Gauge and Averaging in Gravitational Self-force

TL;DR

This paper introduces a gauge-invariant method to compute the gravitational self-force using angle-averages, simplifying calculations and extending applicability across different gauges in black hole spacetimes.

Contribution

It demonstrates that the gravitational self-force can be expressed as an angle-average in gauges satisfying a parity condition, simplifying computations and avoiding complex expansions.

Findings

Self-force given by angle-average in certain gauges

Rest mass of particle remains unchanged at first order

Mode sum regularization applies across all parity-regular gauges

Abstract

A difficulty with previous treatments of the gravitational self-force is that an explicit formula for the force is available only in a particular gauge (Lorenz gauge), where the force in other gauges must be found through a transformation law once the Lorenz gauge force is known. For a class of gauges satisfying a ``parity condition'' ensuring that the Hamiltonian center of mass of the particle is well-defined, I show that the gravitational self-force is always given by the angle-average of the bare gravitational force. To derive this result I replace the computational strategy of previous work with a new approach, wherein the form of the force is first fixed up to a gauge-invariant piece by simple manipulations, and then that piece is determined by working in a gauge designed specifically to simplify the computation. This offers significant computational savings over the Lorenz gauge, since the Hadamard expansion is avoided entirely and the metric perturbation takes a very simple form. I also show that the rest mass of the particle does not evolve due to first-order self-force effects. Finally, I consider the ``mode sum regularization'' scheme for computing the self-force in black hole background spacetimes, and use the angle-average form of the force to show that the same mode-by-mode subtraction may be performed in all parity-regular gauges. It appears plausible that suitably modified versions of the Regge-Wheeler and radiation gauges (convenient to Schwarzschild and Kerr, respectively) are in this class.