Electrostatic Potential of a Point Charge in a Brans-Dicke Reissner-Nordstrom Field

TL;DR

This paper analyzes the electrostatic potential of a point charge in Brans-Dicke Reissner-Nordstrom spacetime, deriving a fundamental solution and demonstrating linear stability under electrostatic perturbations.

Contribution

It develops a method based on the Copson approach to solve the Maxwell equation in this spacetime and proves stability for various background cases.

Findings

Derived a closed-form fundamental solution for the electrostatic potential.

Showed the solution is regular except at the charge and its image point.

Proved the spacetime is linearly stable under electrostatic perturbations.

Abstract

We consider the Brans-Dicke Reissner-Nordstrom spacetime in isotropic coordinates and the electrostatic field of an electric point charge placed outside its surface of inversion. We treat the static electric point charge as a linear perturbation on the Brans-Dicke Reissner-Nordstrom background. We develop a method based upon the Copson method to convert the governing Maxwell equation on the electrostatic potential generated by the static electric point charge into a solvable linear second order ordinary differential equation. We obtain a closed form fundamental solution of the curved space Laplace equation arising from the background metric, which is shown to be regular everywhere except at the point charge and its image point inside the surface of inversion. We also develop a method that demonstrates that the solution does not contain any other charge that may creep into the region that lies beyond the surface of inversion and which is not covered by the isotropic coordinates. The Brans-Dicke Reissner-Nordstrom spacetime therefore is linearly stable under electrostatic perturbations. This stability result includes the three degenerate cases of the fundamental solution that correspond to the Brans Type 1, the Reissner-Nordstrom and the Schwarzschild background spacetimes.