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

Abstract

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 [A. Pound, Nonlinear gravitational self-force: Second-order equation of motion, Phys. Rev. D 95, 104056 (2017)PRVDAQ2470-001010.1103/PhysRevD.95.104056] 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 regularization rather than the more cumbersome puncture schemes that have been necessary.

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