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