Analytical high-order post-Newtonian expansions for spinning extreme mass ratio binaries

TL;DR

This paper analytically computes the Detweiler redshift invariant for a small mass orbiting a Kerr black hole, achieving high-order post-Newtonian accuracy without assuming small spin, and employing a rigorous mode-sum regularization method.

Contribution

It provides the first high-order post-Newtonian expansion of the redshift invariant for spinning Kerr black holes without spin magnitude restrictions.

Findings

Results up to 8.5 post-Newtonian order

Expressions include all orders in black hole spin

Method applicable to other gauge-invariant quantities

Abstract

We present an analytic computation of Detweiler's redshift invariant for a point mass in a circular orbit around a Kerr black hole, giving results up to 8.5 post-Newtonian order while making no assumptions on the magnitude of the spin of the black hole. Our calculation is based on the functional series method of Mano, Suzuki and Takasugi, and employs a rigorous mode-sum regularization prescription based on the Detweiler-Whiting singular-regular decomposition. The approximations used in our approach are minimal; we use the standard self-force expansion to linear order in the mass ratio, and the standard post-Newtonian expansion in the separation of the binary. A key advantage of this approach is that it produces expressions that include contributions at all orders in the spin of the Kerr black hole. While this work applies the method to the specific case of Detweiler's redshift invariant, it can be readily extended to other gauge invariant quantities and to higher post-Newtonian orders.