Second-order gravitational self-force in a highly regular gauge

Extreme-mass-ratio inspirals (EMRIs) will be key sources for LISA. However, accurately extracting system parameters from a detected EMRI waveform will require self-force calculations at second order in perturbation theory, which are still in a nascent stage. One major obstacle in these calculations is the strong divergences that are encountered on the worldline of the small object. Previously, it was shown by one of us [Phys. Rev. D 95, 104056 (2017)] that a class of "highly regular" gauges exist in which the singularities have a qualitatively milder form, promising to enable more efficient numerical calculations. Here we derive expressions for the metric perturbation in this class of gauges, in a local expansion in powers of distance $r$ from the worldline, to sufficient order in $r$ for numerical implementation in a puncture scheme. Additionally, we use the highly regular class to rigorously derive a distributional source for the second-order field and a pointlike second-order stress-energy tensor (the Detweiler stress-energy) for the small object. This makes it possible to calculate the second-order self-force using mode-sum regularisation rather than the more cumbersome puncture schemes that have been necessary previously. Although motivated by EMRIs, our calculations are valid in an arbitrary vacuum background, and they may help clarify the interpretation of point masses and skeleton sources in general relativity more broadly.