On the falloff of radiated energy in black hole spacetimes

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, $\dot E$. Conservation laws guarantee that, in asympotically flat spacetimes, $ψ_4 \propto 1/r$ and $\dot E \propto 1/r^2$ as $r \to \infty$. Most calculations extract these quantities at some finite extraction radius. An understanding of finite radius corrections to $ψ_4$ and $\dot E$ 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 ${\cal O}(1/r^3)$, and that to the energy flux is ${\cal O}(1/r^4)$ --- not ${\cal O}(1/r^2)$ and ${\cal O}(1/r^3)$ 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.