Undergrad Questions about the general principle of relativity

[Frank Castle]

One of the founding principles in GR is the principle of general relativity, which loosely states that all coordinate frames (inertial and non-inertial) are equivalent in the sense that the laws of physics are invariant.

My question is, does the justification for this come from Einstein's equivalence principle, i.e. that, locally, inertial acceleration is equivalent to the effects of a gravitational field, and that the physics in a free-fall frame are those of special relativity (SR)?

In special relativity, the motivation for the laws of physics being invariant under Lorentz transformations follows from the fact that Maxwell's equations are not invariant under Galilean transformations, whereas Newtonian mechanics is. However, there was much evidence that Maxwell's equations do hold across different inertial frames of reference, so both theories could not hold simultaneously. The lack of evidence of an aether as a special background against which Maxwell's equations hold suggested that an alternative was needed. The laws of electromagnetism were shown to be invariant under Lorentz transformations which implies that the speed of light is independent. Einstein postulated that all the non-gravitational laws of physics should thus be invariant under Lorentz transformations, otherwise different observers would measure different fundamental physics simply because they are in relative motion. This would lead us back to the notion of a fixed background, i.e. an aether, and back to square one.

Is the motivation in GR that any local observer should be able to transform to a local free-fall frame in which the laws of physics reduce to those of SR. Furthermore, an observer must not be able to distinguish, locally, between the effects of an accelerated frame of reference frame and a gravitational field. This can only be the case if all local observers, regardless of their reference frame, agree on the (formulation) of the laws of physics?

 

[Nugatory]

My question is, does the justification for this come from Einstein's equivalence principle, i.e. that, locally, inertial acceleration is equivalent to the effects of a gravitational field, and that the physics in a free-fall frame are those of special relativity (SR)?

No. The principle of relativity dates back to Galileo, long before Einstein advanced the equivalence principle. One of the key reasons that SR was accepted is that it preserved the principle of relativity.

Is the motivation in GR that any local observer should be able to transform to a local free-fall frame in which the laws of physics reduce to those of SR.

That's not quite right; the word "free-fall" is overly constraining, as the laws of physics reduce to SR in non-inertial frames as well as inertial ones. What matters is that the frame be small enough that tidal effects are not detectable.

Furthermore, an observer must not be able to distinguish, locally, between the effects of an accelerated frame of reference frame and a gravitational field.

The equivalence principle, expressed this way, may be better understood as a heuristic showing that gravitational effects traditionally attributed to a classical force can instead be attributed to the effects of spacetime curvature.

 

[dextercioby]

I understand from your post that you ask if the principle of general covariance is (partially or fully) a logical consequence of some form of the principle of equivalence. I think the only intersection between the two principles is that, mathematically, vierbein fields are possible. I kindly ask @PeterDonis to provide you with his opinion on this matter. He is much more knoweldgeable than me on the physical foundation of relativity.

 

[Frank Castle]

No. The principle of relativity dates back to Galileo, long before Einstein advanced the equivalence principle. One of the key reasons that SR was accepted is that it preserved the principle of relativity.

That's not quite right; the word "free-fall" is overly constraining, as the laws of physics reduce to SR in non-inertial frames as well as inertial ones. What matters is that the frame be small enough that tidal effects are not detectable.

The equivalence principle, expressed this way, may be better understood as a heuristic showing that gravitational effects traditionally attributed to a classical force can instead be attributed to the effects of spacetime curvature.

What is the argument for why the laws of physics should reduce to SR in non-inertial frames too? I thought it was only in local free-fall frames that this happened.

Can one argue that the reason why the laws of physics must be invariant in arbitrary frames of reference is because if they weren't then one would be possible to perform an experiment in a given local frame of reference and be able to determine whether or not one is in a gravitational field, or accelerating uniformly in flat spacetime?

 

[Frank Castle]

I understand from your post that you ask if the principle of general covariance is (partially or fully) a logical consequence of some form of the principle of equivalence. I think the only intersection between the two principles is that, mathematically, vierbein fields are possible. I kindly ask @PeterDonis to provide you with his opinion on this matter. He is much more knoweldgeable than me on the physical foundation of relativity.

How does one argue that the laws of physics should be valid in arbitrary frames of reference though? I thought that one could argue this from the equivalence principle, since if they weren't valid in arbitrary frames of reference, then it would be possible to determine locally whether or not you're in a gravitational field by carrying out local experiments.

 

[Frank Castle]

I’ve had a further think (and read around) about this, and I think it’s the case that although the principle of equivalence and general covariance do not imply one another, it is the case that the principle of equivalence, in particular, the EEP, that gives physical content to the principle of general covariance. The reason being that, by itself, the principle of general covariance is vacuous since any physical theory can be expressed in a general covariant form. What gives it physical content is that the EEP implies that the (non-gravitational) laws of physics reduce to those of special relativity in the presence of an arbitrary gravitational field, so long as one is in a local inertial frame of reference. Given this, we can use the notion of general covariance to re-express the equations in this frame in tensorial form. These equations will then reduce to their special relativistic form whenever one is in a local inertial reference frame. Now that they are in tensorial form, if they hold in one frame of reference, they must hold in all frames of reference (both inertial and non-inertial). This is exactly the requirement of general covariance.