A new method to integrate (2+1)-wave equations with Dirac's delta functions as sources

TL;DR

This paper develops a novel numerical method for solving (2+1)-dimensional wave equations with Dirac delta sources, specifically applied to scalar perturbations around Kerr black holes, facilitating more accurate gravitational wave modeling.

Contribution

It introduces an analytical technique to remove singularities from the source term, enabling stable numerical integration of wave equations with delta-function sources in Kerr backgrounds.

Findings

Successfully derived a regularized effective source for scalar perturbations.

Demonstrated the method's applicability to Kerr black hole perturbations.

Enhanced the numerical stability of wave equation solutions with point-particle sources.

Abstract

Unlike in the Schwarzschild black hole background, gravitational perturbations in a Kerr black hole background can not be decomposed into simple tensor harmonics in the time domain. Here, we make mode decompositions only in the azimuthal direction. As a first step, we discuss the resulting (2+1)-dimensional Klein-Gordon differential equation for scalar perturbations with a two dimensional Dirac's $\delta$-function as a source representing a point particle orbiting a much larger black hole. To make this equation amenable for numerical integrations we explicitly remove analytically the singular behavior of the source and compute a global, well bahaved, effective source for the corresponding waveform.