Detweiler's redshift invariant for spinning particles along circular orbits on a Schwarzschild background

TL;DR

This paper calculates the first-order spin effects on Detweiler's redshift invariant for a spinning particle in circular orbit around a Schwarzschild black hole, using two gauge approaches and extending the post-Newtonian order.

Contribution

It provides the first-order spin correction to the redshift invariant up to 8.5 post-Newtonian order using two different gauge formalisms, confirming gauge invariance and matching known results.

Findings

Gauge-invariant redshift correction computed in two gauges.

Agreement with known 3.5 post-Newtonian results.

Extension of spin effects to 8.5 post-Newtonian order.

Abstract

We study the metric perturbations induced by a classical spinning particle moving along a circular orbit on a Schwarzschild background, limiting the analysis to effects which are first order in spin. The particle is assumed to move on the equatorial plane and has its spin aligned with the $z$-axis. The metric perturbations are obtained by using two different approaches, i.e., by working in two different gauges: the Regge-Wheeler gauge (using the Regge-Wheeler-Zerilli formalism) and a radiation gauge (using the Teukolsky formalism). We then compute the linear-in-spin contribution to the first-order self-force contribution to Detweiler's redshift invariant up to the 8.5 post-Newtonian order. We check that our result is the same in both gauges, as appropriate for a gauge-invariant quantity, and agrees with the currently known 3.5 post-Newtonian results.