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On Gravitational Time Dilation

 

Abstract. In this article we discuss the rates of identical clocks in a uniform gravitational field.

 

Let us suppose that observer A coincides with an emitter of blue light, of frequency f, at some point rA in a uniform field. Assuming A can detect individual waves, he can say that in one second 1/f waves are emitted. Thus he can use the emitter as a clock.

 

At some point rB>rA a second observer, B, is watching A and Aís clock. B has an identical emitter of blue light. We shall suppose that B is the identical twin of A. Owing to red shift the light from Aís clock is observed by B to be running more slowly than his own clock. At Bís location the light from Aís clock has a lower observed frequency than the light from Bís clock.

 

Now if Aís clock is observed to be running more slowly than Bís clock, then every temporal process occurring at rA will be observed to be occurring at a lower rate than the same process at rB. This includes the aging of observer A.

 

The entire scenario is reversible. Owing to blue shift A will observe Bís clock to run faster than his own. Among other things, A will observe B to age faster than he does.

 

Is it an illusion or, like speed-dependent time dilation, is it real? The matter can be resolved if, after a time much greater than (rB-rA)/c, the twins move to a center point at (rB+rA)/2. B will be older than A upon their reunion!

 

The equivalent phenomenon can be observed in Einsteinís celebrated box, uniformly accelerating at a constant rate in gravity-free space. A clock in the bottom of the box should actually run slower than one in the top of the box! Similarly, clocks at the front and rear of a rocket, synchronized at launch, should be out of synch after a period of acceleration. Given the accuracy of modern clocks, the loss of synchronicity should be easily observed.