General Relativity from A to B by Robert Geroch

Author:Robert Geroch [Geroch, Robert]
Language: eng
Format: epub
Publisher: The University of Chicago Press
Published: 1978-07-14T16:00:00+00:00

Fig. 52 Given the world-line of an observer, and an event p on that world-line, the observer can decide, for each nearby event, whether or not he regards it as simultaneous with event p.

Fig. 53 The locus of events deemed simultaneous with p under the construction of figure 52 forms a 3-surface in space-time.

Let us next pick, more or less at random, some time, say, 1 second. If event q should happen to be such that Δt = 1 sec, that is, such that 1/2 (t1 − t2) = 1 sec, or such that t1 − t2 = 2 sec, then our individual would of course say that q occurred 1 second after p (since the elapsed time between p and q is, for him, 1 second). We may now ask him to draw, in space-time, that surface which passes through precisely those events q with Δt = 1 sec. The result will be some other surface in space-time, say that illustrated in figure 54. Event q, for example, is entitled to lie on this surface, since for q we have t1 − t2 = 2 sec, and that is the criterion for qs being on our surface. This surface meets the world-line of our individual at some event, denoted by u in figure 54. What do we know about this u? Well, the surface itself is the locus of events whose apparent elapsed time from p is 1 second, and so, since u is on the surface, the apparent elapsed time from p to u must be 1 second. But both p and u are on the world-line of our observer. Thus, the elapsed time between p and u on this world-line, as measured directly by the clock on that world-line, must be 1 second. Again, we ask our individual what he has to say about this surface. He might answer, “This surface represents the locus of all events which are 1 second later than p, or, what is the same thing, this is the locus of all events simultaneous with u (since u is 1 second later than p). In other words, all the points on this surface are simultaneous with each other, and all 1 second later than the points on the first surface we drew.”

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