# New gravitational self-force analytical results for eccentric equatorial orbits around a Kerr black hole: redshift invariant

## TL;DR

This paper enhances the analytical understanding of the redshift invariant for particles in eccentric equatorial orbits around Kerr black holes by extending calculations to higher eccentricity orders and validating against post-Newtonian predictions.

## Contribution

The authors extend the analytical computation of the redshift invariant to include terms up to fourth order in eccentricity at the same post-Newtonian order, improving previous results.

## Key findings

- Results agree with post-Newtonian expectations.
- Extended the eccentricity expansion to fourth order.
- Validated analytical results against known Hamiltonian calculations.

## Abstract

The Detweiler-Barack-Sago redshift function for particles moving along slightly eccentric equatorial orbits around a Kerr black hole is currently known up to the second order in eccentricity, second order in spin parameter, and the 8.5 post-Newtonian order. We improve the analytical computation of such a gauge-invariant quantity by including terms up to the fourth order in eccentricity at the same post-Newtonian approximation level. We also check that our results agrees with the corresponding post-Newtonian expectation of the same quantity, calculated by using the currently known Hamiltonian for spinning binaries.