Scalar, electromagnetic, and gravitational self-forces in weakly curved spacetimes

Abstract

We calculate the self-force experienced by a point scalar charge $q,$ a point electric charge $e,$ and a point mass m moving in a weakly curved spacetime characterized by a time-independent Newtonian potential $\ensuremath{\Phi}.$ We assume that the matter distribution responsible for this potential is bounded, so that $\ensuremath{\Phi}\ensuremath{\sim}\ensuremath{-}M/r$ at large distances r from the matter, whose total mass is $M;$ otherwise, the Newtonian potential is left unspecified. (We use units in which $G=c=1.)$ The self-forces are calculated by first computing the retarded Green's functions for scalar, electromagnetic, and (linearized) gravitational fields in the weakly curved spacetime, and then evaluating an integral over the particle's past world line. The self-force typically contains both a conservative and a nonconservative (radiation-reaction) part. For the scalar charge, the conservative part of the self-force is equal to $2\ensuremath{\xi}{q}^{2}M\mathit{r}^{/r}^{3},$ where $\ensuremath{\xi}$ is a dimensionless constant measuring the coupling of the scalar field to the spacetime curvature, and $\mathit{r}^$ is a unit vector pointing in the radial direction. For the electric charge, the conservative part of the self-force is ${e}^{2}M\mathit{r}^{/r}^{3}.$ For the massive particle, the conservative force vanishes. For the scalar charge, the radiation-reaction force is $\frac{1}{3}{q}^{2}d\mathit{g}/dt,$ where $\mathit{g}=\ensuremath{-}\mathbf{\ensuremath{\nabla}}\ensuremath{\Phi}$ is the Newtonian gravitational field. For the electric charge, the radiation-reaction force is $\frac{2}{3}{e}^{2}d\mathit{g}/dt.$ For the massive particle, the radiation-reaction force is $\ensuremath{-}\frac{11}{3}{m}^{2}d\mathit{g}/dt.$ Our result for the gravitational self-force is disturbing: a radiation-reaction force should not appear in the equations of motion at this level of approximation, and it should certainly not give rise to radiation antidamping. In the last section of the paper we prove that while a massive particle in a vacuum spacetime is subjected only to its self-force, it is also subjected to a matter-mediated force when it moves in a spacetime that contains matter; this force originates from the changes in the matter distribution that are induced by the presence of the particle. We show that the matter-mediated force contains a radiation-damping term that precisely cancels out the antidamping contribution from the gravitational self-force. When both forces are combined, the equations of motion are conservative, and they agree with the appropriate limit of the standard post-Newtonian equations of motion.