00.820> like<00:22:01.600> like<00:22:01.869> directions - -00:22:02.369 --> 00:22:02.379 align:start position:0% -they correspond to like like directions - - -00:22:02.379 --> 00:22:04.889 align:start position:0% -they correspond to like like directions -and<00:22:02.919> for<00:22:03.610> them<00:22:03.789> the<00:22:03.970> proper<00:22:04.330> time<00:22:04.539> is<00:22:04.779> just - -00:22:04.889 --> 00:22:04.899 align:start position:0% -and for them the proper time is just - - -00:22:04.899 --> 00:22:07.649 align:start position:0% -and for them the proper time is just -plain<00:22:05.169> zero<00:22:05.999> so<00:22:06.999> there<00:22:07.240> are<00:22:07.330> clearly - -00:22:07.649 --> 00:22:07.659 align:start position:0% -plain zero so there are clearly - - -00:22:07.659 --> 00:22:11.220 align:start position:0% -plain zero so there are clearly -directions<00:22:08.470> in<00:22:08.980> which<00:22:09.299> this<00:22:10.299> is<00:22:10.570> not<00:22:10.600> greater - -00:22:11.220 --> 00:22:11.230 align:start position:0% -directions in which this is not greater - - -00:22:11.230 --> 00:22:13.080 align:start position:0% -directions in which this is not greater -than<00:22:11.379> zero<00:22:11.769> these<00:22:12.249> directions<00:22:12.759> are<00:22:12.820> which<00:22:12.970> is - -00:22:13.080 --> 00:22:13.090 align:start position:0% -than zero these directions are which is - - -00:22:13.090 --> 00:22:14.999 align:start position:0% -than zero these directions are which is -zero<00:22:13.389> and<00:22:13.629> directions<00:22:14.259> which<00:22:14.470> is<00:22:14.590> negative - -00:22:14.999 --> 00:22:15.009 align:start position:0% -zero and directions which is negative - - -00:22:15.009 --> 00:22:16.549 align:start position:0% -zero and directions which is negative -and<00:22:15.159> directions<00:22:15.580> in<00:22:15.669> which<00:22:15.700> is<00:22:15.970> positive - -00:22:16.549 --> 00:22:16.559 align:start position:0% -and directions in which is positive - - -00:22:16.559 --> 00:22:23.639 align:start position:0% -and directions in which is positive -right<00:22:19.169> looking<00:22:20.169> at<00:22:20.289> this<00:22:20.440> matron - -00:22:23.639 --> 00:22:23.649 align:start position:0% - - - -00:22:23.649 --> 00:22:27.419 align:start position:0% - -the<00:22:24.129> metric<00:22:24.519> tensor<00:22:25.139> the<00:22:26.139> metric<00:22:26.529> tensor<00:22:26.919> as<00:22:27.369> a - -00:22:27.419 --> 00:22:27.429 align:start position:0% -the metric tensor the metric tensor as a - - -00:22:27.429 --> 00:22:31.080 align:start position:0% -the metric tensor the metric tensor as a -matrix<00:22:27.899> would<00:22:28.899> have<00:22:29.080> a<00:22:29.110> one<00:22:29.559> and<00:22:29.889> a<00:22:30.460> minus<00:22:30.669> one - -00:22:31.080 --> 00:22:31.090 align:start position:0% -matrix would have a one and a minus one - - -00:22:31.090 --> 00:22:33.840 align:start position:0% -matrix would have a one and a minus one -over<00:22:31.330> C<00:22:31.509> squared<00:22:32.190> let's<00:22:33.190> get<00:22:33.399> rid<00:22:33.519> of<00:22:33.580> the<00:22:33.669> C - -00:22:33.840 --> 00:22:33.850 align:start position:0% -over C squared let's get rid of the C - - -00:22:33.850 --> 00:22:35.639 align:start position:0% -over C squared let's get rid of the C -squared<00:22:34.029> now<00:22:34.330> now<00:22:34.509> that<00:22:34.690> I've<00:22:34.809> put<00:22:35.379> the<00:22:35.499> C - -00:22:35.639 --> 00:22:35.649 align:start position:0% -squared now now that I've put the C - - -00:22:35.649 --> 00:22:37.859 align:start position:0% -squared now now that I've put the C -squared<00:22:35.950> and<00:22:36.129> let's<00:22:36.249> take<00:22:36.399> it<00:22:36.519> back<00:22:36.610> out<00:22:36.869> minus - -00:22:37.859 --> 00:22:37.869 align:start position:0% -squared and let's take it back out minus - - -00:22:37.869 --> 00:22:40.590 align:start position:0% -squared and let's take it back out minus -one<00:22:38.080> minus<00:22:38.440> one<00:22:38.649> minus - -00:22:40.590 --> 00:22:40.600 align:start position:0% -one minus one minus - - -00:22:40.600 --> 00:22:43.840 align:start position:0% -one minus one minus -okay<00:22:41.600> how<00:22:42.350> many<00:22:42.409> positive<00:22:42.769> eigenvalues<00:22:43.639> does - -00:22:43.840 --> 00:22:43.850 align:start position:0% -okay how many positive eigenvalues does - - -00:22:43.850 --> 00:22:46.659 align:start position:0% -okay how many positive eigenvalues does -it<00:22:43.970> have<00:22:44.620> one<00:22:45.620> how<00:22:46.279> many<00:22:46.460> negative - -00:22:46.659 --> 00:22:46.669 align:start position:0% -it have one how many negative - - -00:22:46.669 --> 00:22:50.980 align:start position:0% -it have one how many negative -eigenvalues<00:22:47.860> three<00:22:49.360> the<00:22:50.360> three<00:22:50.600> negative - -00:22:50.980 --> 00:22:50.990 align:start position:0% -eigenvalues three the three negative - - -00:22:50.990 --> 00:22:53.049 align:start position:0% -eigenvalues three the three negative -eigenvalues<00:22:51.649> correspond<00:22:52.340> to<00:22:52.610> the<00:22:52.759> idea<00:22:52.999> that - -00:22:53.049 --> 00:22:53.059 align:start position:0% -eigenvalues correspond to the idea that - - -00:22:53.059 --> 00:22:55.389 align:start position:0% -eigenvalues correspond to the idea that -there<00:22:53.269> are<00:22:53.360> three<00:22:53.570> space<00:22:53.840> directions<00:22:54.470> and<00:22:55.249> the - -00:22:55.389 --> 00:22:55.399 align:start position:0% -there are three space directions and the - - -00:22:55.399 --> 00:22:57.610 align:start position:0% -there are three space directions and the -one<00:22:55.580> positive<00:22:56.059> eigenvalue<00:22:56.899> corresponds<00:22:57.559> to - -00:22:57.610 --> 00:22:57.620 align:start position:0% -one positive eigenvalue corresponds to - - -00:22:57.620 --> 00:22:59.680 align:start position:0% -one positive eigenvalue corresponds to -the<00:22:57.769> idea<00:22:58.039> that<00:22:58.249> there's<00:22:58.549> one<00:22:58.730> time<00:22:59.509> like - -00:22:59.680 --> 00:22:59.690 align:start position:0% -the idea that there's one time like - - -00:22:59.690 --> 00:23:02.560 align:start position:0% -the idea that there's one time like -direction<00:23:00.259> the<00:23:01.129> metric<00:23:01.580> and<00:23:01.820> relativity - -00:23:02.560 --> 00:23:02.570 align:start position:0% -direction the metric and relativity - - -00:23:02.570 --> 00:23:06.009 align:start position:0% -direction the metric and relativity -general<00:23:02.809> and<00:23:03.740> special<00:23:04.450> must<00:23:05.450> have<00:23:05.659> a<00:23:05.690> feature - -00:23:06.009 --> 00:23:06.019 align:start position:0% -general and special must have a feature - - -00:23:06.019 --> 00:23:09.009 align:start position:0% -general and special must have a feature -that<00:23:06.350> there<00:23:06.679> are<00:23:06.830> three<00:23:07.309> times<00:23:08.149> three<00:23:08.659> space - -00:23:09.009 --> 00:23:09.019 align:start position:0% -that there are three times three space - - -00:23:09.019 --> 00:23:11.590 align:start position:0% -that there are three times three space -like<00:23:09.289> directions<00:23:09.950> which<00:23:10.820> means<00:23:11.090> three - -00:23:11.590 --> 00:23:11.600 align:start position:0% -like directions which means three - - -00:23:11.600 --> 00:23:14.009 align:start position:0% -like directions which means three -negative<00:23:12.230> eigenvalues<00:23:12.919> and<00:23:13.220> one<00:23:13.399> positive - -00:23:14.009 --> 00:23:14.019 align:start position:0% -negative eigenvalues and one positive - - -00:23:14.019 --> 00:23:17.409 align:start position:0% -negative eigenvalues and one positive -now<00:23:15.019> now<00:23:15.919> I<00:23:15.950> should<00:23:16.309> tell<00:23:16.340> you<00:23:16.519> that<00:23:17.179> if<00:23:17.330> you - -00:23:17.409 --> 00:23:17.419 align:start position:0% -now now I should tell you that if you - - -00:23:17.419 --> 00:23:21.210 align:start position:0% -now now I should tell you that if you -compare<00:23:17.929> with<00:23:18.320> much<00:23:18.710> in<00:23:19.039> the<00:23:19.190> literature - -00:23:21.210 --> 00:23:21.220 align:start position:0% -compare with much in the literature - - -00:23:21.220 --> 00:23:25.080 align:start position:0% -compare with much in the literature -depending<00:23:22.220> on<00:23:22.460> how<00:23:22.730> old<00:23:22.789> the<00:23:23.269> literature<00:23:23.809> is - -00:23:25.080 --> 00:23:25.090 align:start position:0% -depending on how old the literature is - - -00:23:25.090 --> 00:23:27.070 align:start position:0% -depending on how old the literature is -depending<00:23:26.090> on<00:23:26.179> the<00:23:26.299> tastes<00:23:26.690> of<00:23:26.749> the<00:23:26.899> author - -00:23:27.070 --> 00:23:27.080 align:start position:0% -depending on the tastes of the author - - -00:23:27.080 --> 00:23:29.830 align:start position:0% -depending on the tastes of the author -and<00:23:27.649> depending<00:23:28.610> on<00:23:28.700> how<00:23:28.909> quirky<00:23:29.269> the<00:23:29.629> author - -00:23:29.830 --> 00:23:29.840 align:start position:0% -and depending on how quirky the author - - -00:23:29.840 --> 00:23:34.889 align:start position:0% -and depending on how quirky the author -is<00:23:30.669> one<00:23:31.669> sometimes<00:23:32.330> defines<00:23:32.809> instead<00:23:33.320> of<00:23:33.440> the - -00:23:34.889 --> 00:23:34.899 align:start position:0% -is one sometimes defines instead of the - - -00:23:34.899 --> 00:23:39.779 align:start position:0% -is one sometimes defines instead of the -the<00:23:35.899> proper<00:23:36.799> time<00:23:37.100> one<00:23:37.970> defines<00:23:38.419> a<00:23:38.749> proper - -00:23:39.779 --> 00:23:39.789 align:start position:0% -the proper time one defines a proper - - -00:23:39.789 --> 00:23:42.700 align:start position:0% -the proper time one defines a proper -spatial<00:23:40.789> interval<00:23:41.360> and<00:23:41.960> one<00:23:42.080> defines<00:23:42.379> it<00:23:42.559> this - -00:23:42.700 --> 00:23:42.710 align:start position:0% -spatial interval and one defines it this - - -00:23:42.710 --> 00:23:47.399 align:start position:0% -spatial interval and one defines it this -way<00:23:43.269> the<00:23:44.269> s<00:23:44.480> squared<00:23:45.019> s<00:23:45.769> for<00:23:46.249> space<00:23:46.549> and - -00:23:47.399 --> 00:23:47.409 align:start position:0% -way the s squared s for space and - - -00:23:47.409 --> 00:23:51.129 align:start position:0% -way the s squared s for space and -defines<00:23:48.409> it<00:23:48.649> as<00:23:48.830> the<00:23:49.730> x<00:23:49.940> squared<00:23:50.419> plus<00:23:50.929> dy - -00:23:51.129 --> 00:23:51.139 align:start position:0% -defines it as the x squared plus dy - - -00:23:51.139 --> 00:23:56.320 align:start position:0% -defines it as the x squared plus dy -squared<00:23:51.320> plus<00:23:51.980> DZ<00:23:52.429> squared<00:23:54.820> minus<00:23:55.820> C<00:23:56.299> squared - -00:23:56.320 --> 00:23:56.330 align:start position:0% -squared plus DZ squared minus C squared - - -00:23:56.330 --> 00:24:01.269 align:start position:0% -squared plus DZ squared minus C squared -DT<00:23:57.110> squared<00:23:59.019> that's<00:24:00.019> clearly<00:24:00.529> the<00:24:00.679> same<00:24:00.889> thing - -00:24:01.269 --> 00:24:01.279 align:start position:0% -DT squared that's clearly the same thing - - -00:24:01.279 --> 00:24:04.149 align:start position:0% -DT squared that's clearly the same thing -except<00:24:01.820> for<00:24:01.879> a<00:24:01.999> factor<00:24:02.450> of<00:24:02.480> C<00:24:02.869> C<00:24:03.409> squared<00:24:03.769> and<00:24:04.070> a - -00:24:04.149 --> 00:24:04.159 align:start position:0% -except for a factor of C C squared and a - - -00:24:04.159 --> 00:24:06.930 align:start position:0% -except for a factor of C C squared and a -negative<00:24:04.549> sign<00:24:05.049> they<00:24:06.049> are<00:24:06.230> the<00:24:06.409> same<00:24:06.619> thing - -00:24:06.930 --> 00:24:06.940 align:start position:0% -negative sign they are the same thing - - -00:24:06.940 --> 00:24:12.669 align:start position:0% -negative sign they are the same thing -and<00:24:08.440> so<00:24:10.690> so<00:24:11.690> far<00:24:11.720> there's<00:24:12.139> no<00:24:12.259> difference - -00:24:12.669 --> 00:24:12.679 align:start position:0% -and so so far there's no difference - - -00:24:12.679 --> 00:24:14.019 align:start position:0% -and so so far there's no difference -between<00:24:13.039> what<00:24:13.220> I<00:24:13.249> would<00:24:13.460> write<00:24:13.639> down<00:24:13.669> with - -00:24:14.019 --> 00:24:14.029 align:start position:0% -between what I would write down with - - -00:24:14.029 --> 00:24:16.330 align:start position:0% -between what I would write down with -everybody<00:24:14.480> else<00:24:14.600> to<00:24:14.720> write<00:24:14.869> down<00:24:15.129> the<00:24:16.129> tau - -00:24:16.330 --> 00:24:16.340 align:start position:0% -everybody else to write down the tau - - -00:24:16.340 --> 00:24:18.129 align:start position:0% -everybody else to write down the tau -squared<00:24:16.730> is<00:24:16.879> a<00:24:16.909> pape<00:24:17.149> thing<00:24:17.419> the<00:24:17.629> S<00:24:17.720> squared<00:24:17.869> is - -00:24:18.129 --> 00:24:18.139 align:start position:0% -squared is a pape thing the S squared is - - -00:24:18.139 --> 00:24:20.859 align:start position:0% -squared is a pape thing the S squared is -a<00:24:18.169> bottom<00:24:18.499> thing<00:24:18.740> but<00:24:19.429> sometimes<00:24:19.909> the<00:24:20.299> metric - -00:24:20.859 --> 00:24:20.869 align:start position:0% -a bottom thing but sometimes the metric - - -00:24:20.869 --> 00:24:24.700 align:start position:0% -a bottom thing but sometimes the metric -is<00:24:21.279> read<00:24:22.279> off<00:24:22.549> this<00:24:22.999> expression<00:24:23.720> in<00:24:24.440> which - -00:24:24.700 --> 00:24:24.710 align:start position:0% -is read off this expression in which - - -00:24:24.710 --> 00:24:30.149 align:start position:0% -is read off this expression in which -case<00:24:25.039> the<00:24:25.639> metric<00:24:26.059> would<00:24:26.240> be<00:24:26.419> 1<00:24:26.809> 1<00:24:27.499> 1<00:24:28.129> minus<00:24:28.629> 1 - -00:24:30.149 --> 00:24:30.159 align:start position:0% -case the metric would be 1 1 1 minus 1 - - -00:24:30.159 --> 00:24:32.919 align:start position:0% -case the metric would be 1 1 1 minus 1 -now<00:24:31.159> it's<00:24:31.610> not<00:24:31.820> that<00:24:31.879> the<00:24:32.090> met<00:24:32.450> is<00:24:32.600> not<00:24:32.779> that - -00:24:32.919 --> 00:24:32.929 align:start position:0% -now it's not that the met is not that - - -00:24:32.929 --> 00:24:34.450 align:start position:0% -now it's not that the met is not that -the<00:24:33.019> physics<00:24:33.409> is<00:24:33.559> different<00:24:33.950> it's<00:24:34.129> just<00:24:34.190> the - -00:24:34.450 --> 00:24:34.460 align:start position:0% -the physics is different it's just the - - -00:24:34.460 --> 00:24:37.989 align:start position:0% -the physics is different it's just the -definition<00:24:35.299> of<00:24:35.480> what<00:24:36.019> you<00:24:36.139> mean<00:24:36.440> by<00:24:37.429> squared - -00:24:37.989 --> 00:24:37.999 align:start position:0% -definition of what you mean by squared - - -00:24:37.999 --> 00:24:39.159 align:start position:0% -definition of what you mean by squared -length<00:24:38.389> is<00:24:38.629> different - -00:24:39.159 --> 00:24:39.169 align:start position:0% -length is different - - -00:24:39.169 --> 00:24:44.350 align:start position:0% -length is different -in<00:24:39.769> one<00:24:40.100> case<00:24:40.399> you<00:24:41.950> think<00:24:42.950> of<00:24:43.159> squared<00:24:43.610> length - -00:24:44.350 --> 00:24:44.360 align:start position:0% -in one case you think of squared length - - -00:24:44.360 --> 00:24:48.399 align:start position:0% -in one case you think of squared length -as<00:24:44.690> being<00:24:45.590> a<00:24:45.860> time<00:24:46.399> like<00:24:46.730> object<00:24:47.360> proper<00:24:48.169> time - -00:24:48.399 --> 00:24:48.409 align:start position:0% -as being a time like object proper time - - -00:24:48.409 --> 00:24:50.139 align:start position:0% -as being a time like object proper time -in<00:24:48.830> the<00:24:48.950> other<00:24:49.070> case<00:24:49.340> you<00:24:49.369> think<00:24:49.759> of<00:24:49.909> it<00:24:50.059> as - -00:24:50.139 --> 00:24:50.149 align:start position:0% -in the other case you think of it as - - -00:24:50.149 --> 00:24:52.220 align:start position:0% -in the other case you think of it as -being<00:24:50.330> a<00:24:50.480> space<00:24:50.720> like<00:24:50.960> object - -00:24:52.220 --> 00:24:52.230 align:start position:0% -being a space like object - - -00:24:52.230 --> 00:24:57.269 align:start position:0% -being a space like object -and<00:24:53.230> so<00:24:53.350> it<00:24:53.470> depending<00:24:54.159> on<00:24:54.279> notation<00:24:56.279> there - -00:24:57.269 --> 00:24:57.279 align:start position:0% -and so it depending on notation there - - -00:24:57.279 --> 00:25:02.789 align:start position:0% -and so it depending on notation there -are<00:24:57.490> many<00:24:58.480> many<00:24:58.870> many<00:25:01.440> physicists<00:25:02.440> who<00:25:02.559> like - -00:25:02.789 --> 00:25:02.799 align:start position:0% -are many many many physicists who like - - -00:25:02.799 --> 00:25:08.159 align:start position:0% -are many many many physicists who like -the<00:25:03.130> metric<00:25:03.460> to<00:25:03.640> be<00:25:03.669> 1<00:25:03.970> 1<00:25:04.270> 1<00:25:04.330> minus<00:25:05.230> 1<00:25:05.260> I<00:25:07.169> use - -00:25:08.159 --> 00:25:08.169 align:start position:0% -the metric to be 1 1 1 minus 1 I use - - -00:25:08.169 --> 00:25:13.560 align:start position:0% -the metric to be 1 1 1 minus 1 I use -minus<00:25:08.890> 1<00:25:09.100> 1<00:25:09.460> 1<00:25:09.490> 1<00:25:10.029> why<00:25:10.900> I<00:25:11.200> don't<00:25:11.500> know<00:25:11.620> I<00:25:12.570> guess - -00:25:13.560 --> 00:25:13.570 align:start position:0% -minus 1 1 1 1 why I don't know I guess - - -00:25:13.570 --> 00:25:20.159 align:start position:0% -minus 1 1 1 1 why I don't know I guess -it's<00:25:13.750> because<00:25:13.929> I<00:25:14.169> am<00:25:14.590> a<00:25:14.620> maverick<00:25:18.840> and<00:25:19.840> I<00:25:19.929> warn - -00:25:20.159 --> 00:25:20.169 align:start position:0% -it's because I am a maverick and I warn - - -00:25:20.169 --> 00:25:23.490 align:start position:0% -it's because I am a maverick and I warn -you<00:25:20.320> that<00:25:20.529> I<00:25:21.360> almost<00:25:22.360> everybody<00:25:22.929> uses<00:25:23.230> the - -00:25:23.490 --> 00:25:23.500 align:start position:0% -you that I almost everybody uses the - - -00:25:23.500 --> 00:25:24.510 align:start position:0% -you that I almost everybody uses the -other<00:25:23.650> notation - -00:25:24.510 --> 00:25:24.520 align:start position:0% -other notation - - -00:25:24.520 --> 00:25:27.659 align:start position:0% -other notation -I<00:25:25.409> think<00:25:26.409> Einstein<00:25:26.590> used<00:25:27.100> the<00:25:27.220> same<00:25:27.399> as<00:25:27.580> I - -00:25:27.659 --> 00:25:27.669 align:start position:0% -I think Einstein used the same as I - - -00:25:27.669 --> 00:25:34.769 align:start position:0% -I think Einstein used the same as I -don't<00:25:28.270> remember<00:25:32.940> sorry - -00:25:34.769 --> 00:25:34.779 align:start position:0% -don't remember sorry - - -00:25:34.779 --> 00:25:36.630 align:start position:0% -don't remember sorry -yeah<00:25:35.470> you<00:25:35.620> might<00:25:35.710> put<00:25:35.890> the<00:25:35.980> minus<00:25:36.279> 1<00:25:36.460> up<00:25:36.610> here - -00:25:36.630 --> 00:25:36.640 align:start position:0% -yeah you might put the minus 1 up here - - -00:25:36.640 --> 00:25:39.899 align:start position:0% -yeah you might put the minus 1 up here -and<00:25:37.120> one<00:25:37.390> down<00:25:37.419> and<00:25:37.960> one<00:25:38.140> down<00:25:38.320> here<00:25:38.909> doesn't - -00:25:39.899 --> 00:25:39.909 align:start position:0% -and one down and one down here doesn't - - -00:25:39.909 --> 00:25:45.630 align:start position:0% -and one down and one down here doesn't -matter<00:25:39.970> right<00:25:40.779> so<00:25:42.090> so<00:25:43.090> in<00:25:44.190> one<00:25:45.190> set<00:25:45.490> of - -00:25:45.630 --> 00:25:45.640 align:start position:0% -matter right so so in one set of - - -00:25:45.640 --> 00:25:47.190 align:start position:0% -matter right so so in one set of -conventions<00:25:46.029> you<00:25:46.539> would<00:25:46.690> say<00:25:46.899> that<00:25:46.929> the - -00:25:47.190 --> 00:25:47.200 align:start position:0% -conventions you would say that the - - -00:25:47.200 --> 00:25:49.200 align:start position:0% -conventions you would say that the -eigenvalues<00:25:47.649> are<00:25:47.890> three<00:25:48.250> positives<00:25:48.850> for<00:25:49.120> the - -00:25:49.200 --> 00:25:49.210 align:start position:0% -eigenvalues are three positives for the - - -00:25:49.210 --> 00:25:50.639 align:start position:0% -eigenvalues are three positives for the -three<00:25:49.419> directions<00:25:49.570> of<00:25:50.049> space<00:25:50.080> and<00:25:50.529> one - -00:25:50.639 --> 00:25:50.649 align:start position:0% -three directions of space and one - - -00:25:50.649 --> 00:25:52.889 align:start position:0% -three directions of space and one -negative<00:25:50.860> in<00:25:51.789> the<00:25:51.970> other<00:25:52.120> set<00:25:52.390> of<00:25:52.419> conventions - -00:25:52.889 --> 00:25:52.899 align:start position:0% -negative in the other set of conventions - - -00:25:52.899 --> 00:25:55.830 align:start position:0% -negative in the other set of conventions -you<00:25:53.350> would<00:25:53.470> say<00:25:53.679> one<00:25:53.950> positive<00:25:54.909> for<00:25:55.179> time<00:25:55.419> and - -00:25:55.830 --> 00:25:55.840 align:start position:0% -you would say one positive for time and - - -00:25:55.840 --> 00:25:58.919 align:start position:0% -you would say one positive for time and -three<00:25:56.440> negative<00:25:56.830> for<00:25:56.980> space<00:25:57.360> the<00:25:58.360> main<00:25:58.630> thing - -00:25:58.919 --> 00:25:58.929 align:start position:0% -three negative for space the main thing - - -00:25:58.929 --> 00:26:01.080 align:start position:0% -three negative for space the main thing -is<00:25:59.080> to<00:25:59.200> be<00:25:59.320> consistent<00:25:59.500> you<00:26:00.250> can<00:26:00.669> use<00:26:00.909> either - -00:26:01.080 --> 00:26:01.090 align:start position:0% -is to be consistent you can use either - - -00:26:01.090 --> 00:26:04.310 align:start position:0% -is to be consistent you can use either -whichever<00:26:01.960> you<00:26:02.110> like<00:26:02.140> but<00:26:03.130> be<00:26:03.250> consistent - -00:26:04.310 --> 00:26:04.320 align:start position:0% -whichever you like but be consistent - - -00:26:04.320 --> 00:26:06.419 align:start position:0% -whichever you like but be consistent -essentially<00:26:05.320> all<00:26:05.470> the<00:26:05.679> formulas<00:26:06.190> are<00:26:06.340> the - -00:26:06.419 --> 00:26:06.429 align:start position:0% -essentially all the formulas are the - - -00:26:06.429 --> 00:26:08.070 align:start position:0% -essentially all the formulas are the -same<00:26:06.640> no<00:26:06.909> matter<00:26:07.000> which<00:26:07.240> way<00:26:07.390> you<00:26:07.450> do<00:26:07.690> it<00:26:07.840> you - -00:26:08.070 --> 00:26:08.080 align:start position:0% -same no matter which way you do it you - - -00:26:08.080 --> 00:26:15.269 align:start position:0% -same no matter which way you do it you -just<00:26:08.770> have<00:26:09.039> to<00:26:09.159> remember<00:26:09.370> in<00:26:09.640> the<00:26:09.760> end<00:26:14.279> what - -00:26:15.269 --> 00:26:15.279 align:start position:0% -just have to remember in the end what - - -00:26:15.279 --> 00:26:18.210 align:start position:0% -just have to remember in the end what -you're<00:26:15.429> doing<00:26:16.200> it'll<00:26:17.200> always<00:26:17.350> come<00:26:17.649> back<00:26:17.860> to - -00:26:18.210 --> 00:26:18.220 align:start position:0% -you're doing it'll always come back to - - -00:26:18.220 --> 00:26:20.820 align:start position:0% -you're doing it'll always come back to -the<00:26:18.610> special<00:26:19.000> relativity<00:26:19.260> description<00:26:20.260> which - -00:26:20.820 --> 00:26:20.830 align:start position:0% -the special relativity description which - - -00:26:20.830 --> 00:26:22.080 align:start position:0% -the special relativity description which -one<00:26:21.039> you're<00:26:21.190> doing<00:26:21.429> are<00:26:21.549> you<00:26:21.640> talking<00:26:21.909> about - -00:26:22.080 --> 00:26:22.090 align:start position:0% -one you're doing are you talking about - - -00:26:22.090 --> 00:26:24.029 align:start position:0% -one you're doing are you talking about -proper<00:26:22.570> time<00:26:22.840> or<00:26:23.169> are<00:26:23.380> you<00:26:23.470> talking<00:26:23.860> about - -00:26:24.029 --> 00:26:24.039 align:start position:0% -proper time or are you talking about - - -00:26:24.039 --> 00:26:26.610 align:start position:0% -proper time or are you talking about -proper<00:26:24.490> distance<00:26:25.000> proper<00:26:25.960> space<00:26:26.200> like<00:26:26.440> this - -00:26:26.610 --> 00:26:26.620 align:start position:0% -proper distance proper space like this - - -00:26:26.620 --> 00:26:31.350 align:start position:0% -proper distance proper space like this -thinness<00:26:27.240> okay<00:26:28.240> so<00:26:28.299> with<00:26:28.539> that<00:26:28.570> warning<00:26:30.360> again - -00:26:31.350 --> 00:26:31.360 align:start position:0% -thinness okay so with that warning again - - -00:26:31.360 --> 00:26:35.240 align:start position:0% -thinness okay so with that warning again -I<00:26:31.390> can't<00:26:31.720> remember<00:26:31.899> what<00:26:32.140> the<00:26:32.230> question<00:26:32.620> was - -00:26:35.240 --> 00:26:35.250 align:start position:0% - - - -00:26:35.250 --> 00:26:38.190 align:start position:0% - -the<00:26:36.250> properties<00:26:36.700> of<00:26:36.730> the<00:26:36.880> metric<00:26:37.210> yes<00:26:37.450> can<00:26:38.169> be - -00:26:38.190 --> 00:26:38.200 align:start position:0% -the properties of the metric yes can be - - -00:26:38.200 --> 00:26:41.070 align:start position:0% -the properties of the metric yes can be -anything<00:26:38.470> at<00:26:38.649> all<00:26:39.600> within<00:26:40.600> reason - -00:26:41.070 --> 00:26:41.080 align:start position:0% -anything at all within reason - - -00:26:41.080 --> 00:26:50.399 align:start position:0% -anything at all within reason -smooth<00:26:42.010> differentiable<00:26:43.919> or<00:26:46.470> and<00:26:49.200> not<00:26:50.200> always - -00:26:50.399 --> 00:26:50.409 align:start position:0% -smooth differentiable or and not always - - -00:26:50.409 --> 00:26:52.230 align:start position:0% -smooth differentiable or and not always -sometimes<00:26:51.220> it's<00:26:51.429> allowed<00:26:51.669> to<00:26:51.820> have<00:26:51.940> certain - -00:26:52.230 --> 00:26:52.240 align:start position:0% -sometimes it's allowed to have certain - - -00:26:52.240 --> 00:26:55.019 align:start position:0% -sometimes it's allowed to have certain -singularities<00:26:53.250> but<00:26:54.250> mostly<00:26:54.520> you<00:26:54.730> want<00:26:54.820> it<00:26:54.940> to - -00:26:55.019 --> 00:26:55.029 align:start position:0% -singularities but mostly you want it to - - -00:26:55.029 --> 00:26:56.820 align:start position:0% -singularities but mostly you want it to -be<00:26:55.120> smooth<00:26:55.419> differentiable<00:26:56.140> like<00:26:56.470> any<00:26:56.649> other - -00:26:56.820 --> 00:26:56.830 align:start position:0% -be smooth differentiable like any other - - -00:26:56.830 --> 00:26:59.120 align:start position:0% -be smooth differentiable like any other -field - -00:26:59.120 --> 00:26:59.130 align:start position:0% -field - - -00:26:59.130 --> 00:27:14.669 align:start position:0% -field -invertible<00:27:00.130> symmetric<00:27:01.000> symmetric<00:27:13.679> lovely - -00:27:14.669 --> 00:27:14.679 align:start position:0% -invertible symmetric symmetric lovely - - -00:27:14.679 --> 00:27:16.440 align:start position:0% -invertible symmetric symmetric lovely -Romani<00:27:15.100> and<00:27:15.340> simply<00:27:15.700> means<00:27:15.940> it's<00:27:16.150> smooth - -00:27:16.440 --> 00:27:16.450 align:start position:0% -Romani and simply means it's smooth - - -00:27:16.450 --> 00:27:18.779 align:start position:0% -Romani and simply means it's smooth -means<00:27:17.230> that<00:27:17.470> it's<00:27:17.590> continuous<00:27:17.980> and<00:27:18.580> may<00:27:18.730> be - -00:27:18.779 --> 00:27:18.789 align:start position:0% -means that it's continuous and may be - - -00:27:18.789 --> 00:27:20.340 align:start position:0% -means that it's continuous and may be -differentiable<00:27:19.480> but<00:27:19.780> even<00:27:20.080> that's<00:27:20.169> not - -00:27:20.340 --> 00:27:20.350 align:start position:0% -differentiable but even that's not - - -00:27:20.350 --> 00:27:21.240 align:start position:0% -differentiable but even that's not -necessary - -00:27:21.240 --> 00:27:21.250 align:start position:0% -necessary - - -00:27:21.250 --> 00:27:25.020 align:start position:0% -necessary -just<00:27:21.880> that<00:27:22.030> it's<00:27:22.150> a<00:27:22.270> smooth<00:27:22.659> the<00:27:24.030> smooth - -00:27:25.020 --> 00:27:25.030 align:start position:0% -just that it's a smooth the smooth - - -00:27:25.030 --> 00:27:26.820 align:start position:0% -just that it's a smooth the smooth -metric<00:27:25.419> let's<00:27:25.870> write<00:27:26.049> down<00:27:26.080> the<00:27:26.260> conditions - -00:27:26.820 --> 00:27:26.830 align:start position:0% -metric let's write down the conditions - - -00:27:26.830 --> 00:27:28.590 align:start position:0% -metric let's write down the conditions -let's<00:27:27.130> write<00:27:27.309> them<00:27:27.460> down<00:27:27.610> help<00:27:28.330> me<00:27:28.450> out - -00:27:28.590 --> 00:27:28.600 align:start position:0% -let's write them down help me out - - -00:27:28.600 --> 00:27:31.919 align:start position:0% -let's write them down help me out -because<00:27:28.929> they<00:27:29.049> don't<00:27:29.110> remember<00:27:30.360> smooth<00:27:31.360> which - -00:27:31.919 --> 00:27:31.929 align:start position:0% -because they don't remember smooth which - - -00:27:31.929 --> 00:27:35.640 align:start position:0% -because they don't remember smooth which -means<00:27:33.240> continuous<00:27:34.240> and<00:27:34.539> differentiable<00:27:35.230> and - -00:27:35.640 --> 00:27:35.650 align:start position:0% -means continuous and differentiable and - - -00:27:35.650 --> 00:27:37.830 align:start position:0% -means continuous and differentiable and -you<00:27:36.309> can<00:27:36.460> differentiate<00:27:36.880> it<00:27:37.210> as<00:27:37.330> many<00:27:37.510> times - -00:27:37.830 --> 00:27:37.840 align:start position:0% -you can differentiate it as many times - - -00:27:37.840 --> 00:27:41.039 align:start position:0% -you can differentiate it as many times -as<00:27:38.049> is<00:27:38.200> necessary<00:27:38.230> in<00:27:38.980> a<00:27:39.100> calculation<00:27:39.870> so<00:27:40.870> just - -00:27:41.039 --> 00:27:41.049 align:start position:0% -as is necessary in a calculation so just - - -00:27:41.049 --> 00:27:42.060 align:start position:0% -as is necessary in a calculation so just -assume<00:27:41.289> it's<00:27:41.470> smooth<00:27:41.679> enough<00:27:41.890> to - -00:27:42.060 --> 00:27:42.070 align:start position:0% -assume it's smooth enough to - - -00:27:42.070 --> 00:27:45.240 align:start position:0% -assume it's smooth enough to -differentiate<00:27:43.380> as<00:27:44.380> many<00:27:44.650> times<00:27:44.950> as<00:27:45.070> you<00:27:45.190> need - -00:27:45.240 --> 00:27:45.250 align:start position:0% -differentiate as many times as you need - - -00:27:45.250 --> 00:27:53.240 align:start position:0% -differentiate as many times as you need -to<00:27:45.600> smooth<00:27:49.289> ok<00:27:50.289> what<00:27:51.100> was<00:27:51.250> it<00:27:51.370> what<00:27:51.490> was<00:27:51.520> next - -00:27:53.240 --> 00:27:53.250 align:start position:0% -to smooth ok what was it what was next - - -00:27:53.250 --> 00:27:59.700 align:start position:0% -to smooth ok what was it what was next -invertible<00:27:56.610> now<00:27:57.610> that<00:27:58.260> well<00:27:59.260> let's<00:27:59.470> let's - -00:27:59.700 --> 00:27:59.710 align:start position:0% -invertible now that well let's let's - - -00:27:59.710 --> 00:28:02.340 align:start position:0% -invertible now that well let's let's -let's<00:28:00.130> put<00:28:00.549> in<00:28:00.669> before<00:28:00.820> invertible<00:28:01.539> let's<00:28:02.140> put - -00:28:02.340 --> 00:28:02.350 align:start position:0% -let's put in before invertible let's put - - -00:28:02.350 --> 00:28:03.710 align:start position:0% -let's put in before invertible let's put -symmetric - -00:28:03.710 --> 00:28:03.720 align:start position:0% -symmetric - - -00:28:03.720 --> 00:28:06.270 align:start position:0% -symmetric -let's<00:28:04.720> put<00:28:04.929> symmetric<00:28:05.350> now<00:28:05.919> where<00:28:06.070> did<00:28:06.190> the - -00:28:06.270 --> 00:28:06.280 align:start position:0% -let's put symmetric now where did the - - -00:28:06.280 --> 00:28:11.539 align:start position:0% -let's put symmetric now where did the -symmetric<00:28:06.789> come<00:28:07.059> from<00:28:07.299> incidentally - -00:28:11.539 --> 00:28:11.549 align:start position:0% - - - -00:28:11.549 --> 00:28:15.000 align:start position:0% - -remember<00:28:12.549> it's<00:28:12.669> definition<00:28:13.360> it's<00:28:14.260> D<00:28:14.799> tau - -00:28:15.000 --> 00:28:15.010 align:start position:0% -remember it's definition it's D tau - - -00:28:15.010 --> 00:28:20.520 align:start position:0% -remember it's definition it's D tau -squared<00:28:15.520> equals<00:28:16.120> G<00:28:16.419> mu<00:28:16.900> nu<00:28:17.400> DX<00:28:18.400> mu<00:28:18.789> DX<00:28:18.820> nu<00:28:19.750> what - -00:28:20.520 --> 00:28:20.530 align:start position:0% -squared equals G mu nu DX mu DX nu what - - -00:28:20.530 --> 00:28:21.539 align:start position:0% -squared equals G mu nu DX mu DX nu what -would<00:28:20.650> happen<00:28:21.039> if<00:28:21.159> there<00:28:21.280> was<00:28:21.429> an - -00:28:21.539 --> 00:28:21.549 align:start position:0% -would happen if there was an - - -00:28:21.549 --> 00:28:25.049 align:start position:0% -would happen if there was an -anti-symmetric<00:28:22.210> piece<00:28:22.659> to<00:28:22.900> jamia<00:28:23.260> no<00:28:24.059> any - -00:28:25.049 --> 00:28:25.059 align:start position:0% -anti-symmetric piece to jamia no any - - -00:28:25.059 --> 00:28:26.789 align:start position:0% -anti-symmetric piece to jamia no any -matrix<00:28:25.419> can<00:28:25.809> be<00:28:25.929> written<00:28:25.960> as<00:28:26.289> a<00:28:26.350> sum<00:28:26.650> of - -00:28:26.789 --> 00:28:26.799 align:start position:0% -matrix can be written as a sum of - - -00:28:26.799 --> 00:28:29.039 align:start position:0% -matrix can be written as a sum of -symmetric<00:28:27.280> and<00:28:27.460> anti-symmetric<00:28:28.049> the - -00:28:29.039 --> 00:28:29.049 align:start position:0% -symmetric and anti-symmetric the - - -00:28:29.049 --> 00:28:30.480 align:start position:0% -symmetric and anti-symmetric the -anti-symmetric<00:28:29.679> means<00:28:30.100> that<00:28:30.280> when<00:28:30.400> you - -00:28:30.480 --> 00:28:30.490 align:start position:0% -anti-symmetric means that when you - - -00:28:30.490 --> 00:28:32.360 align:start position:0% -anti-symmetric means that when you -interchange<00:28:30.940> the<00:28:31.150> indices<00:28:31.539> it<00:28:31.690> changes<00:28:32.080> sign - -00:28:32.360 --> 00:28:32.370 align:start position:0% -interchange the indices it changes sign - - -00:28:32.370 --> 00:28:34.470 align:start position:0% -interchange the indices it changes sign -what<00:28:33.370> would<00:28:33.490> happen<00:28:33.789> if<00:28:33.880> G<00:28:34.030> mu<00:28:34.179> nu<00:28:34.240> was - -00:28:34.470 --> 00:28:34.480 align:start position:0% -what would happen if G mu nu was - - -00:28:34.480 --> 00:28:39.240 align:start position:0% -what would happen if G mu nu was -anti-symmetric<00:28:37.230> you<00:28:38.230> would<00:28:38.350> get<00:28:38.500> zero<00:28:38.890> from - -00:28:39.240 --> 00:28:39.250 align:start position:0% -anti-symmetric you would get zero from - - -00:28:39.250 --> 00:28:42.450 align:start position:0% -anti-symmetric you would get zero from -this<00:28:40.140> you<00:28:41.140> will<00:28:41.169> get<00:28:41.470> zero<00:28:41.740> from<00:28:42.010> this<00:28:42.159> because - -00:28:42.450 --> 00:28:42.460 align:start position:0% -this you will get zero from this because - - -00:28:42.460 --> 00:28:45.630 align:start position:0% -this you will get zero from this because -the<00:28:42.610> coefficient<00:28:43.299> for<00:28:43.360> example<00:28:43.960> of<00:28:44.200> the<00:28:45.039> x<00:28:45.280> 1 - -00:28:45.630 --> 00:28:45.640 align:start position:0% -the coefficient for example of the x 1 - - -00:28:45.640 --> 00:28:49.649 align:start position:0% -the coefficient for example of the x 1 -DX<00:28:46.030> 2<00:28:46.590> would<00:28:47.590> be<00:28:47.820> opposite<00:28:48.820> to<00:28:49.539> the - -00:28:49.649 --> 00:28:49.659 align:start position:0% -DX 2 would be opposite to the - - -00:28:49.659 --> 00:28:53.159 align:start position:0% -DX 2 would be opposite to the -coefficient<00:28:50.320> of<00:28:50.440> DX<00:28:50.650> 2<00:28:51.100> DX<00:28:51.340> 1<00:28:51.929> but<00:28:52.929> they're<00:28:53.080> the - -00:28:53.159 --> 00:28:53.169 align:start position:0% -coefficient of DX 2 DX 1 but they're the - - -00:28:53.169 --> 00:28:55.610 align:start position:0% -coefficient of DX 2 DX 1 but they're the -same<00:28:53.380> thing<00:28:53.650> the<00:28:53.860> product<00:28:54.250> the<00:28:54.370> x1<00:28:54.760> and<00:28:54.940> the<00:28:55.030> x2 - -00:28:55.610 --> 00:28:55.620 align:start position:0% -same thing the product the x1 and the x2 - - -00:28:55.620 --> 00:28:58.770 align:start position:0% -same thing the product the x1 and the x2 -so<00:28:56.620> if<00:28:57.010> it<00:28:57.340> was<00:28:57.460> anti-symmetric<00:28:58.210> you<00:28:58.539> wouldn't - -00:28:58.770 --> 00:28:58.780 align:start position:0% -so if it was anti-symmetric you wouldn't - - -00:28:58.780 --> 00:29:03.000 align:start position:0% -so if it was anti-symmetric you wouldn't -get<00:28:58.900> anything<00:29:01.110> any<00:29:02.110> matrix<00:29:02.470> can<00:29:02.860> be<00:29:02.980> written - -00:29:03.000 --> 00:29:03.010 align:start position:0% -get anything any matrix can be written - - -00:29:03.010 --> 00:29:04.799 align:start position:0% -get anything any matrix can be written -as<00:29:03.309> a<00:29:03.370> sum<00:29:03.700> of<00:29:03.880> symmetric<00:29:04.360> and<00:29:04.510> anti-symmetric - -00:29:04.799 --> 00:29:04.809 align:start position:0% -as a sum of symmetric and anti-symmetric - - -00:29:04.809 --> 00:29:08.430 align:start position:0% -as a sum of symmetric and anti-symmetric -only<00:29:05.679> the<00:29:05.830> symmetric<00:29:06.460> part<00:29:06.730> of<00:29:07.000> G<00:29:07.270> mu<00:29:07.450> nu<00:29:07.480> will - -00:29:08.430 --> 00:29:08.440 align:start position:0% -only the symmetric part of G mu nu will - - -00:29:08.440 --> 00:29:11.310 align:start position:0% -only the symmetric part of G mu nu will -contribute<00:29:08.799> so<00:29:09.280> you<00:29:09.340> might<00:29:09.640> as<00:29:09.880> well - -00:29:11.310 --> 00:29:11.320 align:start position:0% -contribute so you might as well - - -00:29:11.320 --> 00:29:14.880 align:start position:0% -contribute so you might as well -say<00:29:11.770> the<00:29:12.280> gym<00:29:12.550> you<00:29:12.670> knew<00:29:12.850> is<00:29:13.300> symmetric<00:29:14.200> right - -00:29:14.880 --> 00:29:14.890 align:start position:0% -say the gym you knew is symmetric right - - -00:29:14.890 --> 00:29:24.810 align:start position:0% -say the gym you knew is symmetric right -now<00:29:15.820> any<00:29:16.150> matrix<00:29:23.100> any<00:29:24.100> terms<00:29:24.430> that<00:29:24.580> you<00:29:24.700> would - -00:29:24.810 --> 00:29:24.820 align:start position:0% -now any matrix any terms that you would - - -00:29:24.820 --> 00:29:26.640 align:start position:0% -now any matrix any terms that you would -put<00:29:25.000> in<00:29:25.180> or<00:29:25.510> anti-symmetric<00:29:26.140> wouldn't - -00:29:26.640 --> 00:29:26.650 align:start position:0% -put in or anti-symmetric wouldn't - - -00:29:26.650 --> 00:29:28.650 align:start position:0% -put in or anti-symmetric wouldn't -wouldn't<00:29:27.310> affect<00:29:27.610> anything<00:29:27.910> so<00:29:28.390> there<00:29:28.540> would - -00:29:28.650 --> 00:29:28.660 align:start position:0% -wouldn't affect anything so there would - - -00:29:28.660 --> 00:29:32.550 align:start position:0% -wouldn't affect anything so there would -be<00:29:28.690> sort<00:29:28.900> of<00:29:28.990> spurious<00:29:30.390> no<00:29:31.390> physics<00:29:31.840> to<00:29:32.050> them - -00:29:32.550 --> 00:29:32.560 align:start position:0% -be sort of spurious no physics to them - - -00:29:32.560 --> 00:29:34.590 align:start position:0% -be sort of spurious no physics to them -nothing<00:29:33.400> could<00:29:33.610> determine<00:29:34.000> them<00:29:34.300> since - -00:29:34.590 --> 00:29:34.600 align:start position:0% -nothing could determine them since - - -00:29:34.600 --> 00:29:36.870 align:start position:0% -nothing could determine them since -they're<00:29:34.750> arbitrary<00:29:35.350> so<00:29:35.920> yes<00:29:36.130> it's<00:29:36.370> it<00:29:36.730> is<00:29:36.850> a - -00:29:36.870 --> 00:29:36.880 align:start position:0% -they're arbitrary so yes it's it is a - - -00:29:36.880 --> 00:29:38.850 align:start position:0% -they're arbitrary so yes it's it is a -yes<00:29:37.420> it<00:29:37.570> is<00:29:37.660> a<00:29:37.690> it<00:29:38.110> is<00:29:38.290> a<00:29:38.320> question<00:29:38.740> of - -00:29:38.850 --> 00:29:38.860 align:start position:0% -yes it is a it is a question of - - -00:29:38.860 --> 00:29:42.180 align:start position:0% -yes it is a it is a question of -convenience<00:29:39.070> yes<00:29:39.840> yes<00:29:40.840> it<00:29:41.530> is<00:29:41.680> a<00:29:41.710> question<00:29:41.890> of - -00:29:42.180 --> 00:29:42.190 align:start position:0% -convenience yes yes it is a question of - - -00:29:42.190 --> 00:29:42.930 align:start position:0% -convenience yes yes it is a question of -convenience - -00:29:42.930 --> 00:29:42.940 align:start position:0% -convenience - - -00:29:42.940 --> 00:29:46.800 align:start position:0% -convenience -absolutely<00:29:43.660> but<00:29:44.650> once<00:29:44.860> it's<00:29:45.040> symmetric<00:29:45.810> then - -00:29:46.800 --> 00:29:46.810 align:start position:0% -absolutely but once it's symmetric then - - -00:29:46.810 --> 00:29:51.750 align:start position:0% -absolutely but once it's symmetric then -you<00:29:47.260> can<00:29:47.500> say<00:29:47.790> that<00:29:48.790> it<00:29:49.120> has<00:29:49.150> eigenvalues<00:29:50.760> and - -00:29:51.750 --> 00:29:51.760 align:start position:0% -you can say that it has eigenvalues and - - -00:29:51.760 --> 00:29:55.350 align:start position:0% -you can say that it has eigenvalues and -how<00:29:52.450> many<00:29:52.630> eigenvalues<00:29:53.320> does<00:29:53.530> it<00:29:53.680> have<00:29:54.360> in - -00:29:55.350 --> 00:29:55.360 align:start position:0% -how many eigenvalues does it have in - - -00:29:55.360 --> 00:29:57.990 align:start position:0% -how many eigenvalues does it have in -general<00:29:56.050> it<00:29:56.440> will<00:29:56.560> have<00:29:56.890> a<00:29:57.190> full<00:29:57.610> set<00:29:57.940> of - -00:29:57.990 --> 00:29:58.000 align:start position:0% -general it will have a full set of - - -00:29:58.000 --> 00:30:01.140 align:start position:0% -general it will have a full set of -eigenvalues<00:29:59.220> equal<00:30:00.220> to<00:30:00.430> the<00:30:00.520> dimensionality - -00:30:01.140 --> 00:30:01.150 align:start position:0% -eigenvalues equal to the dimensionality - - -00:30:01.150 --> 00:30:02.940 align:start position:0% -eigenvalues equal to the dimensionality -of<00:30:01.420> the<00:30:01.570> space<00:30:01.600> so<00:30:02.260> there<00:30:02.410> will<00:30:02.500> be<00:30:02.560> for - -00:30:02.940 --> 00:30:02.950 align:start position:0% -of the space so there will be for - - -00:30:02.950 --> 00:30:04.760 align:start position:0% -of the space so there will be for -example<00:30:03.040> in<00:30:03.580> four-dimensional<00:30:04.360> space-time - -00:30:04.760 --> 00:30:04.770 align:start position:0% -example in four-dimensional space-time - - -00:30:04.770 --> 00:30:08.910 align:start position:0% -example in four-dimensional space-time -for<00:30:05.770> eigenvalues<00:30:06.520> so<00:30:06.880> it<00:30:07.690> will<00:30:07.900> have<00:30:08.170> by - -00:30:08.910 --> 00:30:08.920 align:start position:0% -for eigenvalues so it will have by - - -00:30:08.920 --> 00:30:11.520 align:start position:0% -for eigenvalues so it will have by -virtue<00:30:08.950> of<00:30:09.490> it<00:30:09.610> being<00:30:09.640> symmetric<00:30:10.480> they<00:30:11.350> will - -00:30:11.520 --> 00:30:11.530 align:start position:0% -virtue of it being symmetric they will - - -00:30:11.530 --> 00:30:17.780 align:start position:0% -virtue of it being symmetric they will -have<00:30:11.710> a<00:30:11.740> set<00:30:11.980> of<00:30:12.070> eigenvalues - -00:30:17.780 --> 00:30:17.790 align:start position:0% - - - -00:30:17.790 --> 00:30:21.390 align:start position:0% - -what<00:30:18.790> is<00:30:18.910> the<00:30:19.060> condition<00:30:19.270> that<00:30:19.930> the<00:30:20.830> matrix<00:30:21.250> be - -00:30:21.390 --> 00:30:21.400 align:start position:0% -what is the condition that the matrix be - - -00:30:21.400 --> 00:30:26.520 align:start position:0% -what is the condition that the matrix be -invertible<00:30:24.000> nonzero<00:30:25.000> determinant<00:30:25.350> but<00:30:26.350> that - -00:30:26.520 --> 00:30:26.530 align:start position:0% -invertible nonzero determinant but that - - -00:30:26.530 --> 00:30:30.810 align:start position:0% -invertible nonzero determinant but that -simply<00:30:26.920> means<00:30:27.100> no<00:30:27.640> zero<00:30:28.210> eigenvalues<00:30:29.820> all - -00:30:30.810 --> 00:30:30.820 align:start position:0% -simply means no zero eigenvalues all - - -00:30:30.820 --> 00:30:32.340 align:start position:0% -simply means no zero eigenvalues all -right<00:30:30.970> because<00:30:31.120> the<00:30:31.360> determinant<00:30:31.900> is<00:30:32.050> just<00:30:32.260> a - -00:30:32.340 --> 00:30:32.350 align:start position:0% -right because the determinant is just a - - -00:30:32.350 --> 00:30:43.370 align:start position:0% -right because the determinant is just a -product<00:30:32.650> of<00:30:32.950> the<00:30:33.040> eigenvalues<00:30:33.160> so<00:30:37.170> invertible - -00:30:43.370 --> 00:30:43.380 align:start position:0% - - - -00:30:43.380 --> 00:30:48.030 align:start position:0% - -simply<00:30:44.380> precludes<00:30:45.190> zero<00:30:46.150> eigen<00:30:46.780> values<00:30:47.230> eigen - -00:30:48.030 --> 00:30:48.040 align:start position:0% -simply precludes zero eigen values eigen - - -00:30:48.040 --> 00:30:50.910 align:start position:0% -simply precludes zero eigen values eigen -values<00:30:48.640> which<00:30:48.880> are<00:30:49.000> zero<00:30:49.530> that's<00:30:50.530> all<00:30:50.770> this - -00:30:50.910 --> 00:30:50.920 align:start position:0% -values which are zero that's all this - - -00:30:50.920 --> 00:30:52.740 align:start position:0% -values which are zero that's all this -means<00:30:51.190> once<00:30:51.610> it's<00:30:51.790> symmetric<00:30:52.330> it's - -00:30:52.740 --> 00:30:52.750 align:start position:0% -means once it's symmetric it's - - -00:30:52.750 --> 00:30:55.890 align:start position:0% -means once it's symmetric it's -invertible<00:30:53.290> but<00:30:54.010> then<00:30:54.220> on<00:30:54.340> top<00:30:54.580> of<00:30:54.820> that<00:30:55.000> it - -00:30:55.890 --> 00:30:55.900 align:start position:0% -invertible but then on top of that it - - -00:30:55.900 --> 00:31:00.210 align:start position:0% -invertible but then on top of that it -should<00:30:56.260> have<00:30:57.990> depending<00:30:58.990> on<00:30:59.140> the<00:30:59.440> application - -00:31:00.210 --> 00:31:00.220 align:start position:0% -should have depending on the application - - -00:31:00.220 --> 00:31:02.460 align:start position:0% -should have depending on the application -if<00:31:00.970> we're<00:31:01.150> talking<00:31:01.450> about<00:31:01.570> ordinary<00:31:02.140> space - -00:31:02.460 --> 00:31:02.470 align:start position:0% -if we're talking about ordinary space - - -00:31:02.470 --> 00:31:08.670 align:start position:0% -if we're talking about ordinary space -versus<00:31:03.340> space<00:31:03.640> time<00:31:04.000> in<00:31:04.710> ordinary<00:31:05.710> space<00:31:07.680> it - -00:31:08.670 --> 00:31:08.680 align:start position:0% -versus space time in ordinary space it - - -00:31:08.680 --> 00:31:13.050 align:start position:0% -versus space time in ordinary space it -should<00:31:09.010> have<00:31:09.490> all<00:31:09.820> positive<00:31:10.690> eigenvalues<00:31:12.060> and - -00:31:13.050 --> 00:31:13.060 align:start position:0% -should have all positive eigenvalues and - - -00:31:13.060 --> 00:31:14.700 align:start position:0% -should have all positive eigenvalues and -of<00:31:13.180> course<00:31:13.450> we<00:31:13.630> would<00:31:13.810> be<00:31:13.840> talking<00:31:14.140> about<00:31:14.410> the - -00:31:14.700 --> 00:31:14.710 align:start position:0% -of course we would be talking about the - - -00:31:14.710 --> 00:31:17.430 align:start position:0% -of course we would be talking about the -s<00:31:14.890> square<00:31:15.280> then<00:31:15.610> not<00:31:16.450> the<00:31:16.630> tau<00:31:16.810> squared<00:31:17.260> in - -00:31:17.430 --> 00:31:17.440 align:start position:0% -s square then not the tau squared in - - -00:31:17.440 --> 00:31:20.670 align:start position:0% -s square then not the tau squared in -ordinary<00:31:17.800> space<00:31:18.070> all<00:31:19.060> positive<00:31:19.960> eigenvalues - -00:31:20.670 --> 00:31:20.680 align:start position:0% -ordinary space all positive eigenvalues - - -00:31:20.680 --> 00:31:23.460 align:start position:0% -ordinary space all positive eigenvalues -and<00:31:21.370> we<00:31:21.580> would<00:31:21.790> say<00:31:22.000> the<00:31:22.180> language<00:31:22.720> that - -00:31:23.460 --> 00:31:23.470 align:start position:0% -and we would say the language that - - -00:31:23.470 --> 00:31:24.900 align:start position:0% -and we would say the language that -mathematicians<00:31:24.250> use - -00:31:24.900 --> 00:31:24.910 align:start position:0% -mathematicians use - - -00:31:24.910 --> 00:31:27.090 align:start position:0% -mathematicians use -as<00:31:25.270> they<00:31:25.510> would<00:31:25.630> say<00:31:25.840> the<00:31:25.990> signature<00:31:26.650> of<00:31:26.860> the - -00:31:27.090 --> 00:31:27.100 align:start position:0% -as they would say the signature of the - - -00:31:27.100 --> 00:31:29.490 align:start position:0% -as they would say the signature of the -space<00:31:27.370> of<00:31:27.790> the<00:31:27.910> space<00:31:28.150> would<00:31:28.420> be<00:31:28.480> plus<00:31:28.900> plus - -00:31:29.490 --> 00:31:29.500 align:start position:0% -space of the space would be plus plus - - -00:31:29.500 --> 00:31:34.740 align:start position:0% -space of the space would be plus plus -plus<00:31:29.530> plus<00:31:31.620> plus<00:31:32.620> plus<00:31:32.890> is<00:31:33.640> the<00:31:34.150> signature<00:31:34.570> of - -00:31:34.740 --> 00:31:34.750 align:start position:0% -plus plus plus plus is the signature of - - -00:31:34.750 --> 00:31:36.900 align:start position:0% -plus plus plus plus is the signature of -the<00:31:34.930> space<00:31:35.200> and<00:31:35.650> it's<00:31:36.160> simply<00:31:36.460> telling<00:31:36.730> you - -00:31:36.900 --> 00:31:36.910 align:start position:0% -the space and it's simply telling you - - -00:31:36.910 --> 00:31:38.370 align:start position:0% -the space and it's simply telling you -how<00:31:37.120> many<00:31:37.180> positive<00:31:37.930> and<00:31:38.050> negative - -00:31:38.370 --> 00:31:38.380 align:start position:0% -how many positive and negative - - -00:31:38.380 --> 00:31:42.330 align:start position:0% -how many positive and negative -eigenvalues<00:31:39.070> if<00:31:39.850> I<00:31:40.450> use<00:31:40.810> this<00:31:41.080> notation<00:31:41.560> here - -00:31:42.330 --> 00:31:42.340 align:start position:0% -eigenvalues if I use this notation here - - -00:31:42.340 --> 00:31:45.450 align:start position:0% -eigenvalues if I use this notation here -where<00:31:43.270> I<00:31:43.300> study<00:31:43.660> the<00:31:43.840> proper<00:31:44.260> time<00:31:44.500> I<00:31:44.950> would - -00:31:45.450 --> 00:31:45.460 align:start position:0% -where I study the proper time I would - - -00:31:45.460 --> 00:31:49.290 align:start position:0% -where I study the proper time I would -say<00:31:45.670> that<00:31:46.120> the<00:31:46.390> that<00:31:47.320> that<00:31:47.800> the<00:31:48.270> signature<00:31:49.270> of - -00:31:49.290 --> 00:31:49.300 align:start position:0% -say that the that that the signature of - - -00:31:49.300 --> 00:31:52.470 align:start position:0% -say that the that that the signature of -the<00:31:49.480> metric<00:31:49.870> was<00:31:50.530> minus<00:31:51.070> plus<00:31:51.370> plus<00:31:51.430> plus<00:31:51.820> one - -00:31:52.470 --> 00:31:52.480 align:start position:0% -the metric was minus plus plus plus one - - -00:31:52.480 --> 00:31:54.750 align:start position:0% -the metric was minus plus plus plus one -minus<00:31:52.900> and<00:31:53.050> three<00:31:53.200> pluses<00:31:53.490> so<00:31:54.490> for<00:31:54.700> the - -00:31:54.750 --> 00:31:54.760 align:start position:0% -minus and three pluses so for the - - -00:31:54.760 --> 00:31:56.400 align:start position:0% -minus and three pluses so for the -purposes<00:31:55.210> of<00:31:55.270> general<00:31:55.690> relativity<00:31:55.840> or - -00:31:56.400 --> 00:31:56.410 align:start position:0% -purposes of general relativity or - - -00:31:56.410 --> 00:32:02.900 align:start position:0% -purposes of general relativity or -relativity<00:31:56.980> in<00:31:57.070> general<00:31:57.730> the<00:31:58.600> signature<00:31:59.200> of - -00:32:02.900 --> 00:32:02.910 align:start position:0% - - - -00:32:02.910 --> 00:32:09.090 align:start position:0% - -the<00:32:03.910> metric<00:32:04.300> is<00:32:05.070> plus<00:32:06.070> minus<00:32:06.430> minus<00:32:07.300> minus<00:32:08.100> in - -00:32:09.090 --> 00:32:09.100 align:start position:0% -the metric is plus minus minus minus in - - -00:32:09.100 --> 00:32:12.270 align:start position:0% -the metric is plus minus minus minus in -other<00:32:09.340> words<00:32:09.640> this<00:32:10.210> means<00:32:10.930> three<00:32:11.740> negative - -00:32:12.270 --> 00:32:12.280 align:start position:0% -other words this means three negative - - -00:32:12.280 --> 00:32:14.670 align:start position:0% -other words this means three negative -eigenvalues<00:32:13.000> in<00:32:13.240> one<00:32:13.480> positive<00:32:13.750> eigenvalue - -00:32:14.670 --> 00:32:14.680 align:start position:0% -eigenvalues in one positive eigenvalue - - -00:32:14.680 --> 00:32:17.640 align:start position:0% -eigenvalues in one positive eigenvalue -those<00:32:15.670> are<00:32:15.880> the<00:32:15.970> conditions<00:32:16.210> for<00:32:16.810> a<00:32:17.020> metric<00:32:17.440> to - -00:32:17.640 --> 00:32:17.650 align:start position:0% -those are the conditions for a metric to - - -00:32:17.650 --> 00:32:19.980 align:start position:0% -those are the conditions for a metric to -be<00:32:17.680> an<00:32:17.980> allowable<00:32:18.220> metric<00:32:18.990> symmetric - -00:32:19.980 --> 00:32:19.990 align:start position:0% -be an allowable metric symmetric - - -00:32:19.990 --> 00:32:21.460 align:start position:0% -be an allowable metric symmetric -guarantees - -00:32:21.460 --> 00:32:21.470 align:start position:0% -guarantees - - -00:32:21.470 --> 00:32:26.510 align:start position:0% -guarantees -good<00:32:23.240> yes<00:32:24.270> I'm<00:32:24.690> not<00:32:24.870> positive<00:32:25.520> right - -00:32:26.510 --> 00:32:26.520 align:start position:0% -good yes I'm not positive right - - -00:32:26.520 --> 00:32:29.650 align:start position:0% -good yes I'm not positive right -absolutely - -00:32:29.650 --> 00:32:29.660 align:start position:0% - - - -00:32:29.660 --> 00:32:32.390 align:start position:0% - -okay<00:32:30.660> so<00:32:30.720> that's<00:32:31.020> that's<00:32:31.290> the<00:32:31.770> conditions<00:32:32.220> on - -00:32:32.390 --> 00:32:32.400 align:start position:0% -okay so that's that's the conditions on - - -00:32:32.400 --> 00:32:37.600 align:start position:0% -okay so that's that's the conditions on -metric<00:32:32.820> now<00:32:33.300> you<00:32:33.660> might<00:32:33.930> there<00:32:34.470> are<00:32:34.650> important - -00:32:37.600 --> 00:32:37.610 align:start position:0% - - - -00:32:37.610 --> 00:32:41.380 align:start position:0% - -dynamical<00:32:38.610> conditions<00:32:39.270> for<00:32:40.170> example<00:32:40.320> the - -00:32:41.380 --> 00:32:41.390 align:start position:0% -dynamical conditions for example the - - -00:32:41.390 --> 00:32:44.330 align:start position:0% -dynamical conditions for example the -gravitational<00:32:42.390> field<00:32:42.630> is<00:32:43.050> the<00:32:43.290> metric<00:32:43.680> it's - -00:32:44.330 --> 00:32:44.340 align:start position:0% -gravitational field is the metric it's - - -00:32:44.340 --> 00:32:46.250 align:start