Equations of Motion in General Relativity of a Small Charged Black Hole

TL;DR

This paper derives the equations of motion for a small charged black hole in general relativity, accounting for electromagnetic and gravitational interactions without infinities, and introduces a new explicit form for the tail term.

Contribution

It provides a novel derivation of black hole equations of motion from Einstein-Maxwell equations without infinities, including a new explicit tail term expression.

Findings

Derived equations of motion for a charged black hole in external fields.

Introduced a new explicit form for the tail term in the equations.

Demonstrated the model's consistency with established equations after renormalization.

Abstract

We present the details of a model in general relativity of a small charged black hole moving in an external gravitational and electromagnetic field. The importance of our model lies in the fact that we can derive the equations of motion of the black hole from the Einstein-Maxwell vacuum field equations without encountering infinities. The key assumptions which we base our results upon are that (a) the black hole is isolated and (b) near the black hole the wave fronts of the radiation generated by its motion are smoothly deformed spheres. The equations of motion which emerge fit the pattern of the original DeWitt and Brehme equations of motion (after they "renormalise"). Our calculations are carried out in a coordinate system in which the null hypersurface histories of the wave fronts can be specified in a simple way, with the result that we obtain a new explicit form, particular to our model, for the well-known "tail term" in the equations of motion.