Quoting from the abstract of The Motion of Point Particles in Curved Spacetime <https://web.archive.org/web/20151203142700/http://relativity...>
> This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. 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. In the near zone the field acts on the particle and gives rise to a self-force that prevents the particle from moving on a geodesic of the background spacetime. The self-force contains both conservative and dissipative terms, and the latter are responsible for the radiation reaction. The work done by the self-force matches the energy radiated away by the particle.
Note especially that the charged particle won’t actually follow the geodesic. Because the synchrotron radiation carries away some energy, the particle will be continually pushed off of the geodesic; it will spiral inwards instead of orbiting forever.