Scalar-field perturbations from a particle orbiting a black hole using numerical evolution in 2+1 dimensions

TL;DR

This paper introduces a new numerical method for evolving scalar fields caused by particles orbiting black holes in 2+1 dimensions, effectively handling singularities for better perturbation calculations.

Contribution

The paper presents a novel technique that regularizes singularities in 2+1D scalar field evolution around black holes using a puncture scheme and domain splitting.

Findings

Successfully demonstrates the method for circular orbits around Schwarzschild black holes.

Handles the divergence of azimuthal modes at the particle's location.

Provides a framework for efficient self-force computations in black hole perturbation theory.

Abstract

We present a new technique for time-domain numerical evolution of the scalar field generated by a pointlike scalar charge orbiting a black hole. Time-domain evolution offers an efficient way for calculating black hole perturbations, especially as input for computations of the local self force acting on orbiting particles. In Kerr geometry, the field equations are not fully separable in the time domain, and one has to tackle them in 2+1 dimensions (two spatial dimensions and time; the azimuthal dependence is still separable). A technical difficulty arises when the source of the field is a pointlike particle, as the 2+1-dimensional perturbation is then singular: Each of the azimuthal modes diverges logarithmically at the particle. To deal with this problem we split the numerical domain into two regions: Inside a thin worldtube surrounding the particle's worldline we solve for a regularized variable, obtained from the full field by subtracting out a suitable ``puncture'' function, given analytically. Outside this worldtube we solve for the full, original field. The value of the evolution variable is adjusted across the boundary of the worldtube. In this work we demonstrate the applicability of this method in the example of circular orbits around a Schwarzschild black hole (refraining from exploiting the spherical symmetry of the background, and working in 2+1 dimensions).