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, ${\psi }_0^{\mathrm{ret}}$. 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 $h_{\alpha \beta }^{\mathrm{ret}}$; 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 $f_{\alpha }^{\mathrm{S}}=⟨{\nabla }_{\alpha }{\rho }^{-1}⟩$, 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 $\rho$.

Figure 1

The solid curve is a plot of $|E_{0(k_{\max }+1)}-E_{0k_{\max }}|/E_{0k_{\max }}$ vs $k_{\max }$. The dashed curve is a plot of $|E_{0\mathrm{analytic}}-E_{0\tilde{k}}|/(E_{0\mathrm{analytic}})$ as a function of $k_{\max }$. Both are for orbital radius $r_0=10M$.

Figure 2

This figure shows a plot of $\log a_{\ell }^{\mathrm{ret}r}$ (and subsequent subtractions of the singular terms) vs $\log \ell$. The topmost curve is $a_{\ell }^{\mathrm{ret}r}\sim \ell$. The second curve (from the top) is $(a_{\ell }^{\mathrm{ret}r}-A(\ell +1/2))\sim {\ell }^0$. The third curve shows $(a_{\ell }^{\mathrm{ret}r}-A(\ell +1/2)-B)\sim {\ell }^{-2}$. The fourth, fifth and sixth curves show the subsequent cumulative subtractions of $E_2/P_2(\ell )$, $E_4/P_4(\ell )$ and $E_6/P_6(\ell )$ from the third.