2010/06/18 by Jonathan Thornburg, Thornburg, Jonathan
Physics and Astronomy · #Pulsars and Gravitational Waves Research #Astrophysical Phenomena and Observations #Black Holes and Theoretical Physics
paper · pdf · doi:10.48550/arxiv.1006.3788
If a small "particle" of mass μM (with μ≪ 1) orbits a Schwarzschild or Kerr black hole of mass M, the particle is subject to an Ø(μ) 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 ∑ℓ=0^∞ Fℓ,\reg, with Fℓ,\reg (and an "internal" error estimate) computed numerically for ℓ \ltsim 30 and estimated for larger~ℓ by fitting an asymptotic "tail" series. Here I validate the internal error estimates for the individual Fℓ,\reg using a large set of numerical self-force computations of widely-varying accuracies. I present numerical evidence that the actual numerical errors in Fℓ,\reg for different~ℓ 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.