Metric perturbations produced by eccentric equatorial orbits around a Kerr black hole

TL;DR

This paper presents the first numerical calculation of metric perturbations caused by a small object on an eccentric equatorial orbit around a Kerr black hole, improving accuracy and providing new data for gravitational self-force analysis.

Contribution

It introduces a novel numerical method to compute metric perturbations for eccentric equatorial orbits around Kerr black holes, including new results for the redshift invariant in Kerr spacetime.

Findings

Enhanced accuracy over previous results in Schwarzschild case.

New estimates for 4PN and 5PN terms in post-Newtonian expansion.

Novel values of the redshift invariant U for Kerr black holes.

Abstract

We present the first numerical calculation of the (local) metric perturbation produced by a small compact object moving on an eccentric equatorial geodesic around a Kerr black hole, accurate to first order in the mass ratio. The procedure starts by first solving the Teukolsky equation to obtain the Weyl scalar $\psi_4$ using semi-analytical methods. The metric perturbation is then reconstructed from $\psi_4$ in an (outgoing) radiation gauge, adding the appropriate non-radiative contributions arising from the shifts in mass and angular momentum of the spacetime. As a demonstration we calculate the generalized redshift $U$ as a function of the orbital frequencies $\Omega_r$ and $\Omega_\phi$ to linear order in the mass ratio, a gauge invariant measure of the conservative corrections to the orbit due to self-interactions. In Schwarzschild, the results surpass the existing result in the literature in accuracy, and we find new estimates for some of the unknown 4PN and 5PN terms in the post-Newtonian expansion of $U$. In Kerr, we provide completely novel values of $U$ for eccentric equatorial orbits. Calculation of the full self-force will appear in a forthcoming paper.