position:0% -gravitational field is the metric it's -the<00:32:44.520> thing<00:32:44.700> which<00:32:44.850> satisfies<00:32:45.450> equations<00:32:46.110> of - -00:32:46.250 --> 00:32:46.260 align:start position:0% -the thing which satisfies equations of - - -00:32:46.260 --> 00:32:50.270 align:start position:0% -the thing which satisfies equations of -motion<00:32:47.930> how<00:32:48.930> do<00:32:48.990> I<00:32:49.110> know<00:32:49.200> that<00:32:49.620> my<00:32:49.710> equations - -00:32:50.270 --> 00:32:50.280 align:start position:0% -motion how do I know that my equations - - -00:32:50.280 --> 00:32:54.200 align:start position:0% -motion how do I know that my equations -of<00:32:50.430> motion<00:32:50.850> won't<00:32:51.840> suddenly<00:32:52.440> at<00:32:53.400> some<00:32:53.790> point - -00:32:54.200 --> 00:32:54.210 align:start position:0% -of motion won't suddenly at some point - - -00:32:54.210 --> 00:32:58.900 align:start position:0% -of motion won't suddenly at some point -in<00:32:54.450> time<00:32:54.480> change<00:32:55.460> the<00:32:56.460> signature<00:32:57.060> of<00:32:57.240> the<00:32:57.900> a - -00:32:58.900 --> 00:32:58.910 align:start position:0% -in time change the signature of the a - - -00:32:58.910 --> 00:33:02.990 align:start position:0% -in time change the signature of the a -for<00:32:59.910> example<00:33:00.300> how<00:33:00.690> do<00:33:00.750> i<00:33:00.870> know<00:33:01.430> that<00:33:02.430> some<00:33:02.760> g - -00:33:02.990 --> 00:33:03.000 align:start position:0% -for example how do i know that some g - - -00:33:03.000 --> 00:33:09.320 align:start position:0% -for example how do i know that some g -m--<00:33:03.030> you<00:33:03.210> know<00:33:05.450> let's<00:33:06.450> say<00:33:06.690> 1<00:33:07.400> minus<00:33:08.400> 1<00:33:08.610> 1<00:33:09.270> minus - -00:33:09.320 --> 00:33:09.330 align:start position:0% -m-- you know let's say 1 minus 1 1 minus - - -00:33:09.330 --> 00:33:13.700 align:start position:0% -m-- you know let's say 1 minus 1 1 minus -1<00:33:09.870> minus<00:33:10.050> 1<00:33:10.710> minus<00:33:10.740> 1<00:33:11.390> won't<00:33:12.390> evolve<00:33:12.990> in<00:33:13.320> time - -00:33:13.700 --> 00:33:13.710 align:start position:0% -1 minus 1 minus 1 won't evolve in time - - -00:33:13.710 --> 00:33:19.460 align:start position:0% -1 minus 1 minus 1 won't evolve in time -and<00:33:14.370> eventually<00:33:16.340> become<00:33:17.340> 1<00:33:17.670> 1<00:33:18.260> minus<00:33:19.260> 1<00:33:19.440> minus - -00:33:19.460 --> 00:33:19.470 align:start position:0% -and eventually become 1 1 minus 1 minus - - -00:33:19.470 --> 00:33:23.600 align:start position:0% -and eventually become 1 1 minus 1 minus -1<00:33:20.540> that<00:33:21.540> better<00:33:21.750> not<00:33:21.930> happen<00:33:22.110> this<00:33:23.010> is<00:33:23.190> this<00:33:23.430> is - -00:33:23.600 --> 00:33:23.610 align:start position:0% -1 that better not happen this is this is - - -00:33:23.610 --> 00:33:26.240 align:start position:0% -1 that better not happen this is this is -not<00:33:23.790> an<00:33:24.060> allowable<00:33:24.540> metric<00:33:25.140> to<00:33:25.500> describe<00:33:25.950> a - -00:33:26.240 --> 00:33:26.250 align:start position:0% -not an allowable metric to describe a - - -00:33:26.250 --> 00:33:28.610 align:start position:0% -not an allowable metric to describe a -sensible<00:33:26.730> space-time<00:33:27.270> to<00:33:27.630> space<00:33:27.990> directions - -00:33:28.610 --> 00:33:28.620 align:start position:0% -sensible space-time to space directions - - -00:33:28.620 --> 00:33:32.120 align:start position:0% -sensible space-time to space directions -in<00:33:28.770> 1/2<00:33:29.160> time<00:33:29.460> directions<00:33:30.120> bad<00:33:30.660> news<00:33:31.130> that's - -00:33:32.120 --> 00:33:32.130 align:start position:0% -in 1/2 time directions bad news that's - - -00:33:32.130 --> 00:33:36.590 align:start position:0% -in 1/2 time directions bad news that's -not<00:33:32.610> a<00:33:32.640> good<00:33:33.000> thing<00:33:34.280> why<00:33:35.280> doesn't<00:33:35.850> that<00:33:35.880> happen - -00:33:36.590 --> 00:33:36.600 align:start position:0% -not a good thing why doesn't that happen - - -00:33:36.600 --> 00:33:38.930 align:start position:0% -not a good thing why doesn't that happen -what<00:33:37.410> is<00:33:37.500> the<00:33:37.650> dynamics<00:33:38.250> what's<00:33:38.460> what<00:33:38.670> is<00:33:38.790> it - -00:33:38.930 --> 00:33:38.940 align:start position:0% -what is the dynamics what's what is it - - -00:33:38.940 --> 00:33:40.880 align:start position:0% -what is the dynamics what's what is it -in<00:33:39.060> the<00:33:39.180> equations<00:33:39.870> of<00:33:40.020> the<00:33:40.170> theory<00:33:40.590> and<00:33:40.800> the - -00:33:40.880 --> 00:33:40.890 align:start position:0% -in the equations of the theory and the - - -00:33:40.890 --> 00:33:42.830 align:start position:0% -in the equations of the theory and the -evolutionary<00:33:41.610> equations<00:33:42.270> of<00:33:42.420> the<00:33:42.510> theory - -00:33:42.830 --> 00:33:42.840 align:start position:0% -evolutionary equations of the theory - - -00:33:42.840 --> 00:33:44.150 align:start position:0% -evolutionary equations of the theory -which<00:33:42.990> tells<00:33:43.020> things<00:33:43.470> how<00:33:43.650> to<00:33:43.710> change<00:33:44.040> from - -00:33:44.150 --> 00:33:44.160 align:start position:0% -which tells things how to change from - - -00:33:44.160 --> 00:33:46.670 align:start position:0% -which tells things how to change from -moment<00:33:44.460> to<00:33:44.700> moment<00:33:44.900> which<00:33:45.900> precludes<00:33:46.440> this - -00:33:46.670 --> 00:33:46.680 align:start position:0% -moment to moment which precludes this - - -00:33:46.680 --> 00:33:49.430 align:start position:0% -moment to moment which precludes this -from<00:33:46.860> happening<00:33:47.240> well<00:33:48.240> I<00:33:48.570> can<00:33:49.050> tell<00:33:49.140> you<00:33:49.260> one - -00:33:49.430 --> 00:33:49.440 align:start position:0% -from happening well I can tell you one - - -00:33:49.440 --> 00:33:54.820 align:start position:0% -from happening well I can tell you one -thing<00:33:50.630> what's<00:33:51.630> the<00:33:51.900> determinant<00:33:52.140> of<00:33:52.890> a<00:33:53.220> metric - -00:33:54.820 --> 00:33:54.830 align:start position:0% -thing what's the determinant of a metric - - -00:33:54.830 --> 00:33:57.620 align:start position:0% -thing what's the determinant of a metric -with<00:33:55.830> 1<00:33:56.100> minus<00:33:56.460> 1<00:33:56.610> minus<00:33:56.640> 1<00:33:56.940> minus<00:33:57.060> 1<00:33:57.450> these<00:33:57.570> are - -00:33:57.620 --> 00:33:57.630 align:start position:0% -with 1 minus 1 minus 1 minus 1 these are - - -00:33:57.630 --> 00:33:59.270 align:start position:0% -with 1 minus 1 minus 1 minus 1 these are -the<00:33:57.720> eigenvalues<00:33:58.380> i'm<00:33:58.530> specifying<00:33:58.800> the - -00:33:59.270 --> 00:33:59.280 align:start position:0% -the eigenvalues i'm specifying the - - -00:33:59.280 --> 00:34:00.860 align:start position:0% -the eigenvalues i'm specifying the -eigenvalues<00:33:59.820> what's<00:34:00.000> the<00:34:00.120> determinant<00:34:00.330> of - -00:34:00.860 --> 00:34:00.870 align:start position:0% -eigenvalues what's the determinant of - - -00:34:00.870 --> 00:34:03.380 align:start position:0% -eigenvalues what's the determinant of -the<00:34:00.930> metric<00:34:01.230> here<00:34:02.060> minus<00:34:03.060> 1<00:34:03.090> what's<00:34:03.330> the - -00:34:03.380 --> 00:34:03.390 align:start position:0% -the metric here minus 1 what's the - - -00:34:03.390 --> 00:34:05.690 align:start position:0% -the metric here minus 1 what's the -determinant<00:34:03.600> of<00:34:03.960> the<00:34:04.020> metric<00:34:04.320> here<00:34:04.520> plus<00:34:05.520> 1 - -00:34:05.690 --> 00:34:05.700 align:start position:0% -determinant of the metric here plus 1 - - -00:34:05.700 --> 00:34:09.020 align:start position:0% -determinant of the metric here plus 1 -sum<00:34:06.600> where's<00:34:06.870> the<00:34:07.350> metric<00:34:07.800> will<00:34:08.399> have<00:34:08.610> had<00:34:08.850> to - -00:34:09.020 --> 00:34:09.030 align:start position:0% -sum where's the metric will have had to - - -00:34:09.030 --> 00:34:12.110 align:start position:0% -sum where's the metric will have had to -go<00:34:09.179> through<00:34:09.210> having<00:34:09.750> determinant<00:34:10.409> zero<00:34:11.000> so<00:34:12.000> if - -00:34:12.110 --> 00:34:12.120 align:start position:0% -go through having determinant zero so if - - -00:34:12.120 --> 00:34:14.830 align:start position:0% -go through having determinant zero so if -there's<00:34:12.330> something<00:34:12.690> in<00:34:12.780> the<00:34:12.899> dynamics<00:34:13.560> that - -00:34:14.830 --> 00:34:14.840 align:start position:0% -there's something in the dynamics that - - -00:34:14.840 --> 00:34:19.250 align:start position:0% -there's something in the dynamics that -forbids<00:34:16.220> determinant<00:34:17.220> going<00:34:17.490> to<00:34:17.640> 0<00:34:17.960> some<00:34:18.960> sort - -00:34:19.250 --> 00:34:19.260 align:start position:0% -forbids determinant going to 0 some sort - - -00:34:19.260 --> 00:34:22.310 align:start position:0% -forbids determinant going to 0 some sort -of<00:34:19.380> effect<00:34:20.250> in<00:34:20.490> the<00:34:20.580> energy<00:34:21.240> or<00:34:21.540> somehow<00:34:21.929> in - -00:34:22.310 --> 00:34:22.320 align:start position:0% -of effect in the energy or somehow in - - -00:34:22.320 --> 00:34:25.520 align:start position:0% -of effect in the energy or somehow in -the<00:34:22.470> in<00:34:22.740> the<00:34:22.860> dynamics<00:34:23.520> which<00:34:24.149> pushes<00:34:24.600> you - -00:34:25.520 --> 00:34:25.530 align:start position:0% -the in the dynamics which pushes you - - -00:34:25.530 --> 00:34:29.780 align:start position:0% -the in the dynamics which pushes you -away<00:34:26.060> from<00:34:27.060> determinant<00:34:27.960> 0<00:34:28.260> some<00:34:29.100> dynamics - -00:34:29.780 --> 00:34:29.790 align:start position:0% -away from determinant 0 some dynamics - - -00:34:29.790 --> 00:34:32.810 align:start position:0% -away from determinant 0 some dynamics -which<00:34:30.120> is<00:34:30.149> a<00:34:30.390> which<00:34:31.080> repels<00:34:31.649> the<00:34:32.130> metric<00:34:32.520> away - -00:34:32.810 --> 00:34:32.820 align:start position:0% -which is a which repels the metric away - - -00:34:32.820 --> 00:34:34.210 align:start position:0% -which is a which repels the metric away -from<00:34:33.060> determinant<00:34:33.659> 0 - -00:34:34.210 --> 00:34:34.220 align:start position:0% -from determinant 0 - - -00:34:34.220 --> 00:34:38.500 align:start position:0% -from determinant 0 -then<00:34:35.119> it<00:34:35.690> can't<00:34:36.109> change<00:34:36.560> signature<00:34:37.310> and<00:34:38.119> of - -00:34:38.500 --> 00:34:38.510 align:start position:0% -then it can't change signature and of - - -00:34:38.510 --> 00:34:40.180 align:start position:0% -then it can't change signature and of -course<00:34:38.750> in<00:34:38.899> general<00:34:39.079> relativity<00:34:39.319> there<00:34:40.010> is - -00:34:40.180 --> 00:34:40.190 align:start position:0% -course in general relativity there is - - -00:34:40.190 --> 00:34:43.059 align:start position:0% -course in general relativity there is -such<00:34:40.460> an<00:34:40.609> effect<00:34:41.619> passing<00:34:42.619> through<00:34:42.800> zero - -00:34:43.059 --> 00:34:43.069 align:start position:0% -such an effect passing through zero - - -00:34:43.069 --> 00:34:44.950 align:start position:0% -such an effect passing through zero -would<00:34:43.609> cost<00:34:43.849> infinite<00:34:44.389> amounts<00:34:44.750> of<00:34:44.869> energy - -00:34:44.950 --> 00:34:44.960 align:start position:0% -would cost infinite amounts of energy - - -00:34:44.960 --> 00:34:51.339 align:start position:0% -would cost infinite amounts of energy -and<00:34:45.470> and<00:34:46.460> so<00:34:46.669> it<00:34:46.790> doesn't<00:34:47.089> happen<00:34:50.200> or<00:34:51.200> if<00:34:51.290> it - -00:34:51.339 --> 00:34:51.349 align:start position:0% -and and so it doesn't happen or if it - - -00:34:51.349 --> 00:34:53.379 align:start position:0% -and and so it doesn't happen or if it -does<00:34:51.500> happen<00:34:51.710> it's<00:34:51.950> extremely<00:34:52.550> pathological - -00:34:53.379 --> 00:34:53.389 align:start position:0% -does happen it's extremely pathological - - -00:34:53.389 --> 00:34:55.480 align:start position:0% -does happen it's extremely pathological -I<00:34:53.599> don't<00:34:54.230> know<00:34:54.290> any<00:34:54.500> PK<00:34:54.740> any<00:34:54.950> case<00:34:55.190> where<00:34:55.399> it - -00:34:55.480 --> 00:34:55.490 align:start position:0% -I don't know any PK any case where it - - -00:34:55.490 --> 00:35:00.190 align:start position:0% -I don't know any PK any case where it -happens<00:34:58.000> ok<00:34:59.000> so<00:34:59.060> that's<00:34:59.390> those<00:34:59.869> are<00:34:59.930> good - -00:35:00.190 --> 00:35:00.200 align:start position:0% -happens ok so that's those are good - - -00:35:00.200 --> 00:35:00.760 align:start position:0% -happens ok so that's those are good -questions - -00:35:00.760 --> 00:35:00.770 align:start position:0% -questions - - -00:35:00.770 --> 00:35:05.710 align:start position:0% -questions -now<00:35:02.290> we<00:35:03.290> come<00:35:03.589> to<00:35:03.940> Feinstein's<00:35:04.940> equations<00:35:05.540> and - -00:35:05.710 --> 00:35:05.720 align:start position:0% -now we come to Feinstein's equations and - - -00:35:05.720 --> 00:35:08.090 align:start position:0% -now we come to Feinstein's equations and -so<00:35:05.839> forth<00:35:06.079> but<00:35:06.170> before<00:35:06.680> we<00:35:06.770> do - -00:35:08.090 --> 00:35:08.100 align:start position:0% -so forth but before we do - - -00:35:08.100 --> 00:35:12.740 align:start position:0% -so forth but before we do -[Music] - -00:35:12.740 --> 00:35:12.750 align:start position:0% - - - -00:35:12.750 --> 00:35:18.260 align:start position:0% - -no<00:35:15.050> change<00:35:16.250> there's<00:35:17.250> no<00:35:17.400> way<00:35:17.580> that<00:35:17.760> you<00:35:17.910> can<00:35:18.090> go - -00:35:18.260 --> 00:35:18.270 align:start position:0% -no change there's no way that you can go - - -00:35:18.270 --> 00:35:19.910 align:start position:0% -no change there's no way that you can go -from<00:35:18.510> a<00:35:18.690> positive<00:35:19.230> determinant<00:35:19.380> to<00:35:19.890> a - -00:35:19.910 --> 00:35:19.920 align:start position:0% -from a positive determinant to a - - -00:35:19.920 --> 00:35:21.680 align:start position:0% -from a positive determinant to a -negative<00:35:20.099> determinant<00:35:20.940> smoothly - -00:35:21.680 --> 00:35:21.690 align:start position:0% -negative determinant smoothly - - -00:35:21.690 --> 00:35:24.200 align:start position:0% -negative determinant smoothly -remember<00:35:22.320> the<00:35:22.410> metric<00:35:22.680> is<00:35:22.800> smooth<00:35:23.210> so - -00:35:24.200 --> 00:35:24.210 align:start position:0% -remember the metric is smooth so - - -00:35:24.210 --> 00:35:26.330 align:start position:0% -remember the metric is smooth so -supposing<00:35:25.200> over<00:35:25.470> here<00:35:25.619> supposing<00:35:26.099> in<00:35:26.190> this - -00:35:26.330 --> 00:35:26.340 align:start position:0% -supposing over here supposing in this - - -00:35:26.340 --> 00:35:29.660 align:start position:0% -supposing over here supposing in this -point<00:35:26.670> over<00:35:26.849> here<00:35:27.150> in<00:35:27.240> space-time<00:35:28.670> the - -00:35:29.660 --> 00:35:29.670 align:start position:0% -point over here in space-time the - - -00:35:29.670 --> 00:35:31.580 align:start position:0% -point over here in space-time the -determinant<00:35:30.420> of<00:35:30.510> the<00:35:30.660> metric<00:35:31.050> incidentally - -00:35:31.580 --> 00:35:31.590 align:start position:0% -determinant of the metric incidentally - - -00:35:31.590 --> 00:35:33.589 align:start position:0% -determinant of the metric incidentally -the<00:35:31.740> determinant<00:35:32.400> of<00:35:32.490> the<00:35:32.640> metric<00:35:33.030> is<00:35:33.180> usually - -00:35:33.589 --> 00:35:33.599 align:start position:0% -the determinant of the metric is usually - - -00:35:33.599 --> 00:35:37.190 align:start position:0% -the determinant of the metric is usually -just<00:35:33.840> called<00:35:34.140> G<00:35:34.920> like<00:35:35.430> that<00:35:35.840> it's<00:35:36.840> the<00:35:36.900> product - -00:35:37.190 --> 00:35:37.200 align:start position:0% -just called G like that it's the product - - -00:35:37.200 --> 00:35:39.530 align:start position:0% -just called G like that it's the product -of<00:35:37.380> the<00:35:37.500> eigenvalues<00:35:38.240> all<00:35:39.240> right - -00:35:39.530 --> 00:35:39.540 align:start position:0% -of the eigenvalues all right - - -00:35:39.540 --> 00:35:41.750 align:start position:0% -of the eigenvalues all right -supposing<00:35:40.290> the<00:35:40.530> determinant<00:35:41.190> of<00:35:41.310> the<00:35:41.430> metric - -00:35:41.750 --> 00:35:41.760 align:start position:0% -supposing the determinant of the metric - - -00:35:41.760 --> 00:35:45.650 align:start position:0% -supposing the determinant of the metric -over<00:35:42.030> here<00:35:42.390> was<00:35:43.140> -<00:35:44.000> in<00:35:45.000> other<00:35:45.060> words<00:35:45.180> supposing - -00:35:45.650 --> 00:35:45.660 align:start position:0% -over here was - in other words supposing - - -00:35:45.660 --> 00:35:48.620 align:start position:0% -over here was - in other words supposing -the<00:35:45.900> signature<00:35:46.410> was<00:35:46.830> 1<00:35:47.790> minus<00:35:48.000> 1<00:35:48.270> minus<00:35:48.300> 1 - -00:35:48.620 --> 00:35:48.630 align:start position:0% -the signature was 1 minus 1 minus 1 - - -00:35:48.630 --> 00:35:51.859 align:start position:0% -the signature was 1 minus 1 minus 1 -minus<00:35:48.660> 1<00:35:49.109> and<00:35:50.030> then<00:35:51.030> it<00:35:51.330> wants<00:35:51.599> to<00:35:51.720> make<00:35:51.840> a - -00:35:51.859 --> 00:35:51.869 align:start position:0% -minus 1 and then it wants to make a - - -00:35:51.869 --> 00:35:53.630 align:start position:0% -minus 1 and then it wants to make a -transition<00:35:52.470> in<00:35:52.590> some<00:35:52.740> other<00:35:52.859> place<00:35:53.160> over<00:35:53.460> here - -00:35:53.630 --> 00:35:53.640 align:start position:0% -transition in some other place over here - - -00:35:53.640 --> 00:35:55.970 align:start position:0% -transition in some other place over here -it's<00:35:53.849> gonna<00:35:53.940> have<00:35:54.210> the<00:35:54.359> opposite<00:35:54.780> metric<00:35:55.200> of - -00:35:55.970 --> 00:35:55.980 align:start position:0% -it's gonna have the opposite metric of - - -00:35:55.980 --> 00:35:57.980 align:start position:0% -it's gonna have the opposite metric of -the<00:35:56.369> opposite<00:35:56.730> determinant<00:35:57.359> here<00:35:57.810> it's - -00:35:57.980 --> 00:35:57.990 align:start position:0% -the opposite determinant here it's - - -00:35:57.990 --> 00:36:02.180 align:start position:0% -the opposite determinant here it's -positive<00:35:58.470> or<00:35:58.710> negative<00:35:59.609> sorry<00:36:00.300> negative<00:36:01.190> here - -00:36:02.180 --> 00:36:02.190 align:start position:0% -positive or negative sorry negative here - - -00:36:02.190 --> 00:36:07.250 align:start position:0% -positive or negative sorry negative here -it's<00:36:02.310> positive<00:36:04.550> well<00:36:05.550> here's<00:36:06.450> some<00:36:06.660> curves - -00:36:07.250 --> 00:36:07.260 align:start position:0% -it's positive well here's some curves - - -00:36:07.260 --> 00:36:09.980 align:start position:0% -it's positive well here's some curves -smooth<00:36:07.710> curve<00:36:08.010> between<00:36:08.220> them<00:36:08.720> someplace<00:36:09.720> if - -00:36:09.980 --> 00:36:09.990 align:start position:0% -smooth curve between them someplace if - - -00:36:09.990 --> 00:36:11.630 align:start position:0% -smooth curve between them someplace if -the<00:36:10.109> metric<00:36:10.410> is<00:36:10.560> smooth<00:36:10.830> it's<00:36:11.280> going<00:36:11.490> to<00:36:11.550> have - -00:36:11.630 --> 00:36:11.640 align:start position:0% -the metric is smooth it's going to have - - -00:36:11.640 --> 00:36:14.420 align:start position:0% -the metric is smooth it's going to have -to<00:36:11.760> go<00:36:11.880> to<00:36:11.940> 0<00:36:12.560> and<00:36:13.560> those<00:36:13.890> are<00:36:14.099> the<00:36:14.160> kind<00:36:14.369> of - -00:36:14.420 --> 00:36:14.430 align:start position:0% -to go to 0 and those are the kind of - - -00:36:14.430 --> 00:36:18.710 align:start position:0% -to go to 0 and those are the kind of -places<00:36:14.640> where<00:36:15.590> where<00:36:16.590> the<00:36:17.099> equations<00:36:17.720> push - -00:36:18.710 --> 00:36:18.720 align:start position:0% -places where where the equations push - - -00:36:18.720 --> 00:36:23.540 align:start position:0% -places where where the equations push -you<00:36:18.869> away<00:36:18.990> from<00:36:19.290> those<00:36:19.410> places<00:36:22.250> and<00:36:23.250> that's - -00:36:23.540 --> 00:36:23.550 align:start position:0% -you away from those places and that's - - -00:36:23.550 --> 00:36:25.010 align:start position:0% -you away from those places and that's -generally<00:36:23.700> true<00:36:24.089> that<00:36:24.330> a<00:36:24.359> change<00:36:24.690> it's - -00:36:25.010 --> 00:36:25.020 align:start position:0% -generally true that a change it's - - -00:36:25.020 --> 00:36:27.290 align:start position:0% -generally true that a change it's -generally<00:36:25.380> true<00:36:25.619> that<00:36:25.800> the<00:36:25.920> equations<00:36:26.460> push - -00:36:27.290 --> 00:36:27.300 align:start position:0% -generally true that the equations push - - -00:36:27.300 --> 00:36:31.250 align:start position:0% -generally true that the equations push -you<00:36:27.480> away<00:36:27.630> from<00:36:27.839> places<00:36:28.290> where<00:36:28.940> the<00:36:30.260> signature - -00:36:31.250 --> 00:36:31.260 align:start position:0% -you away from places where the signature - - -00:36:31.260 --> 00:36:33.829 align:start position:0% -you away from places where the signature -of<00:36:31.290> the<00:36:31.470> metric<00:36:31.859> might<00:36:32.070> change<00:36:32.430> on<00:36:32.730> you<00:36:32.880> so - -00:36:33.829 --> 00:36:33.839 align:start position:0% -of the metric might change on you so - - -00:36:33.839 --> 00:36:36.500 align:start position:0% -of the metric might change on you so -that's<00:36:33.990> a<00:36:34.050> good<00:36:34.200> thing<00:36:34.970> space-time<00:36:35.970> cannot - -00:36:36.500 --> 00:36:36.510 align:start position:0% -that's a good thing space-time cannot - - -00:36:36.510 --> 00:36:40.070 align:start position:0% -that's a good thing space-time cannot -change<00:36:36.810> into<00:36:37.200> space<00:36:37.500> it<00:36:38.599> space-time<00:36:39.599> cannot - -00:36:40.070 --> 00:36:40.080 align:start position:0% -change into space it space-time cannot - - -00:36:40.080 --> 00:36:43.960 align:start position:0% -change into space it space-time cannot -suddenly<00:36:40.609> become<00:36:41.609> 4<00:36:42.240> dimensions<00:36:42.900> of<00:36:43.050> space - -00:36:43.960 --> 00:36:43.970 align:start position:0% -suddenly become 4 dimensions of space - - -00:36:43.970 --> 00:36:49.310 align:start position:0% -suddenly become 4 dimensions of space -that<00:36:44.970> would<00:36:45.060> be<00:36:45.180> very<00:36:45.359> awkward<00:36:47.990> nor<00:36:48.990> can<00:36:49.290> it - -00:36:49.310 --> 00:36:49.320 align:start position:0% -that would be very awkward nor can it - - -00:36:49.320 --> 00:36:52.880 align:start position:0% -that would be very awkward nor can it -suddenly<00:36:49.890> become<00:36:50.690> 2<00:36:51.690> dimensions<00:36:52.200> of<00:36:52.349> time<00:36:52.589> in - -00:36:52.880 --> 00:36:52.890 align:start position:0% -suddenly become 2 dimensions of time in - - -00:36:52.890 --> 00:36:56.210 align:start position:0% -suddenly become 2 dimensions of time in -two<00:36:53.040> dimensions<00:36:53.550> of<00:36:53.670> space<00:36:54.170> there<00:36:55.170> are<00:36:55.500> there - -00:36:56.210 --> 00:36:56.220 align:start position:0% -two dimensions of space there are there - - -00:36:56.220 --> 00:36:58.370 align:start position:0% -two dimensions of space there are there -are<00:36:56.250> dynamics<00:36:57.030> which<00:36:57.780> prevent<00:36:58.200> that<00:36:58.320> from - -00:36:58.370 --> 00:36:58.380 align:start position:0% -are dynamics which prevent that from - - -00:36:58.380 --> 00:37:05.420 align:start position:0% -are dynamics which prevent that from -happening<00:37:00.320> okay - -00:37:05.420 --> 00:37:05.430 align:start position:0% - - - -00:37:05.430 --> 00:37:11.250 align:start position:0% - -let's<00:37:08.250> let's<00:37:09.250> come<00:37:09.940> to<00:37:10.240> seeing<00:37:10.600> what<00:37:10.870> we<00:37:11.050> can - -00:37:11.250 --> 00:37:11.260 align:start position:0% -let's let's come to seeing what we can - - -00:37:11.260 --> 00:37:13.980 align:start position:0% -let's let's come to seeing what we can -learn<00:37:11.610> about<00:37:12.610> as<00:37:12.970> much<00:37:13.150> as<00:37:13.330> possible<00:37:13.540> that<00:37:13.870> we - -00:37:13.980 --> 00:37:13.990 align:start position:0% -learn about as much as possible that we - - -00:37:13.990 --> 00:37:17.090 align:start position:0% -learn about as much as possible that we -can<00:37:14.140> learn<00:37:14.350> about<00:37:15.000> the<00:37:16.000> equations<00:37:16.720> of<00:37:16.900> motion - -00:37:17.090 --> 00:37:17.100 align:start position:0% -can learn about the equations of motion - - -00:37:17.100 --> 00:37:19.470 align:start position:0% -can learn about the equations of motion -the<00:37:18.100> field<00:37:18.460> equations<00:37:19.090> of<00:37:19.390> general - -00:37:19.470 --> 00:37:19.480 align:start position:0% -the field equations of general - - -00:37:19.480 --> 00:37:22.080 align:start position:0% -the field equations of general -relativity<00:37:20.250> we're<00:37:21.250> not<00:37:21.430> going<00:37:21.640> to<00:37:21.670> go<00:37:21.820> through - -00:37:22.080 --> 00:37:22.090 align:start position:0% -relativity we're not going to go through - - -00:37:22.090 --> 00:37:25.560 align:start position:0% -relativity we're not going to go through -the<00:37:22.270> whole<00:37:22.420> business<00:37:22.750> the<00:37:23.260> whole<00:37:24.010> business<00:37:24.570> is - -00:37:25.560 --> 00:37:25.570 align:start position:0% -the whole business the whole business is - - -00:37:25.570 --> 00:37:41.880 align:start position:0% -the whole business the whole business is -a<00:37:26.080> bit<00:37:26.260> tedious<00:37:40.170> we<00:37:41.170> may<00:37:41.320> or<00:37:41.440> may<00:37:41.530> not<00:37:41.560> get<00:37:41.860> to - -00:37:41.880 --> 00:37:41.890 align:start position:0% -a bit tedious we may or may not get to - - -00:37:41.890 --> 00:37:49.050 align:start position:0% -a bit tedious we may or may not get to -that<00:37:42.040> tonight<00:37:42.280> but<00:37:46.050> no<00:37:47.400> now<00:37:48.400> something - -00:37:49.050 --> 00:37:49.060 align:start position:0% -that tonight but no now something - - -00:37:49.060 --> 00:37:51.150 align:start position:0% -that tonight but no now something -happens<00:37:49.480> at<00:37:49.750> the<00:37:49.990> singularity<00:37:50.350> of<00:37:50.800> a<00:37:50.950> black - -00:37:51.150 --> 00:37:51.160 align:start position:0% -happens at the singularity of a black - - -00:37:51.160 --> 00:37:54.710 align:start position:0% -happens at the singularity of a black -hole<00:37:51.900> expect<00:37:52.900> everything<00:37:53.500> to<00:37:53.740> get<00:37:54.070> nasty<00:37:54.400> and - -00:37:54.710 --> 00:37:54.720 align:start position:0% -hole expect everything to get nasty and - - -00:37:54.720 --> 00:37:58.680 align:start position:0% -hole expect everything to get nasty and -all<00:37:55.720> bets<00:37:56.260> are<00:37:56.440> off<00:37:56.500> when<00:37:57.130> you<00:37:57.880> come<00:37:58.150> to<00:37:58.390> places - -00:37:58.680 --> 00:37:58.690 align:start position:0% -all bets are off when you come to places - - -00:37:58.690 --> 00:38:02.400 align:start position:0% -all bets are off when you come to places -like<00:37:59.220> singularities<00:38:00.220> of<00:38:00.250> black<00:38:00.670> holes<00:38:01.410> it's - -00:38:02.400 --> 00:38:02.410 align:start position:0% -like singularities of black holes it's - - -00:38:02.410 --> 00:38:04.560 align:start position:0% -like singularities of black holes it's -not<00:38:02.740> that<00:38:03.010> the<00:38:03.160> metric<00:38:03.550> changes<00:38:04.030> signature - -00:38:04.560 --> 00:38:04.570 align:start position:0% -not that the metric changes signature - - -00:38:04.570 --> 00:38:07.350 align:start position:0% -not that the metric changes signature -from<00:38:04.960> from<00:38:05.740> space<00:38:06.400> the<00:38:06.640> time<00:38:06.820> or<00:38:07.060> something - -00:38:07.350 --> 00:38:07.360 align:start position:0% -from from space the time or something - - -00:38:07.360 --> 00:38:10.230 align:start position:0% -from from space the time or something -like<00:38:07.450> that<00:38:07.510> just<00:38:08.310> the<00:38:09.310> smoothness<00:38:09.940> breaks - -00:38:10.230 --> 00:38:10.240 align:start position:0% -like that just the smoothness breaks - - -00:38:10.240 --> 00:38:13.460 align:start position:0% -like that just the smoothness breaks -down<00:38:10.540> and<00:38:11.020> components<00:38:11.920> of<00:38:12.010> the<00:38:12.070> metric<00:38:12.390> well - -00:38:13.460 --> 00:38:13.470 align:start position:0% -down and components of the metric well - - -00:38:13.470 --> 00:38:16.050 align:start position:0% -down and components of the metric well -curvatures<00:38:14.470> become<00:38:14.830> infinite<00:38:15.370> bad<00:38:15.760> things - -00:38:16.050 --> 00:38:16.060 align:start position:0% -curvatures become infinite bad things - - -00:38:16.060 --> 00:38:18.240 align:start position:0% -curvatures become infinite bad things -happen<00:38:16.480> but<00:38:16.720> those<00:38:17.440> are<00:38:17.590> the<00:38:17.620> exceptional - -00:38:18.240 --> 00:38:18.250 align:start position:0% -happen but those are the exceptional - - -00:38:18.250 --> 00:38:23.700 align:start position:0% -happen but those are the exceptional -cases<00:38:20.850> which<00:38:21.850> are<00:38:22.420> of<00:38:22.720> course<00:38:22.750> the<00:38:23.020> subject<00:38:23.350> of - -00:38:23.700 --> 00:38:23.710 align:start position:0% -cases which are of course the subject of - - -00:38:23.710 --> 00:38:25.980 align:start position:0% -cases which are of course the subject of -black<00:38:24.010> holes<00:38:24.310> and<00:38:24.610> singularity<00:38:25.330> theorems<00:38:25.600> and - -00:38:25.980 --> 00:38:25.990 align:start position:0% -black holes and singularity theorems and - - -00:38:25.990 --> 00:38:27.560 align:start position:0% -black holes and singularity theorems and -things<00:38:26.170> of<00:38:26.290> that<00:38:26.320> nature - -00:38:27.560 --> 00:38:27.570 align:start position:0% -things of that nature - - -00:38:27.570 --> 00:38:31.410 align:start position:0% -things of that nature -okay<00:38:29.850> and<00:38:30.850> it's<00:38:30.970> not<00:38:31.120> that<00:38:31.210> they<00:38:31.360> were - -00:38:31.410 --> 00:38:31.420 align:start position:0% -okay and it's not that they were - - -00:38:31.420 --> 00:38:34.370 align:start position:0% -okay and it's not that they were -irrelevant<00:38:31.690> they're<00:38:32.140> quite<00:38:32.320> relevant<00:38:32.530> but - -00:38:34.370 --> 00:38:34.380 align:start position:0% -irrelevant they're quite relevant but - - -00:38:34.380 --> 00:38:36.840 align:start position:0% -irrelevant they're quite relevant but -for<00:38:35.380> the<00:38:35.440> moment<00:38:35.860> let's<00:38:36.100> say<00:38:36.250> temporarily - -00:38:36.840 --> 00:38:36.850 align:start position:0% -for the moment let's say temporarily - - -00:38:36.850 --> 00:38:39.930 align:start position:0% -for the moment let's say temporarily -let's<00:38:37.630> just<00:38:37.960> ignore<00:38:38.530> or<00:38:39.010> not<00:38:39.250> ignore<00:38:39.700> but - -00:38:39.930 --> 00:38:39.940 align:start position:0% -let's just ignore or not ignore but - - -00:38:39.940 --> 00:38:41.850 align:start position:0% -let's just ignore or not ignore but -let's<00:38:40.120> pretend<00:38:40.780> that<00:38:41.170> there<00:38:41.470> are<00:38:41.620> no - -00:38:41.850 --> 00:38:41.860 align:start position:0% -let's pretend that there are no - - -00:38:41.860 --> 00:38:44.970 align:start position:0% -let's pretend that there are no -singularities - -00:38:44.970 --> 00:38:44.980 align:start position:0% - - - -00:38:44.980 --> 00:38:48.400 align:start position:0% - -alright<00:38:45.980> let's<00:38:46.160> talk<00:38:46.340> about<00:38:46.400> the<00:38:47.080> begin<00:38:48.080> with - -00:38:48.400 --> 00:38:48.410 align:start position:0% -alright let's talk about the begin with - - -00:38:48.410 --> 00:38:54.130 align:start position:0% -alright let's talk about the begin with -the<00:38:50.290> what<00:38:51.290> we<00:38:51.440> want<00:38:51.650> to<00:38:51.740> compare<00:38:52.190> is<00:38:53.140> Newton - -00:38:54.130 --> 00:38:54.140 align:start position:0% -the what we want to compare is Newton - - -00:38:54.140 --> 00:38:56.400 align:start position:0% -the what we want to compare is Newton -we've<00:38:54.830> done<00:38:55.010> this<00:38:55.190> in<00:38:55.400> various<00:38:55.760> situations - -00:38:56.400 --> 00:38:56.410 align:start position:0% -we've done this in various situations - - -00:38:56.410 --> 00:39:00.550 align:start position:0% -we've done this in various situations -with<00:38:57.410> some<00:38:58.160> ideas<00:38:58.640> from<00:38:58.970> Einstein<00:38:59.570> and<00:38:59.930> try<00:39:00.500> to - -00:39:00.550 --> 00:39:00.560 align:start position:0% -with some ideas from Einstein and try to - - -00:39:00.560 --> 00:39:02.290 align:start position:0% -with some ideas from Einstein and try to -put<00:39:00.920> together<00:39:01.070> or<00:39:01.460> try<00:39:01.580> to<00:39:01.700> guess<00:39:02.000> the - -00:39:02.290 --> 00:39:02.300 align:start position:0% -put together or try to guess the - - -00:39:02.300 --> 00:39:04.450 align:start position:0% -put together or try to guess the -equations<00:39:02.840> the<00:39:03.020> tensor<00:39:03.530> equations<00:39:04.160> that - -00:39:04.450 --> 00:39:04.460 align:start position:0% -equations the tensor equations that - - -00:39:04.460 --> 00:39:08.320 align:start position:0% -equations the tensor equations that -replace<00:39:04.850> the<00:39:04.880> Newtonian<00:39:05.570> equations<00:39:06.970> to<00:39:07.970> do<00:39:08.090> so - -00:39:08.320 --> 00:39:08.330 align:start position:0% -replace the Newtonian equations to do so - - -00:39:08.330 --> 00:39:09.220 align:start position:0% -replace the Newtonian equations to do so -if<00:39:08.540> we're<00:39:08.660> going<00:39:08.750> to<00:39:08.810> think<00:39:08.900> about<00:39:09.020> the - -00:39:09.220 --> 00:39:09.230 align:start position:0% -if we're going to think about the - - -00:39:09.230 --> 00:39:11.560 align:start position:0% -if we're going to think about the -Newtonian<00:39:09.740> equations<00:39:10.280> the<00:39:11.000> Newtonian - -00:39:11.560 --> 00:39:11.570 align:start position:0% -Newtonian equations the Newtonian - - -00:39:11.570 --> 00:39:14.110 align:start position:0% -Newtonian equations the Newtonian -equations<00:39:12.080> are<00:39:12.410> the<00:39:12.830> Einstein<00:39:13.370> equations<00:39:13.940> in - -00:39:14.110 --> 00:39:14.120 align:start position:0% -equations are the Einstein equations in - - -00:39:14.120 --> 00:39:16.690 align:start position:0% -equations are the Einstein equations in -certain<00:39:14.450> limits<00:39:14.900> there<00:39:15.860> are<00:39:16.010> certain<00:39:16.310> limits - -00:39:16.690 --> 00:39:16.700 align:start position:0% -certain limits there are certain limits - - -00:39:16.700 --> 00:39:18.970 align:start position:0% -certain limits there are certain limits -where<00:39:17.060> certain<00:39:17.510> things<00:39:17.540> are<00:39:18.110> big<00:39:18.410> and<00:39:18.710> other - -00:39:18.970 --> 00:39:18.980 align:start position:0% -where certain things are big and other - - -00:39:18.980 --> 00:39:22.510 align:start position:0% -where certain things are big and other -things<00:39:19.310> are<00:39:19.490> small<00:39:20.890> let's<00:39:21.890> see<00:39:22.160> if<00:39:22.250> we<00:39:22.370> can - -00:39:22.510 --> 00:39:22.520 align:start position:0% -things are small let's see if we can - - -00:39:22.520 --> 00:39:24.280 align:start position:0% -things are small let's see if we can -figure<00:39:22.730> out<00:39:22.850> what<00:39:23.180> things<00:39:23.420> are<00:39:23.660> likely<00:39:23.900> to<00:39:24.200> be - -00:39:24.280 --> 00:39:24.290 align:start position:0% -figure out what things are likely to be - - -00:39:24.290 --> 00:39:25.900 align:start position:0% -figure out what things are likely to be -big<00:39:24.590> and<00:39:24.860> what<00:39:25.040> things<00:39:25.250> are<00:39:25.370> likely<00:39:25.550> to<00:39:25.790> be - -00:39:25.900 --> 00:39:25.910 align:start position:0% -big and what things are likely to be - - -00:39:25.910 --> 00:39:30.130 align:start position:0% -big and what things are likely to be -small<00:39:26.240> and<00:39:26.660> then<00:39:27.940> systematically<00:39:28.940> think<00:39:29.930> of - -00:39:30.130 --> 00:39:30.140 align:start position:0% -small and then systematically think of - - -00:39:30.140 --> 00:39:32.890 align:start position:0% -small and then systematically think of -nonrelativistic<00:39:30.620> limit<00:39:31.250> as<00:39:31.790> the<00:39:32.420> limit<00:39:32.720> in - -00:39:32.890 --> 00:39:32.900 align:start position:0% -nonrelativistic limit as the limit in - - -00:39:32.900 --> 00:39:34.390 align:start position:0% -nonrelativistic limit as the limit in -which<00:39:32.930> small<00:39:33.410> things<00:39:33.770> are<00:39:33.980> really<00:39:34.130> much - -00:39:34.390 --> 00:39:34.400 align:start position:0% -which small things are really much - - -00:39:34.400 --> 00:39:38.350 align:start position:0% -which small things are really much -smaller<00:39:34.760> than<00:39:35.030> big<00:39:35.180> things<00:39:36.850> and<00:39:37.850> of<00:39:38.180> course - -00:39:38.350 --> 00:39:38.360 align:start position:0% -smaller than big things and of course - - -00:39:38.360 --> 00:39:39.790 align:start position:0% -smaller than big things and of course -what<00:39:38.540> we're<00:39:38.660> talking<00:39:39.020> about<00:39:39.080> mainly<00:39:39.470> being - -00:39:39.790 --> 00:39:39.800 align:start position:0% -what we're talking about mainly being - - -00:39:39.800 --> 00:39:42.820 align:start position:0% -what we're talking about mainly being -small<00:39:40.250> is<00:39:40.430> velocities<00:39:41.590> velocities<00:39:42.590> on<00:39:42.740> the - -00:39:42.820 --> 00:39:42.830 align:start position:0% -small is velocities velocities on the - - -00:39:42.830 --> 00:39:45.220 align:start position:0% -small is velocities velocities on the -one<00:39:42.980> hand<00:39:43.340> and<00:39:43.610> also<00:39:44.390> the<00:39:44.570> strength<00:39:44.990> of - -00:39:45.220 --> 00:39:45.230 align:start position:0% -one hand and also the strength of - - -00:39:45.230 --> 00:39:48.790 align:start position:0% -one hand and also the strength of -gravitational<00:39:46.040> fields<00:39:47.290> the<00:39:48.290> strength<00:39:48.620> of - -00:39:48.790 --> 00:39:48.800 align:start position:0% -gravitational fields the strength of - - -00:39:48.800 --> 00:39:50.590 align:start position:0% -gravitational fields the strength of -gravitational<00:39:49.250> fields<00:39:49.940> should<00:39:50.120> not<00:39:50.270> be<00:39:50.420> too - -00:39:50.590 --> 00:39:50.600 align:start position:0% -gravitational fields should not be too - - -00:39:50.600 --> 00:39:55.060 align:start position:0% -gravitational fields should not be too -big<00:39:50.630> that<00:39:51.130> entails<00:39:52.130> saying<00:39:52.820> either<00:39:53.800> that<00:39:54.800> no - -00:39:55.060 --> 00:39:55.070 align:start position:0% -big that entails saying either that no - - -00:39:55.070 --> 00:39:57.610 align:start position:0% -big that entails saying either that no -object<00:39:55.550> in<00:39:55.700> your<00:39:55.730> theory<00:39:56.240> is<00:39:56.420> so<00:39:56.720> compact<00:39:57.320> and - -00:39:57.610 --> 00:39:57.620 align:start position:0% -object in your theory is so compact and - - -00:39:57.620 --> 00:40:01.390 align:start position:0% -object in your theory is so compact and -so<00:39:57.830> compressed<00:39:58.430> that<00:39:59.440> that<00:40:00.440> the<00:40:00.680> acceleration - -00:40:01.390 --> 00:40:01.400 align:start position:0% -so compressed that that the acceleration - - -00:40:01.400 --> 00:40:05.080 align:start position:0% -so compressed that that the acceleration -due<00:40:01.760> to<00:40:01.790> gravity<00:40:01.940> becomes<00:40:02.690> enormous<00:40:03.340> or<00:40:04.340> it - -00:40:05.080 --> 00:40:05.090 align:start position:0% -due to gravity becomes enormous or it - - -00:40:05.090 --> 00:40:06.940 align:start position:0% -due to gravity becomes enormous or it -can<00:40:05.300> just<00:40:05.330> mean<00:40:05.660> stay<00:40:06.170> away<00:40:06.440> from<00:40:06.740> such - -00:40:06.940 --> 00:40:06.950 align:start position:0% -can just mean stay away from such - - -00:40:06.950 --> 00:40:09.540 align:start position:0% -can just mean stay away from such -objects<00:40:07.520> stay<00:40:07.700> far<00:40:07.940> away<00:40:08.120> from<00:40:08.270> such<00:40:08.660> objects - -00:40:09.540 --> 00:40:09.550 align:start position:0% -objects stay far away from such objects - - -00:40:09.550 --> 00:40:12.990 align:start position:0% -objects stay far away from such objects -which<00:40:10.550> of<00:40:10.700> course<00:40:10.730> are<00:40:11.030> black<00:40:11.090> holes<00:40:11.830> but - -00:40:12.990 --> 00:40:13.000 align:start position:0% -which of course are black holes but - - -00:40:13.000 --> 00:40:16.900 align:start position:0% -which of course are black holes but -slowly<00:40:14.000> moving<00:40:14.510> things<00:40:14.980> with<00:40:15.980> relatively - -00:40:16.900 --> 00:40:16.910 align:start position:0% -slowly moving things with relatively - - -00:40:16.910 --> 00:40:19.180 align:start position:0% -slowly moving things with relatively -weak<00:40:17.270> gravitational<00:40:17.960> fields<00:40:18.800> are<00:40:18.980> the - -00:40:19.180 --> 00:40:19.190 align:start position:0% -weak gravitational fields are the - - -00:40:19.190 --> 00:40:22.420 align:start position:0% -weak gravitational fields are the -subject<00:40:19.610> of<00:40:19.730> Newton<00:40:20.180> right<00:40:21.020> well<00:40:21.710> first<00:40:22.250> of - -00:40:22.420 --> 00:40:22.430 align:start position:0% -subject of Newton right well first of - - -00:40:22.430 --> 00:40:26.020 align:start position:0% -subject of Newton right well first of -all<00:40:22.580> the<00:40:23.360> first<00:40:23.630> thing<00:40:23.810> to<00:40:23.960> say<00:40:24.170> is<00:40:24.500> that<00:40:25.160> the - -00:40:26.020 --> 00:40:26.030 align:start position:0% -all the first thing to say is that the - - -00:40:26.030 --> 00:40:29.680 align:start position:0% -all the first thing to say is that the -metric<00:40:26.600> under<00:40:27.200> such<00:40:27.550> circumstances<00:40:28.550> in<00:40:29.360> a - -00:40:29.680 --> 00:40:29.690 align:start position:0% -metric under such circumstances in a - - -00:40:29.690 --> 00:40:31.390 align:start position:0% -metric under such circumstances in a -frame<00:40:30.020> of<00:40:30.050> reference<00:40:30.260> where<00:40:30.920> everything<00:40:31.280> is - -00:40:31.390 --> 00:40:31.400 align:start position:0% -frame of reference where everything is - - -00:40:31.400 --> 00:40:35.530 align:start position:0% -frame of reference where everything is -moving<00:40:31.520> slowly<00:40:32.000> is<00:40:32.830> close<00:40:33.910> to<00:40:34.910> the<00:40:35.030> ordinary - -00:40:35.530 --> 00:40:35.540 align:start position:0% -moving slowly is close to the ordinary - - -00:40:35.540 --> 00:40:38.770 align:start position:0% -moving slowly is close to the ordinary -nonrelativistic<00:40:35.960> metric<00:40:36.920> that<00:40:37.730> means<00:40:38.120> that - -00:40:38.770 --> 00:40:38.780 align:start position:0% -nonrelativistic metric that means that - - -00:40:38.780 --> 00:40:42.400 align:start position:0% -nonrelativistic metric that means that -apart<00:40:39.470> from<00:40:39.530> small<00:40:40.100> quantities<00:40:41.140> G<00:40:42.140> naught - -00:40:42.400 --> 00:40:42.410 align:start position:0% -apart from small quantities G naught - - -00:40:42.410 --> 00:40:51.800 align:start position:0% -apart from small quantities G naught -naught<00:40:43.090> is<00:40:44.090> 1<00:40:44.180> I'm<00:40:45.050> gonna<00:40:45.290> write<00:40:45.530> 1<00:40:45.800> plus<00:40:47.470> small - -00:40:51.800 --> 00:40:51.810 align:start position:0% - - - -00:40:51.810 --> 00:40:54.150 align:start position:0% - -and<00:40:52.810> we're<00:40:52.930> gonna<00:40:53.020> keep<00:40:53.290> track<00:40:53.620> of<00:40:53.680> small - -00:40:54.150 --> 00:40:54.160 align:start position:0% -and we're gonna keep track of small - - -00:40:54.160 --> 00:40:55.980 align:start position:0% -and we're gonna keep track of small -things<00:40:54.490> all<00:40:54.730> right<00:40:54.940> but<00:40:55.330> we're<00:40:55.660> gonna<00:40:55.750> keep - -00:40:55.980 --> 00:40:55.990 align:start position:0% -things all right but we're gonna keep - - -00:40:55.990 --> 00:40:57.600 align:start position:0% -things all right but we're gonna keep -track<00:40:56.230> of<00:40:56.290> them<00:40:56.590> to<00:40:56.830> the<00:40:56.950> leading<00:40:57.400> water - -00:40:57.600 --> 00:40:57.610 align:start position:0% -track of them to the leading water - - -00:40:57.610 --> 00:41:00.800 align:start position:0% -track of them to the leading water -basically<00:40:58.270> all<00:40:59.200> right<00:40:59.320> it<00:40:59.650> also<00:40:59.830> means<00:41:00.280> that - -00:41:00.800 --> 00:41:00.810 align:start position:0% -basically all right it also means that - - -00:41:00.810 --> 00:41:05.700 align:start position:0% -basically all right it also means that -what<00:41:01.810> about<00:41:02.250> g11<00:41:03.750> well<00:41:04.750> that's<00:41:04.930> equal<00:41:05.140> to - -00:41:05.700 --> 00:41:05.710 align:start position:0% -what about g11 well that's equal to - - -00:41:05.710 --> 00:41:10.800 align:start position:0% -what about g11 well that's equal to -minus<00:41:06.130> one<00:41:06.690> plus<00:41:07.690> small<00:41:08.020> things<00:41:09.360> what<00:41:10.360> about<00:41:10.570> G - -00:41:10.800 --> 00:41:10.810 align:start position:0% -minus one plus small things what about G - - -00:41:10.810 --> 00:41:15.960 align:start position:0% -minus one plus small things what about G -2<00:41:11.020> -<00:41:11.290> same<00:41:11.800> thing<00:41:12.100> what<00:41:13.030> about<00:41:13.060> G<00:41:13.690> not<00:41:14.260> one<00:41:14.970> well - -00:41:15.960 --> 00:41:15.970 align:start position:0% -2 - same thing what about G not one well - - -00:41:15.970 --> 00:41:18.270 align:start position:0% -2 - same thing what about G not one well -that's<00:41:16.090> something<00:41:16.390> that<00:41:16.660> does<00:41:16.840> not<00:41:16.990> appear<00:41:17.650> in - -00:41:18.270 --> 00:41:18.280 align:start position:0% -that's something that does not appear in - - -00:41:18.280 --> 00:41:22.110 align:start position:0% -that's something that does not appear in -ordinary<00:41:18.760> special<00:41:19.270> relativity<00:41:19.720> and<00:41:20.730> so<00:41:21.730> if - -00:41:22.110 --> 00:41:22.120 align:start position:0% -ordinary special relativity and so if - - -00:41:22.120 --> 00:41:25.980 align:start position:0% -ordinary special relativity and so if -we're<00:41:22.840> when<00:41:23.530> I<00:41:23.560> say<00:41:23.800> I<00:41:24.010> should<00:41:24.700> say<00:41:24.850> when<00:41:25.630> I<00:41:25.660> say - -00:41:25.980 --> 00:41:25.990 align:start position:0% -we're when I say I should say when I say - - -00:41:25.990 --> 00:41:31.770 align:start position:0% -we're when I say I should say when I say -that<00:41:26.520> that<00:41:29.340> there<00:41:30.340> are<00:41:30.400> two<00:41:30.700> limits<00:41:31.150> both<00:41:31.540> of - -00:41:31.770 --> 00:41:31.780 align:start position:0% -that that there are two limits both of - - -00:41:31.780 --> 00:41:33.780 align:start position:0% -that that there are two limits both of -which<00:41:31.870> we<00:41:32.080> should<00:41:32.140> keep<00:41:32.380> track<00:41:32.650> of<00:41:32.890> one<00:41:33.670> of - -00:41:33.780 --> 00:41:33.790 align:start position:0% -which we should keep track of one of - - -00:41:33.790 --> 00:41:35.900 align:start position:0% -which we should keep track of one of -them<00:41:33.940> is<00:41:34.120> things<00:41:34.360> moving<00:41:34.780> very<00:41:34.960> very<00:41:35.110> slowly - -00:41:35.900 --> 00:41:35.910 align:start position:0% -them is things moving very very slowly - - -00:41:35.910 --> 00:41:40.980 align:start position:0% -them is things moving very very slowly -in<00:41:36.910> which<00:41:37.120> case<00:41:37.420> our<00:41:37.900> Newton<00:41:39.180> a<00:41:40.180> week<00:41:40.450> Fields - -00:41:40.980 --> 00:41:40.990 align:start position:0% -in which case our Newton a week Fields - - -00:41:40.990 --> 00:41:43.350 align:start position:0% -in which case our Newton a week Fields -and<00:41:41.140> slowly<00:41:41.500> then<00:41:42.340> Newton<00:41:42.730> is<00:41:43.060> the<00:41:43.210> right - -00:41:43.350 --> 00:41:43.360 align:start position:0% -and slowly then Newton is the right - - -00:41:43.360 --> 00:41:43.890 align:start position:0% -and slowly then Newton is the right -physics - -00:41:43.890 --> 00:41:43.900 align:start position:0% -physics - - -00:41:43.900 --> 00:41:48.950 align:start position:0% -physics -what<00:41:44.590> about<00:41:44.710> week<00:41:45.100> fields<00:41:45.730> but<00:41:46.090> you<00:41:46.210> move<00:41:46.390> fast - -00:41:48.950 --> 00:41:48.960 align:start position:0% - - - -00:41:48.960 --> 00:41:53.010 align:start position:0% - -special<00:41:49.960> relativity<00:41:50.290> all<00:41:51.100> right<00:41:51.190> so<00:41:51.880> we<00:41:52.870> want - -00:41:53.010 --> 00:41:53.020 align:start position:0% -special relativity all right so we want - - -00:41:53.020 --> 00:41:55.470 align:start position:0% -special relativity all right so we want -to<00:41:53.200> we<00:41:53.620> want<00:41:53.680> to<00:41:53.940> deal<00:41:54.940> with<00:41:55.060> both<00:41:55.210> of<00:41:55.270> those - -00:41:55.470 --> 00:41:55.480 align:start position:0% -to we want to deal with both of those - - -00:41:55.480 --> 00:41:58.770 align:start position:0% -to we want to deal with both of those -but<00:41:56.050> in<00:41:56.170> special<00:41:56.350> relativity<00:41:57.180> G<00:41:58.180> naught<00:41:58.480> 1 - -00:41:58.770 --> 00:41:58.780 align:start position:0% -but in special relativity G naught 1 - - -00:41:58.780 --> 00:42:02.400 align:start position:0% -but in special relativity G naught 1 -what<00:41:59.020> is<00:41:59.050> G<00:41:59.260> naught<00:41:59.410> 1<00:42:00.480> at<00:42:01.480> 0<00:42:01.750> that's<00:42:02.050> not<00:42:02.230> there - -00:42:02.400 --> 00:42:02.410 align:start position:0% -what is G naught 1 at 0 that's not there - - -00:42:02.410 --> 00:42:03.990 align:start position:0% -what is G naught 1 at 0 that's not there -in<00:42:02.560> special<00:42:02.890> relativity<00:42:03.250> where<00:42:03.580> a<00:42:03.610> special - -00:42:03.990 --> 00:42:04.000 align:start position:0% -in special relativity where a special - - -00:42:04.000 --> 00:42:05.940 align:start position:0% -in special relativity where a special -relativity<00:42:04.360> here's<00:42:04.810> special<00:42:05.230> relativity<00:42:05.560> and - -00:42:05.940 --> 00:42:05.950 align:start position:0% -relativity here's special relativity and - - -00:42:05.950 --> 00:42:10.400 align:start position:0% -relativity here's special relativity and -there<00:42:06.550> were<00:42:06.640> no<00:42:06.970> terms<00:42:07.300> multiplying<00:42:08.080> DT<00:42:08.530> by<00:42:08.710> DX - -00:42:10.400 --> 00:42:10.410 align:start position:0% -there were no terms multiplying DT by DX - - -00:42:10.410 --> 00:42:13.950 align:start position:0% -there were no terms multiplying DT by DX -now<00:42:11.410> this<00:42:11.620> is<00:42:11.770> equal<00:42:12.010> to<00:42:12.130> zero<00:42:12.660> plus<00:42:13.660> small - -00:42:13.950 --> 00:42:13.960 align:start position:0% -now this is equal to zero plus small - - -00:42:13.960 --> 00:42:16.620 align:start position:0% -now this is equal to zero plus small -things<00:42:14.910> we<00:42:15.910> can<00:42:16.060> write<00:42:16.240> this<00:42:16.360> we<00:42:16.510> can - -00:42:16.620 --> 00:42:16.630 align:start position:0% -things we can write this we can - - -00:42:16.630 --> 00:42:22.800 align:start position:0% -things we can write this we can -generalize<00:42:17.200> this<00:42:20.190> GM<00:42:21.190> n<00:42:21.580> I'm<00:42:22.240> not<00:42:22.450> gonna<00:42:22.600> write - -00:42:22.800 --> 00:42:22.810 align:start position:0% -generalize this GM n I'm not gonna write - - -00:42:22.810 --> 00:42:24.210 align:start position:0% -generalize this GM n I'm not gonna write -the<00:42:22.960> clinic<00:42:23.260> a<00:42:23.350> delta<00:42:23.710> I'm<00:42:23.830> just<00:42:24.010> going<00:42:24.190> to - -00:42:24.210 --> 00:42:24.220 align:start position:0% -the clinic a delta I'm just going to - - -00:42:24.220 --> 00:42:27.390 align:start position:0% -the clinic a delta I'm just going to -write<00:42:24.400> that<00:42:24.640> g11<00:42:25.390> equals<00:42:25.420> G<00:42:25.930> -<00:42:26.140> 2<00:42:26.380> equals<00:42:26.410> G<00:42:26.890> 3<00:42:27.130> 3 - -00:42:27.390 --> 00:42:27.400 align:start position:0% -write that g11 equals G - 2 equals G 3 3 - - -00:42:27.400 --> 00:42:41.570 align:start position:0% -write that g11 equals G - 2 equals G 3 3 -equals<00:42:27.880> and<00:42:28.270> so<00:42:28.480> forth - -00:42:41.570 --> 00:42:41.580 align:start position:0% - - - -00:42:41.580 --> 00:42:46.520 align:start position:0% - -and<00:42:43.910> g12<00:42:44.910> equals<00:42:45.600> blah<00:42:45.870> blah<00:42:46.050> blah<00:42:46.170> blah<00:42:46.320> blah - -00:42:46.520 --> 00:42:46.530 align:start position:0% -and g12 equals blah blah blah blah blah - - -00:42:46.530 --> 00:42:49.130 align:start position:0% -and g12 equals blah blah blah blah blah -equals<00:42:47.040> zero<00:42:47.280> now<00:42:48.060> I<00:42:48.270> don't<00:42:48.390> literally<00:42:48.930> mean - -00:42:49.130 --> 00:42:49.140 align:start position:0% -equals zero now I don't literally mean - - -00:42:49.140 --> 00:42:51.350 align:start position:0% -equals zero now I don't literally mean -this<00:42:49.530> that<00:42:49.770> they're<00:42:49.890> literally<00:42:50.760> equal<00:42:51.240> to - -00:42:51.350 --> 00:42:51.360 align:start position:0% -this that they're literally equal to - - -00:42:51.360 --> 00:42:54.170 align:start position:0% -this that they're literally equal to -zero<00:42:51.600> I<00:42:51.870> mean<00:42:52.170> they're<00:42:52.350> equal<00:42:52.740> to<00:42:52.830> zero<00:42:53.580> plus - -00:42:54.170 --> 00:42:54.180 align:start position:0% -zero I mean they're equal to zero plus - - -00:42:54.180 --> 00:43:01.310 align:start position:0% -zero I mean they're equal to zero plus -small<00:42:56.840> plus<00:42:57.840> small<00:42:59.660> we<00:43:00.660> will<00:43:00.810> keep<00:43:01.020> track<00:43:01.260> of - -00:43:01.310 --> 00:43:01.320 align:start position:0% -small plus small we will keep track of - - -00:43:01.320 --> 00:43:05.950 align:start position:0% -small plus small we will keep track of -small<00:43:01.740> things<00:43:02.030> but<00:43:04.220> the<00:43:05.220> first<00:43:05.490> approximation - -00:43:05.950 --> 00:43:05.960 align:start position:0% -small things but the first approximation - - -00:43:05.960 --> 00:43:08.770 align:start position:0% -small things but the first approximation -is<00:43:06.960> just<00:43:07.290> special<00:43:07.470> relativity<00:43:07.920> now<00:43:08.250> next - -00:43:08.770 --> 00:43:08.780 align:start position:0% -is just special relativity now next - - -00:43:08.780 --> 00:43:11.650 align:start position:0% -is just special relativity now next -let's<00:43:09.780> concentrate<00:43:10.170> on<00:43:10.410> energy<00:43:11.010> and<00:43:11.220> momentum - -00:43:11.650 --> 00:43:11.660 align:start position:0% -let's concentrate on energy and momentum - - -00:43:11.660 --> 00:43:13.580 align:start position:0% -let's concentrate on energy and momentum -we're<00:43:12.660> going<00:43:12.840> to<00:43:12.930> be<00:43:13.020> dealing<00:43:13.410> with<00:43:13.440> the - -00:43:13.580 --> 00:43:13.590 align:start position:0% -we're going to be dealing with the - - -00:43:13.590 --> 00:43:15.920 align:start position:0% -we're going to be dealing with the -energy<00:43:13.890> momentum<00:43:14.040> tensor<00:43:14.850> and<00:43:15.150> the<00:43:15.180> energy - -00:43:15.920 --> 00:43:15.930 align:start position:0% -energy momentum tensor and the energy - - -00:43:15.930 --> 00:43:18.470 align:start position:0% -energy momentum tensor and the energy -momentum<00:43:16.110> tensor<00:43:16.890> is<00:43:17.640> essentially<00:43:18.360> the - -00:43:18.470 --> 00:43:18.480 align:start position:0% -momentum tensor is essentially the - - -00:43:18.480 --> 00:43:21.350 align:start position:0% -momentum tensor is essentially the -right-hand<00:43:19.230> side<00:43:19.680> of<00:43:20.190> Einstein's<00:43:20.760> equations - -00:43:21.350 --> 00:43:21.360 align:start position:0% -right-hand side of Einstein's equations - - -00:43:21.360 --> 00:43:23.450 align:start position:0% -right-hand side of Einstein's equations -it's<00:43:21.540> the<00:43:21.750> sources<00:43:22.350> of<00:43:22.380> a<00:43:22.710> gravitational - -00:43:23.450 --> 00:43:23.460 align:start position:0% -it's the sources of a gravitational - - -00:43:23.460 --> 00:43:26.270 align:start position:0% -it's the sources of a gravitational -field<00:43:24.230> energy<00:43:25.230> of<00:43:25.380> course<00:43:25.620> is<00:43:25.830> the<00:43:25.950> same<00:43:26.100> as - -00:43:26.270 --> 00:43:26.280 align:start position:0% -field energy of course is the same as - - -00:43:26.280 --> 00:43:28.840 align:start position:0% -field energy of course is the same as -mass<00:43:27.020> through<00:43:28.020> the<00:43:28.140> equals<00:43:28.500> mc-squared - -00:43:28.840 --> 00:43:28.850 align:start position:0% -mass through the equals mc-squared - - -00:43:28.850 --> 00:43:31.580 align:start position:0% -mass through the equals mc-squared -relationship<00:43:29.850> mass<00:43:30.840> is<00:43:31.140> the<00:43:31.230> source<00:43:31.470> of - -00:43:31.580 --> 00:43:31.590 align:start position:0% -relationship mass is the source of - - -00:43:31.590 --> 00:43:34.130 align:start position:0% -relationship mass is the source of -gravitational<00:43:32.340> field<00:43:32.610> and<00:43:32.910> so<00:43:33.480> energy<00:43:33.930> is<00:43:33.960> the - -00:43:34.130 --> 00:43:34.140 align:start position:0% -gravitational field and so energy is the - - -00:43:34.140 --> 00:43:36.620 align:start position:0% -gravitational field and so energy is the -source<00:43:34.320> of<00:43:34.530> gravitational<00:43:35.190> field<00:43:35.490> but<00:43:36.180> since - -00:43:36.620 --> 00:43:36.630 align:start position:0% -source of gravitational field but since - - -00:43:36.630 --> 00:43:38.510 align:start position:0% -source of gravitational field but since -we're<00:43:36.780> forced<00:43:37.110> to<00:43:37.170> write<00:43:37.410> tensor<00:43:37.890> equations - -00:43:38.510 --> 00:43:38.520 align:start position:0% -we're forced to write tensor equations - - -00:43:38.520 --> 00:43:41.540 align:start position:0% -we're forced to write tensor equations -in<00:43:39.140> some<00:43:40.140> frame<00:43:40.410> of<00:43:40.620> reference<00:43:41.070> we<00:43:41.250> may<00:43:41.430> have - -00:43:41.540 --> 00:43:41.550 align:start position:0% -in some frame of reference we may have - - -00:43:41.550 --> 00:43:43.790 align:start position:0% -in some frame of reference we may have -an<00:43:41.700> energy<00:43:42.090> in<00:43:42.390> some<00:43:42.960> other<00:43:43.200> frame<00:43:43.620> of - -00:43:43.790 --> 00:43:43.800 align:start position:0% -an energy in some other frame of - - -00:43:43.800 --> 00:43:45.950 align:start position:0% -an energy in some other frame of -reference<00:43:44.220> that<00:43:44.430> same<00:43:44.790> energy<00:43:45.120> turns<00:43:45.720> into - -00:43:45.950 --> 00:43:45.960 align:start position:0% -reference that same energy turns into - - -00:43:45.960 --> 00:43:49.610 align:start position:0% -reference that same energy turns into -energy<00:43:46.350> and<00:43:46.470> the<00:43:46.980> momentum<00:43:48.140> in<00:43:49.140> a<00:43:49.200> frame<00:43:49.440> of - -00:43:49.610 --> 00:43:49.620 align:start position:0% -energy and the momentum in a frame of - - -00:43:49.620 --> 00:43:51.710 align:start position:0% -energy and the momentum in a frame of -reference<00:43:49.650> where<00:43:50.190> there's<00:43:50.370> some<00:43:50.490> energy<00:43:50.720> if - -00:43:51.710 --> 00:43:51.720 align:start position:0% -reference where there's some energy if - - -00:43:51.720 --> 00:43:53.510 align:start position:0% -reference where there's some energy if -we<00:43:51.960> move<00:43:52.170> past<00:43:52.530> that<00:43:52.860> energy<00:43:53.280> that<00:43:53.430> means - -00:43:53.510 --> 00:43:53.520 align:start position:0% -we move past that energy that means - - -00:43:53.520 --> 00:43:56.060 align:start position:0% -we move past that energy that means -there's<00:43:53.730> some<00:43:53.850> mass<00:43:54.060> there<00:43:54.360> if<00:43:55.140> we<00:43:55.350> move<00:43:55.560> past - -00:43:56.060 --> 00:43:56.070 align:start position:0% -there's some mass there if we move past - - -00:43:56.070 --> 00:43:58.130 align:start position:0% -there's some mass there if we move past -that<00:43:56.430> energy<00:43:56.850> then<00:43:57.360> of<00:43:57.450> course<00:43:57.660> we<00:43:57.810> see<00:43:57.960> some - -00:43:58.130 --> 00:43:58.140 align:start position:0% -that energy then of course we see some - - -00:43:58.140 --> 00:44:00.500 align:start position:0% -that energy then of course we see some -momentum<00:43:58.650> so<00:43:58.920> momentum<00:43:59.820> and<00:44:00.000> energy<00:44:00.060> go - -00:44:00.500 --> 00:44:00.510 align:start position:0% -momentum so momentum and energy go - - -00:44:00.510 --> 00:44:02.690 align:start position:0% -momentum so momentum and energy go -together<00:44:00.720> and<00:44:01.410> if<00:44:02.340> we're<00:44:02.430> going<00:44:02.520> to<00:44:02.610> be - -00:44:02.690 --> 00:44:02.700 align:start position:0% -together and if we're going to be - - -00:44:02.700 --> 00:44:04.430 align:start position:0% -together and if we're going to be -writing<00:44:02.880> tensor<00:44:03.360> equations<00:44:03.930> we<00:44:04.170> should - -00:44:04.430 --> 00:44:04.440 align:start position:0% -writing tensor equations we should - - -00:44:04.440 --> 00:44:06.050 align:start position:0% -writing tensor equations we should -expect<00:44:04.860> on<00:44:05.070> the<00:44:05.160> right-hand<00:44:05.550> side<00:44:05.640> of<00:44:05.910> those - -00:44:06.050 --> 00:44:06.060 align:start position:0% -expect on the right-hand side of those - - -00:44:06.060 --> 00:44:08.420 align:start position:0% -expect on the right-hand side of those -equations<00:44:06.680> some<00:44:07.680> of<00:44:07.800> the<00:44:07.860> equations<00:44:08.280> will - -00:44:08.420 --> 00:44:08.430 align:start position:0% -equations some of the equations will - - -00:44:08.430 --> 00:44:11.720 align:start position:0% -equations some of the equations will -have<00:44:08.610> momentum<00:44:08.880> in<00:44:09.240> them<00:44:09.890> as<00:44:10.890> sources<00:44:11.340> of<00:44:11.640> the - -00:44:11.720 --> 00:44:11.730 align:start position:0% -have momentum in them as sources of the - - -00:44:11.730 --> 00:44:14.270 align:start position:0% -have momentum in them as sources of the -gravitational<00:44:12.450> field<00:44:12.630> in<00:44:12.930> other<00:44:13.080> words<00:44:13.290> the - -00:44:14.270 --> 00:44:14.280 align:start position:0% -gravitational field in other words the - - -00:44:14.280 --> 00:44:16.850 align:start position:0% -gravitational field in other words the -energy<00:44:14.790> momentum<00:44:14.970> tensor<00:44:15.780> is<00:44:16.080> the<00:44:16.230> important - -00:44:16.850 --> 00:44:16.860 align:start position:0% -energy momentum tensor is the important - - -00:44:16.860 --> 00:44:20.690 align:start position:0% -energy momentum tensor is the important -thing<00:44:17.750> but<00:44:18.750> which<00:44:19.230> components<00:44:20.100> of<00:44:20.250> the<00:44:20.370> energy - -00:44:20.690 --> 00:44:20.700 align:start position:0% -thing but which components of the energy - - -00:44:20.700 --> 00:44:23.570 align:start position:0% -thing but which components of the energy -momentum<00:44:20.820> tensor<00:44:21.630> are<00:44:22.290> likely<00:44:22.770> to<00:44:22.950> be<00:44:23.070> big<00:44:23.340> and - -00:44:23.570 --> 00:44:23.580 align:start position:0% -momentum tensor are likely to be big and - - -00:44:23.580 --> 00:44:29.330 align:start position:0% -momentum tensor are likely to be big and -small<00:44:24.240> in<00:44:24.720> the<00:44:25.430> slow<00:44:26.430> weak<00:44:27.120> field<00:44:27.510> limit<00:44:28.340> so - -00:44:29.330 --> 00:44:29.340 align:start position:0% -small in the slow weak field limit so - - -00:44:29.340 --> 00:44:32.120 align:start position:0% -small in the slow weak field limit so -let's<00:44:29.610> begin<00:44:29.820> with<00:44:30.060> the<00:44:30.900> ordinary<00:44:31.440> energy<00:44:31.950> of - -00:44:32.120 --> 00:44:32.130 align:start position:0% -let's begin with the ordinary energy of - - -00:44:32.130 --> 00:44:34.910 align:start position:0% -let's begin with the ordinary energy of -a<00:44:32.220> particle<00:44:32.690> what's<00:44:33.690> the<00:44:34.170> energy<00:44:34.380> of<00:44:34.830> a - -00:44:34.910 --> 00:44:34.920 align:start position:0% -a particle what's the energy of a - - -00:44:34.920 --> 00:44:40.280 align:start position:0% -a particle what's the energy of a -particle<00:44:35.280> of<00:44:35.490> mass<00:44:35.550> m<00:44:36.030> and<00:44:36.600> free<00:44:37.110> space<00:44:39.290> MV - -00:44:40.280 --> 00:44:40.290 align:start position:0% -particle of mass m and free space MV - - -00:44:40.290 --> 00:44:42.340 align:start position:0% -particle of mass m and free space MV -squared<00:44:40.590> over<00:44:40.680> 2<00:44:40.950> good - -00:44:42.340 --> 00:44:42.350 align:start position:0% -squared over 2 good - - -00:44:42.350 --> 00:44:49.040 align:start position:0% -squared over 2 good -one-half<00:44:43.350> MV<00:44:44.870> squared<00:44:46.910> over<00:44:47.910> 2<00:44:48.090> no<00:44:48.420> just<00:44:48.810> 1/2 - -00:44:49.040 --> 00:44:49.050 align:start position:0% -one-half MV squared over 2 no just 1/2 - - -00:44:49.050 --> 00:44:50.990 align:start position:0% -one-half MV squared over 2 no just 1/2 -MV<00:44:49.380> square<00:44:49.650> and<00:44:50.070> remember<00:44:50.400> V<00:44:50.550> is<00:44:50.670> small - -00:44:50.990 --> 00:44:51.000 align:start position:0% -MV square and remember V is small - - -00:44:51.000 --> 00:44:53.319 align:start position:0% -MV square and remember V is small -what's<00:44:51.210> the<00:44:51.360> momentum - -00:44:53.319 --> 00:44:53.329 align:start position:0% -what's the momentum - - -00:44:53.329 --> 00:45:02.199 align:start position:0% -what's the momentum -and<00:44:54.329> me<00:44:54.949> this<00:44:55.949> is<00:44:56.099> energy<00:44:56.339> this<00:44:57.349> is<00:44:58.349> momentum - -00:45:02.199 --> 00:45:02.209 align:start position:0% - - - -00:45:02.209 --> 00:45:09.620 align:start position:0% - -momentum<00:45:03.209> this<00:45:03.660> is<00:45:03.839> a<00:45:04.219> P<00:45:05.219> and<00:45:05.519> V<00:45:08.059> alright<00:45:09.059> V<00:45:09.479> is - -00:45:09.620 --> 00:45:09.630 align:start position:0% -momentum this is a P and V alright V is - - -00:45:09.630 --> 00:45:12.439 align:start position:0% -momentum this is a P and V alright V is -small<00:45:09.809> remember<00:45:10.739> that<00:45:10.859> V<00:45:11.039> is<00:45:11.160> small<00:45:11.519> so<00:45:12.239> which - -00:45:12.439 --> 00:45:12.449 align:start position:0% -small remember that V is small so which - - -00:45:12.449 --> 00:45:18.249 align:start position:0% -small remember that V is small so which -one<00:45:12.660> of<00:45:12.689> these<00:45:12.869> is<00:45:13.049> bigger<00:45:16.579> MV<00:45:17.579> right - -00:45:18.249 --> 00:45:18.259 align:start position:0% -one of these is bigger MV right - - -00:45:18.259 --> 00:45:20.689 align:start position:0% -one of these is bigger MV right -yeah<00:45:19.259> it's<00:45:19.529> true<00:45:19.650> MV<00:45:20.069> is<00:45:20.160> bigger<00:45:20.339> than<00:45:20.459> MV - -00:45:20.689 --> 00:45:20.699 align:start position:0% -yeah it's true MV is bigger than MV - - -00:45:20.699 --> 00:45:31.400 align:start position:0% -yeah it's true MV is bigger than MV -squared<00:45:20.999> but<00:45:21.449> we<00:45:22.439> forgot<00:45:22.769> a<00:45:22.799> term<00:45:23.069> here<00:45:30.410> MC - -00:45:31.400 --> 00:45:31.410 align:start position:0% -squared but we forgot a term here MC - - -00:45:31.410 --> 00:45:32.650 align:start position:0% -squared but we forgot a term here MC -squared - -00:45:32.650 --> 00:45:32.660 align:start position:0% -squared - - -00:45:32.660 --> 00:45:36.859 align:start position:0% -squared -every<00:45:33.660> object<00:45:33.989> of<00:45:34.469> mass<00:45:34.680> M<00:45:34.709> has<00:45:35.400> an<00:45:35.579> energy<00:45:36.059> MC - -00:45:36.859 --> 00:45:36.869 align:start position:0% -every object of mass M has an energy MC - - -00:45:36.869 --> 00:45:37.910 align:start position:0% -every object of mass M has an energy MC -squared - -00:45:37.910 --> 00:45:37.920 align:start position:0% -squared - - -00:45:37.920 --> 00:45:42.769 align:start position:0% -squared -C<00:45:38.699> is<00:45:38.910> bigger<00:45:39.209> than<00:45:39.539> anything<00:45:39.959> in<00:45:40.289> sight<00:45:41.779> M - -00:45:42.769 --> 00:45:42.779 align:start position:0% -C is bigger than anything in sight M - - -00:45:42.779 --> 00:45:47.140 align:start position:0% -C is bigger than anything in sight M -times<00:45:43.640> let's<00:45:44.640> write<00:45:44.939> this<00:45:45.180> as<00:45:45.420> M<00:45:45.660> times<00:45:46.259> C - -00:45:47.140 --> 00:45:47.150 align:start position:0% -times let's write this as M times C - - -00:45:47.150 --> 00:45:51.439 align:start position:0% -times let's write this as M times C -times<00:45:48.150> C<00:45:49.039> well<00:45:50.039> MC<00:45:50.549> is<00:45:50.729> certainly<00:45:51.059> bigger<00:45:51.359> than - -00:45:51.439 --> 00:45:51.449 align:start position:0% -times C well MC is certainly bigger than - - -00:45:51.449 --> 00:45:55.009 align:start position:0% -times C well MC is certainly bigger than -MV<00:45:51.930> because<00:45:52.859> C<00:45:53.130> is<00:45:53.189> big<00:45:53.969> 3<00:45:54.329> times<00:45:54.569> 10<00:45:54.809> to<00:45:54.930> the - -00:45:55.009 --> 00:45:55.019 align:start position:0% -MV because C is big 3 times 10 to the - - -00:45:55.019 --> 00:45:58.219 align:start position:0% -MV because C is big 3 times 10 to the -8th<00:45:55.170> meters<00:45:55.199> per<00:45:55.709> second<00:45:56.039> per<00:45:56.130> second<00:45:57.019> 3<00:45:58.019> times - -00:45:58.219 --> 00:45:58.229 align:start position:0% -8th meters per second per second 3 times - - -00:45:58.229 --> 00:46:01.130 align:start position:0% -8th meters per second per second 3 times -10<00:45:58.439> to<00:45:58.559> the<00:45:58.619> 8<00:45:58.739> meters<00:45:58.769> per<00:45:59.249> second<00:45:59.670> and<00:45:59.969> C<00:46:00.809> is - -00:46:01.130 --> 00:46:01.140 align:start position:0% -10 to the 8 meters per second and C is - - -00:46:01.140 --> 00:46:03.979 align:start position:0% -10 to the 8 meters per second and C is -doubly<00:46:01.529> big<00:46:01.769> here<00:46:02.189> so<00:46:02.880> this<00:46:03.179> is<00:46:03.479> typically - -00:46:03.979 --> 00:46:03.989 align:start position:0% -doubly big here so this is typically - - -00:46:03.989 --> 00:46:07.459 align:start position:0% -doubly big here so this is typically -much<00:46:04.439> bigger<00:46:04.829> than<00:46:05.039> MV<00:46:05.599> the<00:46:06.599> biggest<00:46:07.079> thing - -00:46:07.459 --> 00:46:07.469 align:start position:0% -much bigger than MV the biggest thing - - -00:46:07.469 --> 00:46:11.529 align:start position:0% -much bigger than MV the biggest thing -around<00:46:08.039> in<00:46:09.170> terms<00:46:10.170> of<00:46:10.380> energy<00:46:10.619> and<00:46:10.859> momentum - -00:46:11.529 --> 00:46:11.539 align:start position:0% -around in terms of energy and momentum - - -00:46:11.539 --> 00:46:14.539 align:start position:0% -around in terms of energy and momentum -for<00:46:12.539> a<00:46:12.569> massive<00:46:13.079> object<00:46:13.559> you<00:46:13.949> know<00:46:14.039> for<00:46:14.249> star - -00:46:14.539 --> 00:46:14.549 align:start position:0% -for a massive object you know for star - - -00:46:14.549 --> 00:46:17.900 align:start position:0% -for a massive object you know for star -or<00:46:14.999> for<00:46:15.209> a<00:46:15.890> for<00:46:16.890> a<00:46:16.920> thing<00:46:17.130> which<00:46:17.279> gravitates - -00:46:17.900 --> 00:46:17.910 align:start position:0% -or for a for a thing which gravitates - - -00:46:17.910 --> 00:46:19.880 align:start position:0% -or for a for a thing which gravitates -the<00:46:18.209> big<00:46:18.479> term<00:46:18.989> in<00:46:19.259> the<00:46:19.439> energy<00:46:19.529> momentum - -00:46:19.880 --> 00:46:19.890 align:start position:0% -the big term in the energy momentum - - -00:46:19.890 --> 00:46:24.289 align:start position:0% -the big term in the energy momentum -tensor<00:46:20.640> is<00:46:20.849> the<00:46:21.509> energy<00:46:22.019> density<00:46:22.549> all<00:46:23.549> of<00:46:24.179> the - -00:46:24.289 --> 00:46:24.299 align:start position:0% -tensor is the energy density all of the - - -00:46:24.299 --> 00:46:27.709 align:start position:0% -tensor is the energy density all of the -other<00:46:24.479> terms<00:46:24.900> are<00:46:25.229> much<00:46:25.589> smaller<00:46:26.719> because - -00:46:27.709 --> 00:46:27.719 align:start position:0% -other terms are much smaller because - - -00:46:27.719 --> 00:46:30.710 align:start position:0% -other terms are much smaller because -they<00:46:28.170> don't<00:46:28.469> have<00:46:28.559> the<00:46:29.009> MC<00:46:29.400> squared<00:46:29.729> energy - -00:46:30.710 --> 00:46:30.720 align:start position:0% -they don't have the MC squared energy - - -00:46:30.720 --> 00:46:32.540 align:start position:0% -they don't have the MC squared energy -how<00:46:31.170> can<00:46:31.350> you<00:46:31.440> compare<00:46:31.770> thing<00:46:32.040> to<00:46:32.190> different - -00:46:32.540 --> 00:46:32.550 align:start position:0% -how can you compare thing to different - - -00:46:32.550 --> 00:46:36.920 align:start position:0% -how can you compare thing to different -units<00:46:34.790> oh<00:46:35.790> you<00:46:36.180> mean<00:46:36.450> in<00:46:36.540> compared<00:46:36.810> with - -00:46:36.920 --> 00:46:36.930 align:start position:0% -units oh you mean in compared with - - -00:46:36.930 --> 00:46:41.800 align:start position:0% -units oh you mean in compared with -momentum<00:46:37.440> yeah<00:46:37.710> well<00:46:38.520> okay<00:46:38.910> we<00:46:39.450> can<00:46:39.720> we<00:46:40.440> can<00:46:40.680> we - -00:46:41.800 --> 00:46:41.810 align:start position:0% -momentum yeah well okay we can we can we - - -00:46:41.810 --> 00:46:45.200 align:start position:0% -momentum yeah well okay we can we can we -can<00:46:42.810> work<00:46:43.200> out<00:46:43.530> what<00:46:43.650> we<00:46:43.740> can<00:46:43.950> do<00:46:44.160> is<00:46:44.400> work<00:46:45.000> in - -00:46:45.200 --> 00:46:45.210 align:start position:0% -can work out what we can do is work in - - -00:46:45.210 --> 00:46:47.980 align:start position:0% -can work out what we can do is work in -units<00:46:45.600> where<00:46:46.140> the<00:46:46.230> speed<00:46:46.410> of<00:46:46.530> light<00:46:46.710> is<00:46:46.860> warm - -00:46:47.980 --> 00:46:47.990 align:start position:0% -units where the speed of light is warm - - -00:46:47.990 --> 00:46:51.290 align:start position:0% -units where the speed of light is warm -okay<00:46:48.990> then<00:46:49.500> then<00:46:50.400> they<00:46:50.670> will<00:46:50.820> have<00:46:50.850> the<00:46:51.120> same - -00:46:51.290 --> 00:46:51.300 align:start position:0% -okay then then they will have the same - - -00:46:51.300 --> 00:46:55.010 align:start position:0% -okay then then they will have the same -units<00:46:52.580> if<00:46:53.580> we<00:46:53.880> work<00:46:54.060> in<00:46:54.240> units<00:46:54.570> with<00:46:54.720> it<00:46:54.870> yeah - -00:46:55.010 --> 00:46:55.020 align:start position:0% -units if we work in units with it yeah - - -00:46:55.020 --> 00:46:58.160 align:start position:0% -units if we work in units with it yeah -and<00:46:55.850> one<00:46:56.850> of<00:46:56.940> them<00:46:57.060> ISM<00:46:57.330> that<00:46:57.630> means - -00:46:58.160 --> 00:46:58.170 align:start position:0% -and one of them ISM that means - - -00:46:58.170 --> 00:47:03.320 align:start position:0% -and one of them ISM that means -velocities<00:46:58.830> have<00:46:59.100> units<00:47:00.090> of<00:47:00.240> no<00:47:00.690> units<00:47:02.330> right - -00:47:03.320 --> 00:47:03.330 align:start position:0% -velocities have units of no units right - - -00:47:03.330 --> 00:47:06.650 align:start position:0% -velocities have units of no units right -so<00:47:04.050> the<00:47:04.770> the<00:47:05.190> right<00:47:05.370> statement<00:47:05.910> is<00:47:06.120> the<00:47:06.510> right - -00:47:06.650 --> 00:47:06.660 align:start position:0% -so the the right statement is the right - - -00:47:06.660 --> 00:47:08.650 align:start position:0% -so the the right statement is the right -statement<00:47:07.140> is<00:47:07.320> in<00:47:07.590> a<00:47:07.680> dimensionless<00:47:08.190> sense - -00:47:08.650 --> 00:47:08.660 align:start position:0% -statement is in a dimensionless sense - - -00:47:08.660 --> 00:47:11.120 align:start position:0% -statement is in a dimensionless sense -namely<00:47:09.660> the<00:47:10.020> sense<00:47:10.260> in<00:47:10.440> which<00:47:10.560> we<00:47:10.770> choose<00:47:10.950> the - -00:47:11.120 --> 00:47:11.130 align:start position:0% -namely the sense in which we choose the - - -00:47:11.130 --> 00:47:12.770 align:start position:0% -namely the sense in which we choose the -speed<00:47:11.370> of<00:47:11.460> light<00:47:11.640> to<00:47:11.760> be<00:47:11.790> equal<00:47:12.150> to<00:47:12.180> one - -00:47:12.770 --> 00:47:12.780 align:start position:0% -speed of light to be equal to one - - -00:47:12.780 --> 00:47:15.110 align:start position:0% -speed of light to be equal to one -typically<00:47:13.410> momenta<00:47:14.250> are<00:47:14.460> much<00:47:14.730> much<00:47:15.060> smaller - -00:47:15.110 --> 00:47:15.120 align:start position:0% -typically momenta are much much smaller - - -00:47:15.120 --> 00:47:22.730 align:start position:0% -typically momenta are much much smaller -than<00:47:16.520> than<00:47:18.860> than<00:47:19.860> energies<00:47:21.560> energies<00:47:22.560> have - -00:47:22.730 --> 00:47:22.740 align:start position:0% -than than than energies energies have - - -00:47:22.740 --> 00:47:26.350 align:start position:0% -than than than energies energies have -this<00:47:22.830> big<00:47:23.100> whopping<00:47:23.370> MC<00:47:23.910> squared<00:47:24.270> term<00:47:24.540> and - -00:47:26.350 --> 00:47:26.360 align:start position:0% -this big whopping MC squared term and - - -00:47:26.360 --> 00:47:29.750 align:start position:0% -this big whopping MC squared term and -momenta<00:47:27.360> we<00:47:28.140> the<00:47:28.860> other<00:47:28.980> way<00:47:29.160> to<00:47:29.220> say<00:47:29.460> it<00:47:29.610> we - -00:47:29.750 --> 00:47:29.760 align:start position:0% -momenta we the other way to say it we - - -00:47:29.760 --> 00:47:31.850 align:start position:0% -momenta we the other way to say it we -can<00:47:29.910> either<00:47:30.030> say<00:47:30.270> it<00:47:30.480> MC<00:47:31.140> squared<00:47:31.500> is<00:47:31.650> big - -00:47:31.850 --> 00:47:31.860 align:start position:0% -can either say it MC squared is big - - -00:47:31.860 --> 00:47:33.710 align:start position:0% -can either say it MC squared is big -because<00:47:32.190> C<00:47:32.370> is<00:47:32.490> big<00:47:32.730> and<00:47:33.000> this<00:47:33.420> doesn't<00:47:33.600> have - -00:47:33.710 --> 00:47:33.720 align:start position:0% -because C is big and this doesn't have - - -00:47:33.720 --> 00:47:36.490 align:start position:0% -because C is big and this doesn't have -any<00:47:33.960> C<00:47:34.170> in<00:47:34.320> it<00:47:34.500> the<00:47:35.100> other<00:47:35.220> way<00:47:35.430> to<00:47:35.490> say<00:47:35.820> it<00:47:36.000> is - -00:47:36.490 --> 00:47:36.500 align:start position:0% -any C in it the other way to say it is - - -00:47:36.500 --> 00:47:39.880 align:start position:0% -any C in it the other way to say it is -that<00:47:37.500> in<00:47:37.860> units<00:47:38.520> in<00:47:38.640> which<00:47:38.760> C<00:47:39.000> is<00:47:39.150> equal<00:47:39.420> to<00:47:39.540> one - -00:47:39.880 --> 00:47:39.890 align:start position:0% -that in units in which C is equal to one - - -00:47:39.890 --> 00:47:44.810 align:start position:0% -that in units in which C is equal to one -we<00:47:40.890> have<00:47:41.100> an<00:47:41.220> energy<00:47:41.460> which<00:47:42.330> is<00:47:42.540> M<00:47:43.460> but<00:47:44.460> then<00:47:44.640> we - -00:47:44.810 --> 00:47:44.820 align:start position:0% -we have an energy which is M but then we - - -00:47:44.820 --> 00:47:46.700 align:start position:0% -we have an energy which is M but then we -have<00:47:45.030> a<00:47:45.060> momentum<00:47:45.660> which<00:47:45.810> is<00:47:45.960> terribly<00:47:46.380> small - -00:47:46.700 --> 00:47:46.710 align:start position:0% -have a momentum which is terribly small - - -00:47:46.710 --> 00:47:49.270 align:start position:0% -have a momentum which is terribly small -because<00:47:46.920> the<00:47:47.040> velocity<00:47:47.550> is<00:47:47.670> small - -00:47:49.270 --> 00:47:49.280 align:start position:0% -because the velocity is small - - -00:47:49.280 --> 00:47:51.170 align:start position:0% -because the velocity is small -okay<00:47:50.280> comes<00:47:50.520> to<00:47:50.640> the<00:47:50.730> same<00:47:50.910> thing - -00:47:51.170 --> 00:47:51.180 align:start position:0% -okay comes to the same thing - - -00:47:51.180 --> 00:47:53.300 align:start position:0% -okay comes to the same thing -sometimes<00:47:52.110> it's<00:47:52.320> fun<00:47:52.470> to<00:47:52.680> keep<00:47:52.800> track<00:47:53.070> of<00:47:53.100> the - -00:47:53.300 --> 00:47:53.310 align:start position:0% -sometimes it's fun to keep track of the - - -00:47:53.310 --> 00:47:56.420 align:start position:0% -sometimes it's fun to keep track of the -speeds<00:47:53.550> of<00:47:53.670> light<00:47:53.850> just<00:47:54.330> to<00:47:55.290> see<00:47:56.100> what's<00:47:56.280> big - -00:47:56.420 --> 00:47:56.430 align:start position:0% -speeds of light just to see what's big - - -00:47:56.430 --> 00:48:00.050 align:start position:0% -speeds of light just to see what's big -and<00:47:56.460> what's<00:47:56.730> small<00:47:57.080> okay<00:47:58.520> meaning<00:47:59.520> of<00:47:59.610> this<00:47:59.730> is - -00:48:00.050 --> 00:48:00.060 align:start position:0% -and what's small okay meaning of this is - - -00:48:00.060 --> 00:48:04.550 align:start position:0% -and what's small okay meaning of this is -that<00:48:01.310> the<00:48:02.310> first<00:48:02.580> approximation<00:48:03.530> for<00:48:04.530> a - -00:48:04.550 --> 00:48:04.560 align:start position:0% -that the first approximation for a - - -00:48:04.560 --> 00:48:06.560 align:start position:0% -that the first approximation for a -gravitating<00:48:05.340> object<00:48:05.460> for<00:48:05.940> the<00:48:06.030> source<00:48:06.270> of<00:48:06.480> a - -00:48:06.560 --> 00:48:06.570 align:start position:0% -gravitating object for the source of a - - -00:48:06.570 --> 00:48:09.770 align:start position:0% -gravitating object for the source of a -gravitational<00:48:07.260> field<00:48:07.580> if<00:48:08.580> that<00:48:08.970> object<00:48:09.480> is<00:48:09.600> at - -00:48:09.770 --> 00:48:09.780 align:start position:0% -gravitational field if that object is at - - -00:48:09.780 --> 00:48:14.300 align:start position:0% -gravitational field if that object is at -rest<00:48:11.300> or<00:48:12.300> not<00:48:12.360> moving<00:48:13.080> well<00:48:13.620> at<00:48:13.800> rest<00:48:14.070> means - -00:48:14.300 --> 00:48:14.310 align:start position:0% -rest or not moving well at rest means - - -00:48:14.310 --> 00:48:16.310 align:start position:0% -rest or not moving well at rest means -moving<00:48:14.550> slowly<00:48:15.060> compared<00:48:15.720> to<00:48:15.840> the<00:48:15.960> speed<00:48:16.200> of - -00:48:16.310 --> 00:48:16.320 align:start position:0% -moving slowly compared to the speed of - - -00:48:16.320 --> 00:48:18.980 align:start position:0% -moving slowly compared to the speed of -light<00:48:16.530> then<00:48:17.460> the<00:48:17.610> energy<00:48:18.030> momentum<00:48:18.240> tensor - -00:48:18.980 --> 00:48:18.990 align:start position:0% -light then the energy momentum tensor - - -00:48:18.990 --> 00:48:21.680 align:start position:0% -light then the energy momentum tensor -really<00:48:19.620> only<00:48:19.890> has<00:48:20.310> one<00:48:20.820> significant - -00:48:21.680 --> 00:48:21.690 align:start position:0% -really only has one significant - - -00:48:21.690 --> 00:48:24.920 align:start position:0% -really only has one significant -component<00:48:22.380> T<00:48:23.310> naught<00:48:23.610> naught<00:48:23.910> the<00:48:24.360> energy - -00:48:24.920 --> 00:48:24.930 align:start position:0% -component T naught naught the energy - - -00:48:24.930 --> 00:48:28.340 align:start position:0% -component T naught naught the energy -density<00:48:25.760> the<00:48:26.760> momentum<00:48:27.360> that<00:48:27.540> the<00:48:27.930> energy - -00:48:28.340 --> 00:48:28.350 align:start position:0% -density the momentum that the energy - - -00:48:28.350 --> 00:48:31.400 align:start position:0% -density the momentum that the energy -flux<00:48:29.060> the<00:48:30.060> objects<00:48:30.420> not<00:48:30.540> moving<00:48:30.960> very<00:48:31.110> fast - -00:48:31.400 --> 00:48:31.410 align:start position:0% -flux the objects not moving very fast - - -00:48:31.410 --> 00:48:34.790 align:start position:0% -flux the objects not moving very fast -the<00:48:31.920> energy<00:48:32.250> flux<00:48:32.520> will<00:48:32.850> be<00:48:32.880> small<00:48:33.800> the - -00:48:34.790 --> 00:48:34.800 align:start position:0% -the energy flux will be small the - - -00:48:34.800 --> 00:48:37.190 align:start position:0% -the energy flux will be small the -momentum<00:48:35.310> density<00:48:35.490> will<00:48:35.940> be<00:48:35.970> small<00:48:36.390> the<00:48:36.900> flux - -00:48:37.190 --> 00:48:37.200 align:start position:0% -momentum density will be small the flux - - -00:48:37.200 --> 00:48:39.980 align:start position:0% -momentum density will be small the flux -of<00:48:37.380> momentum<00:48:37.500> will<00:48:38.040> be<00:48:38.070> small<00:48:38.780> everything<00:48:39.780> is - -00:48:39.980 --> 00:48:39.990 align:start position:0% -of momentum will be small everything is - - -00:48:39.990 --> 00:48:41.890 align:start position:0% -of momentum will be small everything is -small<00:48:40.290> except<00:48:40.890> T<00:48:41.070> naught<00:48:41.280> naught - -00:48:41.890 --> 00:48:41.900 align:start position:0% -small except T naught naught - - -00:48:41.900 --> 00:48:45.070 align:start position:0% -small except T naught naught -and<00:48:42.770> what<00:48:43.010> does<00:48:43.130> T<00:48:43.340> naught<00:48:43.550> naught<00:48:43.870> T<00:48:44.870> naught - -00:48:45.070 --> 00:48:45.080 align:start position:0% -and what does T naught naught T naught - - -00:48:45.080 --> 00:48:49.390 align:start position:0% -and what does T naught naught T naught -naught<00:48:45.440> is<00:48:45.880> the<00:48:46.880> energy<00:48:47.300> density<00:48:48.100> now<00:48:49.100> I<00:48:49.130> keep - -00:48:49.390 --> 00:48:49.400 align:start position:0% -naught is the energy density now I keep - - -00:48:49.400 --> 00:48:51.430 align:start position:0% -naught is the energy density now I keep -every<00:48:49.760> mob<00:48:49.940> will<00:48:50.090> keep<00:48:50.300> reminding<00:48:50.480> you<00:48:50.960> energy - -00:48:51.430 --> 00:48:51.440 align:start position:0% -every mob will keep reminding you energy - - -00:48:51.440 --> 00:48:53.560 align:start position:0% -every mob will keep reminding you energy -density<00:48:51.890> and<00:48:52.100> mass<00:48:52.340> density<00:48:52.970> are<00:48:53.180> the<00:48:53.330> same - -00:48:53.560 --> 00:48:53.570 align:start position:0% -density and mass density are the same - - -00:48:53.570 --> 00:48:55.960 align:start position:0% -density and mass density are the same -thing<00:48:53.990> if<00:48:54.890> I<00:48:55.040> choose<00:48:55.250> the<00:48:55.430> speed<00:48:55.640> of<00:48:55.760> light - -00:48:55.960 --> 00:48:55.970 align:start position:0% -thing if I choose the speed of light - - -00:48:55.970 --> 00:49:00.520 align:start position:0% -thing if I choose the speed of light -equal<00:48:56.240> to<00:48:56.390> one<00:48:57.070> so<00:48:59.380> to<00:49:00.380> the<00:49:00.500> first - -00:49:00.520 --> 00:49:00.530 align:start position:0% -equal to one so to the first - - -00:49:00.530 --> 00:49:03.700 align:start position:0% -equal to one so to the first -approximation<00:49:01.660> the<00:49:02.660> big<00:49:02.900> chunk<00:49:03.320> and<00:49:03.530> the - -00:49:03.700 --> 00:49:03.710 align:start position:0% -approximation the big chunk and the - - -00:49:03.710 --> 00:49:06.610 align:start position:0% -approximation the big chunk and the -energy<00:49:04.010> momentum<00:49:04.160> tensor<00:49:04.910> is<00:49:05.180> just<00:49:05.990> the<00:49:06.350> mass - -00:49:06.610 --> 00:49:06.620 align:start position:0% -energy momentum tensor is just the mass - - -00:49:06.620 --> 00:49:14.580 align:start position:0% -energy momentum tensor is just the mass -density<00:49:06.920> and<00:49:07.520> everything<00:49:08.120> else<00:49:08.270> being<00:49:08.630> zero - -00:49:14.580 --> 00:49:14.590 align:start position:0% - - - -00:49:14.590 --> 00:49:20.740 align:start position:0% - -plus<00:49:15.590> small<00:49:19.090> okay<00:49:20.090> so<00:49:20.150> that's<00:49:20.390> something<00:49:20.690> we - -00:49:20.740 --> 00:49:20.750 align:start position:0% -plus small okay so that's something we - - -00:49:20.750 --> 00:49:22.330 align:start position:0% -plus small okay so that's something we -should<00:49:20.930> keep<00:49:21.050> in<00:49:21.200> mind<00:49:21.230> when<00:49:21.740> we're<00:49:21.920> trying<00:49:22.070> to - -00:49:22.330 --> 00:49:22.340 align:start position:0% -should keep in mind when we're trying to - - -00:49:22.340 --> 00:49:25.530 align:start position:0% -should keep in mind when we're trying to -compare<00:49:23.620> when<00:49:24.620> we're<00:49:24.740> trying<00:49:24.860> to<00:49:25.100> compare - -00:49:25.530 --> 00:49:25.540 align:start position:0% -compare when we're trying to compare - - -00:49:25.540 --> 00:49:29.350 align:start position:0% -compare when we're trying to compare -with<00:49:26.540> known<00:49:27.260> physics<00:49:27.890> so<00:49:28.040> to<00:49:28.100> speak<00:49:28.430> I<00:49:28.700> mean - -00:49:29.350 --> 00:49:29.360 align:start position:0% -with known physics so to speak I mean - - -00:49:29.360 --> 00:49:31.360 align:start position:0% -with known physics so to speak I mean -what<00:49:29.600> was<00:49:29.690> Einstein's<00:49:30.140> situation<00:49:30.740> he<00:49:31.130> had - -00:49:31.360 --> 00:49:31.370 align:start position:0% -what was Einstein's situation he had - - -00:49:31.370 --> 00:49:34.240 align:start position:0% -what was Einstein's situation he had -these<00:49:31.520> geometric<00:49:32.090> ideas<00:49:32.570> he<00:49:33.380> thought<00:49:33.680> gravity - -00:49:34.240 --> 00:49:34.250 align:start position:0% -these geometric ideas he thought gravity - - -00:49:34.250 --> 00:49:36.430 align:start position:0% -these geometric ideas he thought gravity -had<00:49:34.760> to<00:49:34.910> do<00:49:35.060> with<00:49:35.270> the<00:49:35.360> metric<00:49:35.720> of<00:49:35.840> space-time - -00:49:36.430 --> 00:49:36.440 align:start position:0% -had to do with the metric of space-time - - -00:49:36.440 --> 00:49:39.400 align:start position:0% -had to do with the metric of space-time -but<00:49:37.280> he<00:49:37.370> needed<00:49:37.670> some<00:49:37.850> equations<00:49:38.450> the<00:49:39.140> first - -00:49:39.400 --> 00:49:39.410 align:start position:0% -but he needed some equations the first - - -00:49:39.410 --> 00:49:42.100 align:start position:0% -but he needed some equations the first -thing<00:49:39.650> to<00:49:39.800> do<00:49:39.950> was<00:49:40.130> to<00:49:40.190> make<00:49:40.580> sure<00:49:41.030> that<00:49:41.600> in<00:49:41.930> the - -00:49:42.100 --> 00:49:42.110 align:start position:0% -thing to do was to make sure that in the - - -00:49:42.110 --> 00:49:46.060 align:start position:0% -thing to do was to make sure that in the -limit<00:49:42.500> when<00:49:42.830> Newton<00:49:43.130> ought<00:49:43.490> to<00:49:43.790> apply<00:49:44.470> or<00:49:45.470> in - -00:49:46.060 --> 00:49:46.070 align:start position:0% -limit when Newton ought to apply or in - - -00:49:46.070 --> 00:49:48.160 align:start position:0% -limit when Newton ought to apply or in -the<00:49:46.310> limit<00:49:46.640> where<00:49:46.820> special<00:49:47.240> relativity<00:49:47.630> auto - -00:49:48.160 --> 00:49:48.170 align:start position:0% -the limit where special relativity auto - - -00:49:48.170 --> 00:49:51.580 align:start position:0% -the limit where special relativity auto -apply<00:49:48.470> that<00:49:48.920> they<00:49:49.070> do<00:49:49.130> apply<00:49:49.840> so<00:49:50.840> that<00:49:51.320> was<00:49:51.470> the - -00:49:51.580 --> 00:49:51.590 align:start position:0% -apply that they do apply so that was the - - -00:49:51.590 --> 00:49:53.680 align:start position:0% -apply that they do apply so that was the -first<00:49:51.800> test<00:49:52.460> if<00:49:52.820> you<00:49:52.940> like<00:49:53.180> or<00:49:53.420> the<00:49:53.510> first - -00:49:53.680 --> 00:49:53.690 align:start position:0% -first test if you like or the first - - -00:49:53.690 --> 00:49:57.910 align:start position:0% -first test if you like or the first -criteria<00:49:54.320> a<00:49:55.360> principle<00:49:56.360> of<00:49:56.920> correspondence - -00:49:57.910 --> 00:49:57.920 align:start position:0% -criteria a principle of correspondence - - -00:49:57.920 --> 00:50:02.370 align:start position:0% -criteria a principle of correspondence -with<00:49:58.250> known<00:49:58.910> physics - -00:50:02.370 --> 00:50:02.380 align:start position:0% - - - -00:50:02.380 --> 00:50:08.980 align:start position:0% - -okay<00:50:03.380> that's<00:50:06.070> metric<00:50:07.600> basically<00:50:08.600> special - -00:50:08.980 --> 00:50:08.990 align:start position:0% -okay that's metric basically special - - -00:50:08.990 --> 00:50:13.090 align:start position:0% -okay that's metric basically special -relativity<00:50:10.960> if<00:50:11.960> we<00:50:12.140> keep<00:50:12.350> track<00:50:12.710> of<00:50:13.010> the - -00:50:13.090 --> 00:50:13.100 align:start position:0% -relativity if we keep track of the - - -00:50:13.100 --> 00:50:16.330 align:start position:0% -relativity if we keep track of the -speeds<00:50:13.400> of<00:50:13.580> light<00:50:14.170> then<00:50:15.170> actually<00:50:16.010> the<00:50:16.160> only - -00:50:16.330 --> 00:50:16.340 align:start position:0% -speeds of light then actually the only - - -00:50:16.340 --> 00:50:19.720 align:start position:0% -speeds of light then actually the only -the<00:50:16.940> only<00:50:17.300> big<00:50:17.600> component<00:50:18.380> of<00:50:18.530> the<00:50:18.650> metric<00:50:18.890> is - -00:50:19.720 --> 00:50:19.730 align:start position:0% -the only big component of the metric is - - -00:50:19.730 --> 00:50:23.380 align:start position:0% -the only big component of the metric is -the<00:50:19.790> time<00:50:20.210> time<00:50:20.570> component<00:50:21.370> the<00:50:22.390> space-time - -00:50:23.380 --> 00:50:23.390 align:start position:0% -the time time component the space-time - - -00:50:23.390 --> 00:50:25.720 align:start position:0% -the time time component the space-time -the<00:50:23.660> space<00:50:24.080> space<00:50:24.410> components<00:50:25.130> are<00:50:25.280> smaller - -00:50:25.720 --> 00:50:25.730 align:start position:0% -the space space components are smaller - - -00:50:25.730 --> 00:50:28.210 align:start position:0% -the space space components are smaller -by<00:50:26.060> a<00:50:26.090> factor<00:50:26.210> of<00:50:26.630> one<00:50:26.840> over<00:50:26.870> C<00:50:27.230> squared<00:50:27.590> if<00:50:27.980> we - -00:50:28.210 --> 00:50:28.220 align:start position:0% -by a factor of one over C squared if we - - -00:50:28.220 --> 00:50:31.840 align:start position:0% -by a factor of one over C squared if we -keep<00:50:28.670> track<00:50:28.940> of<00:50:29.000> the<00:50:29.180> speed<00:50:29.450> of<00:50:29.480> light<00:50:30.790> okay<00:50:31.790> if - -00:50:31.840 --> 00:50:31.850 align:start position:0% -keep track of the speed of light okay if - - -00:50:31.850 --> 00:50:35.380 align:start position:0% -keep track of the speed of light okay if -we<00:50:31.940> keep<00:50:32.120> track<00:50:32.330> of<00:50:32.390> C<00:50:32.750> in<00:50:32.990> the<00:50:33.050> equations<00:50:34.390> and - -00:50:35.380 --> 00:50:35.390 align:start position:0% -we keep track of C in the equations and - - -00:50:35.390 --> 00:50:37.240 align:start position:0% -we keep track of C in the equations and -it's<00:50:35.930> useful<00:50:36.260> sometimes<00:50:36.290> to<00:50:36.860> keep<00:50:37.010> track<00:50:37.220> of - -00:50:37.240 --> 00:50:37.250 align:start position:0% -it's useful sometimes to keep track of - - -00:50:37.250 --> 00:50:39.730 align:start position:0% -it's useful sometimes to keep track of -the<00:50:37.430> speed<00:50:37.640> of<00:50:37.670> light<00:50:38.170> okay<00:50:39.170> what<00:50:39.320> else<00:50:39.500> do<00:50:39.680> we - -00:50:39.730 --> 00:50:39.740 align:start position:0% -the speed of light okay what else do we - - -00:50:39.740 --> 00:50:45.460 align:start position:0% -the speed of light okay what else do we -know<00:50:40.870> we<00:50:41.870> know<00:50:42.050> from<00:50:42.980> Newton<00:50:43.400> or<00:50:44.470> generalized - -00:50:45.460 --> 00:50:45.470 align:start position:0% -know we know from Newton or generalized - - -00:50:45.470 --> 00:50:50.460 align:start position:0% -know we know from Newton or generalized -NUI<00:50:46.870> Newton<00:50:47.870> plus<00:50:48.440> Gauss<00:50:48.770> plus - -00:50:50.460 --> 00:50:50.470 align:start position:0% -NUI Newton plus Gauss plus - - -00:50:50.470 --> 00:50:57.960 align:start position:0% -NUI Newton plus Gauss plus -Plus<00:50:50.650> whoever<00:50:51.160> else<00:50:51.400> was<00:50:51.580> involved<00:50:56.820> we<00:50:57.820> have<00:50:57.940> a - -00:50:57.960 --> 00:50:57.970 align:start position:0% -Plus whoever else was involved we have a - - -00:50:57.970 --> 00:51:02.280 align:start position:0% -Plus whoever else was involved we have a -field<00:50:58.270> equation<00:51:00.300> that<00:51:01.300> the<00:51:01.450> gravitational - -00:51:02.280 --> 00:51:02.290 align:start position:0% -field equation that the gravitational - - -00:51:02.290 --> 00:51:07.700 align:start position:0% -field equation that the gravitational -potential<00:51:04.710> satisfies<00:51:05.710> del<00:51:06.310> squared<00:51:06.760> Phi<00:51:07.000> is - -00:51:07.700 --> 00:51:07.710 align:start position:0% -potential satisfies del squared Phi is - - -00:51:07.710 --> 00:51:14.880 align:start position:0% -potential satisfies del squared Phi is -equal<00:51:08.970> to<00:51:09.970> 4<00:51:10.270> pi<00:51:10.950> G<00:51:11.950> Rho<00:51:13.050> this<00:51:14.050> is<00:51:14.109> a<00:51:14.290> disguised - -00:51:14.880 --> 00:51:14.890 align:start position:0% -equal to 4 pi G Rho this is a disguised - - -00:51:14.890 --> 00:51:22.380 align:start position:0% -equal to 4 pi G Rho this is a disguised -form<00:51:15.369> of<00:51:15.700> the<00:51:15.820> inverse<00:51:16.150> square<00:51:16.540> law<00:51:20.700> and<00:51:21.700> we - -00:51:22.380 --> 00:51:22.390 align:start position:0% -form of the inverse square law and we - - -00:51:22.390 --> 00:51:24.420 align:start position:0% -form of the inverse square law and we -also<00:51:22.630> found<00:51:23.170> out<00:51:23.349> by<00:51:23.650> looking<00:51:24.010> at<00:51:24.130> the<00:51:24.220> motion - -00:51:24.420 --> 00:51:24.430 align:start position:0% -also found out by looking at the motion - - -00:51:24.430 --> 00:51:30.020 align:start position:0% -also found out by looking at the motion -of<00:51:24.730> geodesics<00:51:26.220> slowly<00:51:27.220> moving<00:51:27.760> particles - -00:51:30.020 --> 00:51:30.030 align:start position:0% -of geodesics slowly moving particles - - -00:51:30.030 --> 00:51:36.380 align:start position:0% -of geodesics slowly moving particles -satisfy<00:51:31.030> the<00:51:31.570> usual<00:51:34.410> acceleration<00:51:35.410> being - -00:51:36.380 --> 00:51:36.390 align:start position:0% -satisfy the usual acceleration being - - -00:51:36.390 --> 00:51:38.700 align:start position:0% -satisfy the usual acceleration being -acceleration<00:51:37.390> is<00:51:37.599> equal<00:51:37.630> to<00:51:37.960> minus<00:51:38.680> the - -00:51:38.700 --> 00:51:38.710 align:start position:0% -acceleration is equal to minus the - - -00:51:38.710 --> 00:51:41.580 align:start position:0% -acceleration is equal to minus the -derivative<00:51:39.070> of<00:51:39.670> the<00:51:40.200> usual<00:51:41.200> equation - -00:51:41.580 --> 00:51:41.590 align:start position:0% -derivative of the usual equation - - -00:51:41.590 --> 00:51:45.800 align:start position:0% -derivative of the usual equation -acceleration<00:51:43.410> is<00:51:44.410> equal<00:51:44.830> to<00:51:44.950> minus<00:51:45.430> a - -00:51:45.800 --> 00:51:45.810 align:start position:0% -acceleration is equal to minus a - - -00:51:45.810 --> 00:51:51.240 align:start position:0% -acceleration is equal to minus a -gradient<00:51:46.810> of<00:51:47.400> Phi<00:51:48.400> or<00:51:49.180> minus<00:51:49.660> the<00:51:50.260> earth - -00:51:51.240 --> 00:51:51.250 align:start position:0% -gradient of Phi or minus the earth - - -00:51:51.250 --> 00:51:53.370 align:start position:0% -gradient of Phi or minus the earth -component<00:51:52.060> of<00:51:52.240> the<00:51:52.330> M<00:51:52.450> component<00:51:53.170> of<00:51:53.290> the - -00:51:53.370 --> 00:51:53.380 align:start position:0% -component of the M component of the - - -00:51:53.380 --> 00:51:55.800 align:start position:0% -component of the M component of the -acceleration<00:51:53.940> is<00:51:54.940> equal<00:51:55.000> to<00:51:55.420> the<00:51:55.570> derivative - -00:51:55.800 --> 00:51:55.810 align:start position:0% -acceleration is equal to the derivative - - -00:51:55.810 --> 00:52:00.599 align:start position:0% -acceleration is equal to the derivative -with<00:51:56.230> respect<00:51:56.589> to<00:51:56.650> M<00:51:56.980> of<00:51:57.339> v<00:51:57.849> minus<00:51:58.420> a<00:51:58.540> plus<00:51:59.490> -<00:52:00.490> I - -00:52:00.599 --> 00:52:00.609 align:start position:0% -with respect to M of v minus a plus - I - - -00:52:00.609 --> 00:52:06.480 align:start position:0% -with respect to M of v minus a plus - I -think<00:52:03.300> that's<00:52:04.300> F<00:52:04.480> equals<00:52:04.810> MA<00:52:05.109> except<00:52:06.010> dividing - -00:52:06.480 --> 00:52:06.490 align:start position:0% -think that's F equals MA except dividing - - -00:52:06.490 --> 00:52:09.450 align:start position:0% -think that's F equals MA except dividing -by<00:52:06.609> M<00:52:07.380> acceleration<00:52:08.380> is<00:52:08.619> the<00:52:08.830> gradient<00:52:09.250> of<00:52:09.369> a - -00:52:09.450 --> 00:52:09.460 align:start position:0% -by M acceleration is the gradient of a - - -00:52:09.460 --> 00:52:11.150 align:start position:0% -by M acceleration is the gradient of a -potential - -00:52:11.150 --> 00:52:11.160 align:start position:0% -potential - - -00:52:11.160 --> 00:52:15.089 align:start position:0% -potential -okay<00:52:12.160> so<00:52:12.220> we<00:52:12.460> have<00:52:12.609> that<00:52:12.820> and<00:52:13.410> if<00:52:14.410> we<00:52:14.560> compare - -00:52:15.089 --> 00:52:15.099 align:start position:0% -okay so we have that and if we compare - - -00:52:15.099 --> 00:52:17.000 align:start position:0% -okay so we have that and if we compare -that<00:52:15.339> with<00:52:15.550> the<00:52:15.700> equations<00:52:16.180> of<00:52:16.330> a<00:52:16.420> geodesic - -00:52:17.000 --> 00:52:17.010 align:start position:0% -that with the equations of a geodesic - - -00:52:17.010 --> 00:52:19.560 align:start position:0% -that with the equations of a geodesic -that<00:52:18.010> tells<00:52:18.310> us<00:52:18.490> the<00:52:18.640> relationship<00:52:19.150> between - -00:52:19.560 --> 00:52:19.570 align:start position:0% -that tells us the relationship between - - -00:52:19.570 --> 00:52:22.410 align:start position:0% -that tells us the relationship between -the<00:52:19.869> gravitational<00:52:20.680> potential<00:52:21.160> and<00:52:21.820> a - -00:52:22.410 --> 00:52:22.420 align:start position:0% -the gravitational potential and a - - -00:52:22.420 --> 00:52:24.450 align:start position:0% -the gravitational potential and a -certain<00:52:22.750> component<00:52:23.380> of<00:52:23.500> the<00:52:23.589> metric<00:52:23.770> remember - -00:52:24.450 --> 00:52:24.460 align:start position:0% -certain component of the metric remember - - -00:52:24.460 --> 00:52:26.160 align:start position:0% -certain component of the metric remember -we<00:52:24.580> work<00:52:24.760> that<00:52:24.910> out<00:52:24.970> for<00:52:25.150> slowly<00:52:25.720> moving - -00:52:26.160 --> 00:52:26.170 align:start position:0% -we work that out for slowly moving - - -00:52:26.170 --> 00:52:32.310 align:start position:0% -we work that out for slowly moving -geodesics<00:52:27.060> we<00:52:28.060> found<00:52:28.440> that<00:52:31.020> what<00:52:32.020> was<00:52:32.170> it - -00:52:32.310 --> 00:52:32.320 align:start position:0% -geodesics we found that what was it - - -00:52:32.320 --> 00:52:33.450 align:start position:0% -geodesics we found that what was it -anybody<00:52:32.619> remember<00:52:32.859> the<00:52:32.950> formula<00:52:33.310> or<00:52:33.339> in - -00:52:33.450 --> 00:52:33.460 align:start position:0% -anybody remember the formula or in - - -00:52:33.460 --> 00:52:36.180 align:start position:0% -anybody remember the formula or in -Traktor<00:52:33.849> I<00:52:34.089> had<00:52:35.080> things<00:52:35.230> written<00:52:35.500> down<00:52:35.619> but - -00:52:36.180 --> 00:52:36.190 align:start position:0% -Traktor I had things written down but - - -00:52:36.190 --> 00:52:37.710 align:start position:0% -Traktor I had things written down but -unless<00:52:36.520> should<00:52:36.700> I<00:52:36.820> write<00:52:37.000> yeah<00:52:37.390> I<00:52:37.420> think<00:52:37.510> they - -00:52:37.710 --> 00:52:37.720 align:start position:0% -unless should I write yeah I think they - - -00:52:37.720 --> 00:52:44.130 align:start position:0% -unless should I write yeah I think they -are<00:52:37.839> right<00:52:38.670> what<00:52:39.670> we<00:52:39.790> found<00:52:40.030> out<00:52:41.460> well<00:52:43.140> was - -00:52:44.130 --> 00:52:44.140 align:start position:0% -are right what we found out well was - - -00:52:44.140 --> 00:52:49.560 align:start position:0% -are right what we found out well was -that<00:52:44.500> the<00:52:46.440> gravitational<00:52:47.440> potential<00:52:48.570> was - -00:52:49.560 --> 00:52:49.570 align:start position:0% -that the gravitational potential was - - -00:52:49.570 --> 00:52:55.440 align:start position:0% -that the gravitational potential was -like<00:52:50.760> 1/2<00:52:51.760> of<00:52:52.180> G<00:52:52.780> naught<00:52:52.990> naught<00:52:54.270> now<00:52:55.270> that's - -00:52:55.440 --> 00:52:55.450 align:start position:0% -like 1/2 of G naught naught now that's - - -00:52:55.450 --> 00:52:56.910 align:start position:0% -like 1/2 of G naught naught now that's -not<00:52:55.599> right<00:52:55.780> that's<00:52:55.960> not<00:52:56.260> correct<00:52:56.500> that's<00:52:56.770> not - -00:52:56.910 --> 00:52:56.920 align:start position:0% -not right that's not correct that's not - - -00:52:56.920 --> 00:52:58.410 align:start position:0% -not right that's not correct that's not -correct<00:52:57.220> what<00:52:57.369> what<00:52:57.609> actually<00:52:57.910> did<00:52:58.060> we<00:52:58.150> find - -00:52:58.410 --> 00:52:58.420 align:start position:0% -correct what what actually did we find - - -00:52:58.420 --> 00:53:02.870 align:start position:0% -correct what what actually did we find -out - -00:53:02.870 --> 00:53:02.880 align:start position:0% - - - -00:53:02.880 --> 00:53:07.410 align:start position:0% - -see<00:53:04.349> plus<00:53:05.349> a<00:53:05.560> constant<00:53:05.920> what<00:53:06.849> we<00:53:06.940> really<00:53:07.210> found - -00:53:07.410 --> 00:53:07.420 align:start position:0% -see plus a constant what we really found - - -00:53:07.420 --> 00:53:09.839 align:start position:0% -see plus a constant what we really found -out<00:53:07.660> is<00:53:07.900> that<00:53:07.930> the<00:53:08.230> derivatives<00:53:08.859> of<00:53:09.070> Phi<00:53:09.310> were - -00:53:09.839 --> 00:53:09.849 align:start position:0% -out is that the derivatives of Phi were - - -00:53:09.849 --> 00:53:12.990 align:start position:0% -out is that the derivatives of Phi were -related<00:53:10.390> to<00:53:10.420> the<00:53:10.630> derivative<00:53:11.200> of<00:53:11.320> G<00:53:11.820> where<00:53:12.820> did - -00:53:12.990 --> 00:53:13.000 align:start position:0% -related to the derivative of G where did - - -00:53:13.000 --> 00:53:14.820 align:start position:0% -related to the derivative of G where did -the<00:53:13.089> derivatives<00:53:13.599> of<00:53:13.690> G<00:53:13.930> come<00:53:14.200> from<00:53:14.470> they<00:53:14.619> came - -00:53:14.820 --> 00:53:14.830 align:start position:0% -the derivatives of G come from they came - - -00:53:14.830 --> 00:53:18.480 align:start position:0% -the derivatives of G come from they came -from<00:53:15.040> the<00:53:15.220> Christoffel<00:53:15.790> symbols<00:53:17.190> only<00:53:18.190> in<00:53:18.369> the - -00:53:18.480 --> 00:53:18.490 align:start position:0% -from the Christoffel symbols only in the - - -00:53:18.490 --> 00:53:21.030 align:start position:0% -from the Christoffel symbols only in the -equation<00:53:18.970> of<00:53:19.060> motion<00:53:19.119> F<00:53:19.810> equals<00:53:20.349> MA - -00:53:21.030 --> 00:53:21.040 align:start position:0% -equation of motion F equals MA - - -00:53:21.040 --> 00:53:24.240 align:start position:0% -equation of motion F equals MA -only<00:53:21.790> derivatives<00:53:22.359> of<00:53:22.480> Phi<00:53:22.660> appear<00:53:23.140> and<00:53:23.619> the - -00:53:24.240 --> 00:53:24.250 align:start position:0% -only derivatives of Phi appear and the - - -00:53:24.250 --> 00:53:27.510 align:start position:0% -only derivatives of Phi appear and the -Christoffel<00:53:24.820> symbols<00:53:25.330> only<00:53:26.520> only - -00:53:27.510 --> 00:53:27.520 align:start position:0% -Christoffel symbols only only - - -00:53:27.520 --> 00:53:30.420 align:start position:0% -Christoffel symbols only only -derivatives<00:53:28.330> of<00:53:28.450> G<00:53:28.660> appear<00:53:28.990> so<00:53:29.740> it<00:53:29.890> was<00:53:30.040> really - -00:53:30.420 --> 00:53:30.430 align:start position:0% -derivatives of G appear so it was really - - -00:53:30.430 --> 00:53:32.640 align:start position:0% -derivatives of G appear so it was really -that<00:53:30.760> the<00:53:30.940> derivative<00:53:31.480> of<00:53:31.690> Phi<00:53:31.960> in<00:53:32.380> any - -00:53:32.640 --> 00:53:32.650 align:start position:0% -that the derivative of Phi in any - - -00:53:32.650 --> 00:53:35.370 align:start position:0% -that the derivative of Phi in any -direction<00:53:32.859> is<00:53:33.550> the<00:53:34.180> derivative<00:53:34.420> of<00:53:34.869> G<00:53:34.960> naught - -00:53:35.370 --> 00:53:35.380 align:start position:0% -direction is the derivative of G naught - - -00:53:35.380 --> 00:53:37.740 align:start position:0% -direction is the derivative of G naught -north<00:53:35.680> but<00:53:36.460> another<00:53:36.670> way<00:53:36.849> to<00:53:36.880> write<00:53:37.180> that<00:53:37.240> is - -00:53:37.740 --> 00:53:37.750 align:start position:0% -north but another way to write that is - - -00:53:37.750 --> 00:53:40.740 align:start position:0% -north but another way to write that is -the<00:53:38.080> Phi<00:53:38.320> is<00:53:38.650> equal<00:53:38.980> to<00:53:39.099> G<00:53:39.310> naught<00:53:39.520> naught<00:53:39.790> plus - -00:53:40.740 --> 00:53:40.750 align:start position:0% -the Phi is equal to G naught naught plus - - -00:53:40.750 --> 00:53:43.890 align:start position:0% -the Phi is equal to G naught naught plus -some<00:53:41.589> constant<00:53:42.280> which<00:53:43.119> doesn't<00:53:43.480> vary<00:53:43.720> in - -00:53:43.890 --> 00:53:43.900 align:start position:0% -some constant which doesn't vary in - - -00:53:43.900 --> 00:53:48.450 align:start position:0% -some constant which doesn't vary in -position<00:53:44.380> now<00:53:45.160> again<00:53:45.430> I<00:53:45.460> emphasize<00:53:47.339> there's<00:53:48.339> a - -00:53:48.450 --> 00:53:48.460 align:start position:0% -position now again I emphasize there's a - - -00:53:48.460 --> 00:53:52.339 align:start position:0% -position now again I emphasize there's a -G<00:53:48.670> naught<00:53:48.849> naught<00:53:49.060> is<00:53:49.359> equal<00:53:49.390> to<00:53:49.720> Phi<00:53:50.080> 2<00:53:50.410> Phi - -00:53:52.339 --> 00:53:52.349 align:start position:0% -G naught naught is equal to Phi 2 Phi - - -00:53:52.349 --> 00:53:55.410 align:start position:0% -G naught naught is equal to Phi 2 Phi -what<00:53:53.349> was<00:53:53.440> it<00:53:53.589> -<00:53:53.800> all<00:53:54.550> right<00:53:54.640> this<00:53:54.910> design - -00:53:55.410 --> 00:53:55.420 align:start position:0% -what was it - all right this design - - -00:53:55.420 --> 00:54:03.359 align:start position:0% -what was it - all right this design -drives<00:53:55.720> me<00:53:55.900> crazy<00:53:56.170> I<00:53:56.730> believe<00:53:57.730> I<00:54:01.890> do<00:54:02.890> not<00:54:03.160> want - -00:54:03.359 --> 00:54:03.369 align:start position:0% -drives me crazy I believe I do not want - - -00:54:03.369 --> 00:54:12.290 align:start position:0% -drives me crazy I believe I do not want -to<00:54:03.490> use<00:54:03.670> C<00:54:04.560> thank<00:54:05.560> you<00:54:08.010> let's<00:54:09.010> call<00:54:09.310> it<00:54:09.520> C<00:54:10.390> OMS<00:54:11.349> T - -00:54:12.290 --> 00:54:12.300 align:start position:0% -to use C thank you let's call it C OMS T - - -00:54:12.300 --> 00:54:15.990 align:start position:0% -to use C thank you let's call it C OMS T -constant<00:54:14.220> of<00:54:15.220> course<00:54:15.400> the<00:54:15.460> speed<00:54:15.640> of<00:54:15.760> light<00:54:15.880> is - -00:54:15.990 --> 00:54:16.000 align:start position:0% -constant of course the speed of light is - - -00:54:16.000 --> 00:54:21.420 align:start position:0% -constant of course the speed of light is -also<00:54:16.150> a<00:54:16.270> constant<00:54:16.839> but<00:54:17.020> I<00:54:19.410> think<00:54:20.410> that<00:54:20.530> I<00:54:20.560> don't - -00:54:21.420 --> 00:54:21.430 align:start position:0% -also a constant but I think that I don't - - -00:54:21.430 --> 00:54:24.800 align:start position:0% -also a constant but I think that I don't -think<00:54:21.609> it'd<00:54:21.700> be<00:54:21.910> unclear - -00:54:24.800 --> 00:54:24.810 align:start position:0% - - - -00:54:24.810 --> 00:54:30.170 align:start position:0% - -okay<00:54:27.980> the<00:54:28.980> first<00:54:29.220> equation<00:54:29.550> of<00:54:29.760> this<00:54:29.910> Court - -00:54:30.170 --> 00:54:30.180 align:start position:0% -okay the first equation of this Court - - -00:54:30.180 --> 00:54:32.510 align:start position:0% -okay the first equation of this Court -says<00:54:30.390> the<00:54:30.680> gradient<00:54:31.680> of<00:54:31.740> the<00:54:31.860> potential - -00:54:32.510 --> 00:54:32.520 align:start position:0% -says the gradient of the potential - - -00:54:32.520 --> 00:54:34.610 align:start position:0% -says the gradient of the potential -I'm<00:54:32.640> sorry<00:54:32.880> del<00:54:33.570> squared<00:54:33.870> of<00:54:33.990> potential<00:54:34.440> at<00:54:34.560> a - -00:54:34.610 --> 00:54:34.620 align:start position:0% -I'm sorry del squared of potential at a - - -00:54:34.620 --> 00:54:36.590 align:start position:0% -I'm sorry del squared of potential at a -point<00:54:34.800> in<00:54:34.950> space<00:54:35.040> is<00:54:35.550> equal<00:54:36.090> to<00:54:36.240> the<00:54:36.360> density - -00:54:36.590 --> 00:54:36.600 align:start position:0% -point in space is equal to the density - - -00:54:36.600 --> 00:54:39.350 align:start position:0% -point in space is equal to the density -of<00:54:36.870> mass<00:54:37.200> at<00:54:37.530> that<00:54:37.710> point<00:54:37.950> in<00:54:38.070> space<00:54:38.300> so<00:54:39.300> if - -00:54:39.350 --> 00:54:39.360 align:start position:0% -of mass at that point in space so if - - -00:54:39.360 --> 00:54:41.120 align:start position:0% -of mass at that point in space so if -there's<00:54:39.540> no<00:54:39.690> mass<00:54:39.900> the<00:54:40.560> potential<00:54:41.040> is - -00:54:41.120 --> 00:54:41.130 align:start position:0% -there's no mass the potential is - - -00:54:41.130 --> 00:54:46.340 align:start position:0% -there's no mass the potential is -interchangeable<00:54:44.750> be<00:54:45.750> careful<00:54:46.020> be<00:54:46.320> careful - -00:54:46.340 --> 00:54:46.350 align:start position:0% -interchangeable be careful be careful - - -00:54:46.350 --> 00:54:48.770 align:start position:0% -interchangeable be careful be careful -that<00:54:46.980> that<00:54:47.130> is<00:54:47.220> a<00:54:47.250> good<00:54:47.430> question<00:54:47.630> what<00:54:48.630> is - -00:54:48.770 --> 00:54:48.780 align:start position:0% -that that is a good question what is - - -00:54:48.780 --> 00:54:51.710 align:start position:0% -that that is a good question what is -what<00:54:49.230> is<00:54:49.320> this<00:54:49.440> thing<00:54:49.620> here<00:54:49.890> it's<00:54:50.850> the<00:54:51.300> second - -00:54:51.710 --> 00:54:51.720 align:start position:0% -what is this thing here it's the second - - -00:54:51.720 --> 00:54:55.370 align:start position:0% -what is this thing here it's the second -derivative<00:54:52.170> with<00:54:52.620> respect<00:54:53.190> to<00:54:53.550> X<00:54:54.060> plus<00:54:54.570> the - -00:54:55.370 --> 00:54:55.380 align:start position:0% -derivative with respect to X plus the - - -00:54:55.380 --> 00:54:58.070 align:start position:0% -derivative with respect to X plus the -second<00:54:55.920> derivative<00:54:56.310> with<00:54:56.460> respect<00:54:56.850> to<00:54:56.910> Y<00:54:57.180> plus - -00:54:58.070 --> 00:54:58.080 align:start position:0% -second derivative with respect to Y plus - - -00:54:58.080 --> 00:55:00.430 align:start position:0% -second derivative with respect to Y plus -the<00:54:58.230> second<00:54:58.590> derivative<00:54:58.620> with<00:54:59.160> respect<00:54:59.550> to<00:54:59.610> Z - -00:55:00.430 --> 00:55:00.440 align:start position:0% -the second derivative with respect to Z - - -00:55:00.440 --> 00:55:03.500 align:start position:0% -the second derivative with respect to Z -all<00:55:01.440> right<00:55:01.530> well<00:55:01.950> it<00:55:02.040> says<00:55:02.250> what<00:55:03.060> it<00:55:03.180> says<00:55:03.300> is - -00:55:03.500 --> 00:55:03.510 align:start position:0% -all right well it says what it says is - - -00:55:03.510 --> 00:55:04.670 align:start position:0% -all right well it says what it says is -that<00:55:03.540> some<00:55:03.810> are<00:55:03.960> positive<00:55:04.380> and<00:55:04.500> some<00:55:04.560> are - -00:55:04.670 --> 00:55:04.680 align:start position:0% -that some are positive and some are - - -00:55:04.680 --> 00:55:10.880 align:start position:0% -that some are positive and some are -negative<00:55:06.920> it<00:55:07.920> doesn't<00:55:08.340> say<00:55:09.500> it<00:55:10.500> doesn't<00:55:10.590> say - -00:55:10.880 --> 00:55:10.890 align:start position:0% -negative it doesn't say it doesn't say - - -00:55:10.890 --> 00:55:16.250 align:start position:0% -negative it doesn't say it doesn't say -that<00:55:11.160> each<00:55:11.370> one<00:55:11.700> of<00:55:11.850> these<00:55:11.970> is<00:55:12.210> zero<00:55:15.080> first<00:55:16.080> of - -00:55:16.250 --> 00:55:16.260 align:start position:0% -that each one of these is zero first of - - -00:55:16.260 --> 00:55:18.440 align:start position:0% -that each one of these is zero first of -all<00:55:16.410> even<00:55:16.590> if<00:55:16.740> it<00:55:16.890> said<00:55:17.130> that<00:55:17.340> that<00:55:18.120> each<00:55:18.270> one - -00:55:18.440 --> 00:55:18.450 align:start position:0% -all even if it said that that each one - - -00:55:18.450 --> 00:55:20.600 align:start position:0% -all even if it said that that each one -was<00:55:18.630> zero<00:55:18.870> wouldn't<00:55:19.230> say<00:55:19.350> was<00:55:19.500> constant<00:55:20.040> it - -00:55:20.600 --> 00:55:20.610 align:start position:0% -was zero wouldn't say was constant it - - -00:55:20.610 --> 00:55:23.000 align:start position:0% -was zero wouldn't say was constant it -would<00:55:20.760> still<00:55:21.000> allow<00:55:21.120> a<00:55:21.270> first<00:55:21.630> derivative<00:55:22.010> but - -00:55:23.000 --> 00:55:23.010 align:start position:0% -would still allow a first derivative but - - -00:55:23.010 --> 00:55:24.710 align:start position:0% -would still allow a first derivative but -that<00:55:23.280> would<00:55:23.430> be<00:55:23.550> rather<00:55:23.730> limited<00:55:24.060> just<00:55:24.450> having - -00:55:24.710 --> 00:55:24.720 align:start position:0% -that would be rather limited just having - - -00:55:24.720 --> 00:55:26.930 align:start position:0% -that would be rather limited just having -first<00:55:24.900> derivatives<00:55:25.440> what<00:55:26.220> it<00:55:26.310> says<00:55:26.580> is<00:55:26.880> that - -00:55:26.930 --> 00:55:26.940 align:start position:0% -first derivatives what it says is that - - -00:55:26.940 --> 00:55:29.120 align:start position:0% -first derivatives what it says is that -the<00:55:27.750> combination<00:55:28.470> of<00:55:28.560> these<00:55:28.710> second - -00:55:29.120 --> 00:55:29.130 align:start position:0% -the combination of these second - - -00:55:29.130 --> 00:55:31.070 align:start position:0% -the combination of these second -derivatives<00:55:29.190> have<00:55:29.760> to<00:55:29.910> add<00:55:30.150> up<00:55:30.480> to<00:55:30.540> zero - -00:55:31.070 --> 00:55:31.080 align:start position:0% -derivatives have to add up to zero - - -00:55:31.080 --> 00:55:33.860 align:start position:0% -derivatives have to add up to zero -that's<00:55:31.980> quite<00:55:32.430> a<00:55:32.670> different<00:55:33.660> thing<00:55:33.840> than - -00:55:33.860 --> 00:55:33.870 align:start position:0% -that's quite a different thing than - - -00:55:33.870 --> 00:55:35.870 align:start position:0% -that's quite a different thing than -saying<00:55:34.260> it<00:55:34.440> is<00:55:34.590> that<00:55:34.890> they<00:55:35.010> are<00:55:35.190> individually - -00:55:35.870 --> 00:55:35.880 align:start position:0% -saying it is that they are individually - - -00:55:35.880 --> 00:55:38.230 align:start position:0% -saying it is that they are individually -zero<00:55:36.090> so<00:55:36.810> now<00:55:36.930> the<00:55:36.990> gravitational<00:55:37.770> potential - -00:55:38.230 --> 00:55:38.240 align:start position:0% -zero so now the gravitational potential - - -00:55:38.240 --> 00:55:41.270 align:start position:0% -zero so now the gravitational potential -does<00:55:39.240> vary<00:55:39.540> from<00:55:39.720> place<00:55:39.960> to<00:55:40.020> place<00:55:40.280> for - -00:55:41.270 --> 00:55:41.280 align:start position:0% -does vary from place to place for - - -00:55:41.280 --> 00:55:43.340 align:start position:0% -does vary from place to place for -example<00:55:41.400> if<00:55:42.000> you<00:55:42.270> take<00:55:42.510> the<00:55:42.630> gravitational - -00:55:43.340 --> 00:55:43.350 align:start position:0% -example if you take the gravitational - - -00:55:43.350 --> 00:55:45.530 align:start position:0% -example if you take the gravitational -potential<00:55:43.680> one<00:55:44.040> over<00:55:44.250> R<00:55:44.520> one<00:55:44.940> over<00:55:44.970> distance - -00:55:45.530 --> 00:55:45.540 align:start position:0% -potential one over R one over distance - - -00:55:45.540 --> 00:55:48.380 align:start position:0% -potential one over R one over distance -from<00:55:45.660> the<00:55:45.780> origin<00:55:46.250> then<00:55:47.250> except<00:55:47.790> at<00:55:48.060> the - -00:55:48.380 --> 00:55:48.390 align:start position:0% -from the origin then except at the - - -00:55:48.390 --> 00:55:51.590 align:start position:0% -from the origin then except at the -origin<00:55:48.420> itself<00:55:49.260> if<00:55:49.590> you<00:55:49.920> work<00:55:50.280> this<00:55:50.490> out<00:55:50.700> you - -00:55:51.590 --> 00:55:51.600 align:start position:0% -origin itself if you work this out you - - -00:55:51.600 --> 00:55:54.260 align:start position:0% -origin itself if you work this out you -will<00:55:51.720> find<00:55:52.020> out<00:55:52.260> that<00:55:52.560> del<00:55:52.950> squared<00:55:53.340> of<00:55:53.670> 1<00:55:54.060> over - -00:55:54.260 --> 00:55:54.270 align:start position:0% -will find out that del squared of 1 over - - -00:55:54.270 --> 00:55:59.630 align:start position:0% -will find out that del squared of 1 over -R<00:55:54.420> is<00:55:54.630> equal<00:55:54.900> to<00:55:54.990> zero<00:55:57.140> so<00:55:58.340> that<00:55:59.340> is<00:55:59.430> a<00:55:59.460> good - -00:55:59.630 --> 00:55:59.640 align:start position:0% -R is equal to zero so that is a good - - -00:55:59.640 --> 00:56:02.390 align:start position:0% -R is equal to zero so that is a good -question<00:56:00.260> and<00:56:01.260> it<00:56:01.410> has<00:56:01.440> a<00:56:01.560> good<00:56:01.740> answer<00:56:01.950> the - -00:56:02.390 --> 00:56:02.400 align:start position:0% -question and it has a good answer the - - -00:56:02.400 --> 00:56:05.510 align:start position:0% -question and it has a good answer the -answer<00:56:02.520> being<00:56:03.500> that<00:56:04.500> it<00:56:04.650> does<00:56:04.800> not<00:56:04.920> say<00:56:05.280> that - -00:56:05.510 --> 00:56:05.520 align:start position:0% -answer being that it does not say that - - -00:56:05.520 --> 00:56:07.820 align:start position:0% -answer being that it does not say that -the<00:56:05.610> individual<00:56:06.450> derivatives<00:56:07.020> are<00:56:07.140> zero<00:56:07.350> but - -00:56:07.820 --> 00:56:07.830 align:start position:0% -the individual derivatives are zero but - - -00:56:07.830 --> 00:56:10.520 align:start position:0% -the individual derivatives are zero but -just<00:56:07.980> a<00:56:08.040> combination<00:56:08.810> so<00:56:09.810> at<00:56:09.930> any<00:56:10.050> point<00:56:10.140> in - -00:56:10.520 --> 00:56:10.530 align:start position:0% -just a combination so at any point in - - -00:56:10.530 --> 00:56:12.860 align:start position:0% -just a combination so at any point in -space<00:56:10.770> where<00:56:10.800> there<00:56:11.070> is<00:56:11.190> no<00:56:11.340> matter<00:56:11.870> this - -00:56:12.860 --> 00:56:12.870 align:start position:0% -space where there is no matter this - - -00:56:12.870 --> 00:56:14.450 align:start position:0% -space where there is no matter this -combination<00:56:13.530> is<00:56:13.680> up<00:56:13.860> to<00:56:13.980> zero - -00:56:14.450 --> 00:56:14.460 align:start position:0% -combination is up to zero - - -00:56:14.460 --> 00:56:15.980 align:start position:0% -combination is up to zero -but<00:56:14.640> there<00:56:14.760> are<00:56:14.820> plenty<00:56:15.000> of<00:56:15.240> functions<00:56:15.780> with - -00:56:15.980 --> 00:56:15.990 align:start position:0% -but there are plenty of functions with - - -00:56:15.990 --> 00:56:18.110 align:start position:0% -but there are plenty of functions with -that<00:56:16.110> property<00:56:16.760> let's<00:56:17.760> write<00:56:17.940> down<00:56:17.970> some - -00:56:18.110 --> 00:56:18.120 align:start position:0% -that property let's write down some - - -00:56:18.120 --> 00:56:19.550 align:start position:0% -that property let's write down some -functions<00:56:18.720> with<00:56:18.840> that<00:56:18.960> let<00:56:19.140> me<00:56:19.230> just<00:56:19.410> write - -00:56:19.550 --> 00:56:19.560 align:start position:0% -functions with that let me just write - - -00:56:19.560 --> 00:56:22.150 align:start position:0% -functions with that let me just write -down<00:56:19.740> one<00:56:20.100> function<00:56:20.850> with<00:56:21.030> that<00:56:21.060> property - -00:56:22.150 --> 00:56:22.160 align:start position:0% -down one function with that property - - -00:56:22.160 --> 00:56:27.980 align:start position:0% -down one function with that property -take<00:56:24.050> x<00:56:25.050> squared<00:56:25.590> plus<00:56:26.010> y<00:56:26.250> squared<00:56:26.280> minus<00:56:27.120> 2z - -00:56:27.980 --> 00:56:27.990 align:start position:0% -take x squared plus y squared minus 2z - - -00:56:27.990 --> 00:56:31.010 align:start position:0% -take x squared plus y squared minus 2z -squared<00:56:28.730> okay<00:56:29.730> what's<00:56:30.480> the<00:56:30.660> second - -00:56:31.010 --> 00:56:31.020 align:start position:0% -squared okay what's the second - - -00:56:31.020 --> 00:56:32.540 align:start position:0% -squared okay what's the second -derivative<00:56:31.500> that<00:56:31.740> would<00:56:31.860> call<00:56:32.220> us<00:56:32.340> fire - -00:56:32.540 --> 00:56:32.550 align:start position:0% -derivative that would call us fire - - -00:56:32.550 --> 00:56:33.710 align:start position:0% -derivative that would call us fire -what's<00:56:32.820> the<00:56:32.910> second<00:56:33.240> derivative<00:56:33.540> of<00:56:33.600> this - -00:56:33.710 --> 00:56:33.720 align:start position:0% -what's the second derivative of this - - -00:56:33.720 --> 00:56:36.150 align:start position:0% -what's the second derivative of this -with<00:56:33.900> respect<00:56:33.930> to<00:56:34.290> X - -00:56:36.150 --> 00:56:36.160 align:start position:0% -with respect to X - - -00:56:36.160 --> 00:56:39.640 align:start position:0% -with respect to X --<00:56:37.160> how<00:56:37.670> about<00:56:37.730> with<00:56:37.940> respect<00:56:38.089> to<00:56:38.329> why<00:56:38.650> how - -00:56:39.640 --> 00:56:39.650 align:start position:0% -- how about with respect to why how - - -00:56:39.650 --> 00:56:42.790 align:start position:0% -- how about with respect to why how -about<00:56:39.800> with<00:56:39.920> respect<00:56:40.190> to<00:56:40.280> Z<00:56:41.170> minus<00:56:42.170> 4<00:56:42.410> so<00:56:42.740> 2 - -00:56:42.790 --> 00:56:42.800 align:start position:0% -about with respect to Z minus 4 so 2 - - -00:56:42.800 --> 00:56:47.170 align:start position:0% -about with respect to Z minus 4 so 2 -plus<00:56:43.220> 2<00:56:43.369> plus<00:56:43.400> minus<00:56:43.970> 4<00:56:44.390> is<00:56:44.599> equal<00:56:44.809> to<00:56:44.839> 0<00:56:46.119> ok<00:56:47.119> so - -00:56:47.170 --> 00:56:47.180 align:start position:0% -plus 2 plus minus 4 is equal to 0 ok so - - -00:56:47.180 --> 00:56:49.030 align:start position:0% -plus 2 plus minus 4 is equal to 0 ok so -this<00:56:47.450> is<00:56:47.660> a<00:56:47.690> question<00:56:48.050> this<00:56:48.440> is<00:56:48.559> a<00:56:48.589> function - -00:56:49.030 --> 00:56:49.040 align:start position:0% -this is a question this is a function - - -00:56:49.040 --> 00:56:50.559 align:start position:0% -this is a question this is a function -functions<00:56:49.490> with<00:56:49.640> this<00:56:49.790> property<00:56:50.210> of<00:56:50.359> called - -00:56:50.559 --> 00:56:50.569 align:start position:0% -functions with this property of called - - -00:56:50.569 --> 00:56:53.730 align:start position:0% -functions with this property of called -harmonic<00:56:51.050> functions<00:56:51.790> there's<00:56:52.790> lots<00:56:53.030> of<00:56:53.180> them - -00:56:53.730 --> 00:56:53.740 align:start position:0% -harmonic functions there's lots of them - - -00:56:53.740 --> 00:56:56.370 align:start position:0% -harmonic functions there's lots of them -there's<00:56:54.740> no<00:56:54.859> lack<00:56:55.069> of<00:56:55.099> harmonic<00:56:55.670> functions - -00:56:56.370 --> 00:56:56.380 align:start position:0% -there's no lack of harmonic functions - - -00:56:56.380 --> 00:57:04.809 align:start position:0% -there's no lack of harmonic functions -but<00:57:01.390> they're<00:57:02.390> still<00:57:02.599> not<00:57:02.839> as<00:57:03.140> many<00:57:04.099> as<00:57:04.670> there - -00:57:04.809 --> 00:57:04.819 align:start position:0% -but they're still not as many as there - - -00:57:04.819 --> 00:57:07.390 align:start position:0% -but they're still not as many as there -are<00:57:05.000> of<00:57:05.300> all<00:57:05.900> functions<00:57:06.440> so<00:57:07.130> there<00:57:07.280> are - -00:57:07.390 --> 00:57:07.400 align:start position:0% -are of all functions so there are - - -00:57:07.400 --> 00:57:11.920 align:start position:0% -are of all functions so there are -restrictions<00:57:09.700> ok<00:57:10.700> now<00:57:11.059> what<00:57:11.480> is<00:57:11.599> this<00:57:11.750> object - -00:57:11.920 --> 00:57:11.930 align:start position:0% -restrictions ok now what is this object - - -00:57:11.930 --> 00:57:15.040 align:start position:0% -restrictions ok now what is this object -on<00:57:12.290> the<00:57:12.410> on<00:57:12.980> the<00:57:13.130> right<00:57:13.250> hand<00:57:13.579> side<00:57:13.790> we<00:57:14.690> clearly - -00:57:15.040 --> 00:57:15.050 align:start position:0% -on the on the right hand side we clearly - - -00:57:15.050 --> 00:57:18.089 align:start position:0% -on the on the right hand side we clearly -have<00:57:15.410> something<00:57:15.740> involving<00:57:16.369> T<00:57:16.640> naught<00:57:16.880> naught - -00:57:18.089 --> 00:57:18.099 align:start position:0% -have something involving T naught naught - - -00:57:18.099 --> 00:57:20.500 align:start position:0% -have something involving T naught naught -all<00:57:19.099> right<00:57:19.190> the<00:57:19.400> gravitation<00:57:20.059> of<00:57:20.150> the<00:57:20.270> sorry - -00:57:20.500 --> 00:57:20.510 align:start position:0% -all right the gravitation of the sorry - - -00:57:20.510 --> 00:57:23.530 align:start position:0% -all right the gravitation of the sorry -the<00:57:20.720> the<00:57:20.750> mass<00:57:21.440> density<00:57:21.680> on<00:57:22.280> the<00:57:22.940> left<00:57:23.180> hand - -00:57:23.530 --> 00:57:23.540 align:start position:0% -the the mass density on the left hand - - -00:57:23.540 --> 00:57:25.660 align:start position:0% -the the mass density on the left hand -side<00:57:23.780> we<00:57:24.200> have<00:57:24.380> something<00:57:24.740> involving<00:57:25.220> two - -00:57:25.660 --> 00:57:25.670 align:start position:0% -side we have something involving two - - -00:57:25.670 --> 00:57:30.520 align:start position:0% -side we have something involving two -derivatives<00:57:26.599> of<00:57:27.339> what<00:57:28.339> is<00:57:28.550> in<00:57:28.970> essence<00:57:29.569> G - -00:57:30.520 --> 00:57:30.530 align:start position:0% -derivatives of what is in essence G - - -00:57:30.530 --> 00:57:34.079 align:start position:0% -derivatives of what is in essence G -naught<00:57:30.740> naught<00:57:31.720> components<00:57:32.720> of<00:57:32.839> the<00:57:32.900> metric - -00:57:34.079 --> 00:57:34.089 align:start position:0% -naught naught components of the metric - - -00:57:34.089 --> 00:57:35.920 align:start position:0% -naught naught components of the metric -we<00:57:35.089> don't<00:57:35.270> have<00:57:35.359> to<00:57:35.450> worry<00:57:35.599> about<00:57:35.660> this - -00:57:35.920 --> 00:57:35.930 align:start position:0% -we don't have to worry about this - - -00:57:35.930 --> 00:57:38.650 align:start position:0% -we don't have to worry about this -constant<00:57:36.530> the<00:57:37.130> first<00:57:37.430> derivative<00:57:38.059> gets<00:57:38.480> rid - -00:57:38.650 --> 00:57:38.660 align:start position:0% -constant the first derivative gets rid - - -00:57:38.660 --> 00:57:40.660 align:start position:0% -constant the first derivative gets rid -of<00:57:38.780> the<00:57:38.869> constant<00:57:39.230> and<00:57:39.650> the<00:57:40.309> second - -00:57:40.660 --> 00:57:40.670 align:start position:0% -of the constant and the second - - -00:57:40.670 --> 00:57:42.849 align:start position:0% -of the constant and the second -derivative<00:57:41.180> gives<00:57:42.049> us<00:57:42.260> some<00:57:42.500> second - -00:57:42.849 --> 00:57:42.859 align:start position:0% -derivative gives us some second - - -00:57:42.859 --> 00:57:45.220 align:start position:0% -derivative gives us some second -derivative<00:57:43.309> of<00:57:43.430> G<00:57:43.640> naught<00:57:43.819> naught<00:57:44.030> so<00:57:45.020> what's - -00:57:45.220 --> 00:57:45.230 align:start position:0% -derivative of G naught naught so what's - - -00:57:45.230 --> 00:57:48.460 align:start position:0% -derivative of G naught naught so what's -on<00:57:45.410> the<00:57:45.589> left<00:57:45.829> hand<00:57:46.130> side<00:57:46.809> involves<00:57:47.809> second - -00:57:48.460 --> 00:57:48.470 align:start position:0% -on the left hand side involves second - - -00:57:48.470 --> 00:57:51.700 align:start position:0% -on the left hand side involves second -derivatives<00:57:49.130> of<00:57:49.369> G<00:57:49.640> naught<00:57:49.849> naught<00:57:50.589> there<00:57:51.589> are - -00:57:51.700 --> 00:57:51.710 align:start position:0% -derivatives of G naught naught there are - - -00:57:51.710 --> 00:57:54.099 align:start position:0% -derivatives of G naught naught there are -not<00:57:51.920> a<00:57:51.950> lot<00:57:52.339> of<00:57:52.369> tensors<00:57:53.059> that<00:57:53.390> are<00:57:53.690> made<00:57:53.930> up - -00:57:54.099 --> 00:57:54.109 align:start position:0% -not a lot of tensors that are made up - - -00:57:54.109 --> 00:57:56.380 align:start position:0% -not a lot of tensors that are made up -out<00:57:54.410> of<00:57:54.710> second<00:57:55.460> derivatives<00:57:55.910> of<00:57:56.000> G<00:57:56.180> naught - -00:57:56.380 --> 00:57:56.390 align:start position:0% -out of second derivatives of G naught - - -00:57:56.390 --> 00:58:00.299 align:start position:0% -out of second derivatives of G naught -naught<00:57:56.680> there<00:57:57.680> are<00:57:57.890> relatively<00:57:58.819> few<00:57:59.210> of<00:57:59.359> them - -00:58:00.299 --> 00:58:00.309 align:start position:0% -naught there are relatively few of them - - -00:58:00.309 --> 00:58:04.599 align:start position:0% -naught there are relatively few of them -basically<00:58:01.309> only<00:58:01.549> two<00:58:02.410> geometric<00:58:03.609> tensors - -00:58:04.599 --> 00:58:04.609 align:start position:0% -basically only two geometric tensors - - -00:58:04.609 --> 00:58:08.200 align:start position:0% -basically only two geometric tensors -that<00:58:04.760> are<00:58:04.880> made<00:58:05.030> up<00:58:05.180> out<00:58:05.359> of<00:58:05.390> gene<00:58:05.960> or<00:58:06.109> not<00:58:07.210> what - -00:58:08.200 --> 00:58:08.210 align:start position:0% -that are made up out of gene or not what - - -00:58:08.210 --> 00:58:09.760 align:start position:0% -that are made up out of gene or not what -we<00:58:08.329> actually<00:58:08.839> want<00:58:09.049> to<00:58:09.109> have<00:58:09.349> on<00:58:09.530> the<00:58:09.619> right - -00:58:09.760 --> 00:58:09.770 align:start position:0% -we actually want to have on the right - - -00:58:09.770 --> 00:58:14.950 align:start position:0% -we actually want to have on the right -hand<00:58:10.040> side<00:58:12.130> is<00:58:13.130> team<00:58:14.089> you<00:58:14.240> know<00:58:14.420> in<00:58:14.750> the - -00:58:14.950 --> 00:58:14.960 align:start position:0% -hand side is team you know in the - - -00:58:14.960 --> 00:58:17.770 align:start position:0% -hand side is team you know in the -generalized<00:58:15.559> equation<00:58:16.299> I'm<00:58:17.299> not<00:58:17.480> saying<00:58:17.720> that - -00:58:17.770 --> 00:58:17.780 align:start position:0% -generalized equation I'm not saying that - - -00:58:17.780 --> 00:58:20.349 align:start position:0% -generalized equation I'm not saying that -this<00:58:18.020> is<00:58:18.170> equal<00:58:18.380> to<00:58:18.530> this<00:58:19.059> I'm<00:58:20.059> saying<00:58:20.210> the - -00:58:20.349 --> 00:58:20.359 align:start position:0% -this is equal to this I'm saying the - - -00:58:20.359 --> 00:58:22.420 align:start position:0% -this is equal to this I'm saying the -more<00:58:20.450> general<00:58:20.869> equation<00:58:21.470> that<00:58:21.890> we<00:58:22.010> would<00:58:22.160> like - -00:58:22.420 --> 00:58:22.430 align:start position:0% -more general equation that we would like - - -00:58:22.430 --> 00:58:25.390 align:start position:0% -more general equation that we would like -to<00:58:22.670> be<00:58:22.880> the<00:58:23.000> tensor<00:58:23.450> analog<00:58:24.020> of<00:58:24.049> this<00:58:24.400> will - -00:58:25.390 --> 00:58:25.400 align:start position:0% -to be the tensor analog of this will - - -00:58:25.400 --> 00:58:27.039 align:start position:0% -to be the tensor analog of this will -have<00:58:25.549> the<00:58:25.670> energy<00:58:25.819> momentum<00:58:26.119> tensor<00:58:26.839> on<00:58:26.960> the - -00:58:27.039 --> 00:58:27.049 align:start position:0% -have the energy momentum tensor on the - - -00:58:27.049 --> 00:58:29.530 align:start position:0% -have the energy momentum tensor on the -right<00:58:27.200> hand<00:58:27.410> side<00:58:27.530> maybe<00:58:27.950> maybe<00:58:28.520> some<00:58:28.760> 4<00:58:28.970> PI<00:58:29.180> G - -00:58:29.530 --> 00:58:29.540 align:start position:0% -right hand side maybe maybe some 4 PI G - - -00:58:29.540 --> 00:58:32.410 align:start position:0% -right hand side maybe maybe some 4 PI G -maybe<00:58:30.079> some<00:58:30.319> constant<00:58:30.859> let's<00:58:31.520> call<00:58:31.730> it<00:58:31.849> K<00:58:32.059> for - -00:58:32.410 --> 00:58:32.420 align:start position:0% -maybe some constant let's call it K for - - -00:58:32.420 --> 00:58:36.099 align:start position:0% -maybe some constant let's call it K for -constant<00:58:33.700> and<00:58:34.700> on<00:58:34.849> the<00:58:34.940> left<00:58:35.119> hand<00:58:35.450> side<00:58:35.660> we - -00:58:36.099 --> 00:58:36.109 align:start position:0% -constant and on the left hand side we - - -00:58:36.109 --> 00:58:38.980 align:start position:0% -constant and on the left hand side we -need<00:58:36.290> some<00:58:36.500> tensor<00:58:36.950> with<00:58:37.640> two<00:58:37.670> indices<00:58:38.180> which - -00:58:38.980 --> 00:58:38.990 align:start position:0% -need some tensor with two indices which - - -00:58:38.990 --> 00:58:41.710 align:start position:0% -need some tensor with two indices which -is<00:58:39.140> made<00:58:39.380> up<00:58:39.589> out<00:58:39.799> of<00:58:39.859> the<00:58:40.640> second<00:58:41.059> derivatives - -00:58:41.710 --> 00:58:41.720 align:start position:0% -is made up out of the second derivatives - - -00:58:41.720 --> 00:58:46.569 align:start position:0% -is made up out of the second derivatives -of<00:58:42.040> the<00:58:43.040> the<00:58:43.910> metric<00:58:45.099> second<00:58:46.099> derivatives<00:58:46.490> of - -00:58:46.569 --> 00:58:46.579 align:start position:0% -of the the metric second derivatives of - - -00:58:46.579 --> 00:58:47.829 align:start position:0% -of the the metric second derivatives of -the<00:58:46.640> metric<00:58:46.670> that<00:58:47.119> should<00:58:47.329> ring<00:58:47.510> a<00:58:47.599> bell - -00:58:47.829 --> 00:58:47.839 align:start position:0% -the metric that should ring a bell - - -00:58:47.839 --> 00:58:48.790 align:start position:0% -the metric that should ring a bell -sounds<00:58:48.230> like<00:58:48.349> the - -00:58:48.790 --> 00:58:48.800 align:start position:0% -sounds like the - - -00:58:48.800 --> 00:58:51.970 align:start position:0% -sounds like the -which<00:58:48.920> a<00:58:49.040> tensor<00:58:49.930> it<00:58:50.930> can't<00:58:51.290> be<00:58:51.500> the<00:58:51.650> Riemann - -00:58:51.970 --> 00:58:51.980 align:start position:0% -which a tensor it can't be the Riemann - - -00:58:51.980 --> 00:58:54.850 align:start position:0% -which a tensor it can't be the Riemann -tensor<00:58:52.700> because<00:58:52.970> that<00:58:53.120> has<00:58:53.270> four<00:58:53.510> indices<00:58:53.860> it - -00:58:54.850 --> 00:58:54.860 align:start position:0% -tensor because that has four indices it - - -00:58:54.860 --> 00:58:58.210 align:start position:0% -tensor because that has four indices it -can<00:58:55.220> be<00:58:55.460> the<00:58:55.820> Ricci<00:58:56.000> tensor<00:58:56.570> the<00:58:57.440> first<00:58:57.770> thing - -00:58:58.210 --> 00:58:58.220 align:start position:0% -can be the Ricci tensor the first thing - - -00:58:58.220 --> 00:59:05.860 align:start position:0% -can be the Ricci tensor the first thing -that<00:58:58.460> Einstein<00:58:58.760> tried<00:58:59.590> was<00:59:00.590> just<00:59:01.420> mu<00:59:02.420> nu<00:59:04.870> equal - -00:59:05.860 --> 00:59:05.870 align:start position:0% -that Einstein tried was just mu nu equal - - -00:59:05.870 --> 00:59:08.040 align:start position:0% -that Einstein tried was just mu nu equal -that<00:59:06.010> okay - -00:59:08.040 --> 00:59:08.050 align:start position:0% -that okay - - -00:59:08.050 --> 00:59:11.170 align:start position:0% -that okay -he<00:59:09.050> ran<00:59:09.260> into<00:59:09.530> grief<00:59:09.770> the<00:59:10.490> reason<00:59:10.820> that<00:59:10.910> I<00:59:10.970> ran - -00:59:11.170 --> 00:59:11.180 align:start position:0% -he ran into grief the reason that I ran - - -00:59:11.180 --> 00:59:12.880 align:start position:0% -he ran into grief the reason that I ran -into<00:59:11.360> grief<00:59:11.720> had<00:59:12.140> to<00:59:12.290> do<00:59:12.440> with<00:59:12.650> energy - -00:59:12.880 --> 00:59:12.890 align:start position:0% -into grief had to do with energy - - -00:59:12.890 --> 00:59:16.030 align:start position:0% -into grief had to do with energy -momentum<00:59:13.310> conservation<00:59:14.740> let's<00:59:15.740> write<00:59:16.010> down - -00:59:16.030 --> 00:59:16.040 align:start position:0% -momentum conservation let's write down - - -00:59:16.040 --> 00:59:18.640 align:start position:0% -momentum conservation let's write down -what<00:59:16.520> in<00:59:17.120> ordinary<00:59:17.810> non-relative<00:59:18.590> an - -00:59:18.640 --> 00:59:18.650 align:start position:0% -what in ordinary non-relative an - - -00:59:18.650 --> 00:59:22.650 align:start position:0% -what in ordinary non-relative an -ordinary<00:59:19.130> I<00:59:20.230> suppose<00:59:21.230> special<00:59:21.620> relativity - -00:59:22.650 --> 00:59:22.660 align:start position:0% -ordinary I suppose special relativity - - -00:59:22.660 --> 00:59:25.750 align:start position:0% -ordinary I suppose special relativity -the<00:59:23.660> conservation<00:59:24.440> of<00:59:24.560> energy<00:59:24.680> momentum<00:59:25.100> let - -00:59:25.750 --> 00:59:25.760 align:start position:0% -the conservation of energy momentum let - - -00:59:25.760 --> 00:59:27.760 align:start position:0% -the conservation of energy momentum let -me<00:59:25.820> just<00:59:25.970> remind<00:59:26.240> you<00:59:26.390> go<00:59:26.750> slow<00:59:27.020> the - -00:59:27.760 --> 00:59:27.770 align:start position:0% -me just remind you go slow the - - -00:59:27.770 --> 00:59:30.070 align:start position:0% -me just remind you go slow the -conservation<00:59:28.400> of<00:59:28.490> charge<00:59:28.550> the<00:59:29.420> conservation - -00:59:30.070 --> 00:59:30.080 align:start position:0% -conservation of charge the conservation - - -00:59:30.080 --> 00:59:31.780 align:start position:0% -conservation of charge the conservation -of<00:59:30.170> charge<00:59:30.230> was<00:59:30.740> expressed<00:59:31.160> in<00:59:31.370> terms<00:59:31.580> of<00:59:31.700> a - -00:59:31.780 --> 00:59:31.790 align:start position:0% -of charge was expressed in terms of a - - -00:59:31.790 --> 00:59:34.950 align:start position:0% -of charge was expressed in terms of a -current<00:59:32.210> and<00:59:32.480> the<00:59:33.320> current<00:59:33.500> had<00:59:33.830> an<00:59:33.980> index<00:59:34.430> mu - -00:59:34.950 --> 00:59:34.960 align:start position:0% -current and the current had an index mu - - -00:59:34.960 --> 00:59:38.110 align:start position:0% -current and the current had an index mu -the<00:59:35.960> time<00:59:36.230> component<00:59:36.860> was<00:59:36.980> density<00:59:37.490> the<00:59:37.850> space - -00:59:38.110 --> 00:59:38.120 align:start position:0% -the time component was density the space - - -00:59:38.120 --> 00:59:41.010 align:start position:0% -the time component was density the space -component<00:59:38.780> was<00:59:38.960> the<00:59:39.110> flow<00:59:39.320> of<00:59:39.350> charge<00:59:39.820> and - -00:59:41.010 --> 00:59:41.020 align:start position:0% -component was the flow of charge and - - -00:59:41.020 --> 00:59:43.960 align:start position:0% -component was the flow of charge and -anybody<00:59:42.020> remember<00:59:42.470> what<00:59:42.650> the<00:59:42.710> equation<00:59:43.340> is - -00:59:43.960 --> 00:59:43.970 align:start position:0% -anybody remember what the equation is - - -00:59:43.970 --> 00:59:53.470 align:start position:0% -anybody remember what the equation is -for<00:59:44.210> for<00:59:44.840> continuity<00:59:45.380> equation<00:59:52.240> well<00:59:53.240> just - -00:59:53.470 --> 00:59:53.480 align:start position:0% -for for continuity equation well just - - -00:59:53.480 --> 00:59:57.580 align:start position:0% -for for continuity equation well just -derivative<00:59:53.960> ordinary<00:59:54.380> derivative<00:59:54.890> I<00:59:56.590> can - -00:59:57.580 --> 00:59:57.590 align:start position:0% -derivative ordinary derivative I can - - -00:59:57.590 --> 00:59:59.050 align:start position:0% -derivative ordinary derivative I can -also<00:59:57.740> be<00:59:58.010> expressed<00:59:58.430> in<00:59:58.610> terms<00:59:58.640> of<00:59:58.970> a - -00:59:59.050 --> 00:59:59.060 align:start position:0% -also be expressed in terms of a - - -00:59:59.060 --> 01:00:03.730 align:start position:0% -also be expressed in terms of a -covariant<00:59:59.690> derivative<01:00:01.090> but<01:00:02.350> ordinary<01:00:03.350> you - -01:00:03.730 --> 01:00:03.740 align:start position:0% -covariant derivative but ordinary you - - -01:00:03.740 --> 01:00:05.460 align:start position:0% -covariant derivative but ordinary you -know<01:00:03.800> an<01:00:03.950> ordinary<01:00:04.310> flat<01:00:04.580> special<01:00:05.000> relativity - -01:00:05.460 --> 01:00:05.470 align:start position:0% -know an ordinary flat special relativity - - -01:00:05.470 --> 01:00:10.660 align:start position:0% -know an ordinary flat special relativity -it's<01:00:06.470> D<01:00:06.860> mu<01:00:07.100> J<01:00:07.640> mu<01:00:07.670> is<01:00:08.120> equal<01:00:08.390> to<01:00:08.660> zero<01:00:09.490> also<01:00:10.490> in - -01:00:10.660 --> 01:00:10.670 align:start position:0% -it's D mu J mu is equal to zero also in - - -01:00:10.670 --> 01:00:13.090 align:start position:0% -it's D mu J mu is equal to zero also in -special<01:00:11.090> relativity<01:00:12.040> there's<01:00:13.040> a - -01:00:13.090 --> 01:00:13.100 align:start position:0% -special relativity there's a - - -01:00:13.100 --> 01:00:15.820 align:start position:0% -special relativity there's a -conservation<01:00:13.820> of<01:00:13.910> energy<01:00:14.030> and<01:00:14.390> momentum<01:00:14.830> each - -01:00:15.820 --> 01:00:15.830 align:start position:0% -conservation of energy and momentum each - - -01:00:15.830 --> 01:00:17.740 align:start position:0% -conservation of energy and momentum each -component<01:00:16.700> of<01:00:16.850> the<01:00:16.940> energy<01:00:17.090> and<01:00:17.510> the<01:00:17.600> momentum - -01:00:17.740 --> 01:00:17.750 align:start position:0% -component of the energy and the momentum - - -01:00:17.750 --> 01:00:20.350 align:start position:0% -component of the energy and the momentum -are<01:00:18.260> conserved<01:00:18.290> and<01:00:19.190> that<01:00:19.730> became<01:00:20.120> the - -01:00:20.350 --> 01:00:20.360 align:start position:0% -are conserved and that became the - - -01:00:20.360 --> 01:00:25.450 align:start position:0% -are conserved and that became the -equation<01:00:20.870> D<01:00:21.350> mu<01:00:21.620> of<01:00:22.180> T<01:00:23.180> mu<01:00:23.780> nu<01:00:24.020> is<01:00:24.710> equal<01:00:25.340> to - -01:00:25.450 --> 01:00:25.460 align:start position:0% -equation D mu of T mu nu is equal to - - -01:00:25.460 --> 01:00:28.720 align:start position:0% -equation D mu of T mu nu is equal to -zero<01:00:26.590> nu<01:00:27.590> here<01:00:27.890> tells<01:00:28.250> you<01:00:28.370> whether<01:00:28.490> you're - -01:00:28.720 --> 01:00:28.730 align:start position:0% -zero nu here tells you whether you're - - -01:00:28.730 --> 01:00:31.360 align:start position:0% -zero nu here tells you whether you're -talking<01:00:28.910> about<01:00:29.120> energy<01:00:29.600> and<01:00:29.720> momentum<01:00:30.370> mu - -01:00:31.360 --> 01:00:31.370 align:start position:0% -talking about energy and momentum mu - - -01:00:31.370 --> 01:00:32.920 align:start position:0% -talking about energy and momentum mu -tells<01:00:31.790> you<01:00:31.940> whether<01:00:32.120> you're<01:00:32.360> talking<01:00:32.600> about - -01:00:32.920 --> 01:00:32.930 align:start position:0% -tells you whether you're talking about - - -01:00:32.930 --> 01:00:37.240 align:start position:0% -tells you whether you're talking about -the<01:00:33.200> density<01:00:33.440> or<01:00:33.860> the<01:00:34.220> flow<01:00:34.490> and<01:00:35.740> so<01:00:36.740> for<01:00:37.040> each - -01:00:37.240 --> 01:00:37.250 align:start position:0% -the density or the flow and so for each - - -01:00:37.250 --> 01:00:41.550 align:start position:0% -the density or the flow and so for each -new<01:00:37.580> this<01:00:38.180> is<01:00:38.240> a<01:00:38.420> continuity<01:00:39.080> equation<01:00:40.420> the - -01:00:41.550 --> 01:00:41.560 align:start position:0% -new this is a continuity equation the - - -01:00:41.560 --> 01:00:44.380 align:start position:0% -new this is a continuity equation the -correct<01:00:42.560> continuity<01:00:43.370> equation<01:00:43.880> then<01:00:44.120> we'll - -01:00:44.380 --> 01:00:44.390 align:start position:0% -correct continuity equation then we'll - - -01:00:44.390 --> 01:00:47.020 align:start position:0% -correct continuity equation then we'll -come<01:00:44.630> to<01:00:44.810> this<01:00:44.930> I'll<01:00:45.710> explain<01:00:46.160> difference<01:00:46.760> in - -01:00:47.020 --> 01:00:47.030 align:start position:0% -come to this I'll explain difference in - - -01:00:47.030 --> 01:00:48.520 align:start position:0% -come to this I'll explain difference in -a<01:00:47.090> little<01:00:47.300> while<01:00:47.480> but<01:00:48.080> the<01:00:48.230> correct - -01:00:48.520 --> 01:00:48.530 align:start position:0% -a little while but the correct - - -01:00:48.530 --> 01:00:51.460 align:start position:0% -a little while but the correct -continuity<01:00:49.250> equation<01:00:49.730> is<01:00:49.910> that<01:00:50.150> the<01:00:50.230> in<01:00:51.230> this - -01:00:51.460 --> 01:00:51.470 align:start position:0% -continuity equation is that the in this - - -01:00:51.470 --> 01:00:55.540 align:start position:0% -continuity equation is that the in this -form<01:00:52.210> it<01:00:53.210> doesn't<01:00:54.080> really<01:00:54.260> make<01:00:54.530> sense<01:00:54.860> why - -01:00:55.540 --> 01:00:55.550 align:start position:0% -form it doesn't really make sense why - - -01:00:55.550 --> 01:00:57.760 align:start position:0% -form it doesn't really make sense why -not<01:00:55.790> because<01:00:56.750> it's<01:00:56.900> not<01:00:57.050> a<01:00:57.110> covariant - -01:00:57.760 --> 01:00:57.770 align:start position:0% -not because it's not a covariant - - -01:00:57.770 --> 01:01:00.070 align:start position:0% -not because it's not a covariant -derivative<01:00:58.250> incidentally - -01:01:00.070 --> 01:01:00.080 align:start position:0% -derivative incidentally - - -01:01:00.080 --> 01:01:01.390 align:start position:0% -derivative incidentally -we're<01:01:00.260> gonna<01:01:00.380> have<01:01:00.560> to<01:01:00.650> come<01:01:00.830> back<01:01:00.860> to<01:01:01.250> why - -01:01:01.390 --> 01:01:01.400 align:start position:0% -we're gonna have to come back to why - - -01:01:01.400 --> 01:01:05.110 align:start position:0% -we're gonna have to come back to why -this<01:01:01.460> one<01:01:01.790> does<01:01:01.970> make<01:01:02.150> sense<01:01:03.130> but<01:01:04.130> let's<01:01:04.640> let's - -01:01:05.110 --> 01:01:05.120 align:start position:0% -this one does make sense but let's let's - - -01:01:05.120 --> 01:01:07.630 align:start position:0% -this one does make sense but let's let's -just<01:01:05.210> focus<01:01:05.810> on<01:01:06.080> the<01:01:06.530> energy<01:01:06.830> momentum<01:01:06.980> tensor - -01:01:07.630 --> 01:01:07.640 align:start position:0% -just focus on the energy momentum tensor - - -01:01:07.640 --> 01:01:11.950 align:start position:0% -just focus on the energy momentum tensor -the<01:01:08.270> right<01:01:08.450> equation<01:01:09.110> is<01:01:09.760> this<01:01:10.760> one<01:01:11.030> but<01:01:11.840> the - -01:01:11.950 --> 01:01:11.960 align:start position:0% -the right equation is this one but the - - -01:01:11.960 --> 01:01:14.680 align:start position:0% -the right equation is this one but the -covariant<01:01:12.680> thats<01:01:13.430> a<01:01:13.550> nice<01:01:13.700> tensor<01:01:14.180> equation - -01:01:14.680 --> 01:01:14.690 align:start position:0% -covariant thats a nice tensor equation - - -01:01:14.690 --> 01:01:16.750 align:start position:0% -covariant thats a nice tensor equation -if<01:01:15.140> it's<01:01:15.350> true<01:01:15.590> in<01:01:15.710> one<01:01:15.890> frame<01:01:16.190> its<01:01:16.430> true<01:01:16.640> in - -01:01:16.750 --> 01:01:16.760 align:start position:0% -if it's true in one frame its true in - - -01:01:16.760 --> 01:01:20.050 align:start position:0% -if it's true in one frame its true in -every<01:01:17.000> frame<01:01:18.160> covariant<01:01:19.160> derivative<01:01:19.610> of<01:01:19.730> T<01:01:19.910> mu - -01:01:20.050 --> 01:01:20.060 align:start position:0% -every frame covariant derivative of T mu - - -01:01:20.060 --> 01:01:22.810 align:start position:0% -every frame covariant derivative of T mu -nu<01:01:20.270> is<01:01:20.510> equal<01:01:20.660> to<01:01:20.960> zero<01:01:21.430> for<01:01:22.430> the<01:01:22.520> charge - -01:01:22.810 --> 01:01:22.820 align:start position:0% -nu is equal to zero for the charge - - -01:01:22.820 --> 01:01:25.000 align:start position:0% -nu is equal to zero for the charge -density<01:01:23.150> the<01:01:23.600> covariant<01:01:24.170> derivative<01:01:24.200> and<01:01:24.860> the - -01:01:25.000 --> 01:01:25.010 align:start position:0% -density the covariant derivative and the - - -01:01:25.010 --> 01:01:26.710 align:start position:0% -density the covariant derivative and the -ordinary<01:01:25.400> derivative<01:01:25.880> can<01:01:26.060> be<01:01:26.120> taken<01:01:26.510> to<01:01:26.600> be - -01:01:26.710 --> 01:01:26.720 align:start position:0% -ordinary derivative can be taken to be - - -01:01:26.720 --> 01:01:28.690 align:start position:0% -ordinary derivative can be taken to be -the<01:01:26.870> same<01:01:27.050> for<01:01:27.230> technical<01:01:27.590> reasons<01:01:28.100> that<01:01:28.310> I - -01:01:28.690 --> 01:01:28.700 align:start position:0% -the same for technical reasons that I - - -01:01:28.700 --> 01:01:31.060 align:start position:0% -the same for technical reasons that I -may<01:01:29.150> or<01:01:29.300> may<01:01:29.420> not<01:01:29.450> ever<01:01:29.720> get<01:01:29.960> to<01:01:30.020> but<01:01:30.470> let's - -01:01:31.060 --> 01:01:31.070 align:start position:0% -may or may not ever get to but let's - - -01:01:31.070 --> 01:01:32.260 align:start position:0% -may or may not ever get to but let's -concentrate<01:01:31.250> on<01:01:31.670> the<01:01:31.880> energy<01:01:32.150> momentum - -01:01:32.260 --> 01:01:32.270 align:start position:0% -concentrate on the energy momentum - - -01:01:32.270 --> 01:01:36.790 align:start position:0% -concentrate on the energy momentum -tensor<01:01:33.010> it<01:01:34.010> should<01:01:34.280> satisfy<01:01:34.520> Del<01:01:35.360> mu<01:01:35.750> T<01:01:36.620> mu<01:01:36.740> nu - -01:01:36.790 --> 01:01:36.800 align:start position:0% -tensor it should satisfy Del mu T mu nu - - -01:01:36.800 --> 01:01:41.320 align:start position:0% -tensor it should satisfy Del mu T mu nu -equals<01:01:37.220> zero<01:01:38.320> that<01:01:39.320> should<01:01:39.860> be<01:01:40.100> a<01:01:40.130> property<01:01:41.060> of - -01:01:41.320 --> 01:01:41.330 align:start position:0% -equals zero that should be a property of - - -01:01:41.330 --> 01:01:43.840 align:start position:0% -equals zero that should be a property of -the<01:01:42.260> energy<01:01:42.620> momentum<01:01:42.740> tensor<01:01:43.310> without<01:01:43.610> it - -01:01:43.840 --> 01:01:43.850 align:start position:0% -the energy momentum tensor without it - - -01:01:43.850 --> 01:01:46.290 align:start position:0% -the energy momentum tensor without it -there<01:01:44.390> would<01:01:44.510> be<01:01:44.570> no<01:01:44.720> conservation<01:01:45.680> of<01:01:45.770> energy - -01:01:46.290 --> 01:01:46.300 align:start position:0% -there would be no conservation of energy - - -01:01:46.300 --> 01:01:50.350 align:start position:0% -there would be no conservation of energy -we<01:01:47.300> must<01:01:47.540> have<01:01:47.720> in<01:01:48.380> special<01:01:48.890> relativity<01:01:49.360> and - -01:01:50.350 --> 01:01:50.360 align:start position:0% -we must have in special relativity and - - -01:01:50.360 --> 01:01:52.450 align:start position:0% -we must have in special relativity and -in<01:01:50.480> general<01:01:50.810> relativity<01:01:50.870> we<01:01:51.440> also<01:01:51.620> expect<01:01:52.220> us - -01:01:52.450 --> 01:01:52.460 align:start position:0% -in general relativity we also expect us - - -01:01:52.460 --> 01:01:55.510 align:start position:0% -in general relativity we also expect us -expect<01:01:52.940> some<01:01:53.150> kind<01:01:53.180> of<01:01:53.600> local<01:01:54.350> flow<01:01:54.800> of<01:01:55.220> energy - -01:01:55.510 --> 01:01:55.520 align:start position:0% -expect some kind of local flow of energy - - -01:01:55.520 --> 01:01:57.850 align:start position:0% -expect some kind of local flow of energy -and<01:01:55.880> momentum<01:01:56.170> we<01:01:57.170> expect<01:01:57.620> it<01:01:57.740> to<01:01:57.770> be - -01:01:57.850 --> 01:01:57.860 align:start position:0% -and momentum we expect it to be - - -01:01:57.860 --> 01:01:59.830 align:start position:0% -and momentum we expect it to be -conserved<01:01:58.400> we<01:01:58.760> don't<01:01:58.940> expect<01:01:59.240> energy<01:01:59.630> to - -01:01:59.830 --> 01:01:59.840 align:start position:0% -conserved we don't expect energy to - - -01:01:59.840 --> 01:02:02.640 align:start position:0% -conserved we don't expect energy to -disappear<01:02:00.350> and<01:02:00.530> reappear<01:02:00.590> in<01:02:01.190> distant<01:02:01.580> places - -01:02:02.640 --> 01:02:02.650 align:start position:0% -disappear and reappear in distant places - - -01:02:02.650 --> 01:02:07.000 align:start position:0% -disappear and reappear in distant places -and<01:02:03.650> so<01:02:04.420> that<01:02:05.420> says<01:02:05.840> that<01:02:06.530> a<01:02:06.560> certain - -01:02:07.000 --> 01:02:07.010 align:start position:0% -and so that says that a certain - - -01:02:07.010 --> 01:02:09.790 align:start position:0% -and so that says that a certain -covariant<01:02:07.880> derivative<01:02:08.690> of<01:02:08.990> T<01:02:09.320> mu<01:02:09.470> nu<01:02:09.530> is<01:02:09.770> equal - -01:02:09.790 --> 01:02:09.800 align:start position:0% -covariant derivative of T mu nu is equal - - -01:02:09.800 --> 01:02:14.010 align:start position:0% -covariant derivative of T mu nu is equal -to<01:02:10.190> zero<01:02:10.930> we<01:02:11.930> can<01:02:12.110> try<01:02:12.380> that<01:02:12.620> out<01:02:12.650> over<01:02:13.190> here<01:02:13.610> on - -01:02:14.010 --> 01:02:14.020 align:start position:0% -to zero we can try that out over here on - - -01:02:14.020 --> 01:02:16.690 align:start position:0% -to zero we can try that out over here on -the<01:02:15.020> left<01:02:15.200> hand<01:02:15.470> on<01:02:15.680> the<01:02:15.710> right<01:02:15.770> hand<01:02:16.190> side<01:02:16.400> we - -01:02:16.690 --> 01:02:16.700 align:start position:0% -the left hand on the right hand side we - - -01:02:16.700 --> 01:02:21.700 align:start position:0% -the left hand on the right hand side we -can<01:02:16.730> take<01:02:17.120> del<01:02:17.450> mu<01:02:17.750> of<01:02:18.340> T<01:02:19.340> mu<01:02:19.550> nu<01:02:19.580> times<01:02:20.710> Kappa - -01:02:21.700 --> 01:02:21.710 align:start position:0% -can take del mu of T mu nu times Kappa - - -01:02:21.710 --> 01:02:24.970 align:start position:0% -can take del mu of T mu nu times Kappa -times<01:02:22.100> K<01:02:22.670> times<01:02:23.060> this<01:02:23.300> constant<01:02:23.780> K<01:02:24.020> and - -01:02:24.970 --> 01:02:24.980 align:start position:0% -times K times this constant K and - - -01:02:24.980 --> 01:02:27.340 align:start position:0% -times K times this constant K and -according<01:02:25.430> to<01:02:25.460> this<01:02:25.640> equation<01:02:26.240> it<01:02:26.930> would<01:02:27.140> say - -01:02:27.340 --> 01:02:27.350 align:start position:0% -according to this equation it would say - - -01:02:27.350 --> 01:02:33.190 align:start position:0% -according to this equation it would say -that<01:02:27.650> del<01:02:28.130> mu<01:02:28.400> of<01:02:28.730> our<01:02:29.690> mu<01:02:30.200> nu<01:02:30.650> is<01:02:31.810> equal<01:02:32.810> to - -01:02:33.190 --> 01:02:33.200 align:start position:0% -that del mu of our mu nu is equal to - - -01:02:33.200 --> 01:02:37.270 align:start position:0% -that del mu of our mu nu is equal to -zero<01:02:35.020> in<01:02:36.020> other<01:02:36.140> words<01:02:36.380> conservation<01:02:37.190> of - -01:02:37.270 --> 01:02:37.280 align:start position:0% -zero in other words conservation of - - -01:02:37.280 --> 01:02:40.870 align:start position:0% -zero in other words conservation of -energy<01:02:37.400> momentum<01:02:37.760> would<01:02:38.600> demand<01:02:39.640> that<01:02:40.640> this - -01:02:40.870 --> 01:02:40.880 align:start position:0% -energy momentum would demand that this - - -01:02:40.880 --> 01:02:44.680 align:start position:0% -energy momentum would demand that this -property<01:02:41.540> be<01:02:41.720> true<01:02:42.110> of<01:02:43.000> the<01:02:44.000> ramond<01:02:44.360> of<01:02:44.540> the - -01:02:44.680 --> 01:02:44.690 align:start position:0% -property be true of the ramond of the - - -01:02:44.690 --> 01:02:49.750 align:start position:0% -property be true of the ramond of the -Ricci<01:02:44.900> tensor<01:02:46.510> well<01:02:47.510> it<01:02:47.780> turns<01:02:48.080> out<01:02:48.580> that<01:02:49.580> that - -01:02:49.750 --> 01:02:49.760 align:start position:0% -Ricci tensor well it turns out that that - - -01:02:49.760 --> 01:02:53.200 align:start position:0% -Ricci tensor well it turns out that that -property<01:02:50.390> of<01:02:50.630> the<01:02:51.260> Ricci<01:02:51.440> tensor<01:02:52.040> is<01:02:52.610> almost - -01:02:53.200 --> 01:02:53.210 align:start position:0% -property of the Ricci tensor is almost - - -01:02:53.210 --> 01:02:56.410 align:start position:0% -property of the Ricci tensor is almost -never<01:02:53.660> true<01:02:54.640> basically<01:02:55.640> it<01:02:56.120> would<01:02:56.210> require - -01:02:56.410 --> 01:02:56.420 align:start position:0% -never true basically it would require - - -01:02:56.420 --> 01:02:59.380 align:start position:0% -never true basically it would require -it's<01:02:57.320> just<01:02:57.650> a<01:02:57.770> property<01:02:58.250> of<01:02:58.370> the<01:02:58.460> Ricci<01:02:58.970> tensor - -01:02:59.380 --> 01:02:59.390 align:start position:0% -it's just a property of the Ricci tensor - - -01:02:59.390 --> 01:03:02.770 align:start position:0% -it's just a property of the Ricci tensor -it's<01:02:59.570> it's<01:03:00.230> never<01:03:00.410> true<01:03:00.680> and<01:03:01.490> it's<01:03:02.360> it's<01:03:02.660> a - -01:03:02.770 --> 01:03:02.780 align:start position:0% -it's it's never true and it's it's a - - -01:03:02.780 --> 01:03:06.580 align:start position:0% -it's it's never true and it's it's a -false<01:03:03.080> theorem<01:03:04.010> or<01:03:04.250> a<01:03:04.280> false<01:03:04.610> identity<01:03:05.590> where - -01:03:06.580 --> 01:03:06.590 align:start position:0% -false theorem or a false identity where - - -01:03:06.590 --> 01:03:09.160 align:start position:0% -false theorem or a false identity where -Einstein<01:03:06.770> wanted<01:03:07.640> was<01:03:08.330> for<01:03:08.510> the<01:03:08.630> left<01:03:08.870> hand - -01:03:09.160 --> 01:03:09.170 align:start position:0% -Einstein wanted was for the left hand - - -01:03:09.170 --> 01:03:12.319 align:start position:0% -Einstein wanted was for the left hand -side<01:03:09.470> to<01:03:10.040> be<01:03:10.070> an<01:03:10.370> identity - -01:03:12.319 --> 01:03:12.329 align:start position:0% -side to be an identity - - -01:03:12.329 --> 01:03:16.349 align:start position:0% -side to be an identity -automatic<01:03:13.329> so<01:03:14.170> that<01:03:14.500> energy-momentum<01:03:15.490> tensor - -01:03:16.349 --> 01:03:16.359 align:start position:0% -automatic so that energy-momentum tensor - - -01:03:16.359 --> 01:03:17.849 align:start position:0% -automatic so that energy-momentum tensor -the<01:03:16.690> sorry<01:03:16.930> so<01:03:17.230> that<01:03:17.380> energy<01:03:17.530> momentum - -01:03:17.849 --> 01:03:17.859 align:start position:0% -the sorry so that energy momentum - - -01:03:17.859 --> 01:03:21.210 align:start position:0% -the sorry so that energy momentum -conservation<01:03:18.930> would<01:03:19.930> be<01:03:20.140> a<01:03:20.170> consequence<01:03:21.069> of - -01:03:21.210 --> 01:03:21.220 align:start position:0% -conservation would be a consequence of - - -01:03:21.220 --> 01:03:24.960 align:start position:0% -conservation would be a consequence of -his<01:03:21.369> equations<01:03:22.800> he<01:03:23.800> wanted<01:03:24.130> to<01:03:24.250> follow<01:03:24.609> the - -01:03:24.960 --> 01:03:24.970 align:start position:0% -his equations he wanted to follow the - - -01:03:24.970 --> 01:03:29.220 align:start position:0% -his equations he wanted to follow the -pattern<01:03:25.540> of<01:03:26.400> Maxwell's<01:03:27.400> equations<01:03:28.230> Maxwell's - -01:03:29.220 --> 01:03:29.230 align:start position:0% -pattern of Maxwell's equations Maxwell's - - -01:03:29.230 --> 01:03:32.490 align:start position:0% -pattern of Maxwell's equations Maxwell's -equations<01:03:30.010> have<01:03:30.490> the<01:03:30.520> property<01:03:31.329> that<01:03:32.319> their - -01:03:32.490 --> 01:03:32.500 align:start position:0% -equations have the property that their - - -01:03:32.500 --> 01:03:35.099 align:start position:0% -equations have the property that their -connections<01:03:33.160> between<01:03:33.910> the<01:03:34.569> electric<01:03:34.990> and - -01:03:35.099 --> 