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_(αβ)u^(α)u^(β) 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, ψ_(0)^(ret). 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_(αβ)^(ret); we similarly find the leading renormalization coefficients of h_(αβ)u^(α)u^(β) 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^(S)_(α) = <∇_(α)ρ^(-1)>, the part of ∇_(α)ρ^(-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 ρ.

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