Extreme-mass-ratio inspiral corrections to the angular velocity and redshift factor of a mass in circular orbit about a Kerr black hole

Abstract

This is the first of two papers on computing the self-force in a radiation gauge for a particle of mass $\mathfrak{m}$ moving in circular, equatorial orbit about a Kerr black hole. In the extreme-mass-ratio inspiral (EMRI) framework, with mode-sum renormalization, we compute the renormalized value of the quantity $H\ensuremath{\mathrel{:=}}\frac{1}{2}{h}_{\ensuremath{\alpha}\ensuremath{\beta}}{u}^{\ensuremath{\alpha}}{u}^{\ensuremath{\beta}}$, gauge-invariant under gauge transformations generated by a helically symmetric gauge vector; here, ${h}_{\ensuremath{\alpha}\ensuremath{\beta}}$ is the metric perturbation, ${u}^{\ensuremath{\alpha}}$ the particle's 4-velocity. We find the related order $\mathfrak{m}$ correction to the particle's angular velocity at fixed renormalized redshift (and to its redshift at fixed angular velocity), each of which can be written in terms of $H$. The radiative part of the metric perturbation is constructed from a Hertz potential that is extracted from the Weyl scalar by an algebraic inversion T. S. Keidl et al., Phys. Rev. D 82, 124012 (2010). We then write the spin-weighted spheroidal harmonics as a sum over spin-weighted spherical harmonics $_{s}Y_{\ensuremath{\ell}m}$ and use mode-sum renormalization to find the renormalization coefficients by matching a series in $L=\ensuremath{\ell}+1/2$ to the large-$L$ behavior of the expression for $H$. The nonradiative parts of the perturbed metric associated with changes in mass and angular momentum are calculated in the Kerr gauge.