Electrostatic boundary value problems in the Schwarzschild background

TL;DR

This paper derives the electrostatic potential and Green's function in Schwarzschild spacetime, solves boundary value problems, and analyzes the effects of charge distributions near black holes, including self-force calculations.

Contribution

It introduces a method to solve electrostatic boundary value problems in Schwarzschild space, including multipole expansions and self-force analysis, extending previous work to curved spacetime.

Findings

Multipole moments vanish except for the monopole in the black hole field.

Lowering a test charge near the horizon forms a Reissner--Nordstrom black hole.

Electrostatic self-force on a stationary charge is computed in Schwarzschild space.

Abstract

The electrostatic potential of any test charge distribution in Schwarzschild space with boundary values is derived. We calculate the Green's function, generalize the second Green's identity for p-forms and find the general solution. Boundary value problems are solved. With a multipole expansion the asymptotic property for the field of any charge distribution is derived. It is shown that one produces a Reissner--Nordstrom black hole if one lowers a test charge distribution slowly toward the horizon. The symmetry of the distribution is not important. All the multipole moments fade away except the monopole. A calculation of the gravitationally induced electrostatic self-force on a pointlike test charge distribution held stationary outside the black hole is presented.