Does a charged particle accelerating in a gravitational field radiate?

Question

A charged particle undergoing an acceleration radiates photons.

Let's consider a charge in a freely falling frame of reference. In such a frame, the local gravitational field is necessarily zero, and the particle does not accelerate or experience any force. Thus, this charge is free in such a frame. But, a free charge does not emit any photons. There seems to be a paradox. Does a freely falling charge in a gravitational field radiate?

Answer 1

The paradox is, sort of, resolved as follows: the number of photons changes when you switch between non-inertial frames. This is actually a remarkable fact, and holds also for quantum particles, which can be created in pairs-antipairs, and whose number depends on the frame of reference.

Now, a step back. Forget about gravity for a moment, as it is irrelevant here (we are still in GR, though). Imagine a point charge, which is accelerating with respect to a flat empty space. If you switch to the rest frame of the charge, you observe a constant electric field. When you switch back to the inertial frame, you see the field changing with time at each point. This naturally corresponds to appearing magnetic fields, and hence radiation.

In the presence of gravity the case is absolutely similar. To conclude, switching between non-inertial frames makes a static electric field vary and hence represent a radiation flow.

Another point might be: When moving with charge, no energy is emitted, but when standing in the lab frame, there is a flux observed. However, there is no contradiction here as well, as the energy as a quantity is not defined for noninertial frames.

Answer 2

We have F=m1*a where m1 is the mass of the charged particle and a the acceleration.

The gravitational force is F=G*m1*m2/r^2.

Hence a=m2/r^2 where m2 is the mass of the large body (earth) towards which the charged particle is falling and r is the distance from the center of gravity and G the gravitational constant. There is always an acceleration, though when r becomes very large the acceleration is very small and the photons emitted will be very low energy.

What is happening to the freely falling charged particle is that part of the potential energy it is giving up by falling, turns into radiated photon energy, rather than totally to velocity of fall towards the center of gravity, which will happen to an uncharged particle.