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

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(alpha beta)u(alpha)u(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, psi(ret)(0). We use a mode-sum renormalization and find the renormalization coefficients by matching a series in L = l + 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(alpha beta)u(alpha)u(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(alpha)(S) = (del(alpha)rho(-1)), the part del(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.