The inverse spatial Laplacian of spherically symmetric spacetimes

TL;DR

This paper derives the inverse spatial Laplacian for static, spherically symmetric spacetimes, providing explicit solutions for Schwarzschild and de Sitter backgrounds, which are useful in Hamiltonian formulations of these theories.

Contribution

It presents the first explicit closed-form solutions for the inverse spatial Laplacian in Schwarzschild and de Sitter spacetimes, filling a gap in the literature.

Findings

Closed-form solution for Schwarzschild inverse Laplacian

Expression for de Sitter Green function using hypergeometric functions

Clarification of relation to known spacetime Laplacian Green functions

Abstract

In this paper we derive the inverse spatial Laplacian for static, spherically symmetric backgrounds by solving Poisson's equation for a point source. This is different from the electrostatic Green function, which is defined on the four dimensional static spacetime, while the equation we consider is defined on the spatial hypersurface of such spacetimes. This Green function is relevant in the Hamiltonian dynamics of theories defined on spherically symmetric backgrounds, and closed form expressions for the solutions we find are absent in the literature. We derive an expression in terms of elementary functions for the Schwarzschild spacetime, and comment on the relation of this solution with the known Green function of the spacetime Laplacian operator. We also find an expression for the Green function on the static pure de Sitter space in terms of hypergeometric functions.