Kerr metric perturbations for second-order self-force theory

Abstract

We present a calculation of the first-order metric perturbation arising from a point mass moving on a circular, equatorial orbit in Kerr spacetime. The metric perturbation is specifically aimed at meeting the requirements of second-order self-force theory: it is highly accurate; incorporates all contributions, including those associated with perturbations to the black hole's mass and angular momentum; is in Lorenz gauge near the worldline and in (near) Bondi–Sachs gauge at large radius; and can be efficiently evaluated across the entire exterior spacetime. The Lorenz gauge gives a well-understood singular structure at the particle, while the (near) Bondi–Sachs gauge ensures the asymptotic fall-off needed for an asymptotically flat second-order source. Our formalism reconstructs the metric from separable Teukolsky-type equations, decomposed into spin-weighted spheroidal harmonic modes and solved spectrally on hyperboloidal slices. The resulting $(\ell,m)$-mode sums converge rapidly across most of the domain, but slowly near the point mass; we overcome this with a novel use of a puncture to extract accurate $m$-mode data, yielding output accurate to $\sim 10^{-10}$ per $m$-mode and suitable for use at second order. Our metric-perturbation results cover a range of Kerr spins $-0.9 \le a/M \le 0.9$ and orbital radii out to $50M$. In addition to its immediate application in second-order self-force calculations, our results can also be used as input to other calculations involving Kerr metric perturbations, including studies of environmental effects and quasinormal modes.