# Resource for mass-charge interaction

My GR professor mentioned that trajectories of charged particles in GR are not the same as those of non-charged particles (i.e. charged particles don't move on geodesics). As I understood it, the curvature caused by the mass distribution distort the electric field and push the particle to self-interact, effectively creating mass-charge interaction.

Can you refer to any good source that discuss this issue?
(I prefer sources not much above undergraduate level)

• They are not the same in Newtonian mechanics, either, see e.g. Bremsstrahlung losses in accelerators. People who are building accelerators have to deal with that all the time. – CuriousOne Jun 27 '15 at 17:49
• I haven't heard of this. The principle of equivalence says a cannonball and a feather fall at the same rate. And an electron too. Surely if there's no other net charge around, it follows the same geodesic as a hammer, and there is no Bremsstrahlung. Can you ask your GR professor some more about this? – John Duffield Jun 28 '15 at 19:09
• The principle of equivalence says that at the point you can nullify gravitational force, but at the neighbourhood of that point, as close as you want, the spacetime may stay curved. Point-like charge won't feel any gravitational forces, but as I understand it, it's electrical field will distort from spherically symmetric at the neighbourhood of the point. The resultant field, having non trivial divergence at the point (due to "compression" and "looseness" of radial field lines) will apply force back on the charge. – Alexander Jun 28 '15 at 22:30
• Alexander, where are you getting this from? The principle of equivalence does not say you can nullify gravitational force at any point. And it's quantum field theory, not quantum point-particle theory, despite what you might read about electrons looking pointlike in scattering experiments. It's the wave nature of matter, not the point-particle nature of matter. Matter is made of charged particles, which we make out of light in pair production. Charged particles are not immune to gravitational force. Light curves and matter falls down whether it's charged or not. – John Duffield Jun 29 '15 at 18:11

As far as I understood from my so far cursory look into a living review article by Poisson, Pound and Vega on The Motion of Point Particles in Curved Spacetime, it's a bit messy. But I think if you manage to go through GR, this should be manageable, as well. It will probably help if you've dealt with Green's functions before and even better if you've seen (some) renormalization techniques before.

Basically it boils down to the problem that already the path of a point particle is difficult to nail down. A quick look at the Coulomb force (and the associated energy term) tells you that even in flat space-time and a comoving observer - so basically just plain old electrostatics - you get divergences on the world line of the charged particle.

Taking more care to see where this goes then leads to equations of motions that don't look like the geodesics of the space-time you started with. This is where the idea "point particles don't follow geodesics" come from.

On page 21 they refer then to a redefinition of the space-time proposed by Detweiler and Whiting where the particle is included from the beginning and one rearrives at a description of its motion by a geodesic.

I happened to be at seminar on equations of motion in GR, two years ago - sadly, back then most of it went way over my head. But apparently, they finished the proceedings of the seminar (the link is right on the seminar webpage). Some (most? all?) of the chapters appeared on the arxiv. During the seminar, the organizers wanted this book to be a useful introduction to the topic. I didn't get a look at it yet, not even on the arxiv, but that's definitely also worth a look, specifically the chapter contributed by Pound, as he's also behind the LRR article. And the chapter contains the magical word "introduction"...

• Thanks for the answer Seth. But I read this: "In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle". And I have to say it sounds like it's at odds with general relativity. When you drop an electron it doesn't experience a force in the Newtonian sense. It doesn't radiate. If you're free-falling alongside it, you don't see it radiate. Instead it radiates when I stop it. Then the potential energy that was converted into kinetic energy is radiated away and you're left with a mass deficit. – John Duffield Jun 30 '15 at 19:55
• If I'm not mistaken, that sentence refers to the usual situation where you retain your background space-time. It could be understood as a remnant of sticking to the wrong metric - after all, the e.o.m. can be seen as a geodesic with regard to a new metric, given in the first line of page 22. Also: Mass/energy arguments with radiation can be tricky as you have to be very careful what mass (Komar, ADM, ...) you are talking about. Especially since you have (iirc) no local concept for mass. – Wraith of Seth Jul 1 '15 at 13:07
• Again we seem to be at odds with relativity here Seth. The mass of a body is a measure of its energy-content. If a body gives off the energy L in the form of radiation, its mass diminishes. – John Duffield Jul 2 '15 at 7:14