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. 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_{\alpha\beta} u^\alpha u^\beta (= h_{uu}) must have the same value for our radiation gauge as for a Lorenz gauge; and we confirm this numerically to one part in 10^{13}. As outlined in the first paper, the perturbed metric is constructed from a Hertz potential that is in term obtained algebraically from the the retarded perturbed spin-2 Weyl scalar, \psi_0 . We use a mode-sum renormalization and find the renormalization coefficients by matching a series in L = \ell + 1/2 to the large-L behavior of the expression for the self-force in terms of the retarded field h_{\alpha\beta}^{ret}; we similarly find the leading renormalization coefficients of h_{uu} 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_{\alpha} \propto < \nabla_\alpha \rho^{-1}>, the part of \nabla_\alpha \rho^{-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 \rho.