Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2010-06-18
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
27 pages, 12 eps figures (10 of them color, but all are viewable ok in black-and-white), uses RevTeX 4.1
Scientific paper
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.
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