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only space, then the interval squared would be negative and we would have an |
imaginary interval, the square root of a negative number. Intervals can be either |
real or imaginary in the theory. The square of an interval may be either positive |
or negafive, unlike distance, which has a positive square. When an interval is |
imaginary, we say that the two points have a space-like ¿nterual between them |
(instead of imaginary), because the inberval is more like space than like time. Ôn |
the other hang, if two objects are at the same place in a given coordinate system, |
but difer only in time, then the square of the time is positive and the distances |
are zero and the interval squared is positive; this is called a fữne-like ¿mterual. In |
our diagram of space-time, therefore, we would have a representation something |
like this: at 452 there are ©wo lines (actually, in four dimensions these will be |
“cones,” called light cones and points on these lines are all at zero interval from |
the origin. Where light goes from a given point is always separated from it by a |
zero interval, as we see rom Eq. (17.5). Incidentally, we have just proved that |
1f light travels with speed c in one system, i% travels with speed c in another, |
for 1ƒ the Interval is the same in both systems, i.e., zero in one and zero in the |
other, then to state that the propagation speed of light is Invariant is the same |
as saying that the interval is zero. |
17-3 Past, present, and future |
'The space-time region surrounding a given space-time point can be separated |
into three regions, as shown in EFig. 17-3. Ín one region we have space-like |
Intervals, and in two regions, time-like intervals. Physically, these three regions |
into which space-time around a given poïnt is divided have an interesting physical |
relationship to that point: a physical object or a signal can get om a point in |
region 2 to the event @ by moving along at a speed less than the speed of light. |
Therefore events in this region can afect the point Ó, can have an inÑuence |
on i% from the past. In fact, of course, an object at ? on the negative f-axis 1s |
--- Trang 325 --- |
Tàu: LIGHT-CONE |
b % R©) LIGHT-CONE |
Fig. 17-3. The space-time region surrounding a point at the origin. |
precisely in the “past” with respect to ; it is the same space-point as Ó, only |
carlier. What happened there then, affects Ó now. (Unfortunately, that is the |
way life is.) Another object at Q can get to Ó by moving with a certain speed |
less than e, so if this object were in a space ship and moving, it would be, again, |
the past of the same space-point. 'That is, in another coordinate system, the axis |
of time might go through both @Ø and Q. So all points of region 2 are in the |
“past” of Ó, and anything that happens in this region cøn afect Ó. 'Therefore |
region 2 is sometimes called the øƒfectiue past, or affecting past; i is the locus of |
all events which can afect point Ó in any way. |
Region 3, on the other hand, is a region which we can affect from O, we can |
“hit” things by shooting “bullets” out at speeds less than c. So this is the world |
whose future can be afected by us, and we may call that the a[ƒectiue ƒuture. |
Now the interesting thing about all the rest of space-time, I.e., region 1, ¡is that we |
can neither affect it now from O, nor can it affect us now ø‡ Ó, because nothing |
can go faster than the speed of light. Of course, what happens at Ï can affect us |
later; that 1s, 1f the sun is exploding “right now,” it takes eight minutes before |
we know about it, and it cannot possibly affect us before then. |
What we mean by “right now” is a mysterious thing which we cannot deñne |
and we cannot afect, but it can affect us later, or we could have afected it If |
we had done something far enough in the past. When we look at the star Alpha |
Centauri, we see ib as it was Íour years ago; we might wonder what ït is like |
“now.” “NÑow” means at the same time from our special coordinate system. We |
can only see Alpha Centauri by the light that has come from our past, up to four |
years ago, but we do not know what i§ is doïng “now”; it will take Íour years |
before what it is doing “now” can affect us. Alpha Centauri “now” is an idea |
or concept of our mỉnd; it is not something that is really deñnable physically |
at the moment, because we have to wait to observe it; we cannot even defne 1t |
ripght “now.” Purthermore, the “now” depends on the coordinate system. ÏTÝ, for |
--- Trang 326 --- |
example, Alpha Centauri were moving, an observer there would not agree with |
us because he would put his axes at an angle, and his “now” would be a đjƒerent |
time. We have already talked about the fact that simultaneity is not a unique |
thing. |
'There are fortune tellers, or people who tell us they can know the future, and |
there are many wonderful stories about the man who suddenly discovers that |
he has knowledge about the afective future. Well, there are lots of paradoxes |
produced by that because if we know something is going to happen, then we can |
mmake sure we will avoid it by doing the right thing at the right time, and so on. |
But actually there is no fortune teller who can even tell us the øresen#l 'There |
is no one who can tell us what is really happening right now, at any reasonable |
distance, because that is unobservable. We might ask ourselves this question, |
which we leave to the student to try to answer: Would any paradox be produced |
1f it were suddenly to become possible to know things that are in the space-like |
intervals of region 17 |
17-4 More about four-vectors |
Let us now return to our consideration of the analogy of the Lorentz transfor- |
mation and rotations of the space axes. We have learned the utility of collecting |
together other quantities which have the same transformation properties as the |
coordinates, to form what we call øec#ors, directed lines. In the case of ordinary |
rotations, there are many quantities that transform the same way as z, , and z |
under rotation: for example, the velocity has three components, an ø, , and z- |
component; when seen in a diferent coordinate system, none of the components |
1s the same, instead they are all transformed to new values. But, somehow or |
other, the velocity “itself” has a greater reality than do any of its particular |
components, and we represent it by a directed line. |
We therefore ask: Is it or is it not true that there are quantities which |
transform, or which are related, in a moving system and in a nonmoving system, |
in the same way as ø, , z, and #? From our experience with vectors, we know that |
three of the quantities, like ø, , z, would constitute the three components of an |
ordinary space-vector, but the fourth quantity would look like an ordinary scalar |
under space rotation, because it does not change so long as we do not go into |
a moving coordinate system. Is it possible, then, to associate with some of our |
known “three-vectors” a fourth object, that we could call the “time component,” |
in such a manner that the four obJects together would “rotate” the same wawy |
--- Trang 327 --- |
as position and time in space-time? We shall now show that there is, indeed, at |
least one such thing (there are many of them, in fact): the three componenfs oƒ |
momentum, ơnd the cnergụ œs the từne component, transform together to make |
what we call a “four-vector.” In demonstrating this, since it is quite inconvenient |
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