Examining the Field of Static Point Scalar Charge in Schwarzschild Spacetime

TL;DR

This paper derives a closed-form solution for the scalar field of a point charge in Schwarzschild spacetime, compares it to existing local expansions, and corrects a previously identified error in the Riemann tensor term.

Contribution

It provides the first closed-form solution for a scalar point charge in Schwarzschild spacetime and corrects an error in Quinn's local expansion expression.

Findings

Derived a closed-form scalar field solution in Schwarzschild spacetime.

Identified and corrected an error in Quinn's local expansion of the scalar field.

Confirmed the match between the global solution and local expansion terms, except for one corrected term.

Abstract

In this paper we examine some aspects of the field of a scalar point charge in curved spacetimes. First we find the closed form solution for the scalar field due to a point charge in Schwarzschild spacetime. Then we expand it locally in powers of r (coordinate distance from the charge) and compare it to Quinn's local expansion for the field of scalar charge. We show a term by term match, except for a mismatch in one term arising due to an error in a Riemann tensor term in Quinn's expression. We show the correct expression, the detailed derivation of which will be subject of a later paper.