On the falloff of radiated energy in black hole spacetimes

TL;DR

This paper analyzes how gravitational wave quantities like $ ext{psi}_4$ and energy flux decay with distance in black hole spacetimes, revealing that their leading finite-radius corrections are of higher order than previously assumed, aiding more accurate extrapolations.

Contribution

It demonstrates that the leading finite-radius corrections for $ ext{psi}_4$ and energy flux in black hole spacetimes are of order $1/r^3$ and $1/r^4$, respectively, refining previous assumptions.

Findings

Leading correction to $ ext{psi}_4$ is ${ m O}(1/r^3)$.

Leading correction to energy flux is ${ m O}(1/r^4)$.

Finite radius corrections can be accurately computed using black hole perturbation theory.

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 $\psi_4$, 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, $\psi_4 \propto 1/r$ 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 $\psi_4$ 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 $\psi_4$ 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.