Second-order gravitational self-force in a highly regular gauge: Covariant and coordinate punctures

TL;DR

This paper advances gravitational self-force modeling for EMRIs by deriving covariant and coordinate expressions for highly regular gauge metric perturbations, facilitating improved numerical and analytical calculations.

Contribution

It provides the first derivation of covariant and generic coordinate forms of highly regular gauge metric perturbations for second-order self-force calculations.

Findings

Derived covariant expressions for metric perturbations.

Presented generic coordinate expansion of perturbations.

Facilitates implementation of puncture schemes in EMRI modeling.

Abstract

Gravitational self-force theory is the primary way of modelling extreme-mass-ratio inspirals (EMRIs). One difficulty that appears in second-order self-force calculations is the strong divergence at the worldline of the small object, which causes both numerical and analytical issues. Previous work [Phys. Rev. D 95, 104056 (2017); ibid. 103, 124016 (2021)] demonstrated that this could be alleviated within a class of highly regular gauges and presented the metric perturbations in these gauges in a local coordinate form. We build on this previous work by deriving expressions for the highly regular gauge metric perturbations in both fully covariant form and as a generic coordinate expansion. With the metric perturbations in covariant or generic coordinate form, they can easily be expressed in any convenient coordinate system. These results can then be used as input into a puncture scheme in order to solve the field equations describing an EMRI.