GR Journal: Equivalence Principle & Tidal Forces

[DiracPool]

Say we have an elevator moving through deep space at a constant velocity. We have a light emitter at the "top" of the elevator pointing towards a detector at the bottom of the elevator. We shine the light and notice that the frequency at which the light was emitted at the top is the same as that detected at the bottom of the elevator.

Now take the exact same scenario only instead of the elevator moving at constant velocity, it is instead being accelerated at a constant 10 m/s^2 in the direction toward the top of the elevator. We now again shine light toward the bottom of the elevator. What is the result? Does the detector at the bottom measure the exact same frequency as the emitter at the top? Or is it blue shifted as it would be in a gravitational field, say if the elevator were simply resting on the ground on the surface of the Earth?

My intuition tells me that, in deep space, the frequency would be the same since the top of the elevator and the bottom of the elevator are accelerating in the same direction at the exact same rate, therefore there would be no doppler shift.

On the surface of the Earth, however, you would see a small blue shift only because the gravitational effect in the elevator is not uniform, i.e., it is greater nearer the bottom. Is this assessment correct and, if so, does it display an exception to the equivalence principle? Furthermore, would that exception be classified under the "tidal force" category?

 

[A.T.]

My intuition tells me that, in deep space, the frequency would be the same since the top of the elevator and the bottom of the elevator are accelerating in the same direction at the exact same rate, therefore there would be no doppler shift.

Your intuition is wrong. The receiver at reception time is moving in the inertial rest frame of the emitter at emission time. What the emitter does at reception time, or what the receiver does at emission time, is irrelevant.

 

[DiracPool]

Your intuition is wrong. The receiver at reception time is moving in the inertial rest frame of the emitter at emission time. What the emitter does at reception time, or what the receiver does at emission time, is irrelevant.

I don't understand how this explanation is different from the one I gave above. My central question is whether or not the light is doppler shifted at the receiver in the deep space scenario (accelerated), and why or why not that is the case.

 

[A.T.]

My central question is whether or not the light is doppler shifted at the receiver in the deep space scenario (accelerated)

Yes.

and why or why not that is the case.

Because the inertial emission rest frame, and the inertial reception rest frame are in relative movement.

 

[DiracPool]

Yes.

I'm assuming this "Yes" means that you are saying the light is doppler shifted in the deep space scenario. How does this shift differ from the Earth scenario, though? They can't be identical, can they? I mean the gravitational force (acceleration) on the surface of the Earth isn't the same everywhere inside the elevator as the acceleration is in the elevator in the deep space scenario, is it? Doesn't that make a difference in the frequency shift even though the deep space elevator has a 1G force on it?

 

[PAllen]

The difference between Earth and a uniformly accelerating rocket is tidal gravity, or, in technical terms, the difference between the Rindler metric (uniformly accelerating rocket) and the Schwarzschild metric. Within the precision of any experiment over the height of a building (for example) there is no difference between these. That is, experiments on gravitational redshift on Earth are primarily tests of the equivalence principle: a building on Earth is equivalent to a uniformly accelerating rocket. Analyzed in any inertial frame, both redshifts are just ordinary Doppler.

1) Rocket case: In the inertial frame of the floor at time of emission, the rocket is accelerating, so, by the time the signal reaches the ceiling of the rocket, the ceiling is moving in the emission inertial frame, thus a ceiling detector measures a redshift.

2) Earth surface case: in the inertial (free fall) frame of the bottom of a building (assuming the ground weren't in the way), the ceiling is accelerating. You have, then, exactly the same case as (1). The affect of actual tidal gravity is not measurable within any achievable precision.

That is why this is a test of the principle of equivalence between gravity on Earth and an accelerating rocket.

 

[DiracPool]

That is why this is a test of the principle of equivalence between gravity on Earth and an accelerating rocket.

Ok, that makes sense, thanks.

The affect of actual tidal gravity is not measurable within any achievable precision.

Hypothetically then, if we could construct an elevator that extended half way to the center of the Earth, or say the equivalent distance above the surface, we might be able to detect a tidal gravity affect?

 

[PAllen]

Hypothetically then, if we could construct an elevator that extended half way to the center of the Earth, or say the equivalent distance above the surface, we might be able to detect a tidal gravity affect?

Yes. Also, I should clarify that when I spoke of tidal effects being beyond detectability over building height, that is just for redshift (the topic of this thread). There are other ways tidal gravity for Earth versus rocket pseudo-gravity can be distinguished over small distances; it is just that redshift is not one of those.