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I am currently having a hard time solving a problem of GR from Lasenby's book. I can't make it more clear than by quoting the exercise:

7.2 A charged object held stationary in a laboratory on the surface of the Earth does not emit electromagnetic radiation. If the object is then dropped so that it is in free fall, it will begin to radiate. Reconcile these observations with the principle of equivalence. Hint: Consider the spatial extent of the electric field of the charge.

Could someone give me a second hint, currently I am stuck because I try to think about an energetic argument: from the laboratory the particle is losing energy from radiation and potential energy from falling, but in the particle none of them is lost. And I am stuck there.

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Don't think about it from an energetic argument. Think about it from the perspectives of what the time evolution of the electric field looks like in the co-moving frame of reference and then from the laboratory frame of reference. – Jerilyn Franz Feb 26 '12 at 22:07
But, wait, I need a clarifier then, I thought that since in free-fall $\nabla_\mu u^\nu =0$ meant that the acceleration was zero, and hence, there should be no radiation emitted. – kηives Feb 27 '12 at 4:27
I would have thought that the body should emit radiation when it is stationary on Earths surface but not in free-fall (althgouh GR is not my forte to put it mildly). Come to think of it, can this be measured? I guess a charged object accelerating a g does not emit much radiation but still might a cute experiment. Maybe I should make this a q. – Bowler Feb 27 '12 at 10:55
Here is a Java applet that helps visualize what happens when a charge accelerates. Play around with it and it will help. – Jerilyn Franz Feb 27 '12 at 14:12
up vote 1 down vote accepted

When the charge is fixed at the surface of the Earth, it is indeed accelerated. But so are we!

When the charge falls with respect to the surface of the Earth, it gets accelerated with respect to us, and hence emits radiation in our reference frame.

It is relative acceleration that matters, because one can write relativistic Maxwells equations in any reference frame, including the comoving frame of the observer (us). In this frame, near the world line of the observer the space-time is always flat, and the charge at rest with respect to it will create a static field. If the charge gets accelerated in this frame, than, as in flat case, it will emit radiation, as seen by the observer.

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I investigated this question. I found the good analysis of this quastion in the arcticle "EQUIVALENCE PRINCIPLE AND RADIATION BY A UNIFORMLY ACCELERATED CHARGE". Now I adduce some conclusions, which is answed on your quastion:

The notion of radiation in terms of receiving electromagnetic energy by an observer is not absolute, but this relative notion is consistent with the principle of equivalence. That is, in a static spacetime, a supported charge does not radiate according to another supported observer; neither does a freely falling charge according to a freely falling ob-server. Also, a freely falling charge does radiate according to a supported observer, and a supported charge does radiate according to a freely falling observer.

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