Self-force of a scalar field for circular orbits about a Schwarzschild black hole

TL;DR

This paper develops a method to compute the self-force on a scalar particle in circular orbit around a Schwarzschild black hole by decomposing the retarded field into singular and regular parts, enabling accurate numerical calculations.

Contribution

It introduces a Green's function-based approach to isolate the singular field and provides a mode-sum method for calculating the self-force in Schwarzschild spacetime.

Findings

Mode-sum decomposition yields accurate self-force estimates.

Including more terms improves convergence and differentiability.

The method is applicable to scalar particles in black hole orbits.

Abstract

The foundations are laid for the numerical computation of the actual worldline for a particle orbiting a black hole and emitting gravitational waves. The essential practicalities of this computation are here illustrated for a scalar particle of infinitesimal size and small but finite scalar charge. This particle deviates from a geodesic because it interacts with its own retarded field $\psi^\ret$. A recently introduced Green's function $G^\SS$ precisely determines the singular part, $\psi^\SS$, of the retarded field. This part exerts no force on the particle. The remainder of the field $\psi^\R = \psi^\ret - \psi^\SS$ is a vacuum solution of the field equation and is entirely responsible for the self-force. A particular, locally inertial coordinate system is used to determine an expansion of $\psi^\SS$ in the vicinity of the particle. For a particle in a circular orbit in the Schwarzschild geometry, the mode-sum decomposition of the difference between $\psi^\ret$ and the dominant terms in the expansion of $\psi^\SS$ provide a mode-sum decomposition of an approximation for $\psi^\R$ from which the self-force is obtained. When more terms are included in the expansion, the approximation for $\psi^\R$ is increasingly differentiable, and the mode-sum for the self-force converges more rapidly.