Simultaneity in a Uniformly Accelerated System
Einstein, in his 1907 paper (see Genesis of General Relativity (1)), introduced the Equivalence Principle, and considered a uniformly accelerated system in order to inquire the law of gravitation. This was the first step towards the general relativity. In this paper, Einstein obtained the following results: (1) a clock in a strong gravitational field runs slowly, and (2) light in a gravitational field curves. More detailed analysis of a uniformly accelerated system was presented in Landau & Zhukov (Japanese tr., 1977) in terms of a Minkowski space. Utilizing their result, Einstein's original inference can be reconstructed very easily. I will reconstruct only (1), since that is harder for ordinary people to understand.
Now, Einstein used three systems, (a) Lorentz system, (b) Uniformly accelerated system A, and (c) local Lorentzian systems for bridging (a) and (b), for determining simultaneity for A; a local Lorentzian system has a uniform velocity v, which corresponds to an instantaneous velocity of (b). Suppose the uniform acceleration of (b) is a. Then, the three kinds of system can be represented in a Minkowski space as follows (remember that we measure time in units of length):
The trajectory of A (its origin) approaches the light ray originating from -(1/a) on the spatial axis of the Lorentz system, and the trajectory itself is a part of a hyperbola. Suppose a clock is located on this trajectory, and call it Clock 1. Another clock is originally located at a distance of X from the origin of A, along its spatial axis. Suppose there is another clock (identical in constitution with Clock 1), Clock 2, on this location, and it moves with the same acceleration on the system A. Now the question: How are Clock 1 and Clock 2 related? This is the question Einstein first pursued in his paper. The problem boils down to determining the surface of simultaneity for system A, at time 0 , and at time τ, both on the Clock 1. Suppose Clock 1 and 2 are synchronized when A has the velocity 0 (i.e., on the spatial axis of the Lorentz system); thus the first half of the problem is already solved. Then what about the second half?
We can utilize the method for determining simultaneity by means of light signals, together with a calculation of the proper time on A's trajectory. Let me give you just the result: The surface of simultaneity can be determined by a straight line originating from the cross point (-1/a) of the spatial axis of the Lorentz system and the light ray to which the trajectory approaches. Thus the solution is obtained from the following figure.
The last equation holds as an approximation, even though the geometry is Lorentzian, not Euclidean; for the time-part is dominant when X is small (recall that, the square of the proper time = [the square of time-part − the square of space-part]). This perfectly coincides with Einstein's original calculation (approximation when X is small). The result,
T = τ(1 + aX)
means that Clock 2 runs faster than Clock 1 (T>τ). Since this result can be transferred to the gravitational field corresponding to this acceleration (gravitational potential is smaller to the left), a clock placed at a location with smaller gravitational potential runs slower.
In retrospect, this consideration of a uniformuly accelerated system already contains, at least implicitly, the problem of a limit case of gravitational field. For, a clock may be located on the left of the origin of A, and suppose the location goes more and more to the left. The trajectory of that clock can be depicted easily (always a hyperbola, but its shape gradually changes from the original hyperbola), and consider what happens if you continue this process (towards the left; notice that the acceleration is the same, a; we are not demanding any larger acceleration). The clock approaches the point <0, -1/a> but can never reach there. But it is easy to see that the clock becomes slower and slower, towards the pace of zero! That is, if the gravitational potential becomes smaller and smaller, the clock's pace goes towards zero. A similar process was later found after Schwarzschild discovered his solution for field equations. And as Zhukov pointed out, the possibility, or rather the necessity for adopting a curved coordinate system for describing gravitational fields was also already contained in the consideration of a uniformly accelerated system. And in this sense, a uniformly accelerated system may be regarded as a miniature model for the whole general relativity!
Reference
ランダウ+ジューコフ『相対性理論入門』（鳥居・広重・金光訳）東京図書、1977 (see pp. 158, 188-192)。
Last modified, July 15, 2003. (c) Soshichi Uchii
suchii@bun.kyoto-u.ac.jp
