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light path.
a “blob” in a new kind of world, and that we look at this “blob” from difÑferent
points of view when we are moving at diferent velocities. Thhis new world, this
geometrical entity in which the “blobs” exist by occupying position and taking
up a certain amount of time, is called space-tme. A given point (z,,z,É) in
space-time is called an cuenứ. Imagine, for example, that we plot the #-positions
horizontally, and z in two other directions, both mutually at “right angles” and
at “ripht angles” to the paper (P), and time, vertically. Now, how does a moving
particle, say, look on such a diagram? If the particle is standing stiH, then it has
a certain ø, and as time goes on, it has the same ø, the same #, the same 4; sO
its “path” is a line that runs parallel to the f-axis (Eig. 17-1 a). On the other
hang, if it drifts outward, then as the time goes on z# increases (Eig. 17-1 b).
So a particle, for example, which starts to drift out and then slows up should
have a motion something like that shown in Fig. 17-I(c). A particle, in other
words, which is permanent and does not disintegrate is represented by a line in
space-time. Á particle which disintegrates would be represented by a forked line,
because it would turn into 6wo other things which would start from that poïnt.
'What about light? Light travels at the speed e, and that would be represented
by a line having a certain ñxed slope (Eig. 17-1 d).
Now according to our new idea, ïÝ a given event occurs to a particle, say IÝ
it suddenly disintegrates at a certain space-time point into two new ones which
follow some new tracks, and this interesting event occurred at a certain value
of z and a certain value of £, then we would expect that, if this makes any sense,
we just have to take a new pair of axes and turn them, and that will give us the
new ý and the new # in our new system, as shown in Fig. 17-2(a). But this is
wrong, because Eq. (17.1) is not ezacflu the same mathematical transformation
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cíí ct ⁄“
\Y. “
Xx x Xx
(a)NOT CORRECT (b) CORRECT
Fig. 17-2. Two views of a disintegrating particle.
as Bq. (17.2). Note, for example, the difference in sign between the two, and the
fact that one is written in terms of cos Ø and sin Ø, while the other is written with
algebraic quantities. (Of course, iÈ is not impossible that the algebraic quantities
could be written as cosine and sine, but actually they cannot.) But still, the two
expressions øre very similar. As we shall see, i% is not really possible bo think
Of space-time as a real, ordinary geometry because of that diference in sign. In
fact, although we shall not emphasize this point, it turns out that a man who is
moving has to use a set of axes which are inclined equally to the light ray, using
a special kind of projection parallel to the z/- and f/-axes, for his ø“ and #, as
shown in EFig. I7-2(b). We shall not deal with the geometry, since it does not
help much; it is easier to work with the equations.
17-2 Space-time intervals
Although the geometry of space-time is not Buclidean in the ordinary sense,
there 7s a geometry which is very similar, but peculiar in certain respects. If
this idea of geometry is right, there ought to be some functions of coordinates
and time which are independent of the coordinate system. Eor example, under
ordinary rotations, if we take two points, one at the origin, for simplicity, and
the other one somewhere else, both systems would have the same origin, and
the distance from here to the other point is the same in both. That is one
property that is independent of the particular way of measuring it. The square
of the distanece is #2 + 2 + z”. Now what about space-time? It is not hard to
demonstrate that we have here, also, something which stays the same, namely, the
combination e?£2 — #2 — 92 — z is the same before and after the transformation:
c2t2 — „2 — 2 — z2 = c3? — g2 — g2 — z3, (17-3)
This quantity is therefore something which, like the distance, is “real” in some
sense; it is called the 7m‡eruøl between the two space-time points, one of which is,
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in this case, at the origin. (Actually, oŸ course, it is the interval squared, just
as #2 -Ƒ 2 + z2 is the distance squared.) We give it a diferent name because it
1s In a diferent geometry, but the interesting thing is only that some signs are
reversed and there is a c in it.
Let us get rid of the œ; that is an absurdity iŸ we are goïng to have a wonderful
space with zˆ's and #'s that can be interchanged. One of the confusions that could
be caused by someone with no experience would be to measure widths, say, by the
angle subtended at the eye, and measure depth in a difÑferent way, like the strain
on the muscles needed to focus them, so that the depths would be measured in
feet and the widths in meters. Then one would get an enormously complicated
mness oŸ equations in making transformations such as (17.2), and would not be
able to see the clarity and simplicity of the thing for a very simple technical
reason, that the same thing is being measured in two diferent units. Now in Eqs.
(17.1) and (17.3) nature is telling us that time and space are equivalent; tỉme
becomes space; (he should be mmeasured ?ín the same uniis. What distance 1s a
“second”? It is easy to fgure out from (17.3) what it is. It is 3 x 10Ÿ meters, fhe
địstance that light tUould go ?m one second. In other words, iŸ we were to measure
all distances and times in the same units, seconds, then our unit of distance
would be 3 x 10 meters, and the equations would be simpler. Or another way
that we could make the units equal is to measure time in meters. What is a
meter of time? AÁ meter of tỉme is the time it takes for light to go one meter,
and is therefore 1/3 x 107 see, or 3.3 billionths of a second! We would like, in
other words, to put all our equations in a system of units in which c= 1. H time
and space are measured in the same units, as suggested, then the equations are
obviously much simplified. They are
AM...
g0 (17⁄4)
Z =Z,
trằ t— Uuz
t2 —ạt2 — 2 — y2 =12— g2 — 2 — z2, (17.5)
TÍ we are ever unsure or “frightened” that after we have this system with c= l
we shall never be able to get our equations right again, the answer is quite the
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opposite. Ït ¡is much easier to remember them without the c's in them, and ï£ 1s
always easy to put the đs back, by looking after the dimensions. For instance,
in V1 — 2, we know that we cannot subtract a velocity squared, which has units,
from the pure number 1, so we know that we must đivide u2 by e? in order to
make that unitless, and that is the way it goes.
'The diference between space-time and ordinary space, and the character of an
interval as related to the distance, is very interesting. According to formula (17.5),
1ƒ we consider a point which in a given coordinate system had zero time, and