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

This is the first of two papers on computing the self-force in a radiation gauge for a particle of mass 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 : = 1/2 h(alpha beta)u(alpha)u(beta), gauge-invariant under gauge transformations generated by a helically symmetric gauge vector; here, h(alpha beta) is the metric perturbation, u(alpha) the particle's 4-velocity. We find the related order 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 Y-s(lm) and use mode-sum renormalization to find the renormalization coefficients by matching a series in L = l + 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.