The EEP is used to justify that if an observer on the ground shoots a beam of li

Question

The EEP is used to justify that if an observer on the ground shoots a beam of light towards a tower, then when the light reaches the tower, it will be red shifted. This is because of what happens in an accelerating spaceship.

The books seem to say this implies time dilation, but I don't completely see why. Could it not just be like any other doppler effect? If I send a sound wave towards you and you are moving towards me, the frequency you observe will be greater than what I send out, but that doesn't mean your clock ticks slower (assume his speed is non-relativistic). Why does it necessarily imply time dilation?

In the derivation used to derive this redshift, from the light beam on a spaceship, special relativity doesn't even seem to come into play, just newtonian physics is used. Anyway if the physics on the ground is indistinguishable from the physics in the spaceship, would that not mean that at the back of the spaceship a clock ticks slower, rather than just appearing to click slower, relative to someone at the front? Otherwise it would seem to me that the EEP would just say that we can't do any experiment to distinguish the physics between the rocket and the gravitational field, though the physics wouldn't really seem to be the same.

Explanation / Answer

The experiments you describe can all be analyzed in a flat spacetime using SR. We can switch back and forth between an inertial frame A and an accelerated frame B. The equivalence principle says that observers in B see a gravitational field. (That is, we have a gravitational field and a varying gravitational potential, but spacetime is flat.)

Observer Alice in inertial frame A describes the source and receiver as accelerating, so she interprets the observations as a kinematic Doppler effect. Bob in B sees source and receiver as both being at rest, so he interprets the effect as gravitational rather than kinematic. He can call it a gravitational Doppler shift, or he can call it a gravitational time dilation. GR doesn't distinguish between the two.

If you wanted to construct a theory in which the distinction between gravitational Doppler shifts and gravitational time dilation was meaningful, an example of how you could do that would be that your theory could predict that clocks of different types that were rate-matched in one location could become mismatched in rate when moved together to some other gravitational potential. This would clearly be an example of gravitational time dilation, not a Doppler shift, because arguments about Doppler shifts can't explain the difference in behavior between the two clocks. But this theory is not a metric theory and it violates the equivalence principle (because Alice sees no gravitational field and therefore can't explain what's going on). Because GR is a metric theory that incorporates the equivalence principle, the distinction you're trying to make isn't a distinction that GR can make.

So the result is that you can call this effect a kinematic Doppler effect, a gravitational Doppler effect, or gravitational time dilation, and all of these interpretations are equally valid

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