Is the Equivalence Principle Contradictory in General Relativity?

[Jonathan Scott]

I don't need to use it for the whole orbit to see that there is a non-zero proper acceleration. I only need to look at a portion of the motion in a region of size limiting to zero. I only need to ask: what is the proper acceleration of the charged particle at this one point on its worldline.

Since the proper acceleration is non-zero, a freefalling observer at the particle will see some kind of anomalous force that can't be accounted for in SR... despite the region being of size limiting to zero.

So we need a better, more precise statement of the EEP. So far no one has been able to present a rigorous mathematical definition. I think we need that to figure out the subtlety that is going on here.

The EEP simply applies to any region which can be approximately described using an inertial frame of reference. The limit of the approximation mainly depends on how accurately you want your results.

You can't necessarily expand the results of the EEP to cover a larger region, as the approximations involved in the EEP may well be of the same order as the approximations involved in the result.

That means for example that although calculations suggest that charge does not radiate in a uniform gravitational field, you cannot extend that to cover a whole orbit, as that is not uniform.

I'd also guess that if you have a charge sitting on the Earth, it may not radiate due to its gravitational acceleration, but it might well radiate due to the rotation of the Earth (effectively causing a slowing torque on the Earth's rotation) and the Earth's orbit around the sun. Obviously, such effects are many orders of magnitude smaller and negligible in practice.

I think that a charge in a uniform gravitational field would fall just like any other matter, with no proper acceleration relative to a falling inertial frame of reference. However, in orbit, where the field isn't uniform, it would presumably in theory radiate and get a slight back-reaction which would mean that its motion would not be exactly the same as free fall, but these effects would be immeasurably tiny.

 

[atyy]

However, in orbit, where the field isn't uniform, it would presumably in theory radiate and get a slight back-reaction which would mean that its motion would not be exactly the same as free fall, but these effects would be immeasurably tiny.

The backreaction is electromagnetic, which is non-gravitational, so we do not expect it to fall freely. An interesting point is, what if we consider a point mass? Would its "gravitational backreaction" cause it not to fall freely? It turns out that it doesn't fall freely on the "background" metric, but it does fall freely on the "background + gravitational backreaction" metric. http://relativity.livingreviews.org/Articles/lrr-2004-6/

 

[Phrak]

Yes, that essentially is the problem. Can you state the equivalence principle mathematically?

I am behind, in this thread, aren't I? Unfortunately I haven't the time nor wherewithall to keep up, but your probing questions have been well appreciated all around, I'm sure. I'm far more interested in the implications a charged body (or bodies) in orbit, anyway.

That cannot be the equivalence principle for as shown by calculations in the journal papers atyy listed, a charged particle orbitting a neutral mass will feel a proper acceleration. This does not reduce to SR even in the region size -> 0 limit.

That was one of the few ways the equivalence principle is usually quoted. To be sure, I'd forgotten that it bothered me, as well. After all, the stress-energy tensor consists of first and second derivatives of the metric.

I think the above quote can be considered an oversimplification in need of several conditionals. After all, gravity waves are a vacuum solution.

 

[atyy]

The Relation between Physical and Gravitational Geometry
Jacob D. Bekenstein
http://arxiv.org/abs/gr-qc/9211017

Has interesting comments on the weak and strong equivalence principle at the start (rest of paper not relevant to this thread). I suspect most of this is in MTW, which unfortunately is not on arXiv.