EMRI corrections to the angular velocity and redshift factor of a mass in circular orbit about a Kerr black hole

TL;DR

This paper calculates the first-order corrections to the angular velocity and redshift factor for a particle in circular orbit around a Kerr black hole using self-force and mode-sum renormalization techniques.

Contribution

It introduces a novel method for computing gauge-invariant self-force corrections in Kerr spacetime using radiation gauge and mode-sum renormalization.

Findings

Computed the renormalized $h_{etaeta}u^eta u^eta$ for particles in Kerr

Derived the correction to angular velocity at fixed redshift

Established a method for extracting renormalization coefficients

Abstract

This is the first of two papers on computing the self-force in a radiation gauge for a particle moving in circular, equatorial orbit about a Kerr black hole. In the EMRI (extreme-mass-ratio inspiral) framework, with mode-sum renormalization, we compute the renormalized value of the quantity $h_{\alpha\beta}u^\alpha u^\beta$, gauge-invariant under gauge transformations generated by a helically symmetric gauge vector; and we find the related order $\frak{m}$ correction to the particle's angular velocity at fixed renormalized redshift (and to its redshift at fixed angular velocity). The radiative part of the perturbed metric is constructed from the Hertz potential which is extracted from the Weyl scalar by an algebraic inversion\cite{sf2}. We then write the spin-weighted spheroidal harmonics as a sum over spin-weighted spherical harmonics and use mode-sum renormalization to find the renormalization coefficients by matching a series in $L=\ell+1/2$ to the large-$L$ behavior of the expression for $H := \frac12 h_{\alpha\beta}u^\alpha u^\beta $. The non-radiative parts of the perturbed metric associated with changes in mass and angular momentum are calculated in the Kerr gauge.