01:03:35.109 align:start position:0% -connections between the electric and - - -01:03:35.109 --> 01:03:37.950 align:start position:0% -connections between the electric and -magnetic<01:03:35.170> fields<01:03:35.980> on<01:03:36.099> one<01:03:36.309> side<01:03:36.960> something - -01:03:37.950 --> 01:03:37.960 align:start position:0% -magnetic fields on one side something - - -01:03:37.960 --> 01:03:40.380 align:start position:0% -magnetic fields on one side something -involving<01:03:38.650> a<01:03:38.950> and<01:03:39.099> B<01:03:39.309> on<01:03:39.460> one<01:03:39.670> side<01:03:39.940> whatever - -01:03:40.380 --> 01:03:40.390 align:start position:0% -involving a and B on one side whatever - - -01:03:40.390 --> 01:03:42.839 align:start position:0% -involving a and B on one side whatever -it<01:03:40.510> is<01:03:40.710> and<01:03:41.710> on<01:03:41.859> the<01:03:41.950> other<01:03:42.069> side<01:03:42.220> what's<01:03:42.730> on - -01:03:42.839 --> 01:03:42.849 align:start position:0% -it is and on the other side what's on - - -01:03:42.849 --> 01:03:46.940 align:start position:0% -it is and on the other side what's on -the<01:03:42.910> other<01:03:42.970> side<01:03:44.040> the<01:03:45.040> charge<01:03:45.369> density<01:03:45.670> and - -01:03:46.940 --> 01:03:46.950 align:start position:0% -the other side the charge density and - - -01:03:46.950 --> 01:03:50.910 align:start position:0% -the other side the charge density and -the<01:03:47.950> current<01:03:48.720> well<01:03:49.720> Maxwell's<01:03:50.440> equations - -01:03:50.910 --> 01:03:50.920 align:start position:0% -the current well Maxwell's equations - - -01:03:50.920 --> 01:03:52.829 align:start position:0% -the current well Maxwell's equations -have<01:03:51.099> the<01:03:51.190> property<01:03:51.670> that<01:03:51.700> whatever<01:03:52.510> is<01:03:52.690> on - -01:03:52.829 --> 01:03:52.839 align:start position:0% -have the property that whatever is on - - -01:03:52.839 --> 01:03:54.930 align:start position:0% -have the property that whatever is on -the<01:03:52.960> left-hand<01:03:53.260> side<01:03:53.530> automatically<01:03:54.520> as<01:03:54.790> an - -01:03:54.930 --> 01:03:54.940 align:start position:0% -the left-hand side automatically as an - - -01:03:54.940 --> 01:04:00.270 align:start position:0% -the left-hand side automatically as an -identity<01:03:55.569> in<01:03:56.380> other<01:03:56.559> words<01:03:57.030> for<01:03:58.030> every<01:03:59.280> way - -01:04:00.270 --> 01:04:00.280 align:start position:0% -identity in other words for every way - - -01:04:00.280 --> 01:04:02.280 align:start position:0% -identity in other words for every way -for<01:04:00.700> every<01:04:00.970> electric<01:04:01.420> and<01:04:01.599> magnetic<01:04:01.750> field - -01:04:02.280 --> 01:04:02.290 align:start position:0% -for every electric and magnetic field - - -01:04:02.290 --> 01:04:04.829 align:start position:0% -for every electric and magnetic field -that<01:04:03.220> you<01:04:03.339> can<01:04:03.490> write<01:04:03.700> down<01:04:03.970> that<01:04:04.510> you<01:04:04.569> can - -01:04:04.829 --> 01:04:04.839 align:start position:0% -that you can write down that you can - - -01:04:04.839 --> 01:04:07.859 align:start position:0% -that you can write down that you can -build<01:04:05.109> up<01:04:05.290> out<01:04:05.530> of<01:04:05.740> the<01:04:05.920> out<01:04:06.579> of<01:04:06.760> the<01:04:06.880> out<01:04:07.510> of - -01:04:07.859 --> 01:04:07.869 align:start position:0% -build up out of the out of the out of - - -01:04:07.869 --> 01:04:12.799 align:start position:0% -build up out of the out of the out of -the<01:04:08.170> vector<01:04:08.500> potential<01:04:09.270> that<01:04:10.270> the<01:04:11.160> derivative - -01:04:12.799 --> 01:04:12.809 align:start position:0% -the vector potential that the derivative - - -01:04:12.809 --> 01:04:16.799 align:start position:0% -the vector potential that the derivative -is<01:04:13.890> automatically<01:04:14.890> zero<01:04:15.309> and<01:04:15.730> so<01:04:16.569> what<01:04:16.690> you - -01:04:16.799 --> 01:04:16.809 align:start position:0% -is automatically zero and so what you - - -01:04:16.809 --> 01:04:18.599 align:start position:0% -is automatically zero and so what you -can<01:04:16.930> say<01:04:17.109> about<01:04:17.230> Maxwell's<01:04:17.829> equations<01:04:18.400> is - -01:04:18.599 --> 01:04:18.609 align:start position:0% -can say about Maxwell's equations is - - -01:04:18.609 --> 01:04:21.870 align:start position:0% -can say about Maxwell's equations is -that<01:04:18.819> charge<01:04:19.630> conservation<01:04:20.280> follows<01:04:21.280> from - -01:04:21.870 --> 01:04:21.880 align:start position:0% -that charge conservation follows from - - -01:04:21.880 --> 01:04:24.630 align:start position:0% -that charge conservation follows from -the<01:04:22.000> form<01:04:22.270> of<01:04:22.420> Maxwell's<01:04:22.930> equations<01:04:23.460> it's<01:04:24.460> not - -01:04:24.630 --> 01:04:24.640 align:start position:0% -the form of Maxwell's equations it's not - - -01:04:24.640 --> 01:04:26.220 align:start position:0% -the form of Maxwell's equations it's not -something<01:04:25.000> you<01:04:25.089> have<01:04:25.180> to<01:04:25.329> add<01:04:25.510> the<01:04:25.809> Maxwell's - -01:04:26.220 --> 01:04:26.230 align:start position:0% -something you have to add the Maxwell's - - -01:04:26.230 --> 01:04:28.950 align:start position:0% -something you have to add the Maxwell's -equations<01:04:26.349> it<01:04:27.309> follows<01:04:28.240> from<01:04:28.270> Maxwell's - -01:04:28.950 --> 01:04:28.960 align:start position:0% -equations it follows from Maxwell's - - -01:04:28.960 --> 01:04:31.260 align:start position:0% -equations it follows from Maxwell's -equations<01:04:29.530> Einstein<01:04:30.520> wanted<01:04:30.760> the<01:04:31.089> same - -01:04:31.260 --> 01:04:31.270 align:start position:0% -equations Einstein wanted the same - - -01:04:31.270 --> 01:04:33.630 align:start position:0% -equations Einstein wanted the same -pattern<01:04:31.720> he<01:04:32.410> wanted<01:04:32.829> energy<01:04:33.339> and<01:04:33.490> momentum - -01:04:33.630 --> 01:04:33.640 align:start position:0% -pattern he wanted energy and momentum - - -01:04:33.640 --> 01:04:36.630 align:start position:0% -pattern he wanted energy and momentum -conservation<01:04:34.030> to<01:04:35.020> be<01:04:35.140> an<01:04:35.230> automatic<01:04:35.640> Konica - -01:04:36.630 --> 01:04:36.640 align:start position:0% -conservation to be an automatic Konica - - -01:04:36.640 --> 01:04:38.880 align:start position:0% -conservation to be an automatic Konica -consequence<01:04:37.299> of<01:04:37.450> his<01:04:37.540> field<01:04:37.839> equations<01:04:38.440> and - -01:04:38.880 --> 01:04:38.890 align:start position:0% -consequence of his field equations and - - -01:04:38.890 --> 01:04:42.390 align:start position:0% -consequence of his field equations and -for<01:04:39.849> that<01:04:40.020> he<01:04:41.020> would<01:04:41.200> have<01:04:41.290> had<01:04:41.470> to<01:04:41.589> have<01:04:41.829> that - -01:04:42.390 --> 01:04:42.400 align:start position:0% -for that he would have had to have that - - -01:04:42.400 --> 01:04:44.880 align:start position:0% -for that he would have had to have that -dumb<01:04:42.700> you<01:04:42.940> arm<01:04:43.210> you<01:04:43.420> knew<01:04:43.630> the<01:04:44.200> covariant - -01:04:44.880 --> 01:04:44.890 align:start position:0% -dumb you arm you knew the covariant - - -01:04:44.890 --> 01:04:47.670 align:start position:0% -dumb you arm you knew the covariant -gradient<01:04:45.849> of<01:04:46.059> arm<01:04:46.270> you<01:04:46.450> know<01:04:46.680> covariant - -01:04:47.670 --> 01:04:47.680 align:start position:0% -gradient of arm you know covariant - - -01:04:47.680 --> 01:04:49.950 align:start position:0% -gradient of arm you know covariant -divergence<01:04:48.220> of<01:04:48.400> arm<01:04:48.549> you<01:04:48.730> knew<01:04:48.910> should<01:04:49.839> be - -01:04:49.950 --> 01:04:49.960 align:start position:0% -divergence of arm you knew should be - - -01:04:49.960 --> 01:04:53.940 align:start position:0% -divergence of arm you knew should be -zero<01:04:50.349> for<01:04:51.000> every<01:04:52.000> arm<01:04:52.210> you<01:04:52.420> know<01:04:52.770> but<01:04:53.770> just - -01:04:53.940 --> 01:04:53.950 align:start position:0% -zero for every arm you know but just - - -01:04:53.950 --> 01:04:56.089 align:start position:0% -zero for every arm you know but just -isn't<01:04:54.190> true<01:04:54.309> you<01:04:54.700> take<01:04:55.420> a<01:04:55.450> general - -01:04:56.089 --> 01:04:56.099 align:start position:0% -isn't true you take a general - - -01:04:56.099 --> 01:04:59.339 align:start position:0% -isn't true you take a general -differential<01:04:57.099> geometry<01:04:57.490> and<01:04:58.270> you<01:04:59.079> calculate - -01:04:59.339 --> 01:04:59.349 align:start position:0% -differential geometry and you calculate - - -01:04:59.349 --> 01:05:05.880 align:start position:0% -differential geometry and you calculate -the<01:05:02.220> this<01:05:03.569> let's<01:05:04.569> call<01:05:04.780> this<01:05:05.020> the<01:05:05.410> the - -01:05:05.880 --> 01:05:05.890 align:start position:0% -the this let's call this the the - - -01:05:05.890 --> 01:05:08.640 align:start position:0% -the this let's call this the the -covariant<01:05:06.490> grade<01:05:07.150> covariant<01:05:07.809> divergence<01:05:08.349> of - -01:05:08.640 --> 01:05:08.650 align:start position:0% -covariant grade covariant divergence of - - -01:05:08.650 --> 01:05:11.849 align:start position:0% -covariant grade covariant divergence of -arm<01:05:09.099> you<01:05:09.309> know<01:05:09.490> it's<01:05:10.000> not<01:05:10.210> equal<01:05:10.390> to<01:05:10.540> zero<01:05:10.859> when - -01:05:11.849 --> 01:05:11.859 align:start position:0% -arm you know it's not equal to zero when - - -01:05:11.859 --> 01:05:13.890 align:start position:0% -arm you know it's not equal to zero when -he<01:05:11.920> calculated<01:05:12.579> it<01:05:12.760> how<01:05:13.359> do<01:05:13.420> you<01:05:13.540> how<01:05:13.780> do<01:05:13.839> you - -01:05:13.890 --> 01:05:13.900 align:start position:0% -he calculated it how do you how do you - - -01:05:13.900 --> 01:05:16.650 align:start position:0% -he calculated it how do you how do you -calculate<01:05:14.440> it<01:05:14.589> you<01:05:15.250> put<01:05:15.549> in<01:05:15.760> forearm<01:05:16.240> you<01:05:16.359> knew - -01:05:16.650 --> 01:05:16.660 align:start position:0% -calculate it you put in forearm you knew - - -01:05:16.660 --> 01:05:18.390 align:start position:0% -calculate it you put in forearm you knew -its<01:05:17.020> expression<01:05:17.799> in<01:05:17.920> terms<01:05:18.130> of<01:05:18.309> the - -01:05:18.390 --> 01:05:18.400 align:start position:0% -its expression in terms of the - - -01:05:18.400 --> 01:05:21.569 align:start position:0% -its expression in terms of the -Christoffel<01:05:18.970> symbols<01:05:19.420> and<01:05:19.780> then<01:05:20.589> actually<01:05:21.520> I - -01:05:21.569 --> 01:05:21.579 align:start position:0% -Christoffel symbols and then actually I - - -01:05:21.579 --> 01:05:22.859 align:start position:0% -Christoffel symbols and then actually I -think<01:05:21.760> that's<01:05:21.880> all<01:05:22.030> you<01:05:22.059> need<01:05:22.359> I<01:05:22.390> don't<01:05:22.690> think - -01:05:22.859 --> 01:05:22.869 align:start position:0% -think that's all you need I don't think - - -01:05:22.869 --> 01:05:24.150 align:start position:0% -think that's all you need I don't think -you<01:05:22.930> even<01:05:23.200> really<01:05:23.319> need<01:05:23.619> to<01:05:23.829> write<01:05:23.950> the - -01:05:24.150 --> 01:05:24.160 align:start position:0% -you even really need to write the - - -01:05:24.160 --> 01:05:25.300 align:start position:0% -you even really need to write the -Christoffel<01:05:24.670> symbols<01:05:25.059> and - -01:05:25.300 --> 01:05:25.310 align:start position:0% -Christoffel symbols and - - -01:05:25.310 --> 01:05:27.730 align:start position:0% -Christoffel symbols and -of<01:05:25.550> the<01:05:25.640> metric<01:05:26.030> but<01:05:26.870> to<01:05:27.020> use<01:05:27.260> the<01:05:27.440> expression - -01:05:27.730 --> 01:05:27.740 align:start position:0% -of the metric but to use the expression - - -01:05:27.740 --> 01:05:29.680 align:start position:0% -of the metric but to use the expression -for<01:05:28.340> arm<01:05:28.550> you<01:05:28.730> knew<01:05:28.940> in<01:05:29.210> terms<01:05:29.450> of<01:05:29.600> the - -01:05:29.680 --> 01:05:29.690 align:start position:0% -for arm you knew in terms of the - - -01:05:29.690 --> 01:05:31.990 align:start position:0% -for arm you knew in terms of the -Christoffel<01:05:30.230> symbols<01:05:30.680> you<01:05:31.640> know<01:05:31.820> what - -01:05:31.990 --> 01:05:32.000 align:start position:0% -Christoffel symbols you know what - - -01:05:32.000 --> 01:05:33.760 align:start position:0% -Christoffel symbols you know what -covariant<01:05:32.600> derivative<01:05:33.080> means<01:05:33.350> it<01:05:33.650> means - -01:05:33.760 --> 01:05:33.770 align:start position:0% -covariant derivative means it means - - -01:05:33.770 --> 01:05:35.740 align:start position:0% -covariant derivative means it means -ordinary<01:05:34.310> derivative<01:05:34.910> plus<01:05:35.150> a<01:05:35.270> thing<01:05:35.570> with - -01:05:35.740 --> 01:05:35.750 align:start position:0% -ordinary derivative plus a thing with - - -01:05:35.750 --> 01:05:38.110 align:start position:0% -ordinary derivative plus a thing with -some<01:05:35.930> Christoffel<01:05:36.530> symbols<01:05:36.980> and<01:05:37.130> you<01:05:37.520> simply - -01:05:38.110 --> 01:05:38.120 align:start position:0% -some Christoffel symbols and you simply - - -01:05:38.120 --> 01:05:41.650 align:start position:0% -some Christoffel symbols and you simply -work<01:05:38.330> it<01:05:38.510> out<01:05:38.660> and<01:05:39.620> what<01:05:40.130> you<01:05:40.280> find<01:05:40.580> is<01:05:41.210> let's - -01:05:41.650 --> 01:05:41.660 align:start position:0% -work it out and what you find is let's - - -01:05:41.660 --> 01:05:45.100 align:start position:0% -work it out and what you find is let's -see<01:05:41.750> if<01:05:41.840> I<01:05:41.930> can<01:05:42.080> find<01:05:42.350> it - -01:05:45.100 --> 01:05:45.110 align:start position:0% - - - -01:05:45.110 --> 01:05:54.450 align:start position:0% - -damn<01:05:46.070> it<01:05:52.270> yeah - -01:05:54.450 --> 01:05:54.460 align:start position:0% -damn it yeah - - -01:05:54.460 --> 01:06:02.440 align:start position:0% -damn it yeah -here's<01:05:55.460> the<01:05:55.640> identity<01:05:56.180> that<01:05:56.330> you<01:05:56.450> find<01:06:01.450> you - -01:06:02.440 --> 01:06:02.450 align:start position:0% -here's the identity that you find you - - -01:06:02.450 --> 01:06:06.250 align:start position:0% -here's the identity that you find you -find<01:06:02.750> that<01:06:02.900> del<01:06:03.230> mu<01:06:03.470> nu<01:06:03.650> sorry<01:06:04.640> dum<01:06:04.880> you<01:06:05.090> of<01:06:05.390> our - -01:06:06.250 --> 01:06:06.260 align:start position:0% -find that del mu nu sorry dum you of our - - -01:06:06.260 --> 01:06:10.570 align:start position:0% -find that del mu nu sorry dum you of our -mu<01:06:06.530> nu<01:06:06.740> is<01:06:07.220> not<01:06:07.820> equal<01:06:08.180> to<01:06:08.330> zero<01:06:08.950> it<01:06:09.950> happens<01:06:10.460> to - -01:06:10.570 --> 01:06:10.580 align:start position:0% -mu nu is not equal to zero it happens to - - -01:06:10.580 --> 01:06:20.410 align:start position:0% -mu nu is not equal to zero it happens to -be<01:06:10.700> equal<01:06:12.820> to<01:06:13.820> one-half<01:06:14.330> G<01:06:14.930> mu<01:06:15.110> nu<01:06:18.280> times<01:06:19.420> the - -01:06:20.410 --> 01:06:20.420 align:start position:0% -be equal to one-half G mu nu times the - - -01:06:20.420 --> 01:06:23.560 align:start position:0% -be equal to one-half G mu nu times the -derivative<01:06:20.780> with<01:06:21.590> respect<01:06:22.040> to<01:06:22.100> mu<01:06:22.400> of<01:06:22.790> the - -01:06:23.560 --> 01:06:23.570 align:start position:0% -derivative with respect to mu of the - - -01:06:23.570 --> 01:06:25.720 align:start position:0% -derivative with respect to mu of the -scalar<01:06:24.080> curvature<01:06:24.380> everybody<01:06:25.340> remember<01:06:25.700> what - -01:06:25.720 --> 01:06:25.730 align:start position:0% -scalar curvature everybody remember what - - -01:06:25.730 --> 01:06:27.310 align:start position:0% -scalar curvature everybody remember what -the<01:06:25.910> scalar<01:06:26.180> curvature<01:06:26.420> is<01:06:26.840> there<01:06:26.870> was<01:06:27.200> the - -01:06:27.310 --> 01:06:27.320 align:start position:0% -the scalar curvature is there was the - - -01:06:27.320 --> 01:06:30.190 align:start position:0% -the scalar curvature is there was the -Riemann<01:06:27.650> curvature<01:06:28.180> the<01:06:29.180> Ricci<01:06:29.660> curvature - -01:06:30.190 --> 01:06:30.200 align:start position:0% -Riemann curvature the Ricci curvature - - -01:06:30.200 --> 01:06:32.170 align:start position:0% -Riemann curvature the Ricci curvature -which<01:06:30.380> you<01:06:30.500> got<01:06:30.680> by<01:06:30.890> contracting<01:06:31.520> two<01:06:31.730> indices - -01:06:32.170 --> 01:06:32.180 align:start position:0% -which you got by contracting two indices - - -01:06:32.180 --> 01:06:34.450 align:start position:0% -which you got by contracting two indices -and<01:06:32.480> left<01:06:32.660> over<01:06:32.810> two<01:06:33.050> indices<01:06:33.500> and<01:06:33.860> the - -01:06:34.450 --> 01:06:34.460 align:start position:0% -and left over two indices and the - - -01:06:34.460 --> 01:06:37.450 align:start position:0% -and left over two indices and the -curvature<01:06:34.760> scalar<01:06:35.290> you<01:06:36.290> got<01:06:36.530> by<01:06:36.800> contracting - -01:06:37.450 --> 01:06:37.460 align:start position:0% -curvature scalar you got by contracting - - -01:06:37.460 --> 01:06:40.600 align:start position:0% -curvature scalar you got by contracting -the<01:06:37.520> remaining<01:06:38.120> indices<01:06:38.600> of<01:06:39.280> the<01:06:40.280> reach<01:06:40.430> each - -01:06:40.600 --> 01:06:40.610 align:start position:0% -the remaining indices of the reach each - - -01:06:40.610 --> 01:06:43.890 align:start position:0% -the remaining indices of the reach each -of<01:06:40.760> the<01:06:40.940> Ricci<01:06:41.480> tensor<01:06:42.400> that's<01:06:43.400> what<01:06:43.580> he<01:06:43.640> found - -01:06:43.890 --> 01:06:43.900 align:start position:0% -of the Ricci tensor that's what he found - - -01:06:43.900 --> 01:06:48.280 align:start position:0% -of the Ricci tensor that's what he found -now<01:06:44.900> I<01:06:46.300> don't<01:06:47.300> know<01:06:47.510> for<01:06:47.720> how<01:06:47.810> long<01:06:48.050> that - -01:06:48.280 --> 01:06:48.290 align:start position:0% -now I don't know for how long that - - -01:06:48.290 --> 01:06:49.990 align:start position:0% -now I don't know for how long that -puzzled<01:06:48.710> him<01:06:48.860> I<01:06:49.100> don't<01:06:49.220> know<01:06:49.460> how<01:06:49.610> long<01:06:49.790> it - -01:06:49.990 --> 01:06:50.000 align:start position:0% -puzzled him I don't know how long it - - -01:06:50.000 --> 01:06:52.510 align:start position:0% -puzzled him I don't know how long it -took<01:06:50.150> him<01:06:50.210> to<01:06:50.330> put<01:06:50.630> two<01:06:50.780> and<01:06:50.810> two<01:06:51.020> together<01:06:51.520> but - -01:06:52.510 --> 01:06:52.520 align:start position:0% -took him to put two and two together but - - -01:06:52.520 --> 01:06:54.610 align:start position:0% -took him to put two and two together but -after<01:06:53.030> a<01:06:53.060> while<01:06:53.300> he<01:06:53.360> realized<01:06:53.840> the<01:06:54.230> following - -01:06:54.610 --> 01:06:54.620 align:start position:0% -after a while he realized the following - - -01:06:54.620 --> 01:06:58.540 align:start position:0% -after a while he realized the following -facts<01:06:55.240> first<01:06:56.240> of<01:06:56.420> all<01:06:56.600> the<01:06:57.320> Riemann<01:06:57.740> curve<01:06:58.400> it - -01:06:58.540 --> 01:06:58.550 align:start position:0% -facts first of all the Riemann curve it - - -01:06:58.550 --> 01:07:02.130 align:start position:0% -facts first of all the Riemann curve it -the<01:06:59.110> the<01:07:00.110> curvature<01:07:00.710> scalar<01:07:00.740> here<01:07:01.550> is<01:07:01.760> a - -01:07:02.130 --> 01:07:02.140 align:start position:0% -the the curvature scalar here is a - - -01:07:02.140 --> 01:07:03.420 align:start position:0% -the the curvature scalar here is a -scalar - -01:07:03.420 --> 01:07:03.430 align:start position:0% -scalar - - -01:07:03.430 --> 01:07:05.980 align:start position:0% -scalar -what's<01:07:04.430> the<01:07:04.580> covariant<01:07:05.270> derivative<01:07:05.810> of<01:07:05.900> a - -01:07:05.980 --> 01:07:05.990 align:start position:0% -what's the covariant derivative of a - - -01:07:05.990 --> 01:07:11.170 align:start position:0% -what's the covariant derivative of a -scalar<01:07:07.990> no<01:07:08.990> no<01:07:09.260> no<01:07:09.380> that's<01:07:09.680> zero<01:07:10.000> it's<01:07:11.000> just - -01:07:11.170 --> 01:07:11.180 align:start position:0% -scalar no no no that's zero it's just - - -01:07:11.180 --> 01:07:12.460 align:start position:0% -scalar no no no that's zero it's just -the<01:07:11.270> ordinary<01:07:11.600> derivative<01:07:11.780> the<01:07:12.440> partial - -01:07:12.460 --> 01:07:12.470 align:start position:0% -the ordinary derivative the partial - - -01:07:12.470 --> 01:07:14.230 align:start position:0% -the ordinary derivative the partial -derivative<01:07:13.220> yeah<01:07:13.430> the<01:07:13.610> ordinary<01:07:13.880> partial - -01:07:14.230 --> 01:07:14.240 align:start position:0% -derivative yeah the ordinary partial - - -01:07:14.240 --> 01:07:18.240 align:start position:0% -derivative yeah the ordinary partial -derivative<01:07:14.690> so<01:07:15.680> this<01:07:16.130> could<01:07:16.370> be<01:07:16.490> replaced<01:07:16.900> by - -01:07:18.240 --> 01:07:18.250 align:start position:0% -derivative so this could be replaced by - - -01:07:18.250 --> 01:07:22.390 align:start position:0% -derivative so this could be replaced by -covariant<01:07:19.250> derivative<01:07:20.650> now<01:07:21.650> covariant - -01:07:22.390 --> 01:07:22.400 align:start position:0% -covariant derivative now covariant - - -01:07:22.400 --> 01:07:24.160 align:start position:0% -covariant derivative now covariant -derivatives<01:07:22.850> are<01:07:22.970> a<01:07:23.000> little<01:07:23.180> bit<01:07:23.390> strange<01:07:23.780> but - -01:07:24.160 --> 01:07:24.170 align:start position:0% -derivatives are a little bit strange but - - -01:07:24.170 --> 01:07:27.100 align:start position:0% -derivatives are a little bit strange but -they<01:07:24.320> satisfy<01:07:24.860> the<01:07:24.920> usual<01:07:25.250> rules<01:07:25.990> that<01:07:26.990> the - -01:07:27.100 --> 01:07:27.110 align:start position:0% -they satisfy the usual rules that the - - -01:07:27.110 --> 01:07:30.370 align:start position:0% -they satisfy the usual rules that the -covariant<01:07:27.770> derivative<01:07:28.310> of<01:07:28.340> a<01:07:29.030> product<01:07:29.380> could - -01:07:30.370 --> 01:07:30.380 align:start position:0% -covariant derivative of a product could - - -01:07:30.380 --> 01:07:32.560 align:start position:0% -covariant derivative of a product could -be<01:07:30.440> product<01:07:30.860> of<01:07:31.010> any<01:07:31.190> to<01:07:31.490> any<01:07:31.670> tensors<01:07:32.180> I'm<01:07:32.420> not - -01:07:32.560 --> 01:07:32.570 align:start position:0% -be product of any to any tensors I'm not - - -01:07:32.570 --> 01:07:34.540 align:start position:0% -be product of any to any tensors I'm not -even<01:07:32.720> going<01:07:32.840> to<01:07:32.870> write<01:07:32.990> indices<01:07:33.530> for<01:07:33.770> them<01:07:33.980> any - -01:07:34.540 --> 01:07:34.550 align:start position:0% -even going to write indices for them any - - -01:07:34.550 --> 01:07:37.360 align:start position:0% -even going to write indices for them any -two<01:07:34.820> tensors<01:07:35.270> a<01:07:35.420> and<01:07:35.570> B<01:07:36.010> satisfy<01:07:37.010> the<01:07:37.070> usual - -01:07:37.360 --> 01:07:37.370 align:start position:0% -two tensors a and B satisfy the usual - - -01:07:37.370 --> 01:07:38.220 align:start position:0% -two tensors a and B satisfy the usual -rule - -01:07:38.220 --> 01:07:38.230 align:start position:0% -rule - - -01:07:38.230 --> 01:07:41.270 align:start position:0% -rule -a<01:07:38.520> times<01:07:39.520> the<01:07:39.700> covariant<01:07:40.270> derivative<01:07:40.329> or<01:07:40.930> b - -01:07:41.270 --> 01:07:41.280 align:start position:0% -a times the covariant derivative or b - - -01:07:41.280 --> 01:07:45.120 align:start position:0% -a times the covariant derivative or b -plus<01:07:42.450> plus<01:07:43.450> the<01:07:43.630> covariant<01:07:44.230> derivative<01:07:44.710> of<01:07:44.890> a - -01:07:45.120 --> 01:07:45.130 align:start position:0% -plus plus the covariant derivative of a - - -01:07:45.130 --> 01:07:49.050 align:start position:0% -plus plus the covariant derivative of a -times<01:07:45.760> b<01:07:47.160> covariant<01:07:48.160> derivatives<01:07:48.640> do<01:07:48.820> have - -01:07:49.050 --> 01:07:49.060 align:start position:0% -times b covariant derivatives do have - - -01:07:49.060 --> 01:07:50.760 align:start position:0% -times b covariant derivatives do have -that<01:07:49.210> property<01:07:49.660> at<01:07:50.020> least<01:07:50.290> they<01:07:50.470> have<01:07:50.619> that - -01:07:50.760 --> 01:07:50.770 align:start position:0% -that property at least they have that - - -01:07:50.770 --> 01:07:53.880 align:start position:0% -that property at least they have that -property<01:07:51.599> so<01:07:52.599> now<01:07:52.750> we<01:07:53.079> look<01:07:53.290> at<01:07:53.470> this<01:07:53.619> over - -01:07:53.880 --> 01:07:53.890 align:start position:0% -property so now we look at this over - - -01:07:53.890 --> 01:07:54.740 align:start position:0% -property so now we look at this over -here - -01:07:54.740 --> 01:07:54.750 align:start position:0% -here - - -01:07:54.750 --> 01:07:57.510 align:start position:0% -here -number<01:07:55.750> one<01:07:55.900> and<01:07:56.530> we<01:07:56.680> say<01:07:56.829> wait<01:07:57.010> a<01:07:57.130> minute - -01:07:57.510 --> 01:07:57.520 align:start position:0% -number one and we say wait a minute - - -01:07:57.520 --> 01:08:00.089 align:start position:0% -number one and we say wait a minute -remember<01:07:58.030> the<01:07:58.329> fact<01:07:58.630> that<01:07:58.810> the<01:07:59.410> covariant - -01:08:00.089 --> 01:08:00.099 align:start position:0% -remember the fact that the covariant - - -01:08:00.099 --> 01:08:03.800 align:start position:0% -remember the fact that the covariant -derivative<01:08:00.190> of<01:08:00.880> the<01:08:01.000> metric<01:08:01.300> tensor<01:08:01.990> is<01:08:02.200> zero - -01:08:03.800 --> 01:08:03.810 align:start position:0% -derivative of the metric tensor is zero - - -01:08:03.810 --> 01:08:06.569 align:start position:0% -derivative of the metric tensor is zero -the<01:08:04.810> covariant<01:08:05.500> derivative<01:08:05.560> of<01:08:06.190> the<01:08:06.369> metric - -01:08:06.569 --> 01:08:06.579 align:start position:0% -the covariant derivative of the metric - - -01:08:06.579 --> 01:08:08.790 align:start position:0% -the covariant derivative of the metric -tensor<01:08:07.180> with<01:08:07.359> upper<01:08:07.750> or<01:08:07.869> lower<01:08:08.140> it<01:08:08.440> doesn't - -01:08:08.790 --> 01:08:08.800 align:start position:0% -tensor with upper or lower it doesn't - - -01:08:08.800 --> 01:08:11.400 align:start position:0% -tensor with upper or lower it doesn't -matter<01:08:09.030> or<01:08:10.030> doesn't<01:08:10.839> matter<01:08:10.990> where<01:08:11.140> you<01:08:11.260> put - -01:08:11.400 --> 01:08:11.410 align:start position:0% -matter or doesn't matter where you put - - -01:08:11.410 --> 01:08:13.349 align:start position:0% -matter or doesn't matter where you put -the<01:08:11.560> indices<01:08:11.800> the<01:08:12.099> covariant<01:08:12.760> derivative<01:08:12.790> of - -01:08:13.349 --> 01:08:13.359 align:start position:0% -the indices the covariant derivative of - - -01:08:13.359 --> 01:08:15.030 align:start position:0% -the indices the covariant derivative of -the<01:08:13.480> metric<01:08:13.690> tensor<01:08:14.230> is<01:08:14.380> zero<01:08:14.410> so<01:08:14.800> you<01:08:14.890> can - -01:08:15.030 --> 01:08:15.040 align:start position:0% -the metric tensor is zero so you can - - -01:08:15.040 --> 01:08:17.999 align:start position:0% -the metric tensor is zero so you can -also<01:08:15.280> write<01:08:15.490> this<01:08:15.960> as<01:08:16.960> one-half<01:08:17.380> the - -01:08:17.999 --> 01:08:18.009 align:start position:0% -also write this as one-half the - - -01:08:18.009 --> 01:08:22.700 align:start position:0% -also write this as one-half the -covariant<01:08:18.790> derivative<01:08:18.850> of<01:08:20.069> G<01:08:21.069> mu<01:08:21.279> nu<01:08:21.339> times<01:08:22.120> R - -01:08:22.700 --> 01:08:22.710 align:start position:0% -covariant derivative of G mu nu times R - - -01:08:22.710 --> 01:08:24.420 align:start position:0% -covariant derivative of G mu nu times R -why<01:08:23.710> is<01:08:23.770> that - -01:08:24.420 --> 01:08:24.430 align:start position:0% -why is that - - -01:08:24.430 --> 01:08:27.030 align:start position:0% -why is that -well<01:08:24.969> this<01:08:25.270> contains<01:08:25.719> G<01:08:26.140> mu<01:08:26.350> nu<01:08:26.380> times<01:08:26.560> the - -01:08:27.030 --> 01:08:27.040 align:start position:0% -well this contains G mu nu times the - - -01:08:27.040 --> 01:08:29.640 align:start position:0% -well this contains G mu nu times the -covariant<01:08:27.580> derivative<01:08:27.609> R<01:08:28.469> but<01:08:29.469> it<01:08:29.529> also - -01:08:29.640 --> 01:08:29.650 align:start position:0% -covariant derivative R but it also - - -01:08:29.650 --> 01:08:31.470 align:start position:0% -covariant derivative R but it also -contains<01:08:30.160> R<01:08:30.400> times<01:08:30.759> the<01:08:30.940> covariant - -01:08:31.470 --> 01:08:31.480 align:start position:0% -contains R times the covariant - - -01:08:31.480 --> 01:08:34.769 align:start position:0% -contains R times the covariant -derivative<01:08:31.930> of<01:08:32.020> G<01:08:32.259> mu<01:08:32.380> nu<01:08:32.969> the<01:08:33.969> last<01:08:34.270> term - -01:08:34.769 --> 01:08:34.779 align:start position:0% -derivative of G mu nu the last term - - -01:08:34.779 --> 01:08:40.979 align:start position:0% -derivative of G mu nu the last term -being<01:08:35.440> zero<01:08:36.210> so<01:08:37.210> you<01:08:37.920> discovered<01:08:39.779> them<01:08:40.779> you - -01:08:40.979 --> 01:08:40.989 align:start position:0% -being zero so you discovered them you - - -01:08:40.989 --> 01:08:44.090 align:start position:0% -being zero so you discovered them you -knew<01:08:41.170> of<01:08:41.469> arm<01:08:41.980> you<01:08:42.190> know<01:08:42.400> is<01:08:43.330> equal<01:08:43.569> to - -01:08:44.090 --> 01:08:44.100 align:start position:0% -knew of arm you know is equal to - - -01:08:44.100 --> 01:08:48.990 align:start position:0% -knew of arm you know is equal to -one-half<01:08:45.100> del<01:08:45.700> mu<01:08:45.940> nu<01:08:46.120> of<01:08:46.359> G<01:08:47.170> mu<01:08:47.319> nu<01:08:47.380> R<01:08:47.790> now<01:08:48.790> I - -01:08:48.990 --> 01:08:49.000 align:start position:0% -one-half del mu nu of G mu nu R now I - - -01:08:49.000 --> 01:08:51.479 align:start position:0% -one-half del mu nu of G mu nu R now I -don't<01:08:49.330> know<01:08:49.630> if<01:08:50.109> this<01:08:50.380> was<01:08:50.589> known<01:08:50.830> before - -01:08:51.479 --> 01:08:51.489 align:start position:0% -don't know if this was known before - - -01:08:51.489 --> 01:08:55.079 align:start position:0% -don't know if this was known before -Einstein<01:08:51.640> or<01:08:52.210> not<01:08:52.450> I<01:08:53.819> don't<01:08:54.819> know<01:08:54.910> that - -01:08:55.079 --> 01:08:55.089 align:start position:0% -Einstein or not I don't know that - - -01:08:55.089 --> 01:08:57.059 align:start position:0% -Einstein or not I don't know that -anybody<01:08:55.450> ever<01:08:55.630> had<01:08:55.839> any<01:08:56.049> particular<01:08:56.230> reason - -01:08:57.059 --> 01:08:57.069 align:start position:0% -anybody ever had any particular reason - - -01:08:57.069 --> 01:08:59.550 align:start position:0% -anybody ever had any particular reason -to<01:08:57.489> calculate<01:08:58.029> the<01:08:58.180> covariant<01:08:58.750> derivative<01:08:58.810> of - -01:08:59.550 --> 01:08:59.560 align:start position:0% -to calculate the covariant derivative of - - -01:08:59.560 --> 01:09:03.690 align:start position:0% -to calculate the covariant derivative of -arm<01:08:59.799> you<01:09:00.009> know<01:09:00.190> the<01:09:00.819> covariant<01:09:02.700> divergence - -01:09:03.690 --> 01:09:03.700 align:start position:0% -arm you know the covariant divergence - - -01:09:03.700 --> 01:09:06.420 align:start position:0% -arm you know the covariant divergence -like<01:09:03.910> that<01:09:04.150> and<01:09:04.930> how<01:09:05.140> many<01:09:05.350> weeks<01:09:05.650> months<01:09:06.219> days - -01:09:06.420 --> 01:09:06.430 align:start position:0% -like that and how many weeks months days - - -01:09:06.430 --> 01:09:08.760 align:start position:0% -like that and how many weeks months days -hours<01:09:06.759> or<01:09:07.150> seconds<01:09:07.600> it<01:09:07.719> took<01:09:07.930> for<01:09:08.170> n<01:09:08.290> Stein<01:09:08.620> to - -01:09:08.760 --> 01:09:08.770 align:start position:0% -hours or seconds it took for n Stein to - - -01:09:08.770 --> 01:09:10.800 align:start position:0% -hours or seconds it took for n Stein to -say<01:09:08.920> hey<01:09:09.100> wait<01:09:09.279> a<01:09:09.339> minute<01:09:09.430> I<01:09:09.850> can<01:09:10.600> now<01:09:10.750> take - -01:09:10.800 --> 01:09:10.810 align:start position:0% -say hey wait a minute I can now take - - -01:09:10.810 --> 01:09:13.800 align:start position:0% -say hey wait a minute I can now take -this<01:09:11.200> over<01:09:11.410> to<01:09:11.560> the<01:09:11.680> left<01:09:11.920> hand<01:09:12.219> side<01:09:12.810> and - -01:09:13.800 --> 01:09:13.810 align:start position:0% -this over to the left hand side and - - -01:09:13.810 --> 01:09:15.809 align:start position:0% -this over to the left hand side and -write<01:09:14.080> that<01:09:14.290> this<01:09:14.410> is<01:09:14.469> equal<01:09:14.620> to<01:09:14.920> Del<01:09:15.310> mu<01:09:15.609> nu - -01:09:15.809 --> 01:09:15.819 align:start position:0% -write that this is equal to Del mu nu - - -01:09:15.819 --> 01:09:26.749 align:start position:0% -write that this is equal to Del mu nu -del<01:09:16.150> mu<01:09:16.529> times<01:09:17.940> R<01:09:18.940> mu<01:09:19.810> nu<01:09:20.640> minus<01:09:21.640> 1/2<01:09:22.480> G<01:09:23.230> mu<01:09:23.469> nu<01:09:23.500> R - -01:09:26.749 --> 01:09:26.759 align:start position:0% - - - -01:09:26.759 --> 01:09:32.940 align:start position:0% - -is<01:09:27.759> equal<01:09:28.239> to<01:09:28.359> zero<01:09:31.200> that<01:09:32.200> was<01:09:32.589> the<01:09:32.739> big - -01:09:32.940 --> 01:09:32.950 align:start position:0% -is equal to zero that was the big - - -01:09:32.950 --> 01:09:36.240 align:start position:0% -is equal to zero that was the big -breakthrough<01:09:34.650> that<01:09:35.650> was<01:09:35.950> the<01:09:36.069> big - -01:09:36.240 --> 01:09:36.250 align:start position:0% -breakthrough that was the big - - -01:09:36.250 --> 01:09:40.110 align:start position:0% -breakthrough that was the big -breakthrough<01:09:37.259> realizing<01:09:38.259> that<01:09:38.980> there<01:09:39.460> was<01:09:39.759> in - -01:09:40.110 --> 01:09:40.120 align:start position:0% -breakthrough realizing that there was in - - -01:09:40.120 --> 01:09:43.620 align:start position:0% -breakthrough realizing that there was in -fact<01:09:40.150> is<01:09:40.779> and<01:09:41.109> was<01:09:41.440> in<01:09:41.680> fact<01:09:41.950> a<01:09:42.310> certain<01:09:43.150> tensor - -01:09:43.620 --> 01:09:43.630 align:start position:0% -fact is and was in fact a certain tensor - - -01:09:43.630 --> 01:09:45.900 align:start position:0% -fact is and was in fact a certain tensor -that<01:09:43.810> can<01:09:44.650> be<01:09:44.799> made<01:09:45.040> up<01:09:45.219> out<01:09:45.460> of<01:09:45.609> the<01:09:45.700> Ricci - -01:09:45.900 --> 01:09:45.910 align:start position:0% -that can be made up out of the Ricci - - -01:09:45.910 --> 01:09:48.959 align:start position:0% -that can be made up out of the Ricci -tensor<01:09:46.509> and<01:09:46.719> the<01:09:46.750> Ricci<01:09:47.650> scalar<01:09:47.859> times<01:09:48.520> G<01:09:48.790> mu - -01:09:48.959 --> 01:09:48.969 align:start position:0% -tensor and the Ricci scalar times G mu - - -01:09:48.969 --> 01:09:52.670 align:start position:0% -tensor and the Ricci scalar times G mu -nu<01:09:49.000> which<01:09:49.960> has<01:09:50.310> automatically - -01:09:52.670 --> 01:09:52.680 align:start position:0% -nu which has automatically - - -01:09:52.680 --> 01:09:55.380 align:start position:0% -nu which has automatically -automatically<01:09:53.680> for<01:09:54.190> any<01:09:54.220> metric<01:09:54.940> doesn't - -01:09:55.380 --> 01:09:55.390 align:start position:0% -automatically for any metric doesn't - - -01:09:55.390 --> 01:09:58.080 align:start position:0% -automatically for any metric doesn't -matter<01:09:55.540> what<01:09:55.750> the<01:09:55.870> metric<01:09:56.200> is<01:09:56.430> for<01:09:57.430> any<01:09:57.640> metric - -01:09:58.080 --> 01:09:58.090 align:start position:0% -matter what the metric is for any metric - - -01:09:58.090 --> 01:10:02.520 align:start position:0% -matter what the metric is for any metric -whatever<01:09:58.450> the<01:09:59.380> del<01:10:00.040> mu<01:10:00.340> of<01:10:00.720> this<01:10:01.720> tensor<01:10:02.170> here - -01:10:02.520 --> 01:10:02.530 align:start position:0% -whatever the del mu of this tensor here - - -01:10:02.530 --> 01:10:05.010 align:start position:0% -whatever the del mu of this tensor here -is<01:10:02.970> automatically<01:10:03.970> zero<01:10:04.330> it's<01:10:04.480> an<01:10:04.540> identity - -01:10:05.010 --> 01:10:05.020 align:start position:0% -is automatically zero it's an identity - - -01:10:05.020 --> 01:10:07.620 align:start position:0% -is automatically zero it's an identity -if<01:10:05.590> you<01:10:06.130> plug<01:10:06.460> in<01:10:06.640> for<01:10:06.670> R<01:10:06.970> mu<01:10:07.120> nu<01:10:07.420> its - -01:10:07.620 --> 01:10:07.630 align:start position:0% -if you plug in for R mu nu its - - -01:10:07.630 --> 01:10:09.480 align:start position:0% -if you plug in for R mu nu its -expression<01:10:08.080> in<01:10:08.410> terms<01:10:08.620> of<01:10:08.770> the<01:10:08.890> Christoffel - -01:10:09.480 --> 01:10:09.490 align:start position:0% -expression in terms of the Christoffel - - -01:10:09.490 --> 01:10:14.010 align:start position:0% -expression in terms of the Christoffel -symbols<01:10:09.940> you'll<01:10:10.900> find<01:10:11.200> this<01:10:11.350> to<01:10:11.500> be<01:10:11.530> true<01:10:13.020> okay - -01:10:14.010 --> 01:10:14.020 align:start position:0% -symbols you'll find this to be true okay - - -01:10:14.020 --> 01:10:18.060 align:start position:0% -symbols you'll find this to be true okay -and<01:10:14.620> again<01:10:14.710> I<01:10:14.980> emphasize<01:10:15.540> you<01:10:16.540> don't<01:10:16.840> need<01:10:17.230> to - -01:10:18.060 --> 01:10:18.070 align:start position:0% -and again I emphasize you don't need to - - -01:10:18.070 --> 01:10:20.130 align:start position:0% -and again I emphasize you don't need to -put<01:10:18.280> in<01:10:18.370> any<01:10:18.550> special<01:10:19.060> form<01:10:19.360> for<01:10:19.630> the<01:10:19.720> metric - -01:10:20.130 --> 01:10:20.140 align:start position:0% -put in any special form for the metric - - -01:10:20.140 --> 01:10:22.590 align:start position:0% -put in any special form for the metric -all<01:10:20.890> the<01:10:21.160> important<01:10:21.640> thing<01:10:21.850> is<01:10:22.120> that<01:10:22.150> it<01:10:22.450> be - -01:10:22.590 --> 01:10:22.600 align:start position:0% -all the important thing is that it be - - -01:10:22.600 --> 01:10:24.570 align:start position:0% -all the important thing is that it be -differentiable<01:10:23.380> enough<01:10:23.800> to<01:10:24.010> be<01:10:24.100> able<01:10:24.310> to<01:10:24.400> do - -01:10:24.570 --> 01:10:24.580 align:start position:0% -differentiable enough to be able to do - - -01:10:24.580 --> 01:10:27.420 align:start position:0% -differentiable enough to be able to do -all<01:10:24.760> the<01:10:24.970> derivatives<01:10:25.540> other<01:10:26.260> than<01:10:26.560> that<01:10:26.740> it - -01:10:27.420 --> 01:10:27.430 align:start position:0% -all the derivatives other than that it - - -01:10:27.430 --> 01:10:29.840 align:start position:0% -all the derivatives other than that it -will<01:10:27.640> always<01:10:27.850> be<01:10:28.210> equal<01:10:28.240> to<01:10:28.570> zero - -01:10:29.840 --> 01:10:29.850 align:start position:0% -will always be equal to zero - - -01:10:29.850 --> 01:10:34.550 align:start position:0% -will always be equal to zero -so<01:10:31.050> that<01:10:32.310> first<01:10:33.310> of<01:10:33.520> all - -01:10:34.550 --> 01:10:34.560 align:start position:0% -so that first of all - - -01:10:34.560 --> 01:10:38.400 align:start position:0% -so that first of all -Einstein<01:10:35.560> gave<01:10:35.830> us<01:10:36.010> a<01:10:36.160> name<01:10:36.370> very<01:10:37.360> modestly<01:10:37.990> he - -01:10:38.400 --> 01:10:38.410 align:start position:0% -Einstein gave us a name very modestly he - - -01:10:38.410 --> 01:10:40.260 align:start position:0% -Einstein gave us a name very modestly he -called<01:10:38.680> the<01:10:38.950> Einstein<01:10:39.430> tensor<01:10:39.880> I<01:10:39.910> don't<01:10:40.180> know - -01:10:40.260 --> 01:10:40.270 align:start position:0% -called the Einstein tensor I don't know - - -01:10:40.270 --> 01:10:43.080 align:start position:0% -called the Einstein tensor I don't know -who<01:10:40.360> quality<01:10:40.780> it<01:10:40.900> wasn't<01:10:41.140> Einstein<01:10:41.880> he<01:10:42.880> could - -01:10:43.080 --> 01:10:43.090 align:start position:0% -who quality it wasn't Einstein he could - - -01:10:43.090 --> 01:10:45.540 align:start position:0% -who quality it wasn't Einstein he could -have<01:10:43.180> called<01:10:43.360> the<01:10:43.510> mu<01:10:43.960> nu<01:10:44.170> but<01:10:44.380> he<01:10:44.500> didn't<01:10:44.830> he - -01:10:45.540 --> 01:10:45.550 align:start position:0% -have called the mu nu but he didn't he - - -01:10:45.550 --> 01:10:54.420 align:start position:0% -have called the mu nu but he didn't he -called<01:10:45.790> it<01:10:46.000> Jimmy<01:10:46.390> oh<01:10:46.540> no<01:10:51.840> no<01:10:52.840> sorry<01:10:53.230> yeah<01:10:54.160> del - -01:10:54.420 --> 01:10:54.430 align:start position:0% -called it Jimmy oh no no sorry yeah del - - -01:10:54.430 --> 01:11:00.810 align:start position:0% -called it Jimmy oh no no sorry yeah del -mu<01:10:54.580> G<01:10:55.150> mu<01:10:55.330> nu<01:10:56.790> equals<01:10:57.790> zero<01:10:58.710> g<01:10:59.710> mu<01:10:59.920> nu<01:11:00.130> is<01:11:00.370> just - -01:11:00.810 --> 01:11:00.820 align:start position:0% -mu G mu nu equals zero g mu nu is just - - -01:11:00.820 --> 01:11:04.530 align:start position:0% -mu G mu nu equals zero g mu nu is just -the<01:11:01.300> Ricci<01:11:01.540> tensor<01:11:02.160> minus<01:11:03.160> 1/2<01:11:03.490> G<01:11:03.820> mu<01:11:04.030> nu<01:11:04.060> times - -01:11:04.530 --> 01:11:04.540 align:start position:0% -the Ricci tensor minus 1/2 G mu nu times - - -01:11:04.540 --> 01:11:08.490 align:start position:0% -the Ricci tensor minus 1/2 G mu nu times -the<01:11:05.530> curvature<01:11:06.070> scalar<01:11:06.900> once<01:11:07.900> he<01:11:08.080> realized - -01:11:08.490 --> 01:11:08.500 align:start position:0% -the curvature scalar once he realized - - -01:11:08.500 --> 01:11:12.150 align:start position:0% -the curvature scalar once he realized -this<01:11:08.770> he<01:11:09.310> knew<01:11:09.550> what<01:11:10.300> he<01:11:10.450> had<01:11:10.480> to<01:11:10.750> do<01:11:10.900> next<01:11:11.160> what - -01:11:12.150 --> 01:11:12.160 align:start position:0% -this he knew what he had to do next what - - -01:11:12.160 --> 01:11:14.420 align:start position:0% -this he knew what he had to do next what -he<01:11:12.250> had<01:11:12.400> to<01:11:12.550> do<01:11:12.700> next<01:11:12.970> was<01:11:13.450> replace<01:11:13.870> this - -01:11:14.420 --> 01:11:14.430 align:start position:0% -he had to do next was replace this - - -01:11:14.430 --> 01:11:17.040 align:start position:0% -he had to do next was replace this -inconsistent<01:11:15.430> equation<01:11:15.940> was<01:11:16.270> inconsistent - -01:11:17.040 --> 01:11:17.050 align:start position:0% -inconsistent equation was inconsistent - - -01:11:17.050 --> 01:11:19.530 align:start position:0% -inconsistent equation was inconsistent -at<01:11:17.230> least<01:11:17.440> it<01:11:17.590> was<01:11:17.680> inconsistent<01:11:18.040> with<01:11:18.640> energy - -01:11:19.530 --> 01:11:19.540 align:start position:0% -at least it was inconsistent with energy - - -01:11:19.540 --> 01:11:22.740 align:start position:0% -at least it was inconsistent with energy -momentum<01:11:19.720> conservation - -01:11:22.740 --> 01:11:22.750 align:start position:0% - - - -01:11:22.750 --> 01:11:31.480 align:start position:0% - -[Music] - -01:11:31.480 --> 01:11:31.490 align:start position:0% - - - -01:11:31.490 --> 01:11:45.590 align:start position:0% - -still<01:11:32.490> in<01:11:32.850> flunkies<01:11:33.330> yeah<01:11:35.180> yeah<01:11:43.970> six<01:11:44.970> four - -01:11:45.590 --> 01:11:45.600 align:start position:0% -still in flunkies yeah yeah six four - - -01:11:45.600 --> 01:11:49.550 align:start position:0% -still in flunkies yeah yeah six four -equations<01:11:46.050> yes<01:11:46.290> four<01:11:47.100> equations<01:11:47.990> yes<01:11:48.990> for - -01:11:49.550 --> 01:11:49.560 align:start position:0% -equations yes four equations yes for - - -01:11:49.560 --> 01:11:52.130 align:start position:0% -equations yes four equations yes for -continuity<01:11:50.100> equations<01:11:50.760> these<01:11:51.420> are<01:11:51.660> not<01:11:51.840> the - -01:11:52.130 --> 01:11:52.140 align:start position:0% -continuity equations these are not the - - -01:11:52.140 --> 01:11:53.930 align:start position:0% -continuity equations these are not the -field<01:11:52.440> equations<01:11:52.650> of<01:11:53.220> Einstein<01:11:53.610> the<01:11:53.700> field - -01:11:53.930 --> 01:11:53.940 align:start position:0% -field equations of Einstein the field - - -01:11:53.940 --> 01:11:56.300 align:start position:0% -field equations of Einstein the field -equations<01:11:54.210> are<01:11:54.780> the<01:11:55.620> other<01:11:55.710> ones<01:11:55.830> over<01:11:56.130> here - -01:11:56.300 --> 01:11:56.310 align:start position:0% -equations are the other ones over here - - -01:11:56.310 --> 01:11:58.490 align:start position:0% -equations are the other ones over here -these<01:11:56.730> are<01:11:56.790> the<01:11:57.150> continuity<01:11:57.810> equations<01:11:58.320> for - -01:11:58.490 --> 01:11:58.500 align:start position:0% -these are the continuity equations for - - -01:11:58.500 --> 01:12:00.530 align:start position:0% -these are the continuity equations for -energy<01:11:58.800> momentum<01:11:58.970> there's<01:11:59.970> a<01:12:00.060> continuity - -01:12:00.530 --> 01:12:00.540 align:start position:0% -energy momentum there's a continuity - - -01:12:00.540 --> 01:12:02.810 align:start position:0% -energy momentum there's a continuity -equation<01:12:01.170> for<01:12:01.350> energy<01:12:01.500> there's<01:12:02.370> a<01:12:02.520> continuity - -01:12:02.810 --> 01:12:02.820 align:start position:0% -equation for energy there's a continuity - - -01:12:02.820 --> 01:12:05.000 align:start position:0% -equation for energy there's a continuity -equation<01:12:03.600> for<01:12:03.780> the<01:12:03.870> X<01:12:04.020> component<01:12:04.230> of<01:12:04.860> momentum - -01:12:05.000 --> 01:12:05.010 align:start position:0% -equation for the X component of momentum - - -01:12:05.010 --> 01:12:07.460 align:start position:0% -equation for the X component of momentum -another<01:12:05.910> one<01:12:06.090> for<01:12:06.270> the<01:12:06.360> Y<01:12:06.510> component<01:12:06.540> another - -01:12:07.460 --> 01:12:07.470 align:start position:0% -another one for the Y component another - - -01:12:07.470 --> 01:12:09.170 align:start position:0% -another one for the Y component another -one<01:12:07.680> for<01:12:07.830> the<01:12:07.920> Z<01:12:08.070> component<01:12:08.640> and<01:12:08.760> so<01:12:08.910> forth - -01:12:09.170 --> 01:12:09.180 align:start position:0% -one for the Z component and so forth - - -01:12:09.180 --> 01:12:13.730 align:start position:0% -one for the Z component and so forth -four<01:12:09.990> distinct<01:12:10.830> equations<01:12:11.490> that<01:12:12.390> Express<01:12:13.320> the - -01:12:13.730 --> 01:12:13.740 align:start position:0% -four distinct equations that Express the - - -01:12:13.740 --> 01:12:19.090 align:start position:0% -four distinct equations that Express the -local<01:12:14.250> continuity<01:12:15.210> of<01:12:15.830> energy<01:12:16.830> and<01:12:16.980> momentum - -01:12:19.090 --> 01:12:19.100 align:start position:0% - - - -01:12:19.100 --> 01:12:21.470 align:start position:0% - -and<01:12:20.100> they<01:12:20.310> are<01:12:20.400> the<01:12:20.490> closest<01:12:21.000> thing<01:12:21.240> that - -01:12:21.470 --> 01:12:21.480 align:start position:0% -and they are the closest thing that - - -01:12:21.480 --> 01:12:22.820 align:start position:0% -and they are the closest thing that -there<01:12:21.750> is<01:12:21.780> the<01:12:22.320> energy<01:12:22.710> momentum - -01:12:22.820 --> 01:12:22.830 align:start position:0% -there is the energy momentum - - -01:12:22.830 --> 01:12:27.050 align:start position:0% -there is the energy momentum -conservation<01:12:23.190> in<01:12:24.060> in<01:12:24.900> gravity<01:12:25.670> now<01:12:26.670> there's<01:12:27.000> a - -01:12:27.050 --> 01:12:27.060 align:start position:0% -conservation in in gravity now there's a - - -01:12:27.060 --> 01:12:28.820 align:start position:0% -conservation in in gravity now there's a -certain<01:12:27.360> point<01:12:27.750> that<01:12:27.990> I'm<01:12:28.200> going<01:12:28.350> to<01:12:28.380> come<01:12:28.590> to - -01:12:28.820 --> 01:12:28.830 align:start position:0% -certain point that I'm going to come to - - -01:12:28.830 --> 01:12:31.340 align:start position:0% -certain point that I'm going to come to -that's<01:12:29.160> a<01:12:29.250> little<01:12:29.460> bit<01:12:29.610> subtle<01:12:29.850> about<01:12:30.780> why - -01:12:31.340 --> 01:12:31.350 align:start position:0% -that's a little bit subtle about why - - -01:12:31.350 --> 01:12:34.610 align:start position:0% -that's a little bit subtle about why -they<01:12:32.040> don't<01:12:32.340> express<01:12:33.470> conservation<01:12:34.470> the - -01:12:34.610 --> 01:12:34.620 align:start position:0% -they don't express conservation the - - -01:12:34.620 --> 01:12:36.760 align:start position:0% -they don't express conservation the -usual<01:12:35.070> sense<01:12:35.430> but<01:12:35.880> we'll<01:12:36.000> come<01:12:36.210> to<01:12:36.240> it<01:12:36.480> soon - -01:12:36.760 --> 01:12:36.770 align:start position:0% -usual sense but we'll come to it soon - - -01:12:36.770 --> 01:12:39.680 align:start position:0% -usual sense but we'll come to it soon -and<01:12:37.770> we<01:12:38.190> shouldn't<01:12:38.460> expect<01:12:38.820> it<01:12:39.030> to<01:12:39.210> but<01:12:39.390> let's - -01:12:39.680 --> 01:12:39.690 align:start position:0% -and we shouldn't expect it to but let's - - -01:12:39.690 --> 01:12:41.270 align:start position:0% -and we shouldn't expect it to but let's -let's<01:12:39.990> go<01:12:40.140> we'll<01:12:40.290> come<01:12:40.440> to<01:12:40.560> it<01:12:40.680> soon<01:12:40.830> enough - -01:12:41.270 --> 01:12:41.280 align:start position:0% -let's go we'll come to it soon enough - - -01:12:41.280 --> 01:12:44.060 align:start position:0% -let's go we'll come to it soon enough -all<01:12:41.790> right<01:12:41.940> so<01:12:42.180> Einstein<01:12:42.330> said<01:12:42.990> nope<01:12:43.650> let's - -01:12:44.060 --> 01:12:44.070 align:start position:0% -all right so Einstein said nope let's - - -01:12:44.070 --> 01:12:45.340 align:start position:0% -all right so Einstein said nope let's -put<01:12:44.280> G<01:12:44.460> mu<01:12:44.610> nu<01:12:44.670> here - -01:12:45.340 --> 01:12:45.350 align:start position:0% -put G mu nu here - - -01:12:45.350 --> 01:12:52.490 align:start position:0% -put G mu nu here -arm<01:12:46.350> you<01:12:46.680> know<01:12:48.170> minus<01:12:49.170> one<01:12:49.440> half<01:12:50.840> G<01:12:51.840> mu<01:12:51.870> nu<01:12:52.050> oh - -01:12:52.490 --> 01:12:52.500 align:start position:0% -arm you know minus one half G mu nu oh - - -01:12:52.500 --> 01:12:53.180 align:start position:0% -arm you know minus one half G mu nu oh -sorry - -01:12:53.180 --> 01:12:53.190 align:start position:0% -sorry - - -01:12:53.190 --> 01:12:55.540 align:start position:0% -sorry -arm<01:12:53.370> you<01:12:53.520> know<01:12:53.640> with<01:12:53.790> upstairs<01:12:54.180> in<01:12:54.450> the<01:12:54.540> sea - -01:12:55.540 --> 01:12:55.550 align:start position:0% -arm you know with upstairs in the sea - - -01:12:55.550 --> 01:13:01.629 align:start position:0% -arm you know with upstairs in the sea -yes - -01:13:01.629 --> 01:13:01.639 align:start position:0% - - - -01:13:01.639 --> 01:13:06.010 align:start position:0% - -we<01:13:02.639> are - -01:13:06.010 --> 01:13:06.020 align:start position:0% - - - -01:13:06.020 --> 01:13:17.350 align:start position:0% - -the<01:13:14.120> energy-momentum<01:13:15.120> tensor<01:13:16.080> is<01:13:16.199> symmetric - -01:13:17.350 --> 01:13:17.360 align:start position:0% -the energy-momentum tensor is symmetric - - -01:13:17.360 --> 01:13:21.530 align:start position:0% -the energy-momentum tensor is symmetric -no<01:13:18.469> now<01:13:19.469> I<01:13:19.500> I<01:13:19.770> haven't<01:13:20.310> I<01:13:20.699> haven't<01:13:21.030> gone<01:13:21.210> into - -01:13:21.530 --> 01:13:21.540 align:start position:0% -no now I I haven't I haven't gone into - - -01:13:21.540 --> 01:13:23.450 align:start position:0% -no now I I haven't I haven't gone into -the<01:13:21.840> question<01:13:22.590> of<01:13:22.679> the<01:13:22.800> symmetry<01:13:23.250> of<01:13:23.280> the - -01:13:23.450 --> 01:13:23.460 align:start position:0% -the question of the symmetry of the - - -01:13:23.460 --> 01:13:27.200 align:start position:0% -the question of the symmetry of the -energy-momentum<01:13:23.880> tensor<01:13:25.670> the<01:13:26.670> symmetry<01:13:27.179> of - -01:13:27.200 --> 01:13:27.210 align:start position:0% -energy-momentum tensor the symmetry of - - -01:13:27.210 --> 01:13:30.709 align:start position:0% -energy-momentum tensor the symmetry of -the<01:13:27.510> energy-momentum<01:13:28.110> tensor<01:13:28.980> is<01:13:29.719> required - -01:13:30.709 --> 01:13:30.719 align:start position:0% -the energy-momentum tensor is required - - -01:13:30.719 --> 01:13:33.229 align:start position:0% -the energy-momentum tensor is required -by<01:13:31.050> Lorentz<01:13:31.500> invariants<01:13:32.250> and<01:13:32.489> I<01:13:32.850> think<01:13:33.120> it - -01:13:33.229 --> 01:13:33.239 align:start position:0% -by Lorentz invariants and I think it - - -01:13:33.239 --> 01:13:35.570 align:start position:0% -by Lorentz invariants and I think it -would<01:13:33.389> just<01:13:33.420> take<01:13:33.900> us<01:13:34.139> outside<01:13:34.560> the<01:13:34.920> scope<01:13:35.370> of - -01:13:35.570 --> 01:13:35.580 align:start position:0% -would just take us outside the scope of - - -01:13:35.580 --> 01:13:38.180 align:start position:0% -would just take us outside the scope of -these<01:13:35.699> lectures<01:13:35.929> but<01:13:36.929> what<01:13:37.469> consists<01:13:37.830> one<01:13:38.010> can - -01:13:38.180 --> 01:13:38.190 align:start position:0% -these lectures but what consists one can - - -01:13:38.190 --> 01:13:40.250 align:start position:0% -these lectures but what consists one can -say<01:13:38.400> is<01:13:38.699> that<01:13:39.000> the<01:13:39.120> Einstein<01:13:39.600> equations - -01:13:40.250 --> 01:13:40.260 align:start position:0% -say is that the Einstein equations - - -01:13:40.260 --> 01:13:44.320 align:start position:0% -say is that the Einstein equations -require<01:13:40.440> team<01:13:41.159> unit<01:13:41.520> to<01:13:41.670> be<01:13:41.790> symmetric<01:13:42.949> so - -01:13:44.320 --> 01:13:44.330 align:start position:0% -require team unit to be symmetric so - - -01:13:44.330 --> 01:13:47.240 align:start position:0% -require team unit to be symmetric so -just<01:13:45.330> as<01:13:45.449> it<01:13:45.570> requires<01:13:45.989> team<01:13:46.620> you<01:13:46.770> know<01:13:46.949> to<01:13:47.159> be - -01:13:47.240 --> 01:13:47.250 align:start position:0% -just as it requires team you know to be - - -01:13:47.250 --> 01:13:48.169 align:start position:0% -just as it requires team you know to be -conserved - -01:13:48.169 --> 01:13:48.179 align:start position:0% -conserved - - -01:13:48.179 --> 01:13:51.939 align:start position:0% -conserved -it<01:13:48.600> also<01:13:48.900> requires<01:13:49.469> it<01:13:49.739> to<01:13:49.800> be<01:13:50.000> symmetric<01:13:51.000> and - -01:13:51.939 --> 01:13:51.949 align:start position:0% -it also requires it to be symmetric and - - -01:13:51.949 --> 01:13:54.320 align:start position:0% -it also requires it to be symmetric and -if<01:13:52.949> you<01:13:53.040> look<01:13:53.219> at<01:13:53.340> the<01:13:53.460> Tenergy<01:13:53.760> momentum - -01:13:54.320 --> 01:13:54.330 align:start position:0% -if you look at the Tenergy momentum - - -01:13:54.330 --> 01:13:56.149 align:start position:0% -if you look at the Tenergy momentum -tensor<01:13:54.659> for<01:13:54.810> any<01:13:54.960> kind<01:13:55.350> of<01:13:55.469> reasonable<01:13:55.860> field - -01:13:56.149 --> 01:13:56.159 align:start position:0% -tensor for any kind of reasonable field - - -01:13:56.159 --> 01:14:14.570 align:start position:0% -tensor for any kind of reasonable field -theory<01:13:56.550> it's<01:13:56.670> always<01:13:57.090> symmetric<01:14:13.580> tensor - -01:14:14.570 --> 01:14:14.580 align:start position:0% -theory it's always symmetric tensor - - -01:14:14.580 --> 01:14:17.209 align:start position:0% -theory it's always symmetric tensor -equation<01:14:15.120> a<01:14:15.420> tensor<01:14:16.350> or<01:14:16.560> some<01:14:16.920> of<01:14:17.010> what - -01:14:17.209 --> 01:14:17.219 align:start position:0% -equation a tensor or some of what - - -01:14:17.219 --> 01:14:18.560 align:start position:0% -equation a tensor or some of what -there's<01:14:17.550> nothing<01:14:17.639> to<01:14:17.880> sum<01:14:18.090> here - -01:14:18.560 --> 01:14:18.570 align:start position:0% -there's nothing to sum here - - -01:14:18.570 --> 01:14:20.899 align:start position:0% -there's nothing to sum here -there's<01:14:19.530> no<01:14:19.650> repeated<01:14:20.190> indices<01:14:20.489> in<01:14:20.790> those - -01:14:20.899 --> 01:14:20.909 align:start position:0% -there's no repeated indices in those - - -01:14:20.909 --> 01:14:26.680 align:start position:0% -there's no repeated indices in those -equations<01:14:23.630> leaving<01:14:24.630> some<01:14:24.900> of<01:14:24.929> this<01:14:25.110> in<01:14:25.290> this - -01:14:26.680 --> 01:14:26.690 align:start position:0% -equations leaving some of this in this - - -01:14:26.690 --> 01:14:29.160 align:start position:0% -equations leaving some of this in this -sum<01:14:27.690> of<01:14:27.780> what - -01:14:29.160 --> 01:14:29.170 align:start position:0% -sum of what - - -01:14:29.170 --> 01:14:34.040 align:start position:0% -sum of what -and - -01:14:34.040 --> 01:14:34.050 align:start position:0% - - - -01:14:34.050 --> 01:14:39.260 align:start position:0% - -sorry<01:14:35.720> some<01:14:36.720> of<01:14:36.750> what<01:14:37.640> you<01:14:38.640> can't<01:14:38.850> say<01:14:39.060> that - -01:14:39.260 --> 01:14:39.270 align:start position:0% -sorry some of what you can't say that - - -01:14:39.270 --> 01:14:41.960 align:start position:0% -sorry some of what you can't say that -some<01:14:39.480> of<01:14:39.720> and<01:14:40.140> pina<01:14:40.860> on<01:14:41.280> the<01:14:41.400> other<01:14:41.580> side<01:14:41.760> for - -01:14:41.960 --> 01:14:41.970 align:start position:0% -some of and pina on the other side for - - -01:14:41.970 --> 01:14:50.030 align:start position:0% -some of and pina on the other side for -some<01:14:42.150> of<01:14:42.390> what<01:14:42.840> and<01:14:43.110> what - -01:14:50.030 --> 01:14:50.040 align:start position:0% - - - -01:14:50.040 --> 01:14:56.729 align:start position:0% - -no<01:14:52.139> this<01:14:53.139> is<01:14:53.440> 16<01:14:54.159> equations<01:14:54.760> 4<01:14:55.630> times<01:14:55.929> 4<01:14:56.170> 16 - -01:14:56.729 --> 01:14:56.739 align:start position:0% -no this is 16 equations 4 times 4 16 - - -01:14:56.739 --> 01:14:59.700 align:start position:0% -no this is 16 equations 4 times 4 16 -dependent<01:14:57.310> equations<01:14:57.900> so<01:14:58.900> there<01:14:59.080> are<01:14:59.170> 16 - -01:14:59.700 --> 01:14:59.710 align:start position:0% -dependent equations so there are 16 - - -01:14:59.710 --> 01:15:01.890 align:start position:0% -dependent equations so there are 16 -Einstein<01:15:00.250> now<01:15:00.369> they're<01:15:00.610> not<01:15:00.790> all<01:15:01.000> completely - -01:15:01.890 --> 01:15:01.900 align:start position:0% -Einstein now they're not all completely - - -01:15:01.900 --> 01:15:04.229 align:start position:0% -Einstein now they're not all completely -independent<01:15:02.619> there<01:15:03.489> are<01:15:03.670> relationships - -01:15:04.229 --> 01:15:04.239 align:start position:0% -independent there are relationships - - -01:15:04.239 --> 01:15:07.290 align:start position:0% -independent there are relationships -among<01:15:04.780> them<01:15:04.960> but<01:15:05.800> at<01:15:06.400> first<01:15:06.610> sight - -01:15:07.290 --> 01:15:07.300 align:start position:0% -among them but at first sight - - -01:15:07.300 --> 01:15:10.680 align:start position:0% -among them but at first sight -and<01:15:07.510> without<01:15:07.710> being<01:15:08.710> too<01:15:08.949> subtle<01:15:09.270> these<01:15:10.270> are - -01:15:10.680 --> 01:15:10.690 align:start position:0% -and without being too subtle these are - - -01:15:10.690 --> 01:15:14.149 align:start position:0% -and without being too subtle these are -the<01:15:10.989> 16<01:15:11.770> independent<01:15:12.460> Einstein<01:15:13.420> equations - -01:15:14.149 --> 01:15:14.159 align:start position:0% -the 16 independent Einstein equations - - -01:15:14.159 --> 01:15:17.160 align:start position:0% -the 16 independent Einstein equations -just<01:15:15.159> like<01:15:15.400> Maxwell's<01:15:16.150> equations<01:15:16.780> how<01:15:16.989> many - -01:15:17.160 --> 01:15:17.170 align:start position:0% -just like Maxwell's equations how many - - -01:15:17.170 --> 01:15:19.050 align:start position:0% -just like Maxwell's equations how many -Maxwell's<01:15:17.739> equations<01:15:18.310> are<01:15:18.520> there<01:15:18.699> let's<01:15:18.969> see - -01:15:19.050 --> 01:15:19.060 align:start position:0% -Maxwell's equations are there let's see - - -01:15:19.060 --> 01:15:27.260 align:start position:0% -Maxwell's equations are there let's see -Delta<01:15:19.480> L<01:15:19.869> 1<01:15:20.139> to<01:15:20.940> the<01:15:21.940> 8th<01:15:22.210> Maxwell<01:15:22.869> equations - -01:15:27.260 --> 01:15:27.270 align:start position:0% - - - -01:15:27.270 --> 01:15:32.220 align:start position:0% - -there<01:15:28.270> are<01:15:28.770> 69<01:15:29.770> Stine<01:15:30.040> equations<01:15:30.630> right<01:15:31.630> for - -01:15:32.220 --> 01:15:32.230 align:start position:0% -there are 69 Stine equations right for - - -01:15:32.230 --> 01:15:33.959 align:start position:0% -there are 69 Stine equations right for -the<01:15:32.320> various<01:15:32.739> components<01:15:33.460> of<01:15:33.670> the - -01:15:33.959 --> 01:15:33.969 align:start position:0% -the various components of the - - -01:15:33.969 --> 01:15:37.770 align:start position:0% -the various components of the -gravitational<01:15:34.150> field - -01:15:37.770 --> 01:15:37.780 align:start position:0% - - - -01:15:37.780 --> 01:15:39.550 align:start position:0% - -we<01:15:38.289> captured - -01:15:39.550 --> 01:15:39.560 align:start position:0% -we captured - - -01:15:39.560 --> 01:15:41.459 align:start position:0% -we captured -in<01:15:39.650> four<01:15:40.160> dimensions<01:15:40.640> of - -01:15:41.459 --> 01:15:41.469 align:start position:0% -in four dimensions of - - -01:15:41.469 --> 01:15:49.429 align:start position:0% -in four dimensions of -doesn't<01:15:42.219> have<01:15:42.340> to<01:15:42.369> be<01:15:42.519> 0bg<01:15:43.329> -<01:15:44.070> that's<01:15:45.070> correct - -01:15:49.429 --> 01:15:49.439 align:start position:0% - - - -01:15:49.439 --> 01:15:56.270 align:start position:0% - -so<01:15:50.439> again<01:15:51.150> in<01:15:52.150> Texas<01:15:52.570> within<01:15:53.139> this<01:15:53.320> curvature - -01:15:56.270 --> 01:15:56.280 align:start position:0% - - - -01:15:56.280 --> 01:15:58.979 align:start position:0% - -if<01:15:57.280> the<01:15:57.519> Ricci<01:15:57.729> tensor<01:15:58.209> is<01:15:58.329> zero<01:15:58.360> that's<01:15:58.780> right - -01:15:58.979 --> 01:15:58.989 align:start position:0% -if the Ricci tensor is zero that's right - - -01:15:58.989 --> 01:16:03.479 align:start position:0% -if the Ricci tensor is zero that's right -so<01:15:59.679> that<01:15:59.860> means<01:16:00.070> that<01:16:00.400> we<01:16:00.699> get<01:16:01.199> stress<01:16:02.199> energy - -01:16:03.479 --> 01:16:03.489 align:start position:0% -so that means that we get stress energy - - -01:16:03.489 --> 01:16:11.109 align:start position:0% -so that means that we get stress energy -energy-momentum<01:16:04.489> tensor<01:16:09.550> but<01:16:10.550> you<01:16:10.730> don't - -01:16:11.109 --> 01:16:11.119 align:start position:0% -energy-momentum tensor but you don't - - -01:16:11.119 --> 01:16:16.149 align:start position:0% -energy-momentum tensor but you don't -have<01:16:11.300> a<01:16:11.360> flat<01:16:11.690> space<01:16:14.770> let's<01:16:15.770> go<01:16:15.889> through<01:16:16.070> the - -01:16:16.149 --> 01:16:16.159 align:start position:0% -have a flat space let's go through the - - -01:16:16.159 --> 01:16:18.220 align:start position:0% -have a flat space let's go through the -cases<01:16:16.340> first<01:16:17.060> possibility<01:16:17.570> is<01:16:17.840> the<01:16:17.960> energy - -01:16:18.220 --> 01:16:18.230 align:start position:0% -cases first possibility is the energy - - -01:16:18.230 --> 01:16:19.689 align:start position:0% -cases first possibility is the energy -let's<01:16:18.530> take<01:16:18.739> the<01:16:18.829> case<01:16:19.010> where<01:16:19.040> the<01:16:19.250> energy - -01:16:19.689 --> 01:16:19.699 align:start position:0% -let's take the case where the energy - - -01:16:19.699 --> 01:16:23.709 align:start position:0% -let's take the case where the energy -momentum<01:16:19.820> tensor<01:16:20.659> is<01:16:20.869> zero<01:16:22.000> okay<01:16:23.000> energy - -01:16:23.709 --> 01:16:23.719 align:start position:0% -momentum tensor is zero okay energy - - -01:16:23.719 --> 01:16:25.780 align:start position:0% -momentum tensor is zero okay energy -momentum<01:16:23.900> no<01:16:24.380> energy<01:16:24.739> momentum<01:16:25.130> whatever - -01:16:25.780 --> 01:16:25.790 align:start position:0% -momentum no energy momentum whatever - - -01:16:25.790 --> 01:16:28.089 align:start position:0% -momentum no energy momentum whatever -then<01:16:26.659> the<01:16:26.780> right<01:16:26.929> hand<01:16:27.139> side<01:16:27.409> is<01:16:27.590> zero<01:16:27.800> let's - -01:16:28.089 --> 01:16:28.099 align:start position:0% -then the right hand side is zero let's - - -01:16:28.099 --> 01:16:31.419 align:start position:0% -then the right hand side is zero let's -see<01:16:28.280> what<01:16:28.429> we<01:16:28.520> can<01:16:28.699> say<01:16:29.230> that<01:16:30.230> says<01:16:30.619> that<01:16:31.130> arm - -01:16:31.419 --> 01:16:31.429 align:start position:0% -see what we can say that says that arm - - -01:16:31.429 --> 01:16:37.000 align:start position:0% -see what we can say that says that arm -you<01:16:31.790> know<01:16:32.710> minus<01:16:33.710> one<01:16:33.920> half<01:16:34.190> G<01:16:34.579> mu<01:16:34.610> nu<01:16:34.929> R<01:16:36.010> is - -01:16:37.000 --> 01:16:37.010 align:start position:0% -you know minus one half G mu nu R is - - -01:16:37.010 --> 01:16:40.419 align:start position:0% -you know minus one half G mu nu R is -equal<01:16:37.429> to<01:16:37.460> zero<01:16:38.380> so<01:16:39.380> far<01:16:39.590> it's<01:16:39.829> not<01:16:39.949> obvious - -01:16:40.419 --> 01:16:40.429 align:start position:0% -equal to zero so far it's not obvious - - -01:16:40.429 --> 01:16:42.729 align:start position:0% -equal to zero so far it's not obvious -that<01:16:40.460> it<01:16:40.909> says<01:16:41.119> that<01:16:41.329> our<01:16:41.480> mu<01:16:41.719> nu<01:16:41.869> is<01:16:41.900> zero<01:16:42.260> but - -01:16:42.729 --> 01:16:42.739 align:start position:0% -that it says that our mu nu is zero but - - -01:16:42.739 --> 01:16:44.410 align:start position:0% -that it says that our mu nu is zero but -it<01:16:42.860> does<01:16:43.070> or<01:16:43.610> which<01:16:43.790> i<01:16:43.909> think<01:16:43.969> is<01:16:44.179> what<01:16:44.329> you - -01:16:44.410 --> 01:16:44.420 align:start position:0% -it does or which i think is what you - - -01:16:44.420 --> 01:16:53.600 align:start position:0% -it does or which i think is what you -were<01:16:44.480> getting<01:16:44.599> at - -01:16:53.600 --> 01:16:53.610 align:start position:0% - - - -01:16:53.610 --> 01:16:55.839 align:start position:0% - -[Music] - -01:16:55.839 --> 01:16:55.849 align:start position:0% -[Music] - - -01:16:55.849 --> 01:16:58.160 align:start position:0% -[Music] -yes<01:16:56.849> here's<01:16:57.389> your<01:16:57.449> what<01:16:57.810> you're<01:16:57.929> saying<01:16:58.110> is - -01:16:58.160 --> 01:16:58.170 align:start position:0% -yes here's your what you're saying is - - -01:16:58.170 --> 01:17:00.020 align:start position:0% -yes here's your what you're saying is -correct<01:16:58.230> but<01:16:58.650> let<01:16:58.830> me<01:16:58.889> just<01:16:59.099> say<01:16:59.520> it<01:16:59.670> in<01:16:59.849> its - -01:17:00.020 --> 01:17:00.030 align:start position:0% -correct but let me just say it in its - - -01:17:00.030 --> 01:17:03.700 align:start position:0% -correct but let me just say it in its -general<01:17:00.449> form<01:17:00.869> take<01:17:01.830> this<01:17:01.980> equation<01:17:02.159> here<01:17:02.969> and - -01:17:03.700 --> 01:17:03.710 align:start position:0% -general form take this equation here and - - -01:17:03.710 --> 01:17:10.870 align:start position:0% -general form take this equation here and -contract<01:17:04.710> it<01:17:05.179> with<01:17:06.179> G<01:17:06.900> mu<01:17:07.409> nu<01:17:07.550> lower<01:17:08.550> index - -01:17:10.870 --> 01:17:10.880 align:start position:0% -contract it with G mu nu lower index - - -01:17:10.880 --> 01:17:12.970 align:start position:0% -contract it with G mu nu lower index -okay - -01:17:12.970 --> 01:17:12.980 align:start position:0% -okay - - -01:17:12.980 --> 01:17:15.919 align:start position:0% -okay -what<01:17:13.980> is<01:17:14.099> the<01:17:14.250> contraction<01:17:14.699> of<01:17:15.119> G<01:17:15.389> mu<01:17:15.540> nu<01:17:15.570> with - -01:17:15.919 --> 01:17:15.929 align:start position:0% -what is the contraction of G mu nu with - - -01:17:15.929 --> 01:17:20.689 align:start position:0% -what is the contraction of G mu nu with -R<01:17:16.080> mu<01:17:16.290> nu<01:17:16.670> are<01:17:17.670> just<01:17:18.480> to<01:17:18.659> the<01:17:18.810> scalar<01:17:19.440> R<01:17:19.710> so<01:17:20.580> what - -01:17:20.689 --> 01:17:20.699 align:start position:0% -R mu nu are just to the scalar R so what - - -01:17:20.699 --> 01:17:28.250 align:start position:0% -R mu nu are just to the scalar R so what -this<01:17:20.880> is<01:17:21.060> then<01:17:21.270> is<01:17:21.540> its<01:17:21.780> R<01:17:24.440> minus<01:17:25.440> 1/2<01:17:26.540> and<01:17:27.540> what - -01:17:28.250 --> 01:17:28.260 align:start position:0% -this is then is its R minus 1/2 and what - - -01:17:28.260 --> 01:17:31.000 align:start position:0% -this is then is its R minus 1/2 and what -is<01:17:28.380> the<01:17:28.469> contraction<01:17:29.040> of<01:17:29.130> G<01:17:29.340> mu<01:17:29.520> nu<01:17:29.550> Oh<01:17:29.699> G<01:17:29.969> mu<01:17:30.150> nu - -01:17:31.000 --> 01:17:31.010 align:start position:0% -is the contraction of G mu nu Oh G mu nu - - -01:17:31.010 --> 01:17:34.310 align:start position:0% -is the contraction of G mu nu Oh G mu nu -whoa<01:17:32.040> not<01:17:32.429> quite<01:17:32.670> we're<01:17:33.540> summing<01:17:33.840> over<01:17:33.989> both - -01:17:34.310 --> 01:17:34.320 align:start position:0% -whoa not quite we're summing over both - - -01:17:34.320 --> 01:17:37.070 align:start position:0% -whoa not quite we're summing over both -indices<01:17:35.150> okay<01:17:36.150> good<01:17:36.300> let's<01:17:36.449> let's<01:17:36.750> do<01:17:36.929> it - -01:17:37.070 --> 01:17:37.080 align:start position:0% -indices okay good let's let's do it - - -01:17:37.080 --> 01:17:41.810 align:start position:0% -indices okay good let's let's do it -let's<01:17:37.500> work<01:17:37.679> it<01:17:37.830> out<01:17:37.949> okay<01:17:38.449> I<01:17:39.650> want<01:17:40.650> G<01:17:41.190> mu<01:17:41.610> nu - -01:17:41.810 --> 01:17:41.820 align:start position:0% -let's work it out okay I want G mu nu - - -01:17:41.820 --> 01:17:46.250 align:start position:0% -let's work it out okay I want G mu nu -times<01:17:42.510> G<01:17:42.810> mu<01:17:43.020> nu<01:17:43.050> right<01:17:43.760> so<01:17:44.760> let's<01:17:45.270> back<01:17:45.989> off - -01:17:46.250 --> 01:17:46.260 align:start position:0% -times G mu nu right so let's back off - - -01:17:46.260 --> 01:17:49.280 align:start position:0% -times G mu nu right so let's back off -for<01:17:46.739> a<01:17:46.800> moment<01:17:47.219> and<01:17:47.599> do<01:17:48.599> something<01:17:48.989> a<01:17:49.050> little - -01:17:49.280 --> 01:17:49.290 align:start position:0% -for a moment and do something a little - - -01:17:49.290 --> 01:17:56.260 align:start position:0% -for a moment and do something a little -simpler<01:17:50.060> let's<01:17:51.060> take<01:17:51.420> G<01:17:52.110> alpha<01:17:52.409> nu<01:17:53.210> g<01:17:54.210> nu<01:17:54.750> mu - -01:17:56.260 --> 01:17:56.270 align:start position:0% -simpler let's take G alpha nu g nu mu - - -01:17:56.270 --> 01:18:00.939 align:start position:0% -simpler let's take G alpha nu g nu mu -what<01:17:57.270> is<01:17:57.389> that - -01:18:00.939 --> 01:18:00.949 align:start position:0% - - - -01:18:00.949 --> 01:18:03.669 align:start position:0% - -that's<01:18:01.610> the<01:18:01.760> chronica<01:18:02.179> Delta<01:18:02.540> that's<01:18:03.530> the - -01:18:03.669 --> 01:18:03.679 align:start position:0% -that's the chronica Delta that's the - - -01:18:03.679 --> 01:18:08.439 align:start position:0% -that's the chronica Delta that's the -chronica<01:18:04.070> Delta<01:18:05.620> Delta<01:18:06.620> alpha<01:18:07.100> mule<01:18:07.370> but<01:18:08.239> now - -01:18:08.439 --> 01:18:08.449 align:start position:0% -chronica Delta Delta alpha mule but now - - -01:18:08.449 --> 01:18:11.680 align:start position:0% -chronica Delta Delta alpha mule but now -I<01:18:08.780> want<01:18:09.230> to<01:18:09.350> identify<01:18:09.610> alpha<01:18:10.610> with<01:18:10.820> Miu<01:18:11.060> also - -01:18:11.680 --> 01:18:11.690 align:start position:0% -I want to identify alpha with Miu also - - -01:18:11.690 --> 01:18:14.169 align:start position:0% -I want to identify alpha with Miu also -because<01:18:12.290> that's<01:18:12.440> what<01:18:12.620> I<01:18:12.650> have<01:18:12.920> here<01:18:13.219> Jim<01:18:14.030> you - -01:18:14.169 --> 01:18:14.179 align:start position:0% -because that's what I have here Jim you - - -01:18:14.179 --> 01:18:17.770 align:start position:0% -because that's what I have here Jim you -know<01:18:14.360> Jim<01:18:15.020> you<01:18:15.140> know<01:18:15.670> so<01:18:16.670> what<01:18:16.850> happens<01:18:17.239> if<01:18:17.540> I - -01:18:17.770 --> 01:18:17.780 align:start position:0% -know Jim you know so what happens if I - - -01:18:17.780 --> 01:18:22.240 align:start position:0% -know Jim you know so what happens if I -identify<01:18:18.100> alpha<01:18:19.100> with<01:18:19.310> mu<01:18:20.590> then<01:18:21.590> this<01:18:21.830> becomes - -01:18:22.240 --> 01:18:22.250 align:start position:0% -identify alpha with mu then this becomes - - -01:18:22.250 --> 01:18:28.600 align:start position:0% -identify alpha with mu then this becomes -mu<01:18:22.610> and<01:18:24.130> this<01:18:25.130> becomes<01:18:25.610> mu<01:18:25.880> also<01:18:27.460> what<01:18:28.460> is - -01:18:28.600 --> 01:18:28.610 align:start position:0% -mu and this becomes mu also what is - - -01:18:28.610 --> 01:18:34.570 align:start position:0% -mu and this becomes mu also what is -Delta<01:18:29.060> mu<01:18:29.300> mu<01:18:30.219> no<01:18:31.360> no<01:18:32.360> I<01:18:32.800> heard<01:18:33.800> one<01:18:34.040> I<01:18:34.219> heard<01:18:34.370> - - -01:18:34.570 --> 01:18:34.580 align:start position:0% -Delta mu mu no no I heard one I heard - - - -01:18:34.580 --> 01:18:37.689 align:start position:0% -Delta mu mu no no I heard one I heard - -I<01:18:34.610> heard<01:18:34.760> four<01:18:36.010> four<01:18:37.010> is<01:18:37.040> of<01:18:37.250> course<01:18:37.460> the<01:18:37.550> right - -01:18:37.689 --> 01:18:37.699 align:start position:0% -I heard four four is of course the right - - -01:18:37.699 --> 01:18:40.840 align:start position:0% -I heard four four is of course the right -answer<01:18:38.030> why<01:18:38.810> because<01:18:39.679> it's<01:18:39.890> the<01:18:40.040> sum<01:18:40.370> of<01:18:40.400> the - -01:18:40.840 --> 01:18:40.850 align:start position:0% -answer why because it's the sum of the - - -01:18:40.850 --> 01:18:44.380 align:start position:0% -answer why because it's the sum of the -four<01:18:41.150> components<01:18:42.730> it's<01:18:43.730> the<01:18:43.910> one<01:18:44.179> one - -01:18:44.380 --> 01:18:44.390 align:start position:0% -four components it's the one one - - -01:18:44.390 --> 01:18:46.630 align:start position:0% -four components it's the one one -component<01:18:45.140> plus<01:18:45.410> the<01:18:45.590> two<01:18:45.770> two<01:18:45.980> component - -01:18:46.630 --> 01:18:46.640 align:start position:0% -component plus the two two component - - -01:18:46.640 --> 01:18:48.280 align:start position:0% -component plus the two two component -plus<01:18:46.850> the<01:18:46.910> three<01:18:47.210> three<01:18:47.449> component<01:18:48.050> plus<01:18:48.230> the - -01:18:48.280 --> 01:18:48.290 align:start position:0% -plus the three three component plus the - - -01:18:48.290 --> 01:18:50.620 align:start position:0% -plus the three three component plus the -four<01:18:48.590> four<01:18:48.830> component<01:18:49.429> so<01:18:50.090> this<01:18:50.239> number<01:18:50.390> is - -01:18:50.620 --> 01:18:50.630 align:start position:0% -four four component so this number is - - -01:18:50.630 --> 01:18:53.340 align:start position:0% -four four component so this number is -four<01:18:50.980> and<01:18:51.980> put<01:18:52.250> a<01:18:52.280> nice<01:18:52.429> thing<01:18:52.640> to<01:18:52.670> remember - -01:18:53.340 --> 01:18:53.350 align:start position:0% -four and put a nice thing to remember - - -01:18:53.350 --> 01:18:56.729 align:start position:0% -four and put a nice thing to remember -Delta<01:18:54.350> mu<01:18:54.560> mu<01:18:55.030> remember<01:18:56.030> it's<01:18:56.150> summoned - -01:18:56.729 --> 01:18:56.739 align:start position:0% -Delta mu mu remember it's summoned - - -01:18:56.739 --> 01:18:59.590 align:start position:0% -Delta mu mu remember it's summoned -there's<01:18:57.739> four<01:18:58.070> so<01:18:58.969> let's<01:18:59.210> see<01:18:59.330> what<01:18:59.449> we<01:18:59.570> have - -01:18:59.590 --> 01:18:59.600 align:start position:0% -there's four so let's see what we have - - -01:18:59.600 --> 01:19:01.959 align:start position:0% -there's four so let's see what we have -here<01:18:59.900> and<01:18:59.989> we<01:19:00.110> now<01:19:00.260> have<01:19:00.520> minus<01:19:01.520> one<01:19:01.730> half - -01:19:01.959 --> 01:19:01.969 align:start position:0% -here and we now have minus one half - - -01:19:01.969 --> 01:19:06.669 align:start position:0% -here and we now have minus one half -times<01:19:02.630> four<01:19:02.989> times<01:19:03.830> R<01:19:04.570> minus<01:19:05.570> one<01:19:05.810> half<01:19:06.110> times - -01:19:06.669 --> 01:19:06.679 align:start position:0% -times four times R minus one half times - - -01:19:06.679 --> 01:19:09.550 align:start position:0% -times four times R minus one half times -four<01:19:07.040> if<01:19:07.370> I<01:19:07.520> remember<01:19:08.000> correctly<01:19:08.480> is<01:19:08.989> minus - -01:19:09.550 --> 01:19:09.560 align:start position:0% -four if I remember correctly is minus - - -01:19:09.560 --> 01:19:16.479 align:start position:0% -four if I remember correctly is minus -two<01:19:11.290> so<01:19:12.290> this<01:19:12.440> is<01:19:12.620> minus<01:19:13.010> two<01:19:13.400> times<01:19:14.030> R<01:19:15.489> is - -01:19:16.479 --> 01:19:16.489 align:start position:0% -two so this is minus two times R is - - -01:19:16.489 --> 01:19:20.410 align:start position:0% -two so this is minus two times R is -equal<01:19:16.880> to<01:19:16.910> zero<01:19:18.070> in<01:19:19.070> other<01:19:19.310> words<01:19:19.550> minus<01:19:20.030> R<01:19:20.210> is - -01:19:20.410 --> 01:19:20.420 align:start position:0% -equal to zero in other words minus R is - - -01:19:20.420 --> 01:19:22.720 align:start position:0% -equal to zero in other words minus R is -equal<01:19:20.750> to<01:19:20.870> 0<01:19:21.110> which<01:19:21.560> means<01:19:21.710> plus<01:19:22.040> R<01:19:22.250> is<01:19:22.460> equal - -01:19:22.720 --> 01:19:22.730 align:start position:0% -equal to 0 which means plus R is equal - - -01:19:22.730 --> 01:19:27.910 align:start position:0% -equal to 0 which means plus R is equal -to<01:19:22.760> 0<01:19:23.260> r<01:19:24.260> is<01:19:24.320> equal<01:19:24.739> to<01:19:24.949> zero<01:19:25.280> now<01:19:26.770> this<01:19:27.770> is - -01:19:27.910 --> 01:19:27.920 align:start position:0% -to 0 r is equal to zero now this is - - -01:19:27.920 --> 01:19:30.040 align:start position:0% -to 0 r is equal to zero now this is -remember<01:19:28.340> what<01:19:28.460> this<01:19:28.550> is<01:19:28.610> this<01:19:29.540> is<01:19:29.719> the<01:19:29.840> case - -01:19:30.040 --> 01:19:30.050 align:start position:0% -remember what this is this is the case - - -01:19:30.050 --> 01:19:33.070 align:start position:0% -remember what this is this is the case -where<01:19:30.320> there<01:19:30.469> is<01:19:30.590> no<01:19:30.800> energy<01:19:31.219> momentum<01:19:32.080> so - -01:19:33.070 --> 01:19:33.080 align:start position:0% -where there is no energy momentum so - - -01:19:33.080 --> 01:19:35.110 align:start position:0% -where there is no energy momentum so -first<01:19:33.469> of<01:19:33.650> all<01:19:33.860> it<01:19:34.010> says<01:19:34.219> that<01:19:34.280> the<01:19:34.790> curvature - -01:19:35.110 --> 01:19:35.120 align:start position:0% -first of all it says that the curvature - - -01:19:35.120 --> 01:19:37.840 align:start position:0% -first of all it says that the curvature -scalar<01:19:35.420> is<01:19:35.870> equal<01:19:36.170> to<01:19:36.290> zero<01:19:36.530> if<01:19:37.429> there's<01:19:37.699> no - -01:19:37.840 --> 01:19:37.850 align:start position:0% -scalar is equal to zero if there's no - - -01:19:37.850 --> 01:19:40.540 align:start position:0% -scalar is equal to zero if there's no -energy<01:19:38.270> momentum<01:19:38.449> around<01:19:39.010> the<01:19:40.010> curvature - -01:19:40.540 --> 01:19:40.550 align:start position:0% -energy momentum around the curvature - - -01:19:40.550 --> 01:19:43.060 align:start position:0% -energy momentum around the curvature -scalar<01:19:40.580> is<01:19:41.150> equal<01:19:41.449> to<01:19:41.570> zero<01:19:41.840> this<01:19:42.530> is<01:19:42.739> kind<01:19:42.980> of - -01:19:43.060 --> 01:19:43.070 align:start position:0% -scalar is equal to zero this is kind of - - -01:19:43.070 --> 01:19:45.459 align:start position:0% -scalar is equal to zero this is kind of -like<01:19:43.219> del<01:19:43.489> squared<01:19:43.820> Phi<01:19:43.910> equals<01:19:44.540> zero<01:19:44.750> doesn't - -01:19:45.459 --> 01:19:45.469 align:start position:0% -like del squared Phi equals zero doesn't - - -01:19:45.469 --> 01:19:48.490 align:start position:0% -like del squared Phi equals zero doesn't -mean<01:19:45.739> that<01:19:45.980> the<01:19:46.190> that<01:19:46.840> that<01:19:47.840> the<01:19:47.960> metric<01:19:48.320> is - -01:19:48.490 --> 01:19:48.500 align:start position:0% -mean that the that that the metric is - - -01:19:48.500 --> 01:19:49.120 align:start position:0% -mean that the that that the metric is -trivial - -01:19:49.120 --> 01:19:49.130 align:start position:0% -trivial - - -01:19:49.130 --> 01:19:51.280 align:start position:0% -trivial -it<01:19:49.280> just<01:19:49.520> says<01:19:49.699> some<01:19:49.969> particular<01:19:50.449> thing<01:19:50.929> is - -01:19:51.280 --> 01:19:51.290 align:start position:0% -it just says some particular thing is - - -01:19:51.290 --> 01:19:54.280 align:start position:0% -it just says some particular thing is -zero<01:19:51.620> about<01:19:51.949> the<01:19:52.100> metric<01:19:52.460> but<01:19:53.239> if<01:19:53.480> R<01:19:53.900> is<01:19:53.929> equal - -01:19:54.280 --> 01:19:54.290 align:start position:0% -zero about the metric but if R is equal - - -01:19:54.290 --> 01:19:58.300 align:start position:0% -zero about the metric but if R is equal -to<01:19:54.530> zero<01:19:54.890> let's<01:19:55.160> come<01:19:55.370> back<01:19:55.400> now<01:19:57.070> let's<01:19:58.070> come - -01:19:58.300 --> 01:19:58.310 align:start position:0% -to zero let's come back now let's come - - -01:19:58.310 --> 01:20:00.580 align:start position:0% -to zero let's come back now let's come -back<01:19:58.340> now<01:19:58.730> to<01:19:59.239> the<01:19:59.660> original<01:20:00.050> equation<01:20:00.230> before - -01:20:00.580 --> 01:20:00.590 align:start position:0% -back now to the original equation before - - -01:20:00.590 --> 01:20:03.540 align:start position:0% -back now to the original equation before -we<01:20:00.830> contracted<01:20:01.460> it<01:20:01.580> if<01:20:01.790> R<01:20:02.600> is<01:20:02.630> equal<01:20:02.810> to<01:20:03.110> zero - -01:20:03.540 --> 01:20:03.550 align:start position:0% -we contracted it if R is equal to zero - - -01:20:03.550 --> 01:20:08.650 align:start position:0% -we contracted it if R is equal to zero -then<01:20:04.550> we<01:20:05.090> can<01:20:05.270> also<01:20:05.420> say<01:20:05.810> that<01:20:06.260> our<01:20:06.699> mu<01:20:07.699> nu<01:20:08.300> is - -01:20:08.650 --> 01:20:08.660 align:start position:0% -then we can also say that our mu nu is - - -01:20:08.660 --> 01:20:13.989 align:start position:0% -then we can also say that our mu nu is -equal<01:20:09.020> to<01:20:09.350> zero<01:20:10.870> so<01:20:11.870> the<01:20:12.020> right<01:20:12.199> statement<01:20:12.650> is - -01:20:13.989 --> 01:20:13.999 align:start position:0% -equal to zero so the right statement is - - -01:20:13.999 --> 01:20:17.169 align:start position:0% -equal to zero so the right statement is -that<01:20:14.539> in<01:20:14.929> regions<01:20:15.559> of<01:20:15.710> space<01:20:16.039> and<01:20:16.340> time<01:20:16.400> where - -01:20:17.169 --> 01:20:17.179 align:start position:0% -that in regions of space and time where - - -01:20:17.179 --> 01:20:20.950 align:start position:0% -that in regions of space and time where -there<01:20:17.329> is<01:20:17.599> no<01:20:17.840> energy<01:20:18.139> and<01:20:18.619> momentum<01:20:19.960> the - -01:20:20.950 --> 01:20:20.960 align:start position:0% -there is no energy and momentum the - - -01:20:20.960 --> 01:20:26.649 align:start position:0% -there is no energy and momentum the -Ricci<01:20:21.199> tensor<01:20:21.829> is<01:20:22.010> equal<01:20:22.039> to<01:20:22.519> zero<01:20:22.760> now<01:20:25.659> what - -01:20:26.649 --> 01:20:26.659 align:start position:0% -Ricci tensor is equal to zero now what - - -01:20:26.659 --> 01:20:31.709 align:start position:0% -Ricci tensor is equal to zero now what -does<01:20:26.780> that<01:20:26.809> say<01:20:28.150> in<01:20:29.979> three-dimensional<01:20:30.979> space - -01:20:31.709 --> 01:20:31.719 align:start position:0% -does that say in three-dimensional space - - -01:20:31.719 --> 01:20:35.049 align:start position:0% -does that say in three-dimensional space -for<01:20:32.719> two<01:20:32.960> dimensional<01:20:33.530> space<01:20:33.860> the<01:20:34.789> Ricci - -01:20:35.049 --> 01:20:35.059 align:start position:0% -for two dimensional space the Ricci - - -01:20:35.059 --> 01:20:37.930 align:start position:0% -for two dimensional space the Ricci -tensor<01:20:35.630> being<01:20:36.199> 0<01:20:36.619> would<01:20:37.159> say<01:20:37.340> the<01:20:37.519> space<01:20:37.699> is - -01:20:37.930 --> 01:20:37.940 align:start position:0% -tensor being 0 would say the space is - - -01:20:37.940 --> 01:20:43.779 align:start position:0% -tensor being 0 would say the space is -flat<01:20:38.229> in<01:20:40.630> four-dimensional<01:20:41.630> space-time<01:20:42.789> the - -01:20:43.779 --> 01:20:43.789 align:start position:0% -flat in four-dimensional space-time the - - -01:20:43.789 --> 01:20:47.140 align:start position:0% -flat in four-dimensional space-time the -Ricci<01:20:44.269> tensor<01:20:44.840> being<01:20:45.079> 0<01:20:45.409> is<01:20:45.739> not<01:20:46.190> enough<01:20:46.880> to - -01:20:47.140 --> 01:20:47.150 align:start position:0% -Ricci tensor being 0 is not enough to - - -01:20:47.150 --> 01:20:49.149 align:start position:0% -Ricci tensor being 0 is not enough to -tell<01:20:47.360> you<01:20:47.479> the<01:20:47.630> space<01:20:47.869> time<01:20:48.170> is<01:20:48.409> flat<01:20:48.440> there - -01:20:49.149 --> 01:20:49.159 align:start position:0% -tell you the space time is flat there - - -01:20:49.159 --> 01:20:50.739 align:start position:0% -tell you the space time is flat there -are<01:20:49.280> too<01:20:49.460> many<01:20:49.579> components<01:20:50.420> too<01:20:50.570> many - -01:20:50.739 --> 01:20:50.749 align:start position:0% -are too many components too many - - -01:20:50.749 --> 01:20:54.160 align:start position:0% -are too many components too many -independent<01:20:51.320> components<01:20:52.749> what<01:20:53.749> kind<01:20:54.079> of - -01:20:54.160 --> 01:20:54.170 align:start position:0% -independent components what kind of - - -01:20:54.170 --> 01:20:56.439 align:start position:0% -independent components what kind of -things<01:20:54.469> can<01:20:54.769> you<01:20:54.829> imagine<01:20:55.249> can<01:20:55.760> be<01:20:55.789> going<01:20:56.210> on - -01:20:56.439 --> 01:20:56.449 align:start position:0% -things can you imagine can be going on - - -01:20:56.449 --> 01:20:58.540 align:start position:0% -things can you imagine can be going on -let's<01:20:56.809> imagine<01:20:57.110> a<01:20:57.469> spacetime<01:20:57.829> which<01:20:58.400> is - -01:20:58.540 --> 01:20:58.550 align:start position:0% -let's imagine a spacetime which is - - -01:20:58.550 --> 01:21:00.459 align:start position:0% -let's imagine a spacetime which is -completely<01:20:59.059> empty<01:20:59.449> of<01:20:59.900> matter - -01:21:00.459 --> 01:21:00.469 align:start position:0% -completely empty of matter - - -01:21:00.469 --> 01:21:03.339 align:start position:0% -completely empty of matter -meaning<01:21:01.369> mass<01:21:01.610> meaning<01:21:02.210> right<01:21:02.389> hand<01:21:02.659> side<01:21:02.929> a - -01:21:03.339 --> 01:21:03.349 align:start position:0% -meaning mass meaning right hand side a - - -01:21:03.349 --> 01:21:06.189 align:start position:0% -meaning mass meaning right hand side a -spacetime<01:21:04.249> which<01:21:05.059> is<01:21:05.269> everywhere<01:21:05.809> is - -01:21:06.189 --> 01:21:06.199 align:start position:0% -spacetime which is everywhere is - - -01:21:06.199 --> 01:21:10.180 align:start position:0% -spacetime which is everywhere is -completely<01:21:07.159> empty<01:21:08.110> does<01:21:09.110> that<01:21:09.380> mean<01:21:09.710> that - -01:21:10.180 --> 01:21:10.190 align:start position:0% -completely empty does that mean that - - -01:21:10.190 --> 01:21:14.859 align:start position:0% -completely empty does that mean that -it's<01:21:10.340> flat<01:21:11.289> no<01:21:12.289> it<01:21:13.039> means<01:21:13.369> that<01:21:13.849> the<01:21:14.630> Ricci - -01:21:14.859 --> 01:21:14.869 align:start position:0% -it's flat no it means that the Ricci - - -01:21:14.869 --> 01:21:17.279 align:start position:0% -it's flat no it means that the Ricci -tensor<01:21:15.499> is<01:21:15.650> everywhere<01:21:16.099> is<01:21:16.249> zero - -01:21:17.279 --> 01:21:17.289 align:start position:0% -tensor is everywhere is zero - - -01:21:17.289 --> 01:21:20.620 align:start position:0% -tensor is everywhere is zero -are<01:21:18.289> there<01:21:18.530> geometries<01:21:19.280> which<01:21:20.119> have<01:21:20.360> Ricci - -01:21:20.620 --> 01:21:20.630 align:start position:0% -are there geometries which have Ricci - - -01:21:20.630 --> 01:21:23.020 align:start position:0% -are there geometries which have Ricci -tensor<01:21:21.199> equal<01:21:21.590> to<01:21:21.710> zero<01:21:21.979> everywhere<01:21:22.519> is<01:21:22.820> but - -01:21:23.020 --> 01:21:23.030 align:start position:0% -tensor equal to zero everywhere is but - - -01:21:23.030 --> 01:21:26.560 align:start position:0% -tensor equal to zero everywhere is but -aren't<01:21:23.360> flat<01:21:23.570> yes<01:21:24.440> indeed<01:21:24.800> there<01:21:24.979> are<01:21:25.570> anybody - -01:21:26.560 --> 01:21:26.570 align:start position:0% -aren't flat yes indeed there are anybody - - -01:21:26.570 --> 01:21:28.180 align:start position:0% -aren't flat yes indeed there are anybody -have<01:21:26.780> an<01:21:26.869> idea<01:21:26.960> what<01:21:27.320> those<01:21:27.499> geometries - -01:21:28.180 --> 01:21:28.190 align:start position:0% -have an idea what those geometries - - -01:21:28.190 --> 01:21:34.330 align:start position:0% -have an idea what those geometries -correspond<01:21:28.999> to<01:21:29.619> what's<01:21:30.619> that<01:21:32.769> no<01:21:33.769> no<01:21:34.039> these - -01:21:34.330 --> 01:21:34.340 align:start position:0% -correspond to what's that no no these - - -01:21:34.340 --> 01:21:35.919 align:start position:0% -correspond to what's that no no these -are<01:21:34.579> these<01:21:34.789> are<01:21:34.820> geometries<01:21:35.510> with<01:21:35.900> a - -01:21:35.919 --> 01:21:35.929 align:start position:0% -are these are geometries with a - - -01:21:35.929 --> 01:21:38.439 align:start position:0% -are these are geometries with a -signature<01:21:36.499> plus<01:21:37.039> plus<01:21:37.369> plus<01:21:37.639> minus<01:21:37.670> I<01:21:38.269> mean - -01:21:38.439 --> 01:21:38.449 align:start position:0% -signature plus plus plus minus I mean - - -01:21:38.449 --> 01:21:41.319 align:start position:0% -signature plus plus plus minus I mean -the<01:21:38.539> user<01:21:38.719> you<01:21:38.900> know<01:21:39.019> good<01:21:39.619> space-time<01:21:40.300> well - -01:21:41.319 --> 01:21:41.329 align:start position:0% -the user you know good space-time well - - -01:21:41.329 --> 01:21:42.700 align:start position:0% -the user you know good space-time well -oh<01:21:41.539> yes<01:21:42.199> of<01:21:42.409> course - -01:21:42.700 --> 01:21:42.710 align:start position:0% -oh yes of course - - -01:21:42.710 --> 01:21:45.189 align:start position:0% -oh yes of course -no<01:21:43.190> no<01:21:43.449> geometries<01:21:44.449> which<01:21:44.630> are<01:21:44.749> not<01:21:44.960> flat - -01:21:45.189 --> 01:21:45.199 align:start position:0% -no no geometries which are not flat - - -01:21:45.199 --> 01:21:46.899 align:start position:0% -no no geometries which are not flat -Euclidean<01:21:45.409> geometry<01:21:45.800> is<01:21:46.369> flat - -01:21:46.899 --> 01:21:46.909 align:start position:0% -Euclidean geometry is flat - - -01:21:46.909 --> 01:21:50.770 align:start position:0% -Euclidean geometry is flat -are<01:21:47.570> there<01:21:47.869> geometries<01:21:49.449> what<01:21:50.449> might<01:21:50.630> they - -01:21:50.770 --> 01:21:50.780 align:start position:0% -are there geometries what might they - - -01:21:50.780 --> 01:21:56.439 align:start position:0% -are there geometries what might they -correspond<01:21:51.380> to<01:21:51.440> physically<01:21:55.449> gravitational - -01:21:56.439 --> 01:21:56.449 align:start position:0% -correspond to physically gravitational - - -01:21:56.449 --> 01:22:01.080 align:start position:0% -correspond to physically gravitational -waves<01:21:57.610> just<01:21:58.780> gravitational<01:21:59.780> waves<01:22:00.019> moving<01:22:00.650> by - -01:22:01.080 --> 01:22:01.090 align:start position:0% -waves just gravitational waves moving by - - -01:22:01.090 --> 01:22:05.290 align:start position:0% -waves just gravitational waves moving by -waves<01:22:02.090> waves<01:22:03.019> wave-like<01:22:04.239> disturbances<01:22:05.239> of - -01:22:05.290 --> 01:22:05.300 align:start position:0% -waves waves wave-like disturbances of - - -01:22:05.300 --> 01:22:09.160 align:start position:0% -waves waves wave-like disturbances of -space-time<01:22:06.039> which<01:22:07.039> are<01:22:07.190> not<01:22:07.369> trivial<01:22:08.170> for - -01:22:09.160 --> 01:22:09.170 align:start position:0% -space-time which are not trivial for - - -01:22:09.170 --> 01:22:10.649 align:start position:0% -space-time which are not trivial for -example<01:22:09.619> there<01:22:09.829> are<01:22:09.860> plane<01:22:10.280> wave - -01:22:10.649 --> 01:22:10.659 align:start position:0% -example there are plane wave - - -01:22:10.659 --> 01:22:13.390 align:start position:0% -example there are plane wave -gravitational<01:22:11.659> excitations<01:22:12.440> in<01:22:12.710> which<01:22:13.219> the - -01:22:13.390 --> 01:22:13.400 align:start position:0% -gravitational excitations in which the - - -01:22:13.400 --> 01:22:15.640 align:start position:0% -gravitational excitations in which the -various<01:22:13.849> components<01:22:14.539> of<01:22:14.630> the<01:22:14.690> metric<01:22:14.929> wobble - -01:22:15.640 --> 01:22:15.650 align:start position:0% -various components of the metric wobble - - -01:22:15.650 --> 01:22:18.969 align:start position:0% -various components of the metric wobble -and<01:22:15.800> wiggle<01:22:16.159> but<01:22:17.030> where<01:22:17.630> the<01:22:17.869> Ricci<01:22:18.260> tensor<01:22:18.860> is - -01:22:18.969 --> 01:22:18.979 align:start position:0% -and wiggle but where the Ricci tensor is - - -01:22:18.979 --> 01:22:21.489 align:start position:0% -and wiggle but where the Ricci tensor is -everywhere<01:22:19.429> zero<01:22:19.690> but<01:22:20.690> the<01:22:20.809> Riemann<01:22:21.110> tensor - -01:22:21.489 --> 01:22:21.499 align:start position:0% -everywhere zero but the Riemann tensor - - -01:22:21.499 --> 01:22:26.189 align:start position:0% -everywhere zero but the Riemann tensor -is<01:22:21.619> not<01:22:21.909> so<01:22:23.469> there<01:22:24.469> are<01:22:24.710> solutions - -01:22:26.189 --> 01:22:26.199 align:start position:0% -is not so there are solutions - - -01:22:26.199 --> 01:22:28.560 align:start position:0% -is not so there are solutions -empties<01:22:26.830> or<01:22:27.489> we<01:22:27.670> call<01:22:27.850> it<01:22:27.910> they're<01:22:28.449> usually - -01:22:28.560 --> 01:22:28.570 align:start position:0% -empties or we call it they're usually - - -01:22:28.570 --> 01:22:31.380 align:start position:0% -empties or we call it they're usually -called<01:22:28.929> vacuum<01:22:29.290> solutions<01:22:30.190> they're<01:22:31.060> called - -01:22:31.380 --> 01:22:31.390 align:start position:0% -called vacuum solutions they're called - - -01:22:31.390 --> 01:22:33.300 align:start position:0% -called vacuum solutions they're called -vacuum<01:22:31.750> solutions<01:22:32.590> because<01:22:32.980> they<01:22:33.100> don't - -01:22:33.300 --> 01:22:33.310 align:start position:0% -vacuum solutions because they don't - - -01:22:33.310 --> 01:22:35.120 align:start position:0% -vacuum solutions because they don't -contain<01:22:33.489> any<01:22:33.790> ordinary<01:22:34.420> energy<01:22:34.840> and<01:22:34.989> momentum - -01:22:35.120 --> 01:22:35.130 align:start position:0% -contain any ordinary energy and momentum - - -01:22:35.130 --> 01:22:39.979 align:start position:0% -contain any ordinary energy and momentum -but<01:22:36.130> they<01:22:36.310> do<01:22:36.370> contain<01:22:37.080> gravitational<01:22:38.080> waves - -01:22:39.979 --> 01:22:39.989 align:start position:0% -but they do contain gravitational waves - - -01:22:39.989 --> 01:22:48.060 align:start position:0% -but they do contain gravitational waves -disturbances<01:22:40.989> real<01:22:41.260> disturbances<01:22:45.449> yes<01:22:47.070> yes - -01:22:48.060 --> 01:22:48.070 align:start position:0% -disturbances real disturbances yes yes - - -01:22:48.070 --> 01:22:50.070 align:start position:0% -disturbances real disturbances yes yes -okay<01:22:48.670> that's<01:22:49.060> a<01:22:49.179> good<01:22:49.300> that<01:22:49.449> that<01:22:49.570> that<01:22:49.870> is<01:22:50.050> a - -01:22:50.070 --> 01:22:50.080 align:start position:0% -okay that's a good that that that is a - - -01:22:50.080 --> 01:22:51.600 align:start position:0% -okay that's a good that that that is a -very<01:22:50.290> interesting<01:22:50.620> point<01:22:50.980> so<01:22:51.219> let's<01:22:51.429> come - -01:22:51.600 --> 01:22:51.610 align:start position:0% -very interesting point so let's come - - -01:22:51.610 --> 01:22:52.950 align:start position:0% -very interesting point so let's come -let's<01:22:51.790> come<01:22:51.969> to<01:22:52.090> it<01:22:52.210> right<01:22:52.360> now<01:22:52.540> let's<01:22:52.719> face<01:22:52.900> it - -01:22:52.950 --> 01:22:52.960 align:start position:0% -let's come to it right now let's face it - - -01:22:52.960 --> 01:23:06.000 align:start position:0% -let's come to it right now let's face it -head-on<01:23:03.960> here's<01:23:04.960> our<01:23:05.170> equation<01:23:05.739> of<01:23:05.860> energy - -01:23:06.000 --> 01:23:06.010 align:start position:0% -head-on here's our equation of energy - - -01:23:06.010 --> 01:23:09.419 align:start position:0% -head-on here's our equation of energy -momentum<01:23:06.310> conservation<01:23:06.880> the<01:23:07.780> only<01:23:08.190> team<01:23:09.190> you - -01:23:09.419 --> 01:23:09.429 align:start position:0% -momentum conservation the only team you - - -01:23:09.429 --> 01:23:14.130 align:start position:0% -momentum conservation the only team you -knew<01:23:10.350> is<01:23:11.350> equal<01:23:11.679> to<01:23:11.830> zero<01:23:12.090> this<01:23:13.090> is<01:23:13.150> not<01:23:13.719> the - -01:23:14.130 --> 01:23:14.140 align:start position:0% -knew is equal to zero this is not the - - -01:23:14.140 --> 01:23:20.520 align:start position:0% -knew is equal to zero this is not the -same<01:23:14.410> thing<01:23:14.800> as<01:23:15.810> d<01:23:16.810> mu<01:23:17.850> T<01:23:18.850> mu<01:23:19.030> nu<01:23:19.090> is<01:23:20.020> equal<01:23:20.230> to - -01:23:20.520 --> 01:23:20.530 align:start position:0% -same thing as d mu T mu nu is equal to - - -01:23:20.530 --> 01:23:25.770 align:start position:0% -same thing as d mu T mu nu is equal to -zero<01:23:22.350> this<01:23:23.350> equation<01:23:23.460> is<01:23:24.460> the<01:23:25.179> ordinary - -01:23:25.770 --> 01:23:25.780 align:start position:0% -zero this equation is the ordinary - - -01:23:25.780 --> 01:23:28.410 align:start position:0% -zero this equation is the ordinary -continuity<01:23:26.650> equation<01:23:27.190> and<01:23:27.460> it's<01:23:28.300> the - -01:23:28.410 --> 01:23:28.420 align:start position:0% -continuity equation and it's the - - -01:23:28.420 --> 01:23:32.520 align:start position:0% -continuity equation and it's the -equation<01:23:28.960> which<01:23:29.260> expresses<01:23:29.920> the<01:23:30.160> idea<01:23:31.530> that - -01:23:32.520 --> 01:23:32.530 align:start position:0% -equation which expresses the idea that - - -01:23:32.530 --> 01:23:35.669 align:start position:0% -equation which expresses the idea that -if<01:23:32.710> you<01:23:32.800> have<01:23:32.830> a<01:23:33.070> region<01:23:33.340> space<01:23:33.940> the<01:23:34.870> change<01:23:35.290> of - -01:23:35.669 --> 01:23:35.679 align:start position:0% -if you have a region space the change of - - -01:23:35.679 --> 01:23:38.820 align:start position:0% -if you have a region space the change of -quantity<01:23:36.370> for<01:23:37.330> example<01:23:37.390> charge<01:23:38.110> if<01:23:38.500> this<01:23:38.679> were - -01:23:38.820 --> 01:23:38.830 align:start position:0% -quantity for example charge if this were - - -01:23:38.830 --> 01:23:45.120 align:start position:0% -quantity for example charge if this were -d<01:23:39.370> mu<01:23:40.350> J<01:23:41.350> mu<01:23:41.380> is<01:23:42.040> equal<01:23:42.219> to<01:23:42.429> zero<01:23:43.710> this<01:23:44.710> kind<01:23:45.040> of - -01:23:45.120 --> 01:23:45.130 align:start position:0% -d mu J mu is equal to zero this kind of - - -01:23:45.130 --> 01:23:49.590 align:start position:0% -d mu J mu is equal to zero this kind of -equation<01:23:45.670> here<01:23:46.560> expresses<01:23:47.560> the<01:23:47.710> idea<01:23:48.090> that<01:23:49.090> if - -01:23:49.590 --> 01:23:49.600 align:start position:0% -equation here expresses the idea that if - - -01:23:49.600 --> 01:23:52.680 align:start position:0% -equation here expresses the idea that if -the<01:23:49.870> charge<01:23:50.170> changes<01:23:50.860> inside<01:23:51.400> a<01:23:51.429> region<01:23:51.760> then - -01:23:52.680 --> 01:23:52.690 align:start position:0% -the charge changes inside a region then - - -01:23:52.690 --> 01:23:54.450 align:start position:0% -the charge changes inside a region then -the<01:23:52.840> only<01:23:53.050> way<01:23:53.170> it<01:23:53.290> can<01:23:53.440> do<01:23:53.620> so<01:23:53.860> is<01:23:54.040> by<01:23:54.160> flowing - -01:23:54.450 --> 01:23:54.460 align:start position:0% -the only way it can do so is by flowing - - -01:23:54.460 --> 01:23:57.810 align:start position:0% -the only way it can do so is by flowing -outside<01:23:55.710> this<01:23:56.710> equation<01:23:57.219> is<01:23:57.310> different<01:23:57.489> that - -01:23:57.810 --> 01:23:57.820 align:start position:0% -outside this equation is different that - - -01:23:57.820 --> 01:24:00.780 align:start position:0% -outside this equation is different that -has<01:23:58.000> an<01:23:58.150> extra<01:23:58.540> term<01:23:58.810> in<01:23:59.050> it<01:23:59.199> there's<01:24:00.190> a<01:24:00.280> D<01:24:00.550> mu - -01:24:00.780 --> 01:24:00.790 align:start position:0% -has an extra term in it there's a D mu - - -01:24:00.790 --> 01:24:06.300 align:start position:0% -has an extra term in it there's a D mu -has<01:24:01.239> ad<01:24:01.390> mu<01:24:01.630> team<01:24:02.110> u<01:24:02.260> nu<01:24:04.710> plus<01:24:05.710> another<01:24:05.949> term - -01:24:06.300 --> 01:24:06.310 align:start position:0% -has ad mu team u nu plus another term - - -01:24:06.310 --> 01:24:09.270 align:start position:0% -has ad mu team u nu plus another term -with<01:24:06.550> some<01:24:06.699> gammas<01:24:07.150> and<01:24:07.330> so<01:24:07.449> forth<01:24:08.280> what's - -01:24:09.270 --> 01:24:09.280 align:start position:0% -with some gammas and so forth what's - - -01:24:09.280 --> 01:24:13.110 align:start position:0% -with some gammas and so forth what's -going<01:24:09.610> on<01:24:09.790> here<01:24:10.390> is<01:24:10.540> that<01:24:11.260> this<01:24:12.219> here<01:24:12.730> is<01:24:12.880> the - -01:24:13.110 --> 01:24:13.120 align:start position:0% -going on here is that this here is the - - -01:24:13.120 --> 01:24:16.110 align:start position:0% -going on here is that this here is the -energy<01:24:13.660> and<01:24:13.810> momentum<01:24:13.989> of<01:24:14.560> everything<01:24:15.400> except - -01:24:16.110 --> 01:24:16.120 align:start position:0% -energy and momentum of everything except - - -01:24:16.120 --> 01:24:18.950 align:start position:0% -energy and momentum of everything except -for<01:24:16.600> the<01:24:17.140> gravitational<01:24:18.040> waves - -01:24:18.950 --> 01:24:18.960 align:start position:0% -for the gravitational waves - - -01:24:18.960 --> 01:24:22.080 align:start position:0% -for the gravitational waves -it's<01:24:19.960> the<01:24:20.170> right-hand<01:24:20.679> source<01:24:21.040> it's<01:24:21.370> a<01:24:21.460> term - -01:24:22.080 --> 01:24:22.090 align:start position:0% -it's the right-hand source it's a term - - -01:24:22.090 --> 01:24:26.490 align:start position:0% -it's the right-hand source it's a term -which<01:24:24.600> expresses<01:24:25.600> the<01:24:25.750> sources<01:24:26.290> of<01:24:26.410> the - -01:24:26.490 --> 01:24:26.500 align:start position:0% -which expresses the sources of the - - -01:24:26.500 --> 01:24:28.919 align:start position:0% -which expresses the sources of the -gravitational<01:24:26.949> field<01:24:27.520> but<01:24:28.360> it<01:24:28.510> does<01:24:28.660> not - -01:24:28.919 --> 01:24:28.929 align:start position:0% -gravitational field but it does not - - -01:24:28.929 --> 01:24:31.950 align:start position:0% -gravitational field but it does not -contain<01:24:29.530> the<01:24:29.890> energy<01:24:30.190> and<01:24:30.730> the<01:24:30.850> momentum<01:24:30.969> in - -01:24:31.950 --> 01:24:31.960 align:start position:0% -contain the energy and the momentum in - - -01:24:31.960 --> 01:24:36.630 align:start position:0% -contain the energy and the momentum in -the<01:24:32.380> gravitational<01:24:33.190> field<01:24:33.489> that<01:24:34.440> and<01:24:35.640> that's - -01:24:36.630 --> 01:24:36.640 align:start position:0% -the gravitational field that and that's - - -01:24:36.640 --> 01:24:39.630 align:start position:0% -the gravitational field that and that's -why<01:24:36.880> by<01:24:37.270> itself<01:24:37.750> you<01:24:38.620> don't<01:24:38.830> have<01:24:39.219> an - -01:24:39.630 --> 01:24:39.640 align:start position:0% -why by itself you don't have an - - -01:24:39.640 --> 01:24:42.420 align:start position:0% -why by itself you don't have an -conservation<01:24:40.480> and<01:24:40.630> general<01:24:40.870> relativity<01:24:41.430> and - -01:24:42.420 --> 01:24:42.430 align:start position:0% -conservation and general relativity and - - -01:24:42.430 --> 01:24:44.370 align:start position:0% -conservation and general relativity and -general<01:24:42.490> relativity<01:24:42.880> if<01:24:43.420> you<01:24:43.540> take<01:24:43.720> only<01:24:44.230> the - -01:24:44.370 --> 01:24:44.380 align:start position:0% -general relativity if you take only the - - -01:24:44.380 --> 01:24:46.080 align:start position:0% -general relativity if you take only the -energy<01:24:44.560> that<01:24:45.220> appears<01:24:45.610> on<01:24:45.790> the<01:24:45.880> right-hand - -01:24:46.080 --> 01:24:46.090 align:start position:0% -energy that appears on the right-hand - - -01:24:46.090 --> 01:24:49.440 align:start position:0% -energy that appears on the right-hand -side<01:24:46.600> of<01:24:46.630> the<01:24:46.870> equation<01:24:47.010> it's<01:24:48.010> not<01:24:48.370> conserved - -01:24:49.440 --> 01:24:49.450 align:start position:0% -side of the equation it's not conserved - - -01:24:49.450 --> 01:24:52.380 align:start position:0% -side of the equation it's not conserved -why<01:24:50.080> because<01:24:50.920> the<01:24:51.070> gravitational<01:24:51.550> field<01:24:52.030> also - -01:24:52.380 --> 01:24:52.390 align:start position:0% -why because the gravitational field also - - -01:24:52.390 --> 01:24:55.800 align:start position:0% -why because the gravitational field also -has<01:24:52.660> energy<01:24:53.290> and<01:24:53.440> momentum<01:24:54.240> it's<01:24:55.240> different - -01:24:55.800 --> 01:24:55.810 align:start position:0% -has energy and momentum it's different - - -01:24:55.810 --> 01:25:00.140 align:start position:0% -has energy and momentum it's different -than<01:24:56.730> electrodynamics<01:24:57.930> in<01:24:58.930> electrodynamics - -01:25:00.140 --> 01:25:00.150 align:start position:0% -than electrodynamics in electrodynamics - - -01:25:00.150 --> 01:25:02.940 align:start position:0% -than electrodynamics in electrodynamics -charge<01:25:01.150> is<01:25:01.630> the<01:25:01.870> source<01:25:02.140> of<01:25:02.350> the<01:25:02.500> electric<01:25:02.860> and - -01:25:02.940 --> 01:25:02.950 align:start position:0% -charge is the source of the electric and - - -01:25:02.950 --> 01:25:05.010 align:start position:0% -charge is the source of the electric and -magnetic<01:25:03.130> fields<01:25:03.910> the<01:25:04.540> electric<01:25:04.900> and - -01:25:05.010 --> 01:25:05.020 align:start position:0% -magnetic fields the electric and - - -01:25:05.020 --> 01:25:07.470 align:start position:0% -magnetic fields the electric and -magnetic<01:25:05.170> fields<01:25:05.740> do<01:25:05.890> not<01:25:05.920> have<01:25:06.430> charge<01:25:06.910> they - -01:25:07.470 --> 01:25:07.480 align:start position:0% -magnetic fields do not have charge they - - -01:25:07.480 --> 01:25:10.830 align:start position:0% -magnetic fields do not have charge they -are<01:25:07.510> completely<01:25:08.050> uncharged<01:25:09.090> so<01:25:10.090> if<01:25:10.330> electric - -01:25:10.830 --> 01:25:10.840 align:start position:0% -are completely uncharged so if electric - - -01:25:10.840 --> 01:25:12.930 align:start position:0% -are completely uncharged so if electric -charge<01:25:11.140> is<01:25:11.350> conserved<01:25:11.530> it's<01:25:12.250> because<01:25:12.640> the - -01:25:12.930 --> 01:25:12.940 align:start position:0% -charge is conserved it's because the - - -01:25:12.940 --> 01:25:14.970 align:start position:0% -charge is conserved it's because the -electric<01:25:13.360> charge<01:25:13.690> of<01:25:13.990> things<01:25:14.170> like<01:25:14.410> electrons - -01:25:14.970 --> 01:25:14.980 align:start position:0% -electric charge of things like electrons - - -01:25:14.980 --> 01:25:16.860 align:start position:0% -electric charge of things like electrons -and<01:25:15.220> protons<01:25:15.340> and<01:25:15.850> neutrons<01:25:16.210> is<01:25:16.420> conserved - -01:25:16.860 --> 01:25:16.870 align:start position:0% -and protons and neutrons is conserved - - -01:25:16.870 --> 01:25:18.960 align:start position:0% -and protons and neutrons is conserved -and<01:25:17.200> you<01:25:17.680> don't<01:25:17.980> have<01:25:18.160> to<01:25:18.310> worry<01:25:18.520> about<01:25:18.730> the - -01:25:18.960 --> 01:25:18.970 align:start position:0% -and you don't have to worry about the - - -01:25:18.970 --> 01:25:21.870 align:start position:0% -and you don't have to worry about the -electric<01:25:19.360> charge<01:25:19.690> of<01:25:20.670> electromagnetic<01:25:21.670> waves - -01:25:21.870 --> 01:25:21.880 align:start position:0% -electric charge of electromagnetic waves - - -01:25:21.880 --> 01:25:25.050 align:start position:0% -electric charge of electromagnetic waves -there<01:25:22.360> is<01:25:22.480> none<01:25:23.070> but<01:25:24.070> in<01:25:24.220> gravitational - -01:25:25.050 --> 01:25:25.060 align:start position:0% -there is none but in gravitational - - -01:25:25.060 --> 01:25:28.680 align:start position:0% -there is none but in gravitational -theory<01:25:25.450> the<01:25:26.190> wave-like<01:25:27.390> disturbances<01:25:28.390> of<01:25:28.450> the - -01:25:28.680 --> 01:25:28.690 align:start position:0% -theory the wave-like disturbances of the - - -01:25:28.690 --> 01:25:31.590 align:start position:0% -theory the wave-like disturbances of the -gravitational<01:25:29.200> field<01:25:29.650> themselves<01:25:30.640> carry - -01:25:31.590 --> 01:25:31.600 align:start position:0% -gravitational field themselves carry - - -01:25:31.600 --> 01:25:34.500 align:start position:0% -gravitational field themselves carry -energy<01:25:32.020> and<01:25:32.440> momentum<01:25:32.610> and<01:25:33.610> therefore<01:25:34.090> energy - -01:25:34.500 --> 01:25:34.510 align:start position:0% -energy and momentum and therefore energy - - -01:25:34.510 --> 01:25:37.230 align:start position:0% -energy and momentum and therefore energy -and<01:25:34.630> momentum<01:25:34.750> is<01:25:35.110> more<01:25:35.260> complicated<01:25:36.100> if<01:25:36.730> some - -01:25:37.230 --> 01:25:37.240 align:start position:0% -and momentum is more complicated if some - - -01:25:37.240 --> 01:25:39.480 align:start position:0% -and momentum is more complicated if some -of<01:25:37.360> it<01:25:37.450> comes<01:25:37.750> from<01:25:37.900> ordinary<01:25:38.290> stuff<01:25:38.800> and<01:25:39.160> some - -01:25:39.480 --> 01:25:39.490 align:start position:0% -of it comes from ordinary stuff and some - - -01:25:39.490 --> 01:25:40.740 align:start position:0% -of it comes from ordinary stuff and some -of<01:25:39.520> it<01:25:39.730> comes<01:25:39.970> from<01:25:40.060> these<01:25:40.300> gravitational - -01:25:40.740 --> 01:25:40.750 align:start position:0% -of it comes from these gravitational - - -01:25:40.750 --> 01:25:45.430 align:start position:0% -of it comes from these gravitational -waves<01:25:41.020> we'll<01:25:41.320> come<01:25:41.470> back<01:25:41.650> to<01:25:41.830> this - -01:25:45.430 --> 01:25:45.440 align:start position:0% - - - -01:25:45.440 --> 01:25:47.580 align:start position:0% - -regarding<01:25:46.190> the<01:25:46.250> gravitational<01:25:46.940> waves<01:25:47.150> that - -01:25:47.580 --> 01:25:47.590 align:start position:0% -regarding the gravitational waves that - - -01:25:47.590 --> 01:25:51.700 align:start position:0% -regarding the gravitational waves that -this<01:25:48.590> definitely<01:25:48.949> pertain<01:25:49.219> to<01:25:49.250> let's<01:25:50.150> see<01:25:50.710> the - -01:25:51.700 --> 01:25:51.710 align:start position:0% -this definitely pertain to let's see the - - -01:25:51.710 --> 01:25:53.620 align:start position:0% -this definitely pertain to let's see the -universe<01:25:52.130> it<01:25:52.280> has<01:25:52.429> no<01:25:52.610> master<01:25:53.060> intervene<01:25:53.480> it - -01:25:53.620 --> 01:25:53.630 align:start position:0% -universe it has no master intervene it - - -01:25:53.630 --> 01:25:56.320 align:start position:0% -universe it has no master intervene it -somehow<01:25:54.020> has<01:25:54.639> gravitational<01:25:55.639> ways<01:25:55.909> and - -01:25:56.320 --> 01:25:56.330 align:start position:0% -somehow has gravitational ways and - - -01:25:56.330 --> 01:25:59.620 align:start position:0% -somehow has gravitational ways and -therefore<01:25:56.719> a<01:25:57.380> universe<01:25:57.949> that<01:25:57.980> has<01:25:58.429> no<01:25:58.790> nothing - -01:25:59.620 --> 01:25:59.630 align:start position:0% -therefore a universe that has no nothing - - -01:25:59.630 --> 01:26:02.979 align:start position:0% -therefore a universe that has no nothing -but<01:25:59.900> gravitational<01:26:00.770> waves<01:26:01.179> has<01:26:02.179> no<01:26:02.480> team<01:26:02.810> you - -01:26:02.979 --> 01:26:02.989 align:start position:0% -but gravitational waves has no team you - - -01:26:02.989 --> 01:26:07.419 align:start position:0% -but gravitational waves has no team you -knew<01:26:03.170> on<01:26:03.380> the<01:26:03.409> right<01:26:03.620> hand<01:26:03.920> side<01:26:05.860> it<01:26:06.860> does<01:26:07.130> have - -01:26:07.419 --> 01:26:07.429 align:start position:0% -knew on the right hand side it does have - - -01:26:07.429 --> 01:26:10.209 align:start position:0% -knew on the right hand side it does have -these<01:26:07.670> wave-like<01:26:07.960> disturbances<01:26:09.219> the - -01:26:10.209 --> 01:26:10.219 align:start position:0% -these wave-like disturbances the - - -01:26:10.219 --> 01:26:12.459 align:start position:0% -these wave-like disturbances the -wave-like<01:26:10.489> disturbances<01:26:11.420> do<01:26:11.690> have<01:26:11.960> energy - -01:26:12.459 --> 01:26:12.469 align:start position:0% -wave-like disturbances do have energy - - -01:26:12.469 --> 01:26:15.250 align:start position:0% -wave-like disturbances do have energy -and<01:26:12.500> momentum<01:26:12.770> so<01:26:13.550> effectively<01:26:14.360> this<01:26:14.659> cannot - -01:26:15.250 --> 01:26:15.260 align:start position:0% -and momentum so effectively this cannot - - -01:26:15.260 --> 01:26:18.129 align:start position:0% -and momentum so effectively this cannot -be<01:26:15.590> the<01:26:15.620> whole<01:26:16.010> energy<01:26:16.340> in<01:26:16.670> momentum<01:26:17.139> right - -01:26:18.129 --> 01:26:18.139 align:start position:0% -be the whole energy in momentum right - - -01:26:18.139 --> 01:26:20.350 align:start position:0% -be the whole energy in momentum right -Einstein<01:26:18.380> understood<01:26:19.040> that<01:26:19.219> there's<01:26:20.000> another - -01:26:20.350 --> 01:26:20.360 align:start position:0% -Einstein understood that there's another - - -01:26:20.360 --> 01:26:22.959 align:start position:0% -Einstein understood that there's another -thing<01:26:20.600> you<01:26:20.750> can<01:26:20.929> write<01:26:21.139> down<01:26:21.170> which<01:26:22.159> is<01:26:22.400> the - -01:26:22.959 --> 01:26:22.969 align:start position:0% -thing you can write down which is the - - -01:26:22.969 --> 01:26:26.500 align:start position:0% -thing you can write down which is the -total<01:26:23.360> energy<01:26:23.840> and<01:26:23.960> momentum<01:26:24.820> but<01:26:25.820> the<01:26:26.270> thing - -01:26:26.500 --> 01:26:26.510 align:start position:0% -total energy and momentum but the thing - - -01:26:26.510 --> 01:26:28.540 align:start position:0% -total energy and momentum but the thing -that<01:26:26.540> goes<01:26:26.870> on<01:26:27.110> the<01:26:27.199> right-hand<01:26:27.620> side<01:26:27.889> is<01:26:28.190> not - -01:26:28.540 --> 01:26:28.550 align:start position:0% -that goes on the right-hand side is not - - -01:26:28.550 --> 01:26:31.720 align:start position:0% -that goes on the right-hand side is not -the<01:26:29.030> whole<01:26:29.150> total<01:26:29.480> energy<01:26:29.780> momentum<01:26:30.730> it's - -01:26:31.720 --> 01:26:31.730 align:start position:0% -the whole total energy momentum it's - - -01:26:31.730 --> 01:26:34.959 align:start position:0% -the whole total energy momentum it's -almost<01:26:32.239> as<01:26:32.510> if<01:26:32.540> it's<01:26:32.840> analogous<01:26:33.469> to<01:26:33.969> the - -01:26:34.959 --> 01:26:34.969 align:start position:0% -almost as if it's analogous to the - - -01:26:34.969 --> 01:26:38.740 align:start position:0% -almost as if it's analogous to the -situation<01:26:35.630> which<01:26:36.230> would<01:26:36.469> exist<01:26:37.130> if<01:26:37.750> photons - -01:26:38.740 --> 01:26:38.750 align:start position:0% -situation which would exist if photons - - -01:26:38.750 --> 01:26:41.169 align:start position:0% -situation which would exist if photons -are<01:26:39.230> electric<01:26:40.040> and<01:26:40.190> magnetic<01:26:40.340> fields<01:26:41.000> had - -01:26:41.169 --> 01:26:41.179 align:start position:0% -are electric and magnetic fields had - - -01:26:41.179 --> 01:26:45.550 align:start position:0% -are electric and magnetic fields had -charge<01:26:42.400> then<01:26:43.400> the<01:26:44.330> charge<01:26:44.659> of<01:26:44.960> ordinary - -01:26:45.550 --> 01:26:45.560 align:start position:0% -charge then the charge of ordinary - - -01:26:45.560 --> 01:26:49.060 align:start position:0% -charge then the charge of ordinary -charges<01:26:46.219> ordinary<01:26:46.909> charges<01:26:47.710> ordinary<01:26:48.710> I<01:26:48.800> now - -01:26:49.060 --> 01:26:49.070 align:start position:0% -charges ordinary charges ordinary I now - - -01:26:49.070 --> 01:26:52.390 align:start position:0% -charges ordinary charges ordinary I now -mean<01:26:49.310> protons<01:26:50.090> electrons<01:26:50.540> and<01:26:51.290> so<01:26:51.320> forth<01:26:51.679> they - -01:26:52.390 --> 01:26:52.400 align:start position:0% -mean protons electrons and so forth they - - -01:26:52.400 --> 01:26:53.709 align:start position:0% -mean protons electrons and so forth they -would<01:26:52.639> be<01:26:52.760> the<01:26:52.940> sources<01:26:53.420> of<01:26:53.449> the - -01:26:53.709 --> 01:26:53.719 align:start position:0% -would be the sources of the - - -01:26:53.719 --> 01:26:55.709 align:start position:0% -would be the sources of the -electromagnetic<01:26:54.380> field - -01:26:55.709 --> 01:26:55.719 align:start position:0% -electromagnetic field - - -01:26:55.719 --> 01:26:59.080 align:start position:0% -electromagnetic field -but<01:26:56.719> the<01:26:56.929> electromagnetic<01:26:57.620> field<01:26:58.040> when<01:26:58.639> it - -01:26:59.080 --> 01:26:59.090 align:start position:0% -but the electromagnetic field when it - - -01:26:59.090 --> 01:27:01.899 align:start position:0% -but the electromagnetic field when it -was<01:26:59.690> radiated<01:27:00.290> off<01:27:00.469> would<01:27:00.830> carry<01:27:01.159> off<01:27:01.460> charge - -01:27:01.899 --> 01:27:01.909 align:start position:0% -was radiated off would carry off charge - - -01:27:01.909 --> 01:27:05.320 align:start position:0% -was radiated off would carry off charge -if<01:27:02.330> photons<01:27:03.139> were<01:27:03.350> charged<01:27:03.860> and<01:27:04.159> so<01:27:05.120> the - -01:27:05.320 --> 01:27:05.330 align:start position:0% -if photons were charged and so the - - -01:27:05.330 --> 01:27:07.780 align:start position:0% -if photons were charged and so the -electrons<01:27:05.870> plus<01:27:06.170> the<01:27:06.380> protons<01:27:06.830> by<01:27:07.070> themselves - -01:27:07.780 --> 01:27:07.790 align:start position:0% -electrons plus the protons by themselves - - -01:27:07.790 --> 01:27:11.050 align:start position:0% -electrons plus the protons by themselves -would<01:27:08.150> not<01:27:08.389> have<01:27:08.690> a<01:27:08.719> conserved<01:27:09.260> charge<01:27:10.060> in - -01:27:11.050 --> 01:27:11.060 align:start position:0% -would not have a conserved charge in - - -01:27:11.060 --> 01:27:13.540 align:start position:0% -would not have a conserved charge in -fact<01:27:11.810> you<01:27:11.900> could<01:27:12.020> even<01:27:12.080> say<01:27:12.380> more<01:27:12.620> you<01:27:13.370> could - -01:27:13.540 --> 01:27:13.550 align:start position:0% -fact you could even say more you could - - -01:27:13.550 --> 01:27:15.879 align:start position:0% -fact you could even say more you could -even<01:27:13.699> say<01:27:14.030> well<01:27:14.780> if<01:27:15.080> the<01:27:15.199> photons<01:27:15.679> themselves - -01:27:15.879 --> 01:27:15.889 align:start position:0% -even say well if the photons themselves - - -01:27:15.889 --> 01:27:18.760 align:start position:0% -even say well if the photons themselves -are<01:27:16.340> charged<01:27:16.880> they<01:27:17.449> are<01:27:17.690> sources<01:27:18.320> of<01:27:18.350> their - -01:27:18.760 --> 01:27:18.770 align:start position:0% -are charged they are sources of their - - -01:27:18.770 --> 01:27:22.479 align:start position:0% -are charged they are sources of their -own<01:27:18.980> cells<01:27:19.900> when<01:27:20.900> a<01:27:20.929> photon<01:27:21.440> is<01:27:21.679> accelerated - -01:27:22.479 --> 01:27:22.489 align:start position:0% -own cells when a photon is accelerated - - -01:27:22.489 --> 01:27:27.820 align:start position:0% -own cells when a photon is accelerated -it<01:27:23.360> could<01:27:24.550> bounce<01:27:25.550> off<01:27:25.850> a<01:27:25.880> mirror<01:27:26.469> it<01:27:27.469> should - -01:27:27.820 --> 01:27:27.830 align:start position:0% -it could bounce off a mirror it should - - -01:27:27.830 --> 01:27:30.850 align:start position:0% -it could bounce off a mirror it should -emit<01:27:28.550> some<01:27:28.790> more<01:27:28.909> photons<01:27:29.590> that's<01:27:30.590> the<01:27:30.770> way - -01:27:30.850 --> 01:27:30.860 align:start position:0% -emit some more photons that's the way - - -01:27:30.860 --> 01:27:35.790 align:start position:0% -emit some more photons that's the way -gravity<01:27:31.130> is<01:27:31.929> gravity<01:27:33.940> gravity<01:27:34.940> is<01:27:35.120> a<01:27:35.150> field - -01:27:35.790 --> 01:27:35.800 align:start position:0% -gravity is gravity gravity is a field - - -01:27:35.800 --> 01:27:39.100 align:start position:0% -gravity is gravity gravity is a field -which<01:27:36.800> is<01:27:37.010> sourced<01:27:37.610> by<01:27:37.820> energy<01:27:38.239> momentum<01:27:38.389> but - -01:27:39.100 --> 01:27:39.110 align:start position:0% -which is sourced by energy momentum but - - -01:27:39.110 --> 01:27:41.500 align:start position:0% -which is sourced by energy momentum but -it<01:27:39.170> also<01:27:39.409> carries<01:27:39.830> energy<01:27:40.489> momentum<01:27:40.639> not<01:27:41.179> not - -01:27:41.500 --> 01:27:41.510 align:start position:0% -it also carries energy momentum not not - - -01:27:41.510 --> 01:27:42.939 align:start position:0% -it also carries energy momentum not not -on<01:27:41.750> the<01:27:41.870> right-hand<01:27:42.199> side<01:27:42.290> of<01:27:42.530> this<01:27:42.620> equation - -01:27:42.939 --> 01:27:42.949 align:start position:0% -on the right-hand side of this equation - - -01:27:42.949 --> 01:27:47.680 align:start position:0% -on the right-hand side of this equation -but<01:27:44.350> are<01:27:45.350> the<01:27:45.620> pieces<01:27:45.860> of<01:27:46.010> energy<01:27:46.219> momentum<01:27:46.690> so - -01:27:47.680 --> 01:27:47.690 align:start position:0% -but are the pieces of energy momentum so - - -01:27:47.690 --> 01:27:52.209 align:start position:0% -but are the pieces of energy momentum so -that<01:27:47.719> when<01:27:48.170> gravitons<01:27:49.210> are<01:27:50.350> accelerated<01:27:51.350> they - -01:27:52.209 --> 01:27:52.219 align:start position:0% -that when gravitons are accelerated they - - -01:27:52.219 --> 01:27:55.629 align:start position:0% -that when gravitons are accelerated they -will<01:27:52.820> emit<01:27:53.210> other<01:27:53.420> gravitons<01:27:54.219> so<01:27:55.219> it's<01:27:55.400> a<01:27:55.460> more - -01:27:55.629 --> 01:27:55.639 align:start position:0% -will emit other gravitons so it's a more - - -01:27:55.639 --> 01:27:58.609 align:start position:0% -will emit other gravitons so it's a more -complicated<01:27:56.030> kind<01:27:56.600> of<01:27:56.780> store<01:27:57.020> it's<01:27:57.139> nonlinear - -01:27:58.609 --> 01:27:58.619 align:start position:0% -complicated kind of store it's nonlinear - - -01:27:58.619 --> 01:28:01.910 align:start position:0% -complicated kind of store it's nonlinear -all<01:27:59.460> right<01:27:59.580> we're<01:27:59.730> coming<01:27:59.820> will<01:28:00.920> little - -01:28:01.910 --> 01:28:01.920 align:start position:0% -all right we're coming will little - - -01:28:01.920 --> 01:28:04.760 align:start position:0% -all right we're coming will little -discuss<01:28:02.340> this<01:28:02.489> again<01:28:03.330> but<01:28:04.260> let's<01:28:04.440> see<01:28:04.590> now - -01:28:04.760 --> 01:28:04.770 align:start position:0% -discuss this again but let's see now - - -01:28:04.770 --> 01:28:07.609 align:start position:0% -discuss this again but let's see now -what<01:28:05.369> we<01:28:05.489> can<01:28:05.639> learn<01:28:05.790> we<01:28:06.780> have<01:28:06.810> a<01:28:06.989> guests<01:28:07.469> now - -01:28:07.609 --> 01:28:07.619 align:start position:0% -what we can learn we have a guests now - - -01:28:07.619 --> 01:28:09.320 align:start position:0% -what we can learn we have a guests now -this<01:28:08.099> was<01:28:08.250> Einsteins<01:28:08.699> guests<01:28:09.119> for<01:28:09.270> the - -01:28:09.320 --> 01:28:09.330 align:start position:0% -this was Einsteins guests for the - - -01:28:09.330 --> 01:28:12.410 align:start position:0% -this was Einsteins guests for the -gravitational<01:28:09.750> field<01:28:10.170> equations<01:28:11.420> the - -01:28:12.410 --> 01:28:12.420 align:start position:0% -gravitational field equations the - - -01:28:12.420 --> 01:28:14.899 align:start position:0% -gravitational field equations the -hardest<01:28:12.840> part<01:28:13.139> of<01:28:13.320> this<01:28:13.440> little<01:28:13.739> exercise<01:28:13.909> is - -01:28:14.899 --> 01:28:14.909 align:start position:0% -hardest part of this little exercise is - - -01:28:14.909 --> 01:28:18.859 align:start position:0% -hardest part of this little exercise is -to<01:28:15.540> show<01:28:16.730> this<01:28:17.730> identity<01:28:18.389> here - -01:28:18.859 --> 01:28:18.869 align:start position:0% -to show this identity here - - -01:28:18.869 --> 01:28:22.669 align:start position:0% -to show this identity here -I<01:28:18.900> know<01:28:19.830> I<01:28:20.550> know<01:28:20.909> two<01:28:21.210> ways<01:28:21.719> to<01:28:21.960> do<01:28:22.110> it<01:28:22.139> one<01:28:22.530> of - -01:28:22.669 --> 01:28:22.679 align:start position:0% -I know I know two ways to do it one of - - -01:28:22.679 --> 01:28:24.919 align:start position:0% -I know I know two ways to do it one of -them<01:28:22.830> one<01:28:23.040> of<01:28:23.070> which<01:28:23.340> is<01:28:23.610> a<01:28:23.639> one-line<01:28:24.449> proof - -01:28:24.919 --> 01:28:24.929 align:start position:0% -them one of which is a one-line proof - - -01:28:24.929 --> 01:28:26.660 align:start position:0% -them one of which is a one-line proof -but<01:28:25.260> it<01:28:25.349> requires<01:28:25.770> a<01:28:25.920> very<01:28:26.159> sophisticated - -01:28:26.660 --> 01:28:26.670 align:start position:0% -but it requires a very sophisticated - - -01:28:26.670 --> 01:28:29.540 align:start position:0% -but it requires a very sophisticated -stuff<01:28:27.389> which<01:28:27.869> we<01:28:28.050> don't<01:28:28.170> have<01:28:28.290> and<01:28:28.830> the<01:28:29.310> other - -01:28:29.540 --> 01:28:29.550 align:start position:0% -stuff which we don't have and the other - - -01:28:29.550 --> 01:28:34.450 align:start position:0% -stuff which we don't have and the other -is<01:28:29.880> a<01:28:30.780> mini<01:28:31.110> line<01:28:31.380> proof<01:28:31.860> which<01:28:32.340> is<01:28:33.080> mechanical - -01:28:34.450 --> 01:28:34.460 align:start position:0% -is a mini line proof which is mechanical - - -01:28:34.460 --> 01:28:38.209 align:start position:0% -is a mini line proof which is mechanical -straightforward<01:28:35.460> I<01:28:36.469> won't<01:28:37.469> say<01:28:37.650> mindless - -01:28:38.209 --> 01:28:38.219 align:start position:0% -straightforward I won't say mindless - - -01:28:38.219 --> 01:28:43.430 align:start position:0% -straightforward I won't say mindless -that's<01:28:38.820> such<01:28:39.150> a<01:28:39.349> right<01:28:41.750> but<01:28:42.750> in<01:28:42.960> which<01:28:43.290> you - -01:28:43.430 --> 01:28:43.440 align:start position:0% -that's such a right but in which you - - -01:28:43.440 --> 01:28:47.479 align:start position:0% -that's such a right but in which you -simply<01:28:43.980> plug<01:28:44.489> in<01:28:45.530> for<01:28:46.530> the<01:28:46.560> components<01:28:47.340> of - -01:28:47.479 --> 01:28:47.489 align:start position:0% -simply plug in for the components of - - -01:28:47.489 --> 01:28:49.550 align:start position:0% -simply plug in for the components of -their<01:28:47.610> tensors<01:28:48.119> their<01:28:48.510> expression<01:28:49.170> in<01:28:49.320> terms - -01:28:49.550 --> 01:28:49.560 align:start position:0% -their tensors their expression in terms - - -01:28:49.560 --> 01:28:51.919 align:start position:0% -their tensors their expression in terms -of<01:28:49.710> the<01:28:49.830> Christoffel<01:28:50.429> symbols<01:28:50.940> and<01:28:51.239> just - -01:28:51.919 --> 01:28:51.929 align:start position:0% -of the Christoffel symbols and just - - -01:28:51.929 --> 01:28:55.450 align:start position:0% -of the Christoffel symbols and just -check<01:28:52.260> out<01:28:52.469> that<01:28:52.679> this<01:28:52.830> relationship<01:28:53.340> is<01:28:53.580> true - -01:28:55.450 --> 01:28:55.460 align:start position:0% -check out that this relationship is true - - -01:28:55.460 --> 01:28:57.919 align:start position:0% -check out that this relationship is true -only<01:28:56.460> way<01:28:56.580> I<01:28:56.610> know<01:28:56.760> how<01:28:56.880> to<01:28:57.000> do<01:28:57.210> it<01:28:57.360> other<01:28:57.690> than - -01:28:57.919 --> 01:28:57.929 align:start position:0% -only way I know how to do it other than - - -01:28:57.929 --> 01:29:03.589 align:start position:0% -only way I know how to do it other than -to<01:28:58.110> use<01:28:58.290> a<01:28:58.320> fancy<01:28:59.119> sophisticated<01:29:01.219> argument<01:29:02.599> I - -01:29:03.589 --> 01:29:03.599 align:start position:0% -to use a fancy sophisticated argument I - - -01:29:03.599 --> 01:29:08.810 align:start position:0% -to use a fancy sophisticated argument I -started<01:29:04.530> to<01:29:04.619> do<01:29:04.860> it<01:29:05.040> and<01:29:05.989> by<01:29:06.989> the<01:29:07.020> time<01:29:07.199> I<01:29:07.820> this - -01:29:08.810 --> 01:29:08.820 align:start position:0% -started to do it and by the time I this - - -01:29:08.820 --> 01:29:12.229 align:start position:0% -started to do it and by the time I this -is<01:29:08.969> when<01:29:09.119> I<01:29:09.150> fell<01:29:09.360> asleep<01:29:10.760> and<01:29:11.760> I<01:29:11.940> can<01:29:12.119> assure - -01:29:12.229 --> 01:29:12.239 align:start position:0% -is when I fell asleep and I can assure - - -01:29:12.239 --> 01:29:14.089 align:start position:0% -is when I fell asleep and I can assure -you<01:29:12.449> if<01:29:12.719> I<01:29:12.840> were<01:29:12.960> to<01:29:13.080> do<01:29:13.199> it<01:29:13.320> on<01:29:13.440> the<01:29:13.560> blackboard - -01:29:14.089 --> 01:29:14.099 align:start position:0% -you if I were to do it on the blackboard - - -01:29:14.099 --> 01:29:17.330 align:start position:0% -you if I were to do it on the blackboard -you<01:29:14.400> would<01:29:14.550> fall<01:29:14.760> asleep<01:29:15.150> so<01:29:16.050> I<01:29:16.080> won't<01:29:16.340> but - -01:29:17.330 --> 01:29:17.340 align:start position:0% -you would fall asleep so I won't but - - -01:29:17.340 --> 01:29:19.490 align:start position:0% -you would fall asleep so I won't but -it's<01:29:17.460> an<01:29:17.909> exercise<01:29:18.330> you<01:29:18.360> do<01:29:18.750> yourself<01:29:18.869> look<01:29:19.349> it - -01:29:19.490 --> 01:29:19.500 align:start position:0% -it's an exercise you do yourself look it - - -01:29:19.500 --> 01:29:24.859 align:start position:0% -it's an exercise you do yourself look it -up<01:29:19.530> or<01:29:19.770> just<01:29:19.949> believe<01:29:20.219> it<01:29:21.590> okay<01:29:23.659> alright<01:29:24.659> so - -01:29:24.859 --> 01:29:24.869 align:start position:0% -up or just believe it okay alright so - - -01:29:24.869 --> 01:29:29.790 align:start position:0% -up or just believe it okay alright so -now<01:29:24.989> let's<01:29:25.020> see<01:29:25.380> what<01:29:25.560> more<01:29:25.770> we<01:29:25.980> can<01:29:26.159> learn - -01:29:29.790 --> 01:29:29.800 align:start position:0% - - - -01:29:29.800 --> 01:29:33.879 align:start position:0% - -yes<01:29:30.800> variational<01:29:31.550> principle<01:29:32.500> I'll<01:29:33.500> tell<01:29:33.770> you - -01:29:33.879 --> 01:29:33.889 align:start position:0% -yes variational principle I'll tell you - - -01:29:33.889 --> 01:29:35.050 align:start position:0% -yes variational principle I'll tell you -all<01:29:34.190> right<01:29:34.219> if<01:29:34.369> you<01:29:34.429> want<01:29:34.610> to<01:29:34.670> know<01:29:34.760> I'll<01:29:34.909> tell - -01:29:35.050 --> 01:29:35.060 align:start position:0% -all right if you want to know I'll tell - - -01:29:35.060 --> 01:29:36.729 align:start position:0% -all right if you want to know I'll tell -you<01:29:35.179> what<01:29:35.329> comes<01:29:35.570> from<01:29:35.690> there<01:29:35.929> is<01:29:36.110> an<01:29:36.320> action - -01:29:36.729 --> 01:29:36.739 align:start position:0% -you what comes from there is an action - - -01:29:36.739 --> 01:29:38.439 align:start position:0% -you what comes from there is an action -formulation<01:29:37.550> of<01:29:37.639> the<01:29:37.730> general<01:29:38.060> theory<01:29:38.420> of - -01:29:38.439 --> 01:29:38.449 align:start position:0% -formulation of the general theory of - - -01:29:38.449 --> 01:29:42.149 align:start position:0% -formulation of the general theory of -relativity<01:29:38.860> maybe<01:29:39.860> we'll<01:29:40.070> get<01:29:40.219> to<01:29:40.400> it<01:29:40.550> and - -01:29:42.149 --> 01:29:42.159 align:start position:0% -relativity maybe we'll get to it and - - -01:29:42.159 --> 01:29:45.580 align:start position:0% -relativity maybe we'll get to it and -just<01:29:43.159> what<01:29:44.060> it<01:29:44.150> says<01:29:44.360> is<01:29:44.570> the<01:29:44.750> action<01:29:45.290> is - -01:29:45.580 --> 01:29:45.590 align:start position:0% -just what it says is the action is - - -01:29:45.590 --> 01:29:47.890 align:start position:0% -just what it says is the action is -stationary<01:29:46.250> with<01:29:46.520> respect<01:29:46.550> to<01:29:47.090> variation<01:29:47.480> of - -01:29:47.890 --> 01:29:47.900 align:start position:0% -stationary with respect to variation of - - -01:29:47.900 --> 01:29:52.060 align:start position:0% -stationary with respect to variation of -all<01:29:48.050> the<01:29:48.199> fields<01:29:48.790> in<01:29:49.790> particular<01:29:51.070> stationary - -01:29:52.060 --> 01:29:52.070 align:start position:0% -all the fields in particular stationary - - -01:29:52.070 --> 01:29:54.939 align:start position:0% -all the fields in particular stationary -with<01:29:52.310> respect<01:29:52.369> to<01:29:53.030> the<01:29:53.179> variation<01:29:53.869> of<01:29:54.079> G<01:29:54.710> mu<01:29:54.889> nu - -01:29:54.939 --> 01:29:54.949 align:start position:0% -with respect to the variation of G mu nu - - -01:29:54.949 --> 01:29:58.600 align:start position:0% -with respect to the variation of G mu nu -itself<01:29:55.570> and<01:29:56.570> it<01:29:56.719> turns<01:29:56.929> out<01:29:57.110> that<01:29:57.409> the<01:29:57.800> that - -01:29:58.600 --> 01:29:58.610 align:start position:0% -itself and it turns out that the that - - -01:29:58.610 --> 01:30:01.089 align:start position:0% -itself and it turns out that the that -the<01:29:58.820> variation<01:29:59.449> of<01:29:59.719> the<01:30:00.469> action<01:30:00.500> with<01:30:00.949> respect - -01:30:01.089 --> 01:30:01.099 align:start position:0% -the variation of the action with respect - - -01:30:01.099 --> 01:30:04.270 align:start position:0% -the variation of the action with respect -to<01:30:01.400> G<01:30:01.579> mu<01:30:01.760> nu<01:30:01.790> is<01:30:02.780> basically<01:30:03.679> the<01:30:03.860> Einstein - -01:30:04.270 --> 01:30:04.280 align:start position:0% -to G mu nu is basically the Einstein - - -01:30:04.280 --> 01:30:06.490 align:start position:0% -to G mu nu is basically the Einstein -tensor<01:30:04.699> so<01:30:05.210> yeah<01:30:05.599> this<01:30:05.780> is<01:30:05.840> a<01:30:05.869> fancy<01:30:06.170> way<01:30:06.260> to<01:30:06.320> do - -01:30:06.490 --> 01:30:06.500 align:start position:0% -tensor so yeah this is a fancy way to do - - -01:30:06.500 --> 01:30:10.120 align:start position:0% -tensor so yeah this is a fancy way to do -it<01:30:06.650> but<01:30:07.900> there's<01:30:08.900> also<01:30:09.050> a<01:30:09.170> non<01:30:09.440> fancy<01:30:09.949> way<01:30:10.070> to - -01:30:10.120 --> 01:30:10.130 align:start position:0% -it but there's also a non fancy way to - - -01:30:10.130 --> 01:30:18.339 align:start position:0% -it but there's also a non fancy way to -do<01:30:10.310> it<01:30:16.060> nobody<01:30:17.060> that<01:30:17.449> I<01:30:17.540> know<01:30:17.599> of<01:30:17.900> I've<01:30:18.050> asked - -01:30:18.339 --> 01:30:18.349 align:start position:0% -do it nobody that I know of I've asked - - -01:30:18.349 --> 01:30:19.629 align:start position:0% -do it nobody that I know of I've asked -I've<01:30:18.530> asked<01:30:18.739> a<01:30:18.800> lot<01:30:18.889> of<01:30:19.040> people<01:30:19.280> I<01:30:19.340> mean<01:30:19.489> I'm - -01:30:19.629 --> 01:30:19.639 align:start position:0% -I've asked a lot of people I mean I'm - - -01:30:19.639 --> 01:30:22.240 align:start position:0% -I've asked a lot of people I mean I'm -not<01:30:19.790> an<01:30:19.940> expert<01:30:20.150> at<01:30:20.389> general<01:30:20.630> relativity<01:30:21.250> you - -01:30:22.240 --> 01:30:22.250 align:start position:0% -not an expert at general relativity you - - -01:30:22.250 --> 01:30:24.010 align:start position:0% -not an expert at general relativity you -need<01:30:22.369> to<01:30:22.489> say<01:30:22.670> you<01:30:23.449> know<01:30:23.510> I'm<01:30:23.659> not<01:30:23.690> an<01:30:23.929> expert - -01:30:24.010 --> 01:30:24.020 align:start position:0% -need to say you know I'm not an expert - - -01:30:24.020 --> 01:30:25.780 align:start position:0% -need to say you know I'm not an expert -at<01:30:24.409> the<01:30:24.500> technology<01:30:25.190> of<01:30:25.219> solving<01:30:25.699> the - -01:30:25.780 --> 01:30:25.790 align:start position:0% -at the technology of solving the - - -01:30:25.790 --> 01:30:29.919 align:start position:0% -at the technology of solving the -equations<01:30:26.270> and<01:30:26.540> stuff<01:30:28.540> you're<01:30:29.540> people<01:30:29.840> who - -01:30:29.919 --> 01:30:29.929 align:start position:0% -equations and stuff you're people who - - -01:30:29.929 --> 01:30:32.260 align:start position:0% -equations and stuff you're people who -are<01:30:30.020> very<01:30:30.230> very<01:30:30.469> good<01:30:30.739> at<01:30:30.889> it<01:30:31.099> they<01:30:31.880> know<01:30:32.060> a<01:30:32.090> lot - -01:30:32.260 --> 01:30:32.270 align:start position:0% -are very very good at it they know a lot - - -01:30:32.270 --> 01:30:37.089 align:start position:0% -are very very good at it they know a lot -of<01:30:32.329> fancy<01:30:33.550> techniques<01:30:34.550> technology<01:30:35.920> and<01:30:36.920> ask - -01:30:37.089 --> 01:30:37.099 align:start position:0% -of fancy techniques technology and ask - - -01:30:37.099 --> 01:30:39.330 align:start position:0% -of fancy techniques technology and ask -them<01:30:37.250> do<01:30:37.369> you<01:30:37.429> have<01:30:37.520> a<01:30:37.760> simple<01:30:38.360> geometric - -01:30:39.330 --> 01:30:39.340 align:start position:0% -them do you have a simple geometric - - -01:30:39.340 --> 01:30:42.330 align:start position:0% -them do you have a simple geometric -understanding<01:30:40.340> of<01:30:40.460> the<01:30:40.550> Einstein<01:30:41.030> tensor<01:30:41.570> or - -01:30:42.330 --> 01:30:42.340 align:start position:0% -understanding of the Einstein tensor or - - -01:30:42.340 --> 01:30:45.220 align:start position:0% -understanding of the Einstein tensor or -even<01:30:43.340> the<01:30:43.820> Ricci<01:30:44.030> tensor<01:30:44.510> itself<01:30:44.840> and - -01:30:45.220 --> 01:30:45.230 align:start position:0% -even the Ricci tensor itself and - - -01:30:45.230 --> 01:30:47.950 align:start position:0% -even the Ricci tensor itself and -generally<01:30:45.710> speaking<01:30:45.980> the<01:30:46.190> answer<01:30:46.219> is<01:30:46.429> no<01:30:46.960> no - -01:30:47.950 --> 01:30:47.960 align:start position:0% -generally speaking the answer is no no - - -01:30:47.960 --> 01:30:49.689 align:start position:0% -generally speaking the answer is no no -don't<01:30:48.349> say<01:30:48.530> anything<01:30:48.710> specially<01:30:49.190> simple<01:30:49.550> and - -01:30:49.689 --> 01:30:49.699 align:start position:0% -don't say anything specially simple and - - -01:30:49.699 --> 01:30:52.540 align:start position:0% -don't say anything specially simple and -especially<01:30:49.969> geometric<01:30:50.659> about<01:30:50.960> it<01:30:51.340> it<01:30:52.340> just - -01:30:52.540 --> 01:30:52.550 align:start position:0% -especially geometric about it it just - - -01:30:52.550 --> 01:30:54.189 align:start position:0% -especially geometric about it it just -happens<01:30:52.940> to<01:30:53.030> be<01:30:53.119> the<01:30:53.329> thing<01:30:53.540> that<01:30:53.570> satisfies - -01:30:54.189 --> 01:30:54.199 align:start position:0% -happens to be the thing that satisfies - - -01:30:54.199 --> 01:30:58.089 align:start position:0% -happens to be the thing that satisfies -this<01:30:54.380> identity<01:30:54.949> and<01:30:55.960> I<01:30:56.960> have<01:30:57.469> never<01:30:57.710> been<01:30:57.949> able - -01:30:58.089 --> 01:30:58.099 align:start position:0% -this identity and I have never been able - - -01:30:58.099 --> 01:31:03.060 align:start position:0% -this identity and I have never been able -to<01:30:58.300> find<01:30:59.300> anybody<01:30:59.599> who<01:30:59.929> could<01:31:01.719> reduce<01:31:02.719> this - -01:31:03.060 --> 01:31:03.070 align:start position:0% -to find anybody who could reduce this - - -01:31:03.070 --> 01:31:08.589 align:start position:0% -to find anybody who could reduce this -equation<01:31:04.119> to<01:31:05.119> a<01:31:05.150> simple<01:31:05.869> explanation<01:31:07.599> other - -01:31:08.589 --> 01:31:08.599 align:start position:0% -equation to a simple explanation other - - -01:31:08.599 --> 01:31:13.010 align:start position:0% -equation to a simple explanation other -than<01:31:08.840> the<01:31:09.260> fancy<01:31:10.869> France<01:31:11.869> is<01:31:12.020> ever<01:31:12.199> as<01:31:12.440> well - -01:31:13.010 --> 01:31:13.020 align:start position:0% -than the fancy France is ever as well - - -01:31:13.020 --> 01:31:14.390 align:start position:0% -than the fancy France is ever as well -was<01:31:13.170> a<01:31:13.200> similar<01:31:13.770> question<01:31:13.950> about<01:31:14.250> the - -01:31:14.390 --> 01:31:14.400 align:start position:0% -was a similar question about the - - -01:31:14.400 --> 01:31:18.380 align:start position:0% -was a similar question about the -fragrance<01:31:15.300> and<01:31:16.400> friends<01:31:17.400> as<01:31:17.490> everett<01:31:17.820> says - -01:31:18.380 --> 01:31:18.390 align:start position:0% -fragrance and friends as everett says - - -01:31:18.390 --> 01:31:19.820 align:start position:0% -fragrance and friends as everett says -basically<01:31:18.690> the<01:31:18.810> same<01:31:18.990> thing<01:31:19.290> asked<01:31:19.590> about<01:31:19.710> the - -01:31:19.820 --> 01:31:19.830 align:start position:0% -basically the same thing asked about the - - -01:31:19.830 --> 01:31:21.950 align:start position:0% -basically the same thing asked about the -frame<01:31:20.040> dragging<01:31:20.250> terms<01:31:20.670> is<01:31:21.270> there<01:31:21.510> a<01:31:21.540> simple - -01:31:21.950 --> 01:31:21.960 align:start position:0% -frame dragging terms is there a simple - - -01:31:21.960 --> 01:31:23.510 align:start position:0% -frame dragging terms is there a simple -geometric<01:31:22.200> explanation<01:31:22.590> he<01:31:23.040> basically<01:31:23.250> said - -01:31:23.510 --> 01:31:23.520 align:start position:0% -geometric explanation he basically said - - -01:31:23.520 --> 01:31:26.780 align:start position:0% -geometric explanation he basically said -no<01:31:23.900> well<01:31:25.130> think<01:31:26.130> about<01:31:26.220> a<01:31:26.400> little<01:31:26.580> more<01:31:26.760> I - -01:31:26.780 --> 01:31:26.790 align:start position:0% -no well think about a little more I - - -01:31:26.790 --> 01:31:27.800 align:start position:0% -no well think about a little more I -think<01:31:26.910> I<01:31:27.120> could<01:31:27.300> probably<01:31:27.450> go<01:31:27.630> a<01:31:27.660> little - -01:31:27.800 --> 01:31:27.810 align:start position:0% -think I could probably go a little - - -01:31:27.810 --> 01:31:29.770 align:start position:0% -think I could probably go a little -better<01:31:28.050> than<01:31:28.200> that<01:31:28.380> for<01:31:28.530> the<01:31:28.620> frame<01:31:28.830> dragging - -01:31:29.770 --> 01:31:29.780 align:start position:0% -better than that for the frame dragging - - -01:31:29.780 --> 01:31:33.200 align:start position:0% -better than that for the frame dragging -but<01:31:30.780> this<01:31:31.260> very<01:31:31.680> particular<01:31:32.010> equation<01:31:32.760> here - -01:31:33.200 --> 01:31:33.210 align:start position:0% -but this very particular equation here - - -01:31:33.210 --> 01:31:35.180 align:start position:0% -but this very particular equation here -I<01:31:33.240> don't<01:31:33.930> know<01:31:34.020> and<01:31:34.230> Einstein<01:31:34.650> himself<01:31:34.770> just - -01:31:35.180 --> 01:31:35.190 align:start position:0% -I don't know and Einstein himself just - - -01:31:35.190 --> 01:31:36.530 align:start position:0% -I don't know and Einstein himself just -humped<01:31:35.430> it<01:31:35.490> around<01:31:35.760> until<01:31:36.120> you<01:31:36.210> found<01:31:36.420> the - -01:31:36.530 --> 01:31:36.540 align:start position:0% -humped it around until you found the - - -01:31:36.540 --> 01:31:39.260 align:start position:0% -humped it around until you found the -right<01:31:36.660> thing<01:31:36.900> I<01:31:37.620> wish<01:31:38.370> I<01:31:38.520> had<01:31:38.670> a<01:31:38.700> simple - -01:31:39.260 --> 01:31:39.270 align:start position:0% -right thing I wish I had a simple - - -01:31:39.270 --> 01:31:40.910 align:start position:0% -right thing I wish I had a simple -explanation<01:31:39.660> for<01:31:40.170> it<01:31:40.290> I<01:31:40.350> never<01:31:40.530> could<01:31:40.710> find - -01:31:40.910 --> 01:31:40.920 align:start position:0% -explanation for it I never could find - - -01:31:40.920 --> 01:31:43.130 align:start position:0% -explanation for it I never could find -one<01:31:41.130> and<01:31:41.400> I<01:31:41.910> never<01:31:42.120> know<01:31:42.300> anybody<01:31:42.780> who<01:31:42.930> knew - -01:31:43.130 --> 01:31:43.140 align:start position:0% -one and I never know anybody who knew - - -01:31:43.140 --> 01:31:52.430 align:start position:0% -one and I never know anybody who knew -one<01:31:46.460> so<01:31:50.000> the<01:31:51.000> worst<01:31:51.180> part<01:31:51.450> of<01:31:51.570> it<01:31:51.690> is<01:31:51.870> it's<01:31:52.260> not - -01:31:52.430 --> 01:31:52.440 align:start position:0% -one so the worst part of it is it's not - - -01:31:52.440 --> 01:31:55.550 align:start position:0% -one so the worst part of it is it's not -a<01:31:52.500> lot<01:31:52.830> of<01:31:52.860> fun<01:31:53.190> to<01:31:53.400> check<01:31:54.290> it's<01:31:55.290> rather - -01:31:55.550 --> 01:31:55.560 align:start position:0% -a lot of fun to check it's rather - - -01:31:55.560 --> 01:31:59.810 align:start position:0% -a lot of fun to check it's rather -tedious<01:31:55.910> but<01:31:56.910> it's<01:31:57.120> true<01:31:57.980> okay<01:31:58.980> so<01:31:59.430> let's<01:31:59.700> see - -01:31:59.810 --> 01:31:59.820 align:start position:0% -tedious but it's true okay so let's see - - -01:31:59.820 --> 01:32:09.880 align:start position:0% -tedious but it's true okay so let's see -where<01:31:59.970> we're<01:32:00.120> going - -01:32:09.880 --> 01:32:09.890 align:start position:0% - - - -01:32:09.890 --> 01:32:13.660 align:start position:0% - -special<01:32:10.670> relativity<01:32:11.060> is<01:32:11.480> basically<01:32:12.520> stuff<01:32:13.520> on - -01:32:13.660 --> 01:32:13.670 align:start position:0% -special relativity is basically stuff on - - -01:32:13.670 --> 01:32:16.690 align:start position:0% -special relativity is basically stuff on -a<01:32:13.700> flat<01:32:14.000> geometry<01:32:15.010> frankly<01:32:16.010> nice<01:32:16.340> time<01:32:16.610> to - -01:32:16.690 --> 01:32:16.700 align:start position:0% -a flat geometry frankly nice time to - - -01:32:16.700 --> 01:32:19.680 align:start position:0% -a flat geometry frankly nice time to -figure<01:32:16.850> out<01:32:17.120> who<01:32:17.240> you<01:32:17.330> curb<01:32:17.690> it<01:32:17.900> that's - -01:32:19.680 --> 01:32:19.690 align:start position:0% -figure out who you curb it that's - - -01:32:19.690 --> 01:32:23.020 align:start position:0% -figure out who you curb it that's -gravity<01:32:20.690> therefore<01:32:21.170> acceleration<01:32:22.030> curvature - -01:32:23.020 --> 01:32:23.030 align:start position:0% -gravity therefore acceleration curvature - - -01:32:23.030 --> 01:32:25.720 align:start position:0% -gravity therefore acceleration curvature -is<01:32:23.270> that<01:32:23.480> is<01:32:23.660> a<01:32:23.690> rhymin<01:32:24.110> vector<01:32:24.530> that<01:32:25.310> doesn't - -01:32:25.720 --> 01:32:25.730 align:start position:0% -is that is a rhymin vector that doesn't - - -01:32:25.730 --> 01:32:27.550 align:start position:0% -is that is a rhymin vector that doesn't -fit<01:32:25.910> because<01:32:26.090> there's<01:32:26.480> too<01:32:26.630> many<01:32:26.750> components - -01:32:27.550 --> 01:32:27.560 align:start position:0% -fit because there's too many components - - -01:32:27.560 --> 01:32:29.290 align:start position:0% -fit because there's too many components -so<01:32:27.770> he<01:32:27.980> shrunk<01:32:28.250> it<01:32:28.400> down<01:32:28.430> to<01:32:28.730> a<01:32:28.760> reaching - -01:32:29.290 --> 01:32:29.300 align:start position:0% -so he shrunk it down to a reaching - - -01:32:29.300 --> 01:32:31.900 align:start position:0% -so he shrunk it down to a reaching -vector<01:32:29.600> now<01:32:29.950> what<01:32:30.950> I'm<01:32:31.100> seeing<01:32:31.490> with<01:32:31.790> this - -01:32:31.900 --> 01:32:31.910 align:start position:0% -vector now what I'm seeing with this - - -01:32:31.910 --> 01:32:33.610 align:start position:0% -vector now what I'm seeing with this -gravitational<01:32:32.390> wave<01:32:32.840> is<01:32:33.050> that<01:32:33.230> there's<01:32:33.500> a - -01:32:33.610 --> 01:32:33.620 align:start position:0% -gravitational wave is that there's a - - -01:32:33.620 --> 01:32:35.470 align:start position:0% -gravitational wave is that there's a -there's<01:32:34.070> a<01:32:34.160> difference<01:32:34.190> in<01:32:34.700> information - -01:32:35.470 --> 01:32:35.480 align:start position:0% -there's a difference in information - - -01:32:35.480 --> 01:32:38.500 align:start position:0% -there's a difference in information -between<01:32:35.690> the<01:32:36.620> tensor<01:32:37.070> with<01:32:37.310> four<01:32:37.820> components - -01:32:38.500 --> 01:32:38.510 align:start position:0% -between the tensor with four components - - -01:32:38.510 --> 01:32:40.420 align:start position:0% -between the tensor with four components -or<01:32:38.660> the<01:32:38.690> two<01:32:38.960> components<01:32:39.590> to<01:32:39.740> run<01:32:39.890> adventure - -01:32:40.420 --> 01:32:40.430 align:start position:0% -or the two components to run adventure - - -01:32:40.430 --> 01:32:42.310 align:start position:0% -or the two components to run adventure -effect<01:32:41.060> to<01:32:41.240> the<01:32:41.330> tensor<01:32:41.720> around<01:32:41.900> this<01:32:42.050> more - -01:32:42.310 --> 01:32:42.320 align:start position:0% -effect to the tensor around this more - - -01:32:42.320 --> 01:32:44.050 align:start position:0% -effect to the tensor around this more -information<01:32:42.980> of<01:32:43.100> the<01:32:43.190> four<01:32:43.400> components<01:32:43.970> of - -01:32:44.050 --> 01:32:44.060 align:start position:0% -information of the four components of - - -01:32:44.060 --> 01:32:47.920 align:start position:0% -information of the four components of -that<01:32:44.840> information<01:32:45.610> that's<01:32:46.610> missing<01:32:46.850> is<01:32:47.480> the - -01:32:47.920 --> 01:32:47.930 align:start position:0% -that information that's missing is the - - -01:32:47.930 --> 01:32:50.320 align:start position:0% -that information that's missing is the -manifest<01:32:48.560> in<01:32:48.740> the<01:32:48.890> form<01:32:49.430> of<01:32:49.580> gravitational - -01:32:50.320 --> 01:32:50.330 align:start position:0% -manifest in the form of gravitational - - -01:32:50.330 --> 01:32:56.590 align:start position:0% -manifest in the form of gravitational -waves<01:32:52.420> that's<01:32:53.420> absolutely<01:32:53.960> correct<01:32:54.490> now<01:32:55.600> in - -01:32:56.590 --> 01:32:56.600 align:start position:0% -waves that's absolutely correct now in - - -01:32:56.600 --> 01:33:00.640 align:start position:0% -waves that's absolutely correct now in -three<01:32:57.410> dimensions<01:32:58.960> in<01:32:59.960> three<01:33:00.230> dimensions - -01:33:00.640 --> 01:33:00.650 align:start position:0% -three dimensions in three dimensions - - -01:33:00.650 --> 01:33:02.200 align:start position:0% -three dimensions in three dimensions -let's<01:33:00.800> remember<01:33:01.070> for<01:33:01.340> a<01:33:01.370> moment<01:33:01.670> what<01:33:01.970> the - -01:33:02.200 --> 01:33:02.210 align:start position:0% -let's remember for a moment what the - - -01:33:02.210 --> 01:33:08.680 align:start position:0% -let's remember for a moment what the -Riemann<01:33:02.690> tensor<01:33:03.170> is<01:33:03.410> the<01:33:04.540> Riemann<01:33:05.540> tensor<01:33:07.690> has - -01:33:08.680 --> 01:33:08.690 align:start position:0% -Riemann tensor is the Riemann tensor has - - -01:33:08.690 --> 01:33:10.900 align:start position:0% -Riemann tensor is the Riemann tensor has -to<01:33:08.960> do<01:33:09.140> basically<01:33:09.890> with<01:33:10.130> two<01:33:10.190> distinct - -01:33:10.900 --> 01:33:10.910 align:start position:0% -to do basically with two distinct - - -01:33:10.910 --> 01:33:14.560 align:start position:0% -to do basically with two distinct -rotations<01:33:11.510> one<01:33:12.350> of<01:33:12.500> the<01:33:12.620> rotations<01:33:13.250> or<01:33:14.060> two - -01:33:14.560 --> 01:33:14.570 align:start position:0% -rotations one of the rotations or two - - -01:33:14.570 --> 01:33:18.070 align:start position:0% -rotations one of the rotations or two -axes<01:33:15.080> one<01:33:15.920> axis<01:33:16.370> is<01:33:16.610> you<01:33:16.940> parallel<01:33:17.450> transport - -01:33:18.070 --> 01:33:18.080 align:start position:0% -axes one axis is you parallel transport - - -01:33:18.080 --> 01:33:20.530 align:start position:0% -axes one axis is you parallel transport -around<01:33:18.470> that<01:33:18.680> axis<01:33:18.970> the<01:33:19.970> result<01:33:20.390> of - -01:33:20.530 --> 01:33:20.540 align:start position:0% -around that axis the result of - - -01:33:20.540 --> 01:33:22.480 align:start position:0% -around that axis the result of -parallelly<01:33:21.050> transporting<01:33:21.800> something<01:33:22.040> around - -01:33:22.480 --> 01:33:22.490 align:start position:0% -parallelly transporting something around - - -01:33:22.490 --> 01:33:25.270 align:start position:0% -parallelly transporting something around -that<01:33:22.640> axis<01:33:23.000> is<01:33:23.540> a<01:33:24.290> rotation<01:33:24.980> of<01:33:25.160> that - -01:33:25.270 --> 01:33:25.280 align:start position:0% -that axis is a rotation of that - - -01:33:25.280 --> 01:33:28.810 align:start position:0% -that axis is a rotation of that -something<01:33:25.990> about<01:33:26.990> some<01:33:27.290> other<01:33:27.440> axis<01:33:27.980> so<01:33:28.700> when - -01:33:28.810 --> 01:33:28.820 align:start position:0% -something about some other axis so when - - -01:33:28.820 --> 01:33:30.340 align:start position:0% -something about some other axis so when -you<01:33:28.910> go<01:33:29.060> around<01:33:29.180> that<01:33:29.450> axis<01:33:29.810> with<01:33:30.230> your - -01:33:30.340 --> 01:33:30.350 align:start position:0% -you go around that axis with your - - -01:33:30.350 --> 01:33:33.040 align:start position:0% -you go around that axis with your -gyroscope<01:33:30.970> the<01:33:31.970> gyroscope<01:33:32.540> gets<01:33:32.750> rotated - -01:33:33.040 --> 01:33:33.050 align:start position:0% -gyroscope the gyroscope gets rotated - - -01:33:33.050 --> 01:33:34.960 align:start position:0% -gyroscope the gyroscope gets rotated -about<01:33:33.650> some<01:33:33.860> other<01:33:34.010> axis<01:33:34.370> so<01:33:34.460> it<01:33:34.580> has<01:33:34.700> to<01:33:34.850> do - -01:33:34.960 --> 01:33:34.970 align:start position:0% -about some other axis so it has to do - - -01:33:34.970 --> 01:33:38.680 align:start position:0% -about some other axis so it has to do -with<01:33:35.090> two<01:33:35.330> axes<01:33:36.400> all<01:33:37.400> right<01:33:37.490> in<01:33:37.690> 3-dimensional - -01:33:38.680 --> 01:33:38.690 align:start position:0% -with two axes all right in 3-dimensional - - -01:33:38.690 --> 01:33:40.360 align:start position:0% -with two axes all right in 3-dimensional -space<01:33:38.720> rotation<01:33:39.590> in<01:33:39.650> three-dimensional - -01:33:40.360 --> 01:33:40.370 align:start position:0% -space rotation in three-dimensional - - -01:33:40.370 --> 01:33:42.970 align:start position:0% -space rotation in three-dimensional -space<01:33:40.400> rotations<01:33:41.390> have<01:33:41.660> to<01:33:41.870> do<01:33:42.020> with<01:33:42.230> axes<01:33:42.770> in - -01:33:42.970 --> 01:33:42.980 align:start position:0% -space rotations have to do with axes in - - -01:33:42.980 --> 01:33:45.190 align:start position:0% -space rotations have to do with axes in -space<01:33:43.280> that's<01:33:43.970> not<01:33:44.150> true<01:33:44.330> in<01:33:44.450> four<01:33:44.660> dimensions - -01:33:45.190 --> 01:33:45.200 align:start position:0% -space that's not true in four dimensions - - -01:33:45.200 --> 01:33:47.230 align:start position:0% -space that's not true in four dimensions -and<01:33:45.440> four<01:33:46.130> dimensions<01:33:46.610> they<01:33:46.760> have<01:33:46.880> to<01:33:47.000> do<01:33:47.120> with - -01:33:47.230 --> 01:33:47.240 align:start position:0% -and four dimensions they have to do with - - -01:33:47.240 --> 01:33:49.690 align:start position:0% -and four dimensions they have to do with -planes<01:33:47.570> in<01:33:48.200> fact<01:33:48.980> they<01:33:49.070> always<01:33:49.310> have<01:33:49.460> to<01:33:49.520> do - -01:33:49.690 --> 01:33:49.700 align:start position:0% -planes in fact they always have to do - - -01:33:49.700 --> 01:33:51.640 align:start position:0% -planes in fact they always have to do -with<01:33:49.850> planes<01:33:50.180> it's<01:33:50.990> just<01:33:51.230> in<01:33:51.380> three - -01:33:51.640 --> 01:33:51.650 align:start position:0% -with planes it's just in three - - -01:33:51.650 --> 01:33:53.230 align:start position:0% -with planes it's just in three -dimensions<01:33:52.190> there's<01:33:52.370> a<01:33:52.430> correspondence - -01:33:53.230 --> 01:33:53.240 align:start position:0% -dimensions there's a correspondence - - -01:33:53.240 --> 01:33:56.020 align:start position:0% -dimensions there's a correspondence -between<01:33:53.330> vectors<01:33:54.110> and<01:33:54.320> planes<01:33:54.760> namely<01:33:55.760> every - -01:33:56.020 --> 01:33:56.030 align:start position:0% -between vectors and planes namely every - - -01:33:56.030 --> 01:33:58.150 align:start position:0% -between vectors and planes namely every -plane<01:33:56.300> has<01:33:56.900> the<01:33:56.930> perpendicular<01:33:57.860> vector - -01:33:58.150 --> 01:33:58.160 align:start position:0% -plane has the perpendicular vector - - -01:33:58.160 --> 01:34:02.020 align:start position:0% -plane has the perpendicular vector -associated<01:33:58.460> with<01:33:58.880> it<01:33:59.180> so<01:34:00.140> if<01:34:01.100> the<01:34:01.640> Riemann - -01:34:02.020 --> 01:34:02.030 align:start position:0% -associated with it so if the Riemann - - -01:34:02.030 --> 01:34:05.430 align:start position:0% -associated with it so if the Riemann -tensor<01:34:02.510> has<01:34:02.720> to<01:34:02.900> do<01:34:03.050> with<01:34:03.140> pairs<01:34:03.620> of<01:34:04.360> rotations - -01:34:05.430 --> 01:34:05.440 align:start position:0% -tensor has to do with pairs of rotations - - -01:34:05.440 --> 01:34:09.340 align:start position:0% -tensor has to do with pairs of rotations -pairs<01:34:06.440> of<01:34:06.710> rotational<01:34:07.340> planes<01:34:07.810> then<01:34:08.810> in<01:34:08.960> three - -01:34:09.340 --> 01:34:09.350 align:start position:0% -pairs of rotational planes then in three - - -01:34:09.350 --> 01:34:11.200 align:start position:0% -pairs of rotational planes then in three -spatial<01:34:09.890> dimensions<01:34:09.920> it<01:34:10.730> can<01:34:10.880> be<01:34:10.970> thought<01:34:11.150> of - -01:34:11.200 --> 01:34:11.210 align:start position:0% -spatial dimensions it can be thought of - - -01:34:11.210 --> 01:34:14.710 align:start position:0% -spatial dimensions it can be thought of -as<01:34:11.420> having<01:34:11.450> to<01:34:11.840> do<01:34:12.020> with<01:34:12.320> pairs<01:34:12.680> of<01:34:12.890> axes<01:34:13.720> pairs - -01:34:14.710 --> 01:34:14.720 align:start position:0% -as having to do with pairs of axes pairs - - -01:34:14.720 --> 01:34:18.700 align:start position:0% -as having to do with pairs of axes pairs -of<01:34:14.960> axes<01:34:15.500> and<01:34:15.740> axis<01:34:16.250> is<01:34:16.520> a<01:34:16.850> vector<01:34:17.530> right<01:34:18.530> if<01:34:18.620> it - -01:34:18.700 --> 01:34:18.710 align:start position:0% -of axes and axis is a vector right if it - - -01:34:18.710 --> 01:34:21.100 align:start position:0% -of axes and axis is a vector right if it -has<01:34:18.830> to<01:34:18.920> do<01:34:19.010> with<01:34:19.100> pairs<01:34:19.460> of<01:34:19.670> axes<01:34:20.120> it<01:34:20.720> means - -01:34:21.100 --> 01:34:21.110 align:start position:0% -has to do with pairs of axes it means - - -01:34:21.110 --> 01:34:23.050 align:start position:0% -has to do with pairs of axes it means -really<01:34:21.590> it<01:34:21.950> can<01:34:22.130> be<01:34:22.280> represented - -01:34:23.050 --> 01:34:23.060 align:start position:0% -really it can be represented - - -01:34:23.060 --> 01:34:25.740 align:start position:0% -really it can be represented -something<01:34:23.390> with<01:34:23.510> only<01:34:23.660> two<01:34:24.020> indices<01:34:24.590> in - -01:34:25.740 --> 01:34:25.750 align:start position:0% -something with only two indices in - - -01:34:25.750 --> 01:34:30.570 align:start position:0% -something with only two indices in -general<01:34:26.750> it<01:34:27.140> requires<01:34:27.560> four<01:34:28.010> indices<01:34:28.900> but<01:34:29.900> in - -01:34:30.570 --> 01:34:30.580 align:start position:0% -general it requires four indices but in - - -01:34:30.580 --> 01:34:32.560 align:start position:0% -general it requires four indices but in -let<01:34:31.580> me<01:34:31.610> tell<01:34:31.850> you<01:34:31.910> another<01:34:32.000> reason<01:34:32.180> well - -01:34:32.560 --> 01:34:32.570 align:start position:0% -let me tell you another reason well - - -01:34:32.570 --> 01:34:33.990 align:start position:0% -let me tell you another reason well -another<01:34:32.900> way<01:34:33.050> to<01:34:33.110> explain<01:34:33.560> it - -01:34:33.990 --> 01:34:34.000 align:start position:0% -another way to explain it - - -01:34:34.000 --> 01:34:36.340 align:start position:0% -another way to explain it -supposing<01:34:35.000> something<01:34:35.180> has<01:34:35.450> a<01:34:35.480> one<01:34:35.720> two<01:34:35.960> index - -01:34:36.340 --> 01:34:36.350 align:start position:0% -supposing something has a one two index - - -01:34:36.350 --> 01:34:37.660 align:start position:0% -supposing something has a one two index -in<01:34:36.590> three<01:34:36.770> dimensions<01:34:37.220> suppose<01:34:37.400> we<01:34:37.550> need - -01:34:37.660 --> 01:34:37.670 align:start position:0% -in three dimensions suppose we need - - -01:34:37.670 --> 01:34:38.980 align:start position:0% -in three dimensions suppose we need -three<01:34:37.820> dimensions<01:34:37.940> supposing<01:34:38.570> something<01:34:38.780> has - -01:34:38.980 --> 01:34:38.990 align:start position:0% -three dimensions supposing something has - - -01:34:38.990 --> 01:34:42.010 align:start position:0% -three dimensions supposing something has -a<01:34:39.020> one<01:34:39.230> two<01:34:39.410> index<01:34:40.240> then<01:34:41.240> I<01:34:41.330> can<01:34:41.480> I<01:34:41.630> can<01:34:41.810> just - -01:34:42.010 --> 01:34:42.020 align:start position:0% -a one two index then I can I can just - - -01:34:42.020 --> 01:34:43.930 align:start position:0% -a one two index then I can I can just -use<01:34:42.410> a<01:34:42.440> new<01:34:42.710> notation<01:34:43.280> in<01:34:43.370> which<01:34:43.400> I<01:34:43.610> call<01:34:43.820> it<01:34:43.850> a - -01:34:43.930 --> 01:34:43.940 align:start position:0% -use a new notation