Multidomain spectral method for self-force calculations

Abstract

Second-order self-force calculations will be critical for modeling extreme-mass-ratio inspirals, and they are now known to have high accuracy even for binaries with mass ratios ∼1:10. Many of the challenges facing these calculations are related to slow convergence of spherical-harmonic (or spheroidal harmonic) mode sums in a region containing the small companion. In this paper, we begin to develop a multidomain framework that can evade those problems. Building on recent work by Osburn and Nishimura, in the problematic region of spacetime, we use a puncture scheme and decompose the punctured field equations into a basis of Fourier and azimuthal m modes, avoiding a harmonic decomposition in the θ direction. Outside the problematic region, we allow for a complete spherical- or spheroidal-harmonic decomposition. As a demonstration, we implement this framework in the simple context of a scalar charge in circular orbit around a Schwarzschild black hole. Our implementation utilizes several recent advances: a spectral method in each region, hyperboloidal compactification, and an extremely high-order puncture.

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