Conservative, gravitational self-force for a particle in circular orbit around a Schwarzschild black hole in a radiation gauge

Abstract

This is the second of two companion papers on computing the self-force in a radiation gauge; more precisely, the method uses a radiation gauge for the radiative part of the metric perturbation, together with an arbitrarily chosen gauge for the parts of the perturbation associated with changes in black-hole mass and spin and with a shift in the center of mass. In a test of the method delineated in the first paper, we compute the conservative part of the self-force for a particle in circular orbit around a Schwarzschild black hole. The gauge vector relating our radiation gauge to a Lorenz gauge is helically symmetric, implying that the quantity ${h}_{\ensuremath{\alpha}\ensuremath{\beta}}{u}^{\ensuremath{\alpha}}{u}^{\ensuremath{\beta}}$ must have the same value for our radiation gauge as for a Lorenz gauge; and we confirm this numerically to one part in ${10}^{14}$. As outlined in the first paper, the perturbed metric is constructed from a Hertz potential that is in a term obtained algebraically from the retarded perturbed spin-2 Weyl scalar, ${\ensuremath{\psi}}_{0}^{\mathrm{ret}}$. We use a mode-sum renormalization and find the renormalization coefficients by matching a series in $L=\ensuremath{\ell}+1/2$ to the large-$L$ behavior of the expression for the self-force in terms of the retarded field ${h}_{\ensuremath{\alpha}\ensuremath{\beta}}^{\mathrm{ret}}$; we similarly find the leading renormalization coefficients of ${h}_{\ensuremath{\alpha}\ensuremath{\beta}}{u}^{\ensuremath{\alpha}}{u}^{\ensuremath{\beta}}$ and the related change in the angular velocity of the particle due to its self-force. We show numerically that the singular part of the self-force has the form ${f}_{\ensuremath{\alpha}}^{\mathrm{S}}=⟨{\ensuremath{\nabla}}_{\ensuremath{\alpha}}{\ensuremath{\rho}}^{\ensuremath{-}1}⟩$, the part of ${\ensuremath{\nabla}}_{\ensuremath{\alpha}}{\ensuremath{\rho}}^{\ensuremath{-}1}$ that is axisymmetric about a radial line through the particle. This differs only by a constant from its form for a Lorenz gauge. It is because we do not use a radiation gauge to describe the change in black-hole mass that the singular part of the self-force has no singularity along a radial line through the particle and, at least in this example, is spherically symmetric to subleading order in $\ensuremath{\rho}$.