Does a charged particle accelerating in a gravitational field radiate?
The paradox is resolved as follows: the number of photons changes when you switch between non-inertial frames. This is actually a remarkable fact and it holds also for quantum particles, which can be created in pairs of particles and antiparticles, 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 and carrying away radiation from the charge.
In the presence of gravity the case is absolutely similar. To conclude, switching between non-inertial frames makes a static electric field variable and corresponds to a radiation flow.
Another relevant point: 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.
The charge accelerates. This is proven in a paper written by Bryce DeWitt and Robert Brehme in the '60s, cited in the paper at this link:
https://www.sciencedirect.com/science/article/pii/0003491660900300
Radiation Damping in a Gravitational Field, Bryce S. DeWitt, Robert W. Brehme, Annals of Physics: 9, 220-259 (1960) The charged particle tries to do its best to satisfy the equivalence principle, and on a local basis, in fact, does so. In the absence of an externally applied electromagnetic field the motion of the particle deviates from geodetic motion only because of the unavoidable tail in the propagation function of the electromagnetic field, which enters into the picture nonlocally by appearing in an integral over the past history of the particles.
The article is out of print, and I had to look it up at a university library to read it. The interesting part of the result is that the acceleration of the particle picks up a non-local term that depends on a path integral over the particle's path.
A recent answer by John Rennie linked this question as 'definitive' yet there are issues with the accepted answer by Alexey Bobrick.
The 'number of photons' mentioned makes one think that this is about a purely quantum effect. It is not. Motion and radiation of a point charge in curved space could be handled classically.
While it is generally true that in a curved background it is difficult to define invariantly what is radiated, there is a whole class of setups where it could be done globally and with ease (at least at conceptual level). Let us consider asymptotically flat spacetimes with time-like Killing vector field. Now consider the point charge that starts moving from infinity with constant velocity at $t=-\infty$ interacts with the nontrivial part of the metric around $t=0$ and flies away at $t=+\infty$. Killing v.f. gives us conservation of energy for the system 'charge ${}+{}$ electromagnetic field' and so the difference between initial and final kinetic energy would be well defined, and it had to be the energy radiated away. (Of course, if there is black hole horizon, 'away' may also mean into the black hole). Even for a charge in a bound motion the decay of bound orbits is such a case also would be observer independent. So if a charge's radius of orbit after, say, $10^{20}$ revolutions in a Schwartzschild metric becomes a half of its initial radius then no coordinate transform could erase the radiation at infinity.
One more note: this question is quite different from the question whether a uniformly accelerating charge radiates: there the motion is given, the charge is being dragged by an external force, while here we are dealing with a charge and its field interacting with gravitational field without interference from additional forces, how the charge moves is unknown beforehand. And while for uniformly accelerated charge the main problem is how to extract radiation from the known EM field, in the current question we could sidestep this issue by restricting ourselves to asymptotically flat spacetimes. The main problem now would be to find the motion of the charge and by applying conservation of energy we would then know the energy of that was radiated away.
Jerry Schirmer's answer does link to a correct paper but does not provide an detailed explanation for the 'paradox'.
So here is the paradox resolution in terms of the question: in a freely falling frame gravitational field acting on a charge is indeed zero. However the charge would not be moving along a geodesic. Instead charge would be moving under the DeWitt-Brehme radiation reaction force: $$ m{a^\mu} = f_{{\rm{ext}}}^\mu + {e^2}(\delta _{\;\nu}^\mu + {u^\mu}{u_\nu})\left({{2 \over {3m}}{{Df_{{\rm{ext}}}^\nu} \over {d\tau}} + {1 \over 3}R_{\;\lambda}^\nu {u^\lambda}} \right) + \\{} + 2{e^2}{u_\nu}\int\nolimits_{- \infty}^{{\tau ^ -}} {{\nabla ^{\left[ \mu \right.}}} G_{+ \,\lambda^{'}}^{\left. {\;\nu} \right]}(z(\tau),z(\tau^{'})){u^{\lambda^{'}}}\,d\tau^{'}, \tag{*} $$ where $G_{+ \,\lambda^{'}}^{ {\;\nu}}(x,x^{'})$ is (retarded) Green's function of EM field (it is a bitensor: so indices $\lambda^{'}$ and $\nu$ correspond to (co)tangent bundles at different points), and integration is carried along the past motion of the charge. In the case of absent external force ($f_{\rm ext}=0$) and in a Ricci-flat metric this force is given only by a non-local integral. Incidentally, original 1960 DeWitt & Brehme paper did not include the term with Ricci tensor. This was corrected in 1968 by Hobbs.
Green's functions in GR for massless fields have a richer structure than in flat space: it is generally nonzero inside the future lightcone of a point $x^{'}$, and so the integral would be nonzero. This property reflects the fact that in curved spacetime, electromagnetic waves propagate not just at the speed of light, but at all speeds smaller than or equal to the speed of light, the delay is caused by an interaction between the radiation and the spacetime curvature. So the integral would generally be nonzero and the charge would radiate EM waves.
There is nothing mysterious in the non-local character of the force (*). It is the result of going from the system with infinite degrees of freedom 'charge ${}+{}$ elecromagnetic field' to a finite dimensional description in terms of charge motion alone. Locally point charge is acted upon by electromagnetic field. This electromagnetic field originated on the same charge in the past, been scattered by a gravitational field some distance away from it and produced a potentially non-zero force in the present.
Situation might be easier to understand if we consider the following flat space situation: a point charge and a small dielectric ball at some distance $d$ from it. The ball gains dipole moment in the field of a charge and exerts a certain force on it. Now let us wiggle the ball a little around the moment $t=0$, then perturbations of EM field from this wiggle would be propagating and the original charge would feel additional force at time $t=d/c$. This is a usual Lorentz force, but since the only source of EM field is our charge we can write it as an integral of Green's function over the past worldline of the charge. This Green's function encodes all information about ball and its wiggle. And since there is diffraction on the ball, this function would be nonzero inside the future lightcone of its argument $x^{'}$. The force felt by the charge (both constant contribution and signal from wiggling) now would be written as an integral over the charge past, an expression similar to the DeWitt-Brehme force.
The actual calculations of Green's function in GR are quite complicated and for a sample I would recommend the review
- Poisson, E., Pound, A., & Vega, I. (2011). The motion of point particles in curved spacetime. Living Reviews in Relativity, 14(1), 7, open access web.
Once that has been done the work done by the DeWitt-Brehme force allows us to calculate the energy of radiated EM wave. This has been demonstrated by Quinn & Wald:
Quinn, T. C., & Wald, R. M. (1999). Energy conservation for point particles undergoing radiation reaction. Physical Review D, 60(6), 064009, doi, arXiv.
who have proven that the net energy radiated to infinity equals to minus the net work done on the particle by the DeWitt-Brehme radiation reaction force.