Uniqueness of the electrostatic solution in Schwarzschild space

TL;DR

This paper proves the uniqueness of electrostatic solutions in Schwarzschild space, demonstrating that static charge distributions lead to a single solution regardless of electric or magnetic configurations, using Green's identity on Riemannian manifolds.

Contribution

It provides a rigorous proof of the uniqueness of electrostatic solutions in Schwarzschild space, applicable to electric and magnetic fields, independent of spacetime geometry.

Findings

Electrostatic solutions in Schwarzschild space are unique.

The proof uses Green's identity with p-forms on (pseudo) Riemannian manifolds.

Uniqueness holds for both electric and magnetic configurations.

Abstract

In this Brief Report we give the proof that the solution of any static test charge distribution in Schwarzschild space is unique. In order to give the proof we derive the first Green's identity written with p-forms on (pseudo) Riemannian manifolds. Moreover, the proof of uniqueness can be shown for either any purely electric or purely magnetic field configuration. The spacetime geometry is not crucial for the proof.