Regularized calculation of the retarded Green function in Schwarzschild spacetime

TL;DR

This paper introduces a regularization method for calculating the retarded Green function in Schwarzschild spacetime by isolating and subtracting the direct null geodesic divergence, facilitating more accurate computations.

Contribution

The paper presents a novel multipolar mode subtraction technique to handle the Dirac delta divergence in the retarded Green function for Schwarzschild black hole perturbations.

Findings

Effective separation of divergence improves Green function calculations.

Method demonstrated with scalar field and self-force examples.

Enhances computational approaches for black hole perturbation analysis.

Abstract

The retarded Green function for linear field perturbations of black hole spacetimes is notoriously difficult to calculate. One of the difficulties is due to a Dirac-$\delta$ divergence that the Green function possesses when the two spacetime points are connected by a "direct" null geodesic. We present a procedure which notably aids its calculation in the case of Schwarzschild spacetime by separating this direct $\delta$-divergence from the remainder of the retarded Green function. More precisely, the method consists of calculating the multipolar $\ell$-modes of the direct $\delta$-divergence and subtracting them from the corresponding modes of the retarded Green function. We illustrate the usefulness of the method with some specific calculations in the case of the scalar Green function and self-field for a point scalar charge in Schwarzschild spacetime.