A source-free integration method for black hole perturbations and self-force computation: Radial fall

TL;DR

This paper introduces a novel source-free numerical method for integrating black hole perturbation equations in the time domain, effectively handling discontinuities and singularities for radial infall scenarios, with potential applications to EMRI orbits.

Contribution

It proposes a new finite element integration approach that uses jump conditions to avoid direct handling of singular source terms in black hole perturbation calculations.

Findings

Successfully recovers waveforms at infinity.

Demonstrates accurate wave function at particle position.

Applicable to generic EMRI orbits.

Abstract

Perturbations of Schwarzschild-Droste black holes in the Regge-Wheeler gauge benefit from the availability of a wave equation and from the gauge invariance of the wave function, but lack smoothness. Nevertheless, the even perturbations belong to the C\textsuperscript{0} continuity class, if the wave function and its derivatives satisfy specific conditions on the discontinuities, known as jump conditions, at the particle position. These conditions suggest a new way for dealing with finite element integration in time domain. The forward time value in the upper node of the $(t, r^*$) grid cell is obtained by the linear combination of the three preceding node values and of analytic expressions based on the jump conditions. The numerical integration does not deal directly with the source term, the associated singularities and the potential. This amounts to an indirect integration of the wave equation. The known wave forms at infinity are recovered and the wave function at the particle position is shown. In this series of papers, the radial trajectory is dealt with first, being this method of integration applicable to generic orbits of EMRI (Extreme Mass Ratio Inspiral).