Charge orbiting gravitational body

Question

I am currently rather uneducated on the subject, but I was thinking of a general relativity thought experiment. Say I take a charge from infinity and give it velocity to orbit a planet in a circle. Now from the earth's perspective the charge is accelerating thus it should radiate EM waves. But Einstein says there is no experiment you can perform (locally) to determine if you are in free fall in a gravitational field as opposed to floating in free space. But if what I described was the case then there would be an experiment you could perform; you could check for the emission of a photon. Where is the problem?

Answer 1

In a consistent theory the electromagnetic potential should be coupled to gravity. Then the body does not radiate on geodesics.

It can be easily shown. Consider the usual Minkowski space and a charge at rest. Surely it doesn't radiate. But there are coordinates in which its worldline is curved. The radiation is still absent, because affine connection components appear in the Maxwell equations. Same goes for arbitrary spacetimes.

So no, the orbiting charge does not emit radiation.

Answer 2

The equivalence principle does not apply to the local rest frame of the orbiting charge, at least not in the way you are applying it, because such an application of the equivalence principle clearly involves taking into account not only the charge but also the electromagnetic field it carries. However the EM field is not an object that is localized to some neighborhood of the charge worldline, it extends out to spatial infinity. The equivalence principle does not apply to such a configuration-it only applies to those which are confined to local regions of space-time.