Gravitational reaction force on a particle in the Schwarzschild background

TL;DR

This paper introduces a new method to compute the gravitational reaction force on a particle orbiting a black hole, using the tail part of the Green function at the Teukolsky equation level, with initial calculations for Schwarzschild black holes.

Contribution

A novel formalism for calculating the gravitational reaction force using the tail part of the Green function at the Teukolsky equation level, enabling systematic post-Minkowskian expansion.

Findings

First post-Newtonian tail part of the Green function calculated.

Method allows systematic post-Minkowskian expansion.

Discussion of unresolved issues for reaction force evaluation.

Abstract

We formulate a new method to calculate the gravitational reaction force on a particle of mass $\mu$ orbiting a massive black hole of mass $M$. In this formalism, the tail part of the retarded Green function, which is responsible for the reaction force, is calculated at the level of the Teukolsky equation. Our method naturally allows a systematic post-Minkowskian (PM) expansion of the tail part at short distances. As a first step, we consider the case of a Schwarzschild black hole and explicitly calculate the first post-Newtonian (1PN) tail part of the Green function. There are, however, a couple of issues to be resolved before explicitly evaluating the reaction force by applying the present method. We discuss possible resolutions of these issues.