On the falloff of radiated energy in black hole spacetimes

Lior M. Burko; Scott A. Hughes

doi:10.1103/PhysRevD.82.104029

paper

On the falloff of radiated energy in black hole spacetimes

arXiv:1007.4596 · doi:10.1103/PhysRevD.82.104029

Abstract

The goal of much research in relativity is to understand gravitational waves generated by a strong-field dynamical spacetime. Quantities of particular interest for many calculations are the Weyl scalar $ψ_{4}$ , which is simply related to the flux of gravitational waves far from the source, and the flux of energy carried to distant observers, $E ˙$ . Conservation laws guarantee that, in asympotically flat spacetimes, $ψ_{4} \propto 1/ r$ and $E ˙ \propto 1/ r^{2}$ as $r \to \infty$ . Most calculations extract these quantities at some finite extraction radius. An understanding of finite radius corrections to $ψ_{4}$ and $E ˙$ allows us to more accurately infer their asymptotic values from a computation. In this paper, we show that, if the final state of the system is a black hole, then the leading correction to $ψ_{4}$ is $O (1/ r^{3})$ , and that to the energy flux is $O (1/ r^{4})$ --- not $O (1/ r^{2})$ and $O (1/ r^{3})$ as one might naively guess. Our argument only relies on the behavior of the curvature scalars for black hole spacetimes. Using black hole perturbation theory, we calculate the corrections to the leading falloff, showing that it is quite easy to correct for finite extraction radius effects.

arXiv comment5 pages, no figures, accepted to Phys. Rev. D. This version corrects several typos and minor errors in the earlier submission

On the falloff of radiated energy in black hole spacetimes

Abstract

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