in which I call it a - - -01:34:43.940 --> 01:34:46.960 align:start position:0% -use a new notation in which I call it a -three<01:34:44.090> index<01:34:44.560> one<01:34:45.560> two<01:34:45.770> replaced<01:34:46.640> by<01:34:46.790> three - -01:34:46.960 --> 01:34:46.970 align:start position:0% -three index one two replaced by three - - -01:34:46.970 --> 01:34:51.670 align:start position:0% -three index one two replaced by three -two<01:34:47.690> three<01:34:47.870> replaced<01:34:48.530> by<01:34:48.710> one<01:34:49.510> pairs<01:34:50.620> can<01:34:51.620> be - -01:34:51.670 --> 01:34:51.680 align:start position:0% -two three replaced by one pairs can be - - -01:34:51.680 --> 01:34:54.610 align:start position:0% -two three replaced by one pairs can be -replaced<01:34:52.130> by<01:34:52.430> single<01:34:53.020> indices<01:34:54.020> in<01:34:54.260> terms<01:34:54.470> of - -01:34:54.610 --> 01:34:54.620 align:start position:0% -replaced by single indices in terms of - - -01:34:54.620 --> 01:34:58.200 align:start position:0% -replaced by single indices in terms of -information<01:34:55.190> content<01:34:55.400> so<01:34:55.910> to<01:34:55.970> speak<01:34:56.410> so<01:34:57.410> a - -01:34:58.200 --> 01:34:58.210 align:start position:0% -information content so to speak so a - - -01:34:58.210 --> 01:35:00.670 align:start position:0% -information content so to speak so a -plane<01:34:59.210> has<01:34:59.570> associated<01:34:59.870> with<01:35:00.230> it<01:35:00.350> a<01:35:00.500> vector - -01:35:00.670 --> 01:35:00.680 align:start position:0% -plane has associated with it a vector - - -01:35:00.680 --> 01:35:02.920 align:start position:0% -plane has associated with it a vector -that's<01:35:01.550> the<01:35:01.670> same<01:35:01.850> as<01:35:02.030> saying<01:35:02.180> two<01:35:02.480> indices - -01:35:02.920 --> 01:35:02.930 align:start position:0% -that's the same as saying two indices - - -01:35:02.930 --> 01:35:04.960 align:start position:0% -that's the same as saying two indices -can<01:35:03.200> be<01:35:03.260> replaced<01:35:03.710> by<01:35:03.890> a<01:35:03.920> third<01:35:04.220> index<01:35:04.640> the - -01:35:04.960 --> 01:35:04.970 align:start position:0% -can be replaced by a third index the - - -01:35:04.970 --> 01:35:07.240 align:start position:0% -can be replaced by a third index the -other<01:35:05.180> one<01:35:05.480> one<01:35:06.080> two<01:35:06.320> gets<01:35:06.560> replaced<01:35:06.830> by<01:35:06.980> three - -01:35:07.240 --> 01:35:07.250 align:start position:0% -other one one two gets replaced by three - - -01:35:07.250 --> 01:35:09.910 align:start position:0% -other one one two gets replaced by three -two<01:35:07.550> three<01:35:07.730> gets<01:35:08.000> replaced<01:35:08.300> by<01:35:08.710> you<01:35:09.710> know<01:35:09.800> what - -01:35:09.910 --> 01:35:09.920 align:start position:0% -two three gets replaced by you know what - - -01:35:09.920 --> 01:35:10.360 align:start position:0% -two three gets replaced by you know what -I<01:35:09.950> mean - -01:35:10.360 --> 01:35:10.370 align:start position:0% -I mean - - -01:35:10.370 --> 01:35:13.270 align:start position:0% -I mean -okay<01:35:11.120> so<01:35:11.660> the<01:35:12.230> answer<01:35:12.260> is<01:35:12.740> in<01:35:12.920> three - -01:35:13.270 --> 01:35:13.280 align:start position:0% -okay so the answer is in three - - -01:35:13.280 --> 01:35:16.060 align:start position:0% -okay so the answer is in three -dimensional<01:35:13.910> space<01:35:14.270> or<01:35:15.170> even<01:35:15.200> three - -01:35:16.060 --> 01:35:16.070 align:start position:0% -dimensional space or even three - - -01:35:16.070 --> 01:35:17.890 align:start position:0% -dimensional space or even three -dimensional<01:35:16.550> space-time<01:35:16.850> if<01:35:17.450> there<01:35:17.600> was<01:35:17.720> such - -01:35:17.890 --> 01:35:17.900 align:start position:0% -dimensional space-time if there was such - - -01:35:17.900 --> 01:35:22.240 align:start position:0% -dimensional space-time if there was such -a<01:35:17.930> thing<01:35:18.520> the<01:35:19.520> Ricci<01:35:20.120> tensor<01:35:20.780> would<01:35:21.440> be<01:35:21.650> a<01:35:21.680> full - -01:35:22.240 --> 01:35:22.250 align:start position:0% -a thing the Ricci tensor would be a full - - -01:35:22.250 --> 01:35:25.090 align:start position:0% -a thing the Ricci tensor would be a full -description<01:35:22.760> and<01:35:23.450> if<01:35:24.080> the<01:35:24.200> Ricci<01:35:24.440> tensor<01:35:24.950> was - -01:35:25.090 --> 01:35:25.100 align:start position:0% -description and if the Ricci tensor was - - -01:35:25.100 --> 01:35:30.220 align:start position:0% -description and if the Ricci tensor was -zero<01:35:25.400> the<01:35:25.880> space<01:35:26.150> is<01:35:26.450> really<01:35:26.720> flat<01:35:27.760> okay<01:35:29.230> there - -01:35:30.220 --> 01:35:30.230 align:start position:0% -zero the space is really flat okay there - - -01:35:30.230 --> 01:35:32.830 align:start position:0% -zero the space is really flat okay there -is<01:35:30.440> no<01:35:30.470> other<01:35:30.830> information<01:35:31.100> and<01:35:31.700> therefore<01:35:32.330> in - -01:35:32.830 --> 01:35:32.840 align:start position:0% -is no other information and therefore in - - -01:35:32.840 --> 01:35:36.700 align:start position:0% -is no other information and therefore in -a<01:35:33.260> world<01:35:33.860> of<01:35:34.250> two<01:35:34.700> plus<01:35:35.090> one<01:35:35.480> two<01:35:35.960> space<01:35:36.290> one - -01:35:36.700 --> 01:35:36.710 align:start position:0% -a world of two plus one two space one - - -01:35:36.710 --> 01:35:39.910 align:start position:0% -a world of two plus one two space one -time<01:35:37.040> three<01:35:37.490> dimensions<01:35:38.410> there<01:35:39.410> are<01:35:39.650> no - -01:35:39.910 --> 01:35:39.920 align:start position:0% -time three dimensions there are no - - -01:35:39.920 --> 01:35:43.330 align:start position:0% -time three dimensions there are no -gravitational<01:35:40.460> waves<01:35:41.230> any<01:35:42.230> planes<01:35:42.740> place<01:35:43.160> in - -01:35:43.330 --> 01:35:43.340 align:start position:0% -gravitational waves any planes place in - - -01:35:43.340 --> 01:35:46.360 align:start position:0% -gravitational waves any planes place in -space<01:35:43.640> where<01:35:44.150> there<01:35:44.270> is<01:35:44.330> no<01:35:44.600> material<01:35:45.370> no - -01:35:46.360 --> 01:35:46.370 align:start position:0% -space where there is no material no - - -01:35:46.370 --> 01:35:49.360 align:start position:0% -space where there is no material no -energy<01:35:46.760> and<01:35:46.880> momentum<01:35:47.000> of<01:35:47.900> this<01:35:48.170> type<01:35:48.410> space - -01:35:49.360 --> 01:35:49.370 align:start position:0% -energy and momentum of this type space - - -01:35:49.370 --> 01:35:50.670 align:start position:0% -energy and momentum of this type space -time<01:35:49.790> is<01:35:50.090> flat - -01:35:50.670 --> 01:35:50.680 align:start position:0% -time is flat - - -01:35:50.680 --> 01:35:54.520 align:start position:0% -time is flat -not<01:35:51.680> just<01:35:52.220> Ricci<01:35:52.820> flat<01:35:53.360> Ricci<01:35:53.720> flap<01:35:54.140> is<01:35:54.410> the - -01:35:54.520 --> 01:35:54.530 align:start position:0% -not just Ricci flat Ricci flap is the - - -01:35:54.530 --> 01:35:56.380 align:start position:0% -not just Ricci flat Ricci flap is the -statement<01:35:54.800> that<01:35:55.160> the<01:35:55.550> Ricci<01:35:55.760> tensor<01:35:56.210> is<01:35:56.360> zero - -01:35:56.380 --> 01:35:56.390 align:start position:0% -statement that the Ricci tensor is zero - - -01:35:56.390 --> 01:35:59.980 align:start position:0% -statement that the Ricci tensor is zero -but<01:35:57.200> just<01:35:57.410> plain<01:35:57.710> flat<01:35:58.450> no<01:35:59.450> gravitational - -01:35:59.980 --> 01:35:59.990 align:start position:0% -but just plain flat no gravitational - - -01:35:59.990 --> 01:36:03.190 align:start position:0% -but just plain flat no gravitational -waves<01:36:00.470> in<01:36:00.920> four<01:36:01.910> dimensions<01:36:02.570> by<01:36:02.720> the<01:36:02.780> time<01:36:03.050> you - -01:36:03.190 --> 01:36:03.200 align:start position:0% -waves in four dimensions by the time you - - -01:36:03.200 --> 01:36:06.550 align:start position:0% -waves in four dimensions by the time you -get<01:36:03.350> up<01:36:03.440> to<01:36:03.470> four<01:36:03.830> dimensions<01:36:04.840> then<01:36:05.840> the<01:36:06.290> Ricci - -01:36:06.550 --> 01:36:06.560 align:start position:0% -get up to four dimensions then the Ricci - - -01:36:06.560 --> 01:36:09.280 align:start position:0% -get up to four dimensions then the Ricci -tensor<01:36:07.130> is<01:36:07.310> not<01:36:07.580> a<01:36:07.850> full<01:36:08.090> description<01:36:08.480> of<01:36:09.080> the - -01:36:09.280 --> 01:36:09.290 align:start position:0% -tensor is not a full description of the - - -01:36:09.290 --> 01:36:11.470 align:start position:0% -tensor is not a full description of the -geometry<01:36:09.950> though<01:36:10.580> the<01:36:10.700> Ricci<01:36:10.850> tensor<01:36:11.270> being - -01:36:11.470 --> 01:36:11.480 align:start position:0% -geometry though the Ricci tensor being - - -01:36:11.480 --> 01:36:15.090 align:start position:0% -geometry though the Ricci tensor being -zero<01:36:11.810> is<01:36:12.260> not<01:36:12.980> enough<01:36:13.280> to<01:36:13.430> tell<01:36:13.640> you<01:36:13.850> that<01:36:14.060> all - -01:36:15.090 --> 01:36:15.100 align:start position:0% -zero is not enough to tell you that all - - -01:36:15.100 --> 01:36:18.460 align:start position:0% -zero is not enough to tell you that all -whatever<01:36:16.100> the<01:36:16.340> number<01:36:16.640> of<01:36:16.760> components<01:36:17.120> of<01:36:17.660> the - -01:36:18.460 --> 01:36:18.470 align:start position:0% -whatever the number of components of the - - -01:36:18.470 --> 01:36:20.740 align:start position:0% -whatever the number of components of the -Riemann<01:36:18.830> tensor<01:36:19.190> are<01:36:19.340> zero<01:36:19.700> and<01:36:20.060> so<01:36:20.570> there's - -01:36:20.740 --> 01:36:20.750 align:start position:0% -Riemann tensor are zero and so there's - - -01:36:20.750 --> 01:36:23.200 align:start position:0% -Riemann tensor are zero and so there's -some<01:36:20.870> leftover<01:36:21.470> information<01:36:22.160> that<01:36:22.790> leftover - -01:36:23.200 --> 01:36:23.210 align:start position:0% -some leftover information that leftover - - -01:36:23.210 --> 01:36:27.160 align:start position:0% -some leftover information that leftover -information<01:36:23.990> is<01:36:24.430> physically<01:36:25.430> in<01:36:26.030> the<01:36:26.720> form<01:36:27.020> of - -01:36:27.160 --> 01:36:27.170 align:start position:0% -information is physically in the form of - - -01:36:27.170 --> 01:36:28.420 align:start position:0% -information is physically in the form of -the<01:36:27.290> gravitational<01:36:27.710> waves - -01:36:28.420 --> 01:36:28.430 align:start position:0% -the gravitational waves - - -01:36:28.430 --> 01:36:33.430 align:start position:0% -the gravitational waves -okay<01:36:29.320> so<01:36:30.320> that's<01:36:30.950> that's<01:36:31.340> the<01:36:31.610> story - -01:36:33.430 --> 01:36:33.440 align:start position:0% -okay so that's that's the story - - -01:36:33.440 --> 01:36:38.029 align:start position:0% -okay so that's that's the story -okay<01:36:35.960> let's<01:36:36.960> see<01:36:37.079> how<01:36:37.170> we're<01:36:37.320> doing<01:36:37.349> we<01:36:37.619> still - -01:36:38.029 --> 01:36:38.039 align:start position:0% -okay let's see how we're doing we still - - -01:36:38.039 --> 01:36:43.189 align:start position:0% -okay let's see how we're doing we still -have<01:36:38.159> a<01:36:38.190> few<01:36:38.369> minutes<01:36:41.780> I'm<01:36:42.780> gonna<01:36:42.900> try<01:36:43.139> to - -01:36:43.189 --> 01:36:43.199 align:start position:0% -have a few minutes I'm gonna try to - - -01:36:43.199 --> 01:36:44.779 align:start position:0% -have a few minutes I'm gonna try to -leave<01:36:43.380> a<01:36:43.409> little<01:36:43.710> early<01:36:44.039> tonight<01:36:44.250> because<01:36:44.550> I - -01:36:44.779 --> 01:36:44.789 align:start position:0% -leave a little early tonight because I - - -01:36:44.789 --> 01:36:46.220 align:start position:0% -leave a little early tonight because I -have<01:36:44.909> a<01:36:44.940> very<01:36:45.329> early<01:36:45.539> morning<01:36:45.719> flight - -01:36:46.220 --> 01:36:46.230 align:start position:0% -have a very early morning flight - - -01:36:46.230 --> 01:37:06.200 align:start position:0% -have a very early morning flight -tomorrow<01:37:04.130> there's<01:37:05.130> a<01:37:05.190> trick<01:37:05.460> to<01:37:05.489> rewrite<01:37:06.030> this - -01:37:06.200 --> 01:37:06.210 align:start position:0% -tomorrow there's a trick to rewrite this - - -01:37:06.210 --> 01:37:08.330 align:start position:0% -tomorrow there's a trick to rewrite this -equation<01:37:06.480> in<01:37:06.900> another<01:37:07.170> form<01:37:07.530> it's<01:37:07.980> a<01:37:08.099> kind<01:37:08.280> of - -01:37:08.330 --> 01:37:08.340 align:start position:0% -equation in another form it's a kind of - - -01:37:08.340 --> 01:37:10.850 align:start position:0% -equation in another form it's a kind of -neat<01:37:08.519> form<01:37:08.849> in<01:37:09.449> which<01:37:09.840> it<01:37:09.900> is<01:37:10.320> an<01:37:10.500> equation - -01:37:10.850 --> 01:37:10.860 align:start position:0% -neat form in which it is an equation - - -01:37:10.860 --> 01:37:13.040 align:start position:0% -neat form in which it is an equation -where<01:37:11.250> R<01:37:11.489> mu<01:37:11.670> nu<01:37:12.000> is<01:37:12.030> set<01:37:12.420> equal<01:37:12.539> to<01:37:12.809> something - -01:37:13.040 --> 01:37:13.050 align:start position:0% -where R mu nu is set equal to something - - -01:37:13.050 --> 01:37:15.830 align:start position:0% -where R mu nu is set equal to something -else<01:37:13.639> to<01:37:14.639> see<01:37:14.820> how<01:37:14.940> it<01:37:15.119> works<01:37:15.360> I'm<01:37:15.630> going<01:37:15.750> to - -01:37:15.830 --> 01:37:15.840 align:start position:0% -else to see how it works I'm going to - - -01:37:15.840 --> 01:37:18.020 align:start position:0% -else to see how it works I'm going to -show<01:37:15.929> you<01:37:16.019> how<01:37:16.139> it<01:37:16.260> works<01:37:16.489> again<01:37:17.489> take<01:37:17.849> this - -01:37:18.020 --> 01:37:18.030 align:start position:0% -show you how it works again take this - - -01:37:18.030 --> 01:37:19.819 align:start position:0% -show you how it works again take this -equation<01:37:18.300> not<01:37:19.260> this<01:37:19.440> one - -01:37:19.819 --> 01:37:19.829 align:start position:0% -equation not this one - - -01:37:19.829 --> 01:37:21.979 align:start position:0% -equation not this one -but<01:37:20.039> the<01:37:20.130> full<01:37:20.309> equation<01:37:20.610> and<01:37:21.059> contract<01:37:21.809> it - -01:37:21.979 --> 01:37:21.989 align:start position:0% -but the full equation and contract it - - -01:37:21.989 --> 01:37:24.919 align:start position:0% -but the full equation and contract it -with<01:37:22.170> G<01:37:22.349> mu<01:37:22.530> nu<01:37:23.300> contract<01:37:24.300> it<01:37:24.449> with<01:37:24.570> G<01:37:24.750> mu<01:37:24.900> nu - -01:37:24.919 --> 01:37:24.929 align:start position:0% -with G mu nu contract it with G mu nu - - -01:37:24.929 --> 01:37:27.049 align:start position:0% -with G mu nu contract it with G mu nu -what<01:37:25.260> do<01:37:25.320> you<01:37:25.409> get<01:37:25.650> on<01:37:25.889> the<01:37:26.519> left-hand<01:37:26.940> side - -01:37:27.049 --> 01:37:27.059 align:start position:0% -what do you get on the left-hand side - - -01:37:27.059 --> 01:37:28.640 align:start position:0% -what do you get on the left-hand side -from<01:37:27.449> the<01:37:27.599> first<01:37:27.809> term<01:37:28.050> you<01:37:28.079> just<01:37:28.260> get<01:37:28.559> the - -01:37:28.640 --> 01:37:28.650 align:start position:0% -from the first term you just get the - - -01:37:28.650 --> 01:37:35.720 align:start position:0% -from the first term you just get the -scalar<01:37:29.039> R<01:37:33.500> the<01:37:34.500> second<01:37:34.949> term<01:37:35.159> remember<01:37:35.489> what - -01:37:35.720 --> 01:37:35.730 align:start position:0% -scalar R the second term remember what - - -01:37:35.730 --> 01:37:41.089 align:start position:0% -scalar R the second term remember what -this<01:37:35.849> was<01:37:37.190> my<01:37:38.719> minus<01:37:39.719> 2r<01:37:40.079> so<01:37:40.590> that<01:37:40.769> gives<01:37:40.980> you - -01:37:41.089 --> 01:37:41.099 align:start position:0% -this was my minus 2r so that gives you - - -01:37:41.099 --> 01:37:49.819 align:start position:0% -this was my minus 2r so that gives you -all<01:37:41.130> together<01:37:41.639> -<01:37:41.909> are<01:37:43.309> running<01:37:44.309> out<01:37:44.429> of<01:37:44.460> ink<01:37:48.829> - - -01:37:49.819 --> 01:37:49.829 align:start position:0% -all together - are running out of ink - - - -01:37:49.829 --> 01:37:55.990 align:start position:0% -all together - are running out of ink - -R - -01:37:55.990 --> 01:37:56.000 align:start position:0% - - - -01:37:56.000 --> 01:37:58.650 align:start position:0% - -and<01:37:56.090> on<01:37:56.660> the<01:37:56.750> right-hand<01:37:56.960> side<01:37:57.440> you<01:37:57.680> get<01:37:57.890> K - -01:37:58.650 --> 01:37:58.660 align:start position:0% -and on the right-hand side you get K - - -01:37:58.660 --> 01:38:09.910 align:start position:0% -and on the right-hand side you get K -times<01:37:59.660> G<01:38:00.550> mu<01:38:01.550> nu<01:38:02.860> team<01:38:03.860> you<01:38:04.040> know<01:38:08.110> that's<01:38:09.110> this - -01:38:09.910 --> 01:38:09.920 align:start position:0% -times G mu nu team you know that's this - - -01:38:09.920 --> 01:38:12.880 align:start position:0% -times G mu nu team you know that's this -that's<01:38:10.610> some<01:38:11.150> scalar<01:38:11.840> that's<01:38:12.350> the<01:38:12.469> same<01:38:12.680> as - -01:38:12.880 --> 01:38:12.890 align:start position:0% -that's some scalar that's the same as - - -01:38:12.890 --> 01:38:17.010 align:start position:0% -that's some scalar that's the same as -incidentally<01:38:13.580> this<01:38:13.699> is<01:38:13.760> the<01:38:13.969> same<01:38:14.180> as<01:38:14.420> P<01:38:15.320> mu<01:38:16.160> mu - -01:38:17.010 --> 01:38:17.020 align:start position:0% -incidentally this is the same as P mu mu - - -01:38:17.020 --> 01:38:19.930 align:start position:0% -incidentally this is the same as P mu mu -contract<01:38:18.020> the<01:38:18.469> left<01:38:18.680> or<01:38:18.830> the<01:38:18.860> one<01:38:19.160> to<01:38:19.430> contract - -01:38:19.930 --> 01:38:19.940 align:start position:0% -contract the left or the one to contract - - -01:38:19.940 --> 01:38:23.320 align:start position:0% -contract the left or the one to contract -the<01:38:20.060> two<01:38:20.210> indices<01:38:21.280> the<01:38:22.280> right-hand<01:38:22.730> side<01:38:23.000> here - -01:38:23.320 --> 01:38:23.330 align:start position:0% -the two indices the right-hand side here - - -01:38:23.330 --> 01:38:26.260 align:start position:0% -the two indices the right-hand side here -is<01:38:23.510> some<01:38:23.750> scalar<01:38:24.230> that's<01:38:25.190> made<01:38:25.550> out<01:38:25.850> of<01:38:25.940> the - -01:38:26.260 --> 01:38:26.270 align:start position:0% -is some scalar that's made out of the - - -01:38:26.270 --> 01:38:29.080 align:start position:0% -is some scalar that's made out of the -energy<01:38:26.630> momentum<01:38:26.780> tensor<01:38:27.670> it's<01:38:28.670> a<01:38:28.760> certain - -01:38:29.080 --> 01:38:29.090 align:start position:0% -energy momentum tensor it's a certain - - -01:38:29.090 --> 01:38:30.850 align:start position:0% -energy momentum tensor it's a certain -scalar<01:38:29.540> that's<01:38:29.750> made<01:38:29.960> out<01:38:30.199> of<01:38:30.440> the<01:38:30.530> energy - -01:38:30.850 --> 01:38:30.860 align:start position:0% -scalar that's made out of the energy - - -01:38:30.860 --> 01:38:36.880 align:start position:0% -scalar that's made out of the energy -momentum<01:38:30.980> tensor<01:38:33.910> and<01:38:35.140> the<01:38:36.140> metric<01:38:36.500> tensor - -01:38:36.880 --> 01:38:36.890 align:start position:0% -momentum tensor and the metric tensor - - -01:38:36.890 --> 01:38:39.880 align:start position:0% -momentum tensor and the metric tensor -and<01:38:37.130> it's<01:38:37.250> just<01:38:37.520> called<01:38:37.880> T<01:38:38.420> in<01:38:38.780> the<01:38:39.380> same<01:38:39.650> way - -01:38:39.880 --> 01:38:39.890 align:start position:0% -and it's just called T in the same way - - -01:38:39.890 --> 01:38:42.820 align:start position:0% -and it's just called T in the same way -that<01:38:40.580> when<01:38:40.730> you<01:38:40.760> contract<01:38:41.360> R<01:38:41.600> mu<01:38:41.840> nu<01:38:41.989> with<01:38:42.590> gym - -01:38:42.820 --> 01:38:42.830 align:start position:0% -that when you contract R mu nu with gym - - -01:38:42.830 --> 01:38:46.990 align:start position:0% -that when you contract R mu nu with gym -you<01:38:42.949> know<01:38:43.130> you<01:38:44.060> get<01:38:44.210> just<01:38:44.390> plain<01:38:44.600> old<01:38:44.780> r<01:38:45.850> it's<01:38:46.850> a - -01:38:46.990 --> 01:38:47.000 align:start position:0% -you know you get just plain old r it's a - - -01:38:47.000 --> 01:38:48.970 align:start position:0% -you know you get just plain old r it's a -standard<01:38:47.420> notation<01:38:47.989> when<01:38:48.230> you<01:38:48.320> contract<01:38:48.800> the - -01:38:48.970 --> 01:38:48.980 align:start position:0% -standard notation when you contract the - - -01:38:48.980 --> 01:38:51.880 align:start position:0% -standard notation when you contract the -tensor<01:38:49.370> with<01:38:49.550> G<01:38:49.730> mu<01:38:49.880> nu<01:38:49.910> the<01:38:50.810> resulting<01:38:51.350> scalar - -01:38:51.880 --> 01:38:51.890 align:start position:0% -tensor with G mu nu the resulting scalar - - -01:38:51.890 --> 01:38:54.970 align:start position:0% -tensor with G mu nu the resulting scalar -carries<01:38:52.880> the<01:38:53.060> same<01:38:53.300> letter<01:38:53.780> that<01:38:54.469> the<01:38:54.590> tensor - -01:38:54.970 --> 01:38:54.980 align:start position:0% -carries the same letter that the tensor - - -01:38:54.980 --> 01:38:57.610 align:start position:0% -carries the same letter that the tensor -started<01:38:55.430> with<01:38:55.840> alright<01:38:56.840> so<01:38:57.020> let's<01:38:57.230> see<01:38:57.440> so - -01:38:57.610 --> 01:38:57.620 align:start position:0% -started with alright so let's see so - - -01:38:57.620 --> 01:39:00.060 align:start position:0% -started with alright so let's see so -what<01:38:57.739> do<01:38:57.800> we<01:38:57.860> have<01:38:57.920> then<01:38:58.280> we<01:38:58.460> have<01:38:58.699> minus<01:38:59.690> R - -01:39:00.060 --> 01:39:00.070 align:start position:0% -what do we have then we have minus R - - -01:39:00.070 --> 01:39:07.030 align:start position:0% -what do we have then we have minus R -equals<01:39:01.070> kapor<01:39:02.410> times<01:39:03.410> T<01:39:05.290> where<01:39:06.290> T<01:39:06.320> is<01:39:06.680> a<01:39:06.710> thing - -01:39:07.030 --> 01:39:07.040 align:start position:0% -equals kapor times T where T is a thing - - -01:39:07.040 --> 01:39:08.320 align:start position:0% -equals kapor times T where T is a thing -made<01:39:07.400> up<01:39:07.550> out<01:39:07.699> of<01:39:07.820> the<01:39:07.910> energy<01:39:08.210> momentum - -01:39:08.320 --> 01:39:08.330 align:start position:0% -made up out of the energy momentum - - -01:39:08.330 --> 01:39:11.979 align:start position:0% -made up out of the energy momentum -tensor<01:39:09.050> by<01:39:09.230> contracting<01:39:09.830> it<01:39:09.980> with<01:39:10.190> the<01:39:10.780> with<01:39:11.780> G - -01:39:11.979 --> 01:39:11.989 align:start position:0% -tensor by contracting it with the with G - - -01:39:11.989 --> 01:39:15.400 align:start position:0% -tensor by contracting it with the with G -mu<01:39:12.140> nu<01:39:12.170> now<01:39:12.920> we<01:39:12.980> can<01:39:13.250> rewrite<01:39:13.430> this<01:39:14.050> plugin<01:39:15.050> for - -01:39:15.400 --> 01:39:15.410 align:start position:0% -mu nu now we can rewrite this plugin for - - -01:39:15.410 --> 01:39:16.840 align:start position:0% -mu nu now we can rewrite this plugin for -R<01:39:15.650> over<01:39:16.250> here - -01:39:16.840 --> 01:39:16.850 align:start position:0% -R over here - - -01:39:16.850 --> 01:39:18.970 align:start position:0% -R over here -that's<01:39:17.420> expression<01:39:17.960> over<01:39:18.050> here<01:39:18.380> let's<01:39:18.860> see - -01:39:18.970 --> 01:39:18.980 align:start position:0% -that's expression over here let's see - - -01:39:18.980 --> 01:39:28.240 align:start position:0% -that's expression over here let's see -what<01:39:19.100> we<01:39:19.190> get<01:39:19.430> we<01:39:20.330> get<01:39:20.540> R<01:39:20.780> mu<01:39:20.960> nu<01:39:26.949> minus<01:39:27.949> one - -01:39:28.240 --> 01:39:28.250 align:start position:0% -what we get we get R mu nu minus one - - -01:39:28.250 --> 01:39:35.260 align:start position:0% -what we get we get R mu nu minus one -half<01:39:29.290> G<01:39:30.290> mu<01:39:30.320> nu<01:39:31.870> times<01:39:32.870> R<01:39:33.230> but<01:39:34.160> R<01:39:34.460> is<01:39:34.760> equal<01:39:35.239> to - -01:39:35.260 --> 01:39:35.270 align:start position:0% -half G mu nu times R but R is equal to - - -01:39:35.270 --> 01:39:40.660 align:start position:0% -half G mu nu times R but R is equal to -minus<01:39:36.020> K<01:39:38.440> so<01:39:39.440> that'll<01:39:39.680> eat<01:39:39.830> up<01:39:39.980> the<01:39:40.100> minus<01:39:40.430> sign - -01:39:40.660 --> 01:39:40.670 align:start position:0% -minus K so that'll eat up the minus sign - - -01:39:40.670 --> 01:39:45.700 align:start position:0% -minus K so that'll eat up the minus sign -here<01:39:41.380> there'll<01:39:42.380> be<01:39:42.410> a<01:39:42.500> K<01:39:42.969> here<01:39:44.469> times<01:39:45.469> the - -01:39:45.700 --> 01:39:45.710 align:start position:0% -here there'll be a K here times the - - -01:39:45.710 --> 01:39:49.300 align:start position:0% -here there'll be a K here times the -scalar<01:39:46.370> of<01:39:46.760> energy<01:39:47.210> and<01:39:47.330> momentum<01:39:47.480> T<01:39:48.310> that's - -01:39:49.300 --> 01:39:49.310 align:start position:0% -scalar of energy and momentum T that's - - -01:39:49.310 --> 01:39:52.600 align:start position:0% -scalar of energy and momentum T that's -not<01:39:49.550> the<01:39:49.670> energy<01:39:50.000> density<01:39:50.150> incidentally<01:39:51.610> more - -01:39:52.600 --> 01:39:52.610 align:start position:0% -not the energy density incidentally more - - -01:39:52.610 --> 01:39:55.570 align:start position:0% -not the energy density incidentally more -complicated<01:39:53.300> has<01:39:53.680> T<01:39:54.680> naught<01:39:54.860> naught<01:39:55.130> that<01:39:55.460> has - -01:39:55.570 --> 01:39:55.580 align:start position:0% -complicated has T naught naught that has - - -01:39:55.580 --> 01:39:58.780 align:start position:0% -complicated has T naught naught that has -T<01:39:55.820> 1<01:39:56.000> 1<01:39:56.210> that<01:39:56.420> has<01:39:56.570> T<01:39:56.780> 2<01:39:57.020> 2<01:39:57.320> and<01:39:57.530> so<01:39:57.650> forth<01:39:57.790> that's - -01:39:58.780 --> 01:39:58.790 align:start position:0% -T 1 1 that has T 2 2 and so forth that's - - -01:39:58.790 --> 01:40:04.660 align:start position:0% -T 1 1 that has T 2 2 and so forth that's -equal<01:39:59.360> decay<01:40:01.060> Tim<01:40:02.060> you<01:40:02.210> know - -01:40:04.660 --> 01:40:04.670 align:start position:0% - - - -01:40:04.670 --> 01:40:07.609 align:start position:0% - -now<01:40:05.670> I<01:40:06.030> can<01:40:06.329> take<01:40:06.510> this<01:40:06.809> over<01:40:07.110> to<01:40:07.349> the<01:40:07.590> other - -01:40:07.609 --> 01:40:07.619 align:start position:0% -now I can take this over to the other - - -01:40:07.619 --> 01:40:12.459 align:start position:0% -now I can take this over to the other -side<01:40:08.190> and<01:40:09.030> write<01:40:09.630> it<01:40:09.840> that<01:40:10.050> the<01:40:10.440> Ricci<01:40:10.679> tensor - -01:40:12.459 --> 01:40:12.469 align:start position:0% -side and write it that the Ricci tensor - - -01:40:12.469 --> 01:40:20.689 align:start position:0% -side and write it that the Ricci tensor -is<01:40:13.469> equal<01:40:13.619> to<01:40:14.010> K<01:40:14.809> times<01:40:15.809> T<01:40:16.440> mu<01:40:17.429> nu<01:40:19.010> minus<01:40:20.010> one - -01:40:20.689 --> 01:40:20.699 align:start position:0% -is equal to K times T mu nu minus one - - -01:40:20.699 --> 01:40:28.100 align:start position:0% -is equal to K times T mu nu minus one -half<01:40:22.190> G<01:40:23.190> mu<01:40:23.219> nu<01:40:24.139> times<01:40:25.139> T<01:40:26.090> so<01:40:27.090> Einstein<01:40:27.300> wasn't - -01:40:28.100 --> 01:40:28.110 align:start position:0% -half G mu nu times T so Einstein wasn't - - -01:40:28.110 --> 01:40:31.279 align:start position:0% -half G mu nu times T so Einstein wasn't -so<01:40:28.289> far<01:40:28.590> off<01:40:29.179> there<01:40:30.179> is<01:40:30.449> a<01:40:30.479> form<01:40:30.989> of<01:40:31.139> the - -01:40:31.279 --> 01:40:31.289 align:start position:0% -so far off there is a form of the - - -01:40:31.289 --> 01:40:32.989 align:start position:0% -so far off there is a form of the -Einstein<01:40:31.739> equations<01:40:32.369> where<01:40:32.550> it's<01:40:32.639> just<01:40:32.849> a - -01:40:32.989 --> 01:40:32.999 align:start position:0% -Einstein equations where it's just a - - -01:40:32.999 --> 01:40:36.799 align:start position:0% -Einstein equations where it's just a -Ricci<01:40:33.269> tensor<01:40:33.960> is<01:40:34.829> equal<01:40:35.280> to<01:40:35.489> something<01:40:35.880> but - -01:40:36.799 --> 01:40:36.809 align:start position:0% -Ricci tensor is equal to something but - - -01:40:36.809 --> 01:40:38.899 align:start position:0% -Ricci tensor is equal to something but -it's<01:40:36.960> not<01:40:37.170> just<01:40:37.499> the<01:40:37.650> energy<01:40:37.800> momentum<01:40:38.130> tensor - -01:40:38.899 --> 01:40:38.909 align:start position:0% -it's not just the energy momentum tensor - - -01:40:38.909 --> 01:40:42.140 align:start position:0% -it's not just the energy momentum tensor -it's<01:40:39.719> the<01:40:39.900> energy<01:40:40.079> momentum<01:40:40.650> tensor<01:40:41.400> minus - -01:40:42.140 --> 01:40:42.150 align:start position:0% -it's the energy momentum tensor minus - - -01:40:42.150 --> 01:40:45.830 align:start position:0% -it's the energy momentum tensor minus -one<01:40:42.389> half<01:40:42.690> G<01:40:43.199> mu<01:40:43.229> nu<01:40:43.380> times<01:40:43.860> T<01:40:44.190> it<01:40:44.369> looks<01:40:44.840> it - -01:40:45.830 --> 01:40:45.840 align:start position:0% -one half G mu nu times T it looks it - - -01:40:45.840 --> 01:40:47.299 align:start position:0% -one half G mu nu times T it looks it -sort<01:40:46.139> of<01:40:46.199> looks<01:40:46.469> kind<01:40:46.769> of<01:40:46.860> like<01:40:47.010> the<01:40:47.190> Einstein - -01:40:47.299 --> 01:40:47.309 align:start position:0% -sort of looks kind of like the Einstein - - -01:40:47.309 --> 01:40:49.790 align:start position:0% -sort of looks kind of like the Einstein -tensor<01:40:48.179> except<01:40:48.570> made<01:40:48.840> up<01:40:49.019> out<01:40:49.260> of<01:40:49.289> the<01:40:49.499> energy - -01:40:49.790 --> 01:40:49.800 align:start position:0% -tensor except made up out of the energy - - -01:40:49.800 --> 01:40:53.020 align:start position:0% -tensor except made up out of the energy -momentum<01:40:49.949> tensor<01:40:50.579> instead<01:40:50.909> of<01:40:50.999> the<01:40:51.059> metric - -01:40:53.020 --> 01:40:53.030 align:start position:0% -momentum tensor instead of the metric - - -01:40:53.030 --> 01:40:55.370 align:start position:0% -momentum tensor instead of the metric -the<01:40:54.030> funny<01:40:54.239> kind<01:40:54.479> of<01:40:54.539> duality<01:40:54.719> to<01:40:55.260> the - -01:40:55.370 --> 01:40:55.380 align:start position:0% -the funny kind of duality to the - - -01:40:55.380 --> 01:40:59.209 align:start position:0% -the funny kind of duality to the -equations<01:40:56.479> one<01:40:57.479> form<01:40:57.780> the<01:40:57.989> equations<01:40:58.499> are<01:40:58.800> a - -01:40:59.209 --> 01:40:59.219 align:start position:0% -equations one form the equations are a - - -01:40:59.219 --> 01:41:01.910 align:start position:0% -equations one form the equations are a -kind<01:40:59.909> of<01:40:59.969> Einstein<01:41:00.570> tensor<01:41:01.050> made<01:41:01.320> up<01:41:01.469> out<01:41:01.619> of<01:41:01.650> G - -01:41:01.910 --> 01:41:01.920 align:start position:0% -kind of Einstein tensor made up out of G - - -01:41:01.920 --> 01:41:04.640 align:start position:0% -kind of Einstein tensor made up out of G -mu<01:41:02.070> nu<01:41:02.159> is<01:41:02.550> equal<01:41:02.880> to<01:41:03.300> team<01:41:03.510> you<01:41:03.539> know<01:41:03.840> in<01:41:04.469> the - -01:41:04.640 --> 01:41:04.650 align:start position:0% -mu nu is equal to team you know in the - - -01:41:04.650 --> 01:41:06.589 align:start position:0% -mu nu is equal to team you know in the -other<01:41:04.829> form<01:41:05.219> it's<01:41:05.489> a<01:41:05.579> kind<01:41:05.789> of<01:41:05.849> Einstein<01:41:06.389> ish - -01:41:06.589 --> 01:41:06.599 align:start position:0% -other form it's a kind of Einstein ish - - -01:41:06.599 --> 01:41:09.680 align:start position:0% -other form it's a kind of Einstein ish -looking<01:41:07.699> penser<01:41:08.699> made<01:41:08.940> up<01:41:09.059> out<01:41:09.179> of<01:41:09.210> team<01:41:09.510> you - -01:41:09.680 --> 01:41:09.690 align:start position:0% -looking penser made up out of team you - - -01:41:09.690 --> 01:41:13.700 align:start position:0% -looking penser made up out of team you -nu<01:41:09.840> is<01:41:10.739> equal<01:41:10.829> to<01:41:11.219> R<01:41:11.340> mu<01:41:11.550> nu<01:41:12.320> these<01:41:13.320> are<01:41:13.499> the<01:41:13.559> two - -01:41:13.700 --> 01:41:13.710 align:start position:0% -nu is equal to R mu nu these are the two - - -01:41:13.710 --> 01:41:21.140 align:start position:0% -nu is equal to R mu nu these are the two -forms<01:41:14.070> of<01:41:14.190> Einstein's<01:41:14.670> equations<01:41:17.960> and<01:41:20.150> this - -01:41:21.140 --> 01:41:21.150 align:start position:0% -forms of Einstein's equations and this - - -01:41:21.150 --> 01:41:23.180 align:start position:0% -forms of Einstein's equations and this -form<01:41:21.449> happens<01:41:21.690> to<01:41:21.989> be<01:41:22.110> convenient<01:41:22.559> for<01:41:22.829> some - -01:41:23.180 --> 01:41:23.190 align:start position:0% -form happens to be convenient for some - - -01:41:23.190 --> 01:41:26.479 align:start position:0% -form happens to be convenient for some -for<01:41:23.760> for<01:41:24.559> sometimes<01:41:25.559> solving<01:41:25.949> the<01:41:26.039> equations - -01:41:26.479 --> 01:41:26.489 align:start position:0% -for for sometimes solving the equations - - -01:41:26.489 --> 01:41:31.910 align:start position:0% -for for sometimes solving the equations -alright<01:41:27.530> let's<01:41:28.530> see<01:41:28.710> should<01:41:28.949> we<01:41:30.860> should<01:41:31.860> we - -01:41:31.910 --> 01:41:31.920 align:start position:0% -alright let's see should we should we - - -01:41:31.920 --> 01:41:33.290 align:start position:0% -alright let's see should we should we -quit<01:41:32.159> at<01:41:32.280> this<01:41:32.429> point<01:41:32.699> that<01:41:32.880> should<01:41:32.999> we<01:41:33.059> go<01:41:33.179> on - -01:41:33.290 --> 01:41:33.300 align:start position:0% -quit at this point that should we go on - - -01:41:33.300 --> 01:41:36.529 align:start position:0% -quit at this point that should we go on -a<01:41:33.329> little<01:41:33.510> more<01:41:34.340> yeah<01:41:35.340> that's<01:41:35.519> a<01:41:35.550> class<01:41:35.760> well - -01:41:36.529 --> 01:41:36.539 align:start position:0% -a little more yeah that's a class well - - -01:41:36.539 --> 01:41:40.069 align:start position:0% -a little more yeah that's a class well -what's<01:41:36.869> what's<01:41:37.050> next<01:41:37.289> week<01:41:38.929> well<01:41:39.929> that's - -01:41:40.069 --> 01:41:40.079 align:start position:0% -what's what's next week well that's - - -01:41:40.079 --> 01:41:42.020 align:start position:0% -what's what's next week well that's -because<01:41:40.289> the<01:41:40.860> students<01:41:41.369> are<01:41:41.550> away<01:41:41.969> for - -01:41:42.020 --> 01:41:42.030 align:start position:0% -because the students are away for - - -01:41:42.030 --> 01:41:47.270 align:start position:0% -because the students are away for -Thanksgiving<01:41:42.449> and<01:41:43.639> but<01:41:44.639> but<01:41:45.479> I'm<01:41:45.749> here<01:41:46.289> let's - -01:41:47.270 --> 01:41:47.280 align:start position:0% -Thanksgiving and but but I'm here let's - - -01:41:47.280 --> 01:41:49.100 align:start position:0% -Thanksgiving and but but I'm here let's -see<01:41:47.429> that's<01:41:47.760> you<01:41:48.539> know<01:41:48.599> we<01:41:48.719> don't<01:41:48.900> meet<01:41:49.019> on - -01:41:49.100 --> 01:41:49.110 align:start position:0% -see that's you know we don't meet on - - -01:41:49.110 --> 01:41:52.040 align:start position:0% -see that's you know we don't meet on -Thursday<01:41:49.559> right<01:41:49.909> we<01:41:50.909> meet<01:41:51.090> on<01:41:51.210> two<01:41:51.420> we<01:41:51.780> meet<01:41:51.929> on - -01:41:52.040 --> 01:41:52.050 align:start position:0% -Thursday right we meet on two we meet on - - -01:41:52.050 --> 01:41:53.420 align:start position:0% -Thursday right we meet on two we meet on -Monday<01:41:52.380> we<01:41:52.650> meet<01:41:52.829> on<01:41:52.949> Monday - -01:41:53.420 --> 01:41:53.430 align:start position:0% -Monday we meet on Monday - - -01:41:53.430 --> 01:41:56.840 align:start position:0% -Monday we meet on Monday -you<01:41:53.610> know<01:41:55.070> all<01:41:56.070> right<01:41:56.130> the<01:41:56.370> next<01:41:56.490> the<01:41:56.820> next - -01:41:56.840 --> 01:41:56.850 align:start position:0% -you know all right the next the next - - -01:41:56.850 --> 01:41:59.810 align:start position:0% -you know all right the next the next -step<01:41:57.270> is<01:41:57.480> to<01:41:57.630> actually<01:41:58.400> let<01:41:59.400> me<01:41:59.460> just<01:41:59.490> tell<01:41:59.760> you - -01:41:59.810 --> 01:41:59.820 align:start position:0% -step is to actually let me just tell you - - -01:41:59.820 --> 01:42:01.460 align:start position:0% -step is to actually let me just tell you -what<01:41:59.940> the<01:41:59.970> next<01:42:00.210> step<01:42:00.390> is<01:42:00.420> the<01:42:00.810> next<01:42:01.080> step<01:42:01.230> is - -01:42:01.460 --> 01:42:01.470 align:start position:0% -what the next step is the next step is - - -01:42:01.470 --> 01:42:03.530 align:start position:0% -what the next step is the next step is -we're<01:42:01.650> gonna<01:42:01.710> make<01:42:01.950> the<01:42:02.280> approximations<01:42:03.030> of - -01:42:03.530 --> 01:42:03.540 align:start position:0% -we're gonna make the approximations of - - -01:42:03.540 --> 01:42:07.130 align:start position:0% -we're gonna make the approximations of -slow<01:42:03.990> things<01:42:04.850> slow<01:42:05.850> weak<01:42:06.180> fields<01:42:06.660> on<01:42:06.840> both - -01:42:07.130 --> 01:42:07.140 align:start position:0% -slow things slow weak fields on both - - -01:42:07.140 --> 01:42:10.160 align:start position:0% -slow things slow weak fields on both -sides<01:42:07.500> of<01:42:07.740> this<01:42:07.860> equation<01:42:08.130> here<01:42:09.050> what<01:42:10.050> we're - -01:42:10.160 --> 01:42:10.170 align:start position:0% -sides of this equation here what we're - - -01:42:10.170 --> 01:42:11.990 align:start position:0% -sides of this equation here what we're -gonna<01:42:10.260> find<01:42:10.560> out<01:42:10.860> is<01:42:11.100> that<01:42:11.130> the<01:42:11.400> left-hand - -01:42:11.990 --> 01:42:12.000 align:start position:0% -gonna find out is that the left-hand - - -01:42:12.000 --> 01:42:15.230 align:start position:0% -gonna find out is that the left-hand -side<01:42:13.160> which<01:42:14.160> is<01:42:14.340> made<01:42:14.550> up<01:42:14.730> out<01:42:14.850> of<01:42:14.910> two - -01:42:15.230 --> 01:42:15.240 align:start position:0% -side which is made up out of two - - -01:42:15.240 --> 01:42:17.780 align:start position:0% -side which is made up out of two -derivatives<01:42:15.570> of<01:42:15.990> the<01:42:16.080> metric<01:42:16.500> the<01:42:17.460> most - -01:42:17.780 --> 01:42:17.790 align:start position:0% -derivatives of the metric the most - - -01:42:17.790 --> 01:42:19.850 align:start position:0% -derivatives of the metric the most -important<01:42:18.360> components<01:42:18.930> of<01:42:19.050> it<01:42:19.170> are<01:42:19.200> just<01:42:19.380> del - -01:42:19.850 --> 01:42:19.860 align:start position:0% -important components of it are just del - - -01:42:19.860 --> 01:42:22.760 align:start position:0% -important components of it are just del -squared<01:42:20.430> Phi<01:42:20.640> or<01:42:21.270> del<01:42:21.540> squared<01:42:21.960> G<01:42:22.530> naught - -01:42:22.760 --> 01:42:22.770 align:start position:0% -squared Phi or del squared G naught - - -01:42:22.770 --> 01:42:25.400 align:start position:0% -squared Phi or del squared G naught -naught<01:42:23.360> we're<01:42:24.360> gonna<01:42:24.450> find<01:42:24.780> out<01:42:24.990> that<01:42:25.200> this<01:42:25.350> is - -01:42:25.400 --> 01:42:25.410 align:start position:0% -naught we're gonna find out that this is - - -01:42:25.410 --> 01:42:30.680 align:start position:0% -naught we're gonna find out that this is -good<01:42:25.710> old<01:42:25.800> Newton's<01:42:26.750> or<01:42:29.150> Laplace's<01:42:30.150> del - -01:42:30.680 --> 01:42:30.690 align:start position:0% -good old Newton's or Laplace's del - - -01:42:30.690 --> 01:42:33.020 align:start position:0% -good old Newton's or Laplace's del -squared<01:42:31.080> Phi<01:42:31.170> on<01:42:31.950> the<01:42:32.100> right<01:42:32.250> hand<01:42:32.490> side<01:42:32.520> and - -01:42:33.020 --> 01:42:33.030 align:start position:0% -squared Phi on the right hand side and - - -01:42:33.030 --> 01:42:36.620 align:start position:0% -squared Phi on the right hand side and -that<01:42:33.060> on<01:42:33.990> the<01:42:34.380> right<01:42:34.560> hand<01:42:34.890> side<01:42:35.510> since<01:42:36.510> the - -01:42:36.620 --> 01:42:36.630 align:start position:0% -that on the right hand side since the - - -01:42:36.630 --> 01:42:39.860 align:start position:0% -that on the right hand side since the -only<01:42:36.950> component<01:42:37.950> of<01:42:38.130> team<01:42:39.090> you<01:42:39.300> knew<01:42:39.480> which<01:42:39.690> is - -01:42:39.860 --> 01:42:39.870 align:start position:0% -only component of team you knew which is - - -01:42:39.870 --> 01:42:42.280 align:start position:0% -only component of team you knew which is -large<01:42:40.230> and<01:42:40.530> significant<01:42:41.280> is<01:42:41.490> T<01:42:41.700> naught<01:42:41.910> naught - -01:42:42.280 --> 01:42:42.290 align:start position:0% -large and significant is T naught naught - - -01:42:42.290 --> 01:42:45.410 align:start position:0% -large and significant is T naught naught -all<01:42:43.290> the<01:42:43.410> other<01:42:43.530> components<01:42:43.920> are<01:42:44.220> small<01:42:44.550> what - -01:42:45.410 --> 01:42:45.420 align:start position:0% -all the other components are small what - - -01:42:45.420 --> 01:42:46.580 align:start position:0% -all the other components are small what -we're<01:42:45.540> gonna<01:42:45.630> get<01:42:45.810> on<01:42:45.930> the<01:42:45.960> right<01:42:46.200> hand<01:42:46.530> side - -01:42:46.580 --> 01:42:46.590 align:start position:0% -we're gonna get on the right hand side - - -01:42:46.590 --> 01:42:49.400 align:start position:0% -we're gonna get on the right hand side -is<01:42:47.460> going<01:42:47.760> to<01:42:47.910> be<01:42:48.030> T<01:42:48.420> naught<01:42:48.630> naught<01:42:48.870> or<01:42:49.290> the - -01:42:49.400 --> 01:42:49.410 align:start position:0% -is going to be T naught naught or the - - -01:42:49.410 --> 01:42:52.730 align:start position:0% -is going to be T naught naught or the -energy<01:42:49.770> density<01:42:50.750> so<01:42:51.750> we're<01:42:51.960> gonna<01:42:52.080> find<01:42:52.440> that - -01:42:52.730 --> 01:42:52.740 align:start position:0% -energy density so we're gonna find that - - -01:42:52.740 --> 01:42:56.030 align:start position:0% -energy density so we're gonna find that -in<01:42:53.250> the<01:42:53.850> approximation<01:42:54.510> where<01:42:55.110> Newton<01:42:55.560> is<01:42:55.860> a - -01:42:56.030 --> 01:42:56.040 align:start position:0% -in the approximation where Newton is a - - -01:42:56.040 --> 01:42:59.480 align:start position:0% -in the approximation where Newton is a -good<01:42:56.340> approximation<01:42:57.290> the<01:42:58.290> time<01:42:59.040> time - -01:42:59.480 --> 01:42:59.490 align:start position:0% -good approximation the time time - - -01:42:59.490 --> 01:43:04.300 align:start position:0% -good approximation the time time -component<01:43:00.180> of<01:43:00.540> this<01:43:00.660> equation<01:43:01.850> two<01:43:02.850> times<01:43:03.180> is - -01:43:04.300 --> 01:43:04.310 align:start position:0% -component of this equation two times is - - -01:43:04.310 --> 01:43:08.120 align:start position:0% -component of this equation two times is -basically<01:43:05.310> del<01:43:05.640> squared<01:43:06.030> Phi<01:43:06.150> is<01:43:07.070> equal<01:43:08.070> to - -01:43:08.120 --> 01:43:08.130 align:start position:0% -basically del squared Phi is equal to - - -01:43:08.130 --> 01:43:10.310 align:start position:0% -basically del squared Phi is equal to -something<01:43:09.060> involving<01:43:09.510> the<01:43:09.600> energy<01:43:09.900> density - -01:43:10.310 --> 01:43:10.320 align:start position:0% -something involving the energy density - - -01:43:10.320 --> 01:43:11.900 align:start position:0% -something involving the energy density -and<01:43:10.350> that's<01:43:10.800> going<01:43:11.160> to<01:43:11.280> tell<01:43:11.520> us<01:43:11.670> the - -01:43:11.900 --> 01:43:11.910 align:start position:0% -and that's going to tell us the - - -01:43:11.910 --> 01:43:15.770 align:start position:0% -and that's going to tell us the -relationship<01:43:12.450> between<01:43:12.900> K<01:43:13.530> or<01:43:13.980> Kappa<01:43:14.490> and<01:43:14.780> four - -01:43:15.770 --> 01:43:15.780 align:start position:0% -relationship between K or Kappa and four - - -01:43:15.780 --> 01:43:20.780 align:start position:0% -relationship between K or Kappa and four -PI<01:43:16.050> G<01:43:17.660> it<01:43:18.660> is<01:43:18.900> not<01:43:19.080> quite<01:43:19.560> that<01:43:19.860> K<01:43:20.190> is<01:43:20.370> equal<01:43:20.670> to - -01:43:20.780 --> 01:43:20.790 align:start position:0% -PI G it is not quite that K is equal to - - -01:43:20.790 --> 01:43:23.930 align:start position:0% -PI G it is not quite that K is equal to -four<01:43:21.000> PI<01:43:21.240> G<01:43:22.310> why<01:43:23.310> not - -01:43:23.930 --> 01:43:23.940 align:start position:0% -four PI G why not - - -01:43:23.940 --> 01:43:33.290 align:start position:0% -four PI G why not -the<01:43:24.750> reason<01:43:25.170> is<01:43:30.080> what's<01:43:31.080> small<01:43:32.000> no<01:43:33.000> no<01:43:33.120> it's - -01:43:33.290 --> 01:43:33.300 align:start position:0% -the reason is what's small no no it's - - -01:43:33.300 --> 01:43:35.450 align:start position:0% -the reason is what's small no no it's -not<01:43:33.390> personal<01:43:33.600> that's<01:43:34.110> not<01:43:34.260> the<01:43:34.350> reason<01:43:34.500> no - -01:43:35.450 --> 01:43:35.460 align:start position:0% -not personal that's not the reason no - - -01:43:35.460 --> 01:43:36.470 align:start position:0% -not personal that's not the reason no -we're<01:43:35.610> just<01:43:35.640> trying<01:43:35.940> to<01:43:36.000> track<01:43:36.180> down<01:43:36.420> a - -01:43:36.470 --> 01:43:36.480 align:start position:0% -we're just trying to track down a - - -01:43:36.480 --> 01:43:38.150 align:start position:0% -we're just trying to track down a -constant<01:43:37.050> now<01:43:37.170> it's<01:43:37.710> this<01:43:37.950> difference - -01:43:38.150 --> 01:43:38.160 align:start position:0% -constant now it's this difference - - -01:43:38.160 --> 01:43:40.540 align:start position:0% -constant now it's this difference -between<01:43:38.580> Phi<01:43:39.030> and<01:43:39.270> G<01:43:39.450> note<01:43:39.600> not<01:43:39.780> the<01:43:39.960> one-half - -01:43:40.540 --> 01:43:40.550 align:start position:0% -between Phi and G note not the one-half - - -01:43:40.550 --> 01:43:43.310 align:start position:0% -between Phi and G note not the one-half -what<01:43:41.550> we're<01:43:41.670> gonna<01:43:41.760> find<01:43:41.970> out<01:43:42.210> as<01:43:42.300> KT<01:43:42.600> as<01:43:42.750> 8<01:43:43.020> pi - -01:43:43.310 --> 01:43:43.320 align:start position:0% -what we're gonna find out as KT as 8 pi - - -01:43:43.320 --> 01:43:47.480 align:start position:0% -what we're gonna find out as KT as 8 pi -G<01:43:44.660> instead<01:43:45.660> of<01:43:45.750> 4<01:43:45.960> PI<01:43:46.140> G<01:43:46.380> it's<01:43:46.560> just<01:43:46.710> as<01:43:46.830> 1/2<01:43:47.280> in - -01:43:47.480 --> 01:43:47.490 align:start position:0% -G instead of 4 PI G it's just as 1/2 in - - -01:43:47.490 --> 01:43:50.870 align:start position:0% -G instead of 4 PI G it's just as 1/2 in -the<01:43:47.580> relationship<01:43:48.150> between<01:43:49.310> the<01:43:50.310> Newtonian - -01:43:50.870 --> 01:43:50.880 align:start position:0% -the relationship between the Newtonian - - -01:43:50.880 --> 01:43:53.570 align:start position:0% -the relationship between the Newtonian -potential<01:43:51.270> and<01:43:51.690> the<01:43:51.840> metric<01:43:52.310> so<01:43:53.310> we'll<01:43:53.460> do - -01:43:53.570 --> 01:43:53.580 align:start position:0% -potential and the metric so we'll do - - -01:43:53.580 --> 01:43:55.670 align:start position:0% -potential and the metric so we'll do -that<01:43:53.700> next<01:43:53.910> time<01:43:54.080> we'll<01:43:55.080> work<01:43:55.290> through<01:43:55.500> that - -01:43:55.670 --> 01:43:55.680 align:start position:0% -that next time we'll work through that - - -01:43:55.680 --> 01:43:59.570 align:start position:0% -that next time we'll work through that -we'll<01:43:55.920> do<01:43:56.040> the<01:43:56.190> approximation<01:43:56.990> on<01:43:58.220> R<01:43:59.220> naught - -01:43:59.570 --> 01:43:59.580 align:start position:0% -we'll do the approximation on R naught - - -01:43:59.580 --> 01:44:01.490 align:start position:0% -we'll do the approximation on R naught -naught<01:43:59.880> we'll<01:44:00.360> see<01:44:00.540> that<01:44:00.690> it's<01:44:00.840> basically<01:44:01.020> del - -01:44:01.490 --> 01:44:01.500 align:start position:0% -naught we'll see that it's basically del - - -01:44:01.500 --> 01:44:03.770 align:start position:0% -naught we'll see that it's basically del -squared<01:44:01.830> Phi<01:44:01.920> and<01:44:02.490> on<01:44:03.180> the<01:44:03.270> right<01:44:03.420> hand<01:44:03.690> side - -01:44:03.770 --> 01:44:03.780 align:start position:0% -squared Phi and on the right hand side - - -01:44:03.780 --> 01:44:05.780 align:start position:0% -squared Phi and on the right hand side -here<01:44:04.260> we'll<01:44:04.620> find<01:44:04.830> that<01:44:04.980> the<01:44:05.130> right-hand<01:44:05.340> side - -01:44:05.780 --> 01:44:05.790 align:start position:0% -here we'll find that the right-hand side - - -01:44:05.790 --> 01:44:07.070 align:start position:0% -here we'll find that the right-hand side -is<01:44:06.000> basically<01:44:06.150> the<01:44:06.600> energy - -01:44:07.070 --> 01:44:07.080 align:start position:0% -is basically the energy - - -01:44:07.080 --> 01:44:10.310 align:start position:0% -is basically the energy -and<01:44:07.650> that'll<01:44:08.610> help<01:44:09.150> us<01:44:09.390> identify<01:44:10.260> the - -01:44:10.310 --> 01:44:10.320 align:start position:0% -and that'll help us identify the - - -01:44:10.320 --> 01:44:12.050 align:start position:0% -and that'll help us identify the -relationship<01:44:10.740> between<01:44:11.310> Newton's<01:44:11.820> constant - -01:44:12.050 --> 01:44:12.060 align:start position:0% -relationship between Newton's constant - - -01:44:12.060 --> 01:44:15.260 align:start position:0% -relationship between Newton's constant -in<01:44:12.660> Kappa<01:44:13.050> but<01:44:14.010> basically<01:44:14.370> Kappa<01:44:14.700> is<01:44:14.910> Newton's - -01:44:15.260 --> 01:44:15.270 align:start position:0% -in Kappa but basically Kappa is Newton's - - -01:44:15.270 --> 01:44:28.820 align:start position:0% -in Kappa but basically Kappa is Newton's -constant<01:44:16.190> x<01:44:17.190> times<01:44:17.730> an<01:44:17.850> eighth<01:44:27.830> cosmological - -01:44:28.820 --> 01:44:28.830 align:start position:0% -constant x times an eighth cosmological - - -01:44:28.830 --> 01:44:31.940 align:start position:0% -constant x times an eighth cosmological -constant<01:44:29.340> goes<01:44:30.470> depending<01:44:31.470> on<01:44:31.530> which<01:44:31.680> side<01:44:31.920> of - -01:44:31.940 --> 01:44:31.950 align:start position:0% -constant goes depending on which side of - - -01:44:31.950 --> 01:44:33.860 align:start position:0% -constant goes depending on which side of -the<01:44:32.160> equation<01:44:32.190> we<01:44:32.760> put<01:44:33.030> it<01:44:33.180> we<01:44:33.360> can<01:44:33.510> put<01:44:33.690> it<01:44:33.780> on - -01:44:33.860 --> 01:44:33.870 align:start position:0% -the equation we put it we can put it on - - -01:44:33.870 --> 01:44:36.340 align:start position:0% -the equation we put it we can put it on -either<01:44:34.080> side<01:44:34.380> of<01:44:34.410> the<01:44:34.650> equation - -01:44:36.340 --> 01:44:36.350 align:start position:0% -either side of the equation - - -01:44:36.350 --> 01:44:39.140 align:start position:0% -either side of the equation -no<01:44:37.350> we<01:44:37.470> haven't<01:44:37.710> addressed<01:44:37.860> it<01:44:38.100> yet<01:44:38.280> we<01:44:39.030> could - -01:44:39.140 --> 01:44:39.150 align:start position:0% -no we haven't addressed it yet we could - - -01:44:39.150 --> 01:44:41.210 align:start position:0% -no we haven't addressed it yet we could -do<01:44:39.240> that<01:44:39.360> very<01:44:39.420> fast<01:44:39.870> but<01:44:40.680> we<01:44:40.830> won't<01:44:41.010> do<01:44:41.100> it - -01:44:41.210 --> 01:44:41.220 align:start position:0% -do that very fast but we won't do it - - -01:44:41.220 --> 01:44:41.930 align:start position:0% -do that very fast but we won't do it -we'll<01:44:41.340> do<01:44:41.430> it<01:44:41.520> next<01:44:41.640> time - -01:44:41.930 --> 01:44:41.940 align:start position:0% -we'll do it next time - - -01:44:41.940 --> 01:44:45.470 align:start position:0% -we'll do it next time -but<01:44:42.660> it's<01:44:42.780> another<01:44:43.020> term<01:44:43.290> in<01:44:43.440> here<01:44:43.650> yeah<01:44:44.480> we - -01:44:45.470 --> 01:44:45.480 align:start position:0% -but it's another term in here yeah we - - -01:44:45.480 --> 01:44:47.930 align:start position:0% -but it's another term in here yeah we -could<01:44:45.630> do<01:44:45.720> that<01:44:45.750> remind<01:44:46.590> me<01:44:46.710> remind<01:44:47.010> me<01:44:47.760> next - -01:44:47.930 --> 01:44:47.940 align:start position:0% -could do that remind me remind me next - - -01:44:47.940 --> 01:44:50.330 align:start position:0% -could do that remind me remind me next -time<01:44:48.150> to<01:44:48.360> do<01:44:48.480> that<01:44:48.630> it's<01:44:49.470> it's<01:44:50.010> really<01:44:50.220> a - -01:44:50.330 --> 01:44:50.340 align:start position:0% -time to do that it's it's really a - - -01:44:50.340 --> 01:44:53.910 align:start position:0% -time to do that it's it's really a -one-line<01:44:50.730> thing - -01:44:53.910 --> 01:44:53.920 align:start position:0% - - - -01:44:53.920 --> 01:44:59.070 align:start position:0% - -[Music] - -01:44:59.070 --> 01:44:59.080 align:start position:0% -[Music] - - -01:44:59.080 --> 01:45:01.689 align:start position:0% -[Music] -the<01:45:00.080> preceding<01:45:00.530> program<01:45:00.680> is<01:45:00.920> copyrighted<01:45:01.100> by - -01:45:01.689 --> 01:45:01.699 align:start position:0% -the preceding program is copyrighted by - - -01:45:01.699 --> 01:45:04.450 align:start position:0% -the preceding program is copyrighted by -Stanford<01:45:01.850> University<01:45:02.390> please<01:45:03.020> visit<01:45:03.710> us<01:45:04.250> at - -01:45:04.450 --> 01:45:04.460 align:start position:0% -Stanford University please visit us at - - -01:45:04.460 --> 01:45:06.910 align:start position:0% -Stanford University please visit us at -stanford.edu - diff --git "a/Physics/Einstein's General Theory of Relativity \357\275\234 Lecture 9 [XG02Yw7u2mc].mp4.part" "b/Physics/Einstein's General Theory of Relativity \357\275\234 Lecture 9 [XG02Yw7u2mc].mp4.part" deleted file mode 100644 index 59ee60d7f1ba29bc41ad89a8e041959576d42396..0000000000000000000000000000000000000000 --- "a/Physics/Einstein's General Theory of Relativity \357\275\234 Lecture 9 [XG02Yw7u2mc].mp4.part" +++ /dev/null @@ -1,3 +0,0 @@ -version https://git-lfs.github.com/spec/v1 -oid sha256:67c63f81a203a60df623cf6b1a863c6aa29fc97de9e6763d1bb944bd468c60b3 -size 3075072 diff --git "a/Physics/Einstein's General Theory of Relativity \357\275\234 Lecture 9 [XG02Yw7u2mc].webp" "b/Physics/Einstein's General Theory of Relativity 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