Gravitational self force and gauge transformations

TL;DR

The paper derives a general gauge transformation law for the gravitational self force and extends the standard harmonic-gauge regularization to arbitrary gauges, showing that the regularization parameters in the mode-sum approach are gauge-independent. It demonstrates that the self force is well-defined in the Regge-Wheeler gauge only for radial orbits in Schwarzschild, while the radiation gauge is generally ill-defined due to gauge-pathology, and it discusses potential remedies such as angular averaging or approximate harmonic transformations. The work provides practical prescriptions for computing the self force in non-harmonic gauges and analyzes the regularity of gauge transformations through detailed examples, with implications for modeling extreme-mass-ratio binaries in gravitational-wave physics.

Abstract

We explore how the gravitational self force (or ``radiation reaction'' force), acting on a pointlike test particle in curved spacetime, is modified in a gauge transformation. We derive the general transformation law, describing the change in the self force in terms of the infinitesimal displacement vector associated with the gauge transformation. Based on this transformation law, we extend the regularization prescription by Mino et al. and Quinn and Wald (originally formulated within the harmonic gauge) to an arbitrary gauge. Then we extend the method of mode-sum regularization (which provides a practical means for calculating the regularized self force and was recently applied to the harmonic-gauge gravitational self force) to an arbitrary gauge. We find that the regularization parameters involved in this method are gauge-independent. We also explore the gauge transformation of the self force from the harmonic gauge to the Regge-Wheeler gauge and to the radiation gauge, focusing attention on the regularity of these gauge transformations. We conclude that the transformation of the self force to the Regge-Wheeler gauge in Schwarzschild spacetime is regular for radial orbits and irregular otherwise, whereas the transformation to the radiation gauge is irregular for all orbits.