Finding fields and self-force in a gauge appropriate to separable wave equations

Abstract

Gravitational waves from the inspiral of a stellar-size black hole to a supermassive black hole can be accurately approximated by a point particle moving in a Kerr background. This paper presents progress on finding the electromagnetic and gravitational field of a point particle in a black-hole spacetime and on computing the self-force in a ``radiation gauge.'' The gauge is chosen to allow one to compute the perturbed metric from a gauge-invariant component ${\ensuremath{\psi}}_{0}$ (or ${\ensuremath{\psi}}_{4}$) of the Weyl tensor and follows earlier work by Chrzanowski, Cohen, and Kegeles (we correct a minor, but propagating, error in the Cohen-Kegeles formalism). The electromagnetic field tensor and vector potential of a static point charge and the perturbed gravitational field of a static point mass in a Schwarzschild geometry are found, surprisingly, to have closed-form expressions. The gravitational field of a static point charge in the Schwarzschild background must have a strut, but ${\ensuremath{\psi}}_{0}$ and ${\ensuremath{\psi}}_{4}$ are smooth except at the particle, and one can find local radiation gauges for which the corresponding spin $\ifmmode\pm\else\textpm\fi{}2$ parts of the perturbed metric are smooth. Finally a method for finding the renormalized self-force from the Teukolsky equation is presented. The method is related to the Mino, Sasaki, Tanaka and Quinn and Wald (MiSaTaQuWa) renormalization and to the Detweiler-Whiting construction of the singular field. It relies on the fact that the renormalized ${\ensuremath{\psi}}_{0}$ (or ${\ensuremath{\psi}}_{4}$) is a source-free solution to the Teukolsky equation; and one can therefore reconstruct a nonsingular renormalized metric in a radiation gauge.