# The Equivalence principle of General Relativity and the Doppler Effect

I am studying General Relativity and am trying to understand the Equivalence Principle more thoroughly. Basically, it is said that if you are in a uniformly accelerated frame of reference in free space away from any gravitational field, let's say an elevator, there is no way for the observer in the elevator to know whether he is in an accelerated 'box' or a uniform gravitational field.

But suppose a person is at rest in free space relative to the surrounding electromagnetic radiation (photons hitting the box are of equal wavelength in all directions). When the box accelerates uniformly, its speed increases in the direction of acceleration and thus the photons on the top of the box (the box is accelerating in the 'top' direction) will hit it with shorter wavelength that photons hitting the bottom. This would create a force imbalance, but we will say there is a rocket on the bottom which increases its thrust in order to maintain uniform acceleration. The person on the inside of the box can detect neither the rocket nor the photons. However, as the speed increases as a result of the acceleration, the box will be squeezed as a result of the speed increase (the force from the redshifted photons plus the rocket thrust always equals the force from the blueshifted photons hitting the top of the box resulting in an ever increasing compressive force). If the observer inside the box continually sends photons from the bottom of the box, which are then reflected back down, the observer will find that the time it takes the photon to make its trip gets increasingly less and thus measures that the box is shrinking (or that time is slowing down at an increasing rate?). The observer in a uniform gravitational field would find no such thing. For small accelerations (or equivalently weak gravitational fields) this difference would be small. For very large accelerations, the effect would be much more noticeable.

Furthermore, Einstein showed that light should fall in a gravitational field because in the equivalent accelerating frame of reference, a beam of light could be emitted from the center of the box perpendicular to the direction of motion and it would 'trace' a parabolic path from the perspective of the internal observer as a result of the acceleration. But for a very large acceleration, again, the box would be squeezed, and the observer should therefore measure the light falling further than the observer in the equivalent gravitational field.

So my question is, shouldn't effect of the equivalence principle depend on the magnitude of the acceleration/gravitational field and the strength of the surrounding electromagnetic field?

Why do you say that the box doesn't shrink in a uniform gravitational field? The photons outside the box also experience the field! Thus, the photons "falling" on top of the box are slightly blue-shifted while the photons hitting it from below are slightly red-shifted. And the box squeezes in the same amount as its rocket analogue.

Let me refer to the thought experiment that you describe as 'the background radiation experiment'. (As I understand it the thought experiment applies for any form of background radiation, not necessarily microwave radiation.)

The thought experiment that you describe is rather laborious. My supposition is that you wanted to avoid a scenario where background radiation is measured directly.

What you describe is a scenario where background radiation impacts the spacecraft, having a physical effect on the spacecraft, and then the scientist inside the spacecraft measures something that happens inside.

That is my supposition as to what your intention is.

Let me first discuss a reduced version of the thought experiment that you proposed.
A spacecraft monitors the Cosmic Microwave Background radiation continuously. Over time the amount of doppler shift of the CMB will change, from which the spacecraft can infer its rate of acceleration relative to the Universe.

Does that constitute a counter-example to the principle of equivalence? It does not. The principle of equivalence applies exclusively to localized phenomena. Obviously the CMB is not local; it's the total opposite of local. The CMB originated from the furthest distance that is observable at all.

Now the thought experiment that you proposed.

The thing is: any form of having a physical effect constitutes a measurement.

In the end it doesn't matter: measuring as directly as possible, or constructing some Rube Goldberg device as go-between; every form of measurement is the observation of a physical effect.

For example, the measurement of the Pound-Rebka experiment. The absorption was not measured directly. Instead they measured the amount of gamma-rays that were not absorbed. The gamma-rays were detected with a scintilation counter.
In actual setups the only way to get what you want is to set up a concatenation of effects. Direct measurement is an idealization that doesn't actually exist.

In your thought experiment most details are interexchangable; the one essential step is the background radiation having a physical effect on the spacecraft. Hence no challenge to the principle of equivalence.