Highly accurate and efficient self-force computations using time-domain methods: Error estimates, validation, and optimization

Abstract

If a small "particle" of mass $\mu M$ (with $\mu \ll 1$) orbits a Schwarzschild or Kerr black hole of mass $M$, the particle is subject to an $\O(\mu)$ radiation-reaction "self-force". Here I argue that it's valuable to compute this self-force highly accurately (relative error of $\ltsim 10^{-6}$) and efficiently, and I describe techniques for doing this and for obtaining and validating error estimates for the computation. I use an adaptive-mesh-refinement (AMR) time-domain numerical integration of the perturbation equations in the Barack-Ori mode-sum regularization formalism; this is efficient, yet allows easy generalization to arbitrary particle orbits. I focus on the model problem of a scalar particle in a circular geodesic orbit in Schwarzschild spacetime. The mode-sum formalism gives the self-force as an infinite sum of regularized spherical-harmonic modes $\sum_{\ell=0}^\infty F_{\ell,\reg}$, with $F_{\ell,\reg}$ (and an "internal" error estimate) computed numerically for $\ell \ltsim 30$ and estimated for larger~$\ell$ by fitting an asymptotic "tail" series. Here I validate the internal error estimates for the individual $F_{\ell,\reg}$ using a large set of numerical self-force computations of widely-varying accuracies. I present numerical evidence that the actual numerical errors in $F_{\ell,\reg}$ for different~$\ell$ are at most weakly correlated, so the usual statistical error estimates are valid for computing the self-force. I show that the tail fit is numerically ill-conditioned, but this can be mostly alleviated by renormalizing the basis functions to have similar magnitudes. Using AMR, fixed mesh refinement, and extended-precision floating-point arithmetic, I obtain the (contravariant) radial component of the self-force for a particle in a circular geodesic orbit of areal radius $r = 10M$ to within $1$~ppm relative error.