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Gravitational self-force theory is the primary way of modellingextreme-mass-ratio inspirals (EMRIs). One difficulty that appears insecond-order self-force calculations is the strong divergence at the worldlineof 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 regulargauges and presented the metric perturbations in these gauges in a localcoordinate form. We build on this previous work by deriving expressions for thehighly regular gauge metric perturbations in both fully covariant form and as ageneric coordinate expansion. With the metric perturbations in covariant orgeneric coordinate form, they can easily be expressed in any convenientcoordinate system. These results can then be used as input into a puncturescheme in order to solve the field equations describing an EMRI.

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