Fast evaluation of asymptotic waveforms from gravitational perturbations

TL;DR

This paper introduces an accurate, easy-to-implement method for evaluating asymptotic gravitational waveforms from black hole perturbations, applicable to both axial and polar sectors, with applications to late-time decay, teleportation, and luminosity calculations.

Contribution

It presents a novel technique for exact and approximate asymptotic waveform evaluation in black hole perturbation theory, including theoretical analysis and practical algorithms.

Findings

Achieved high-accuracy waveform computations with errors below 10^{-12} and 10^{-9}.

Demonstrated late-time decay tails with t^{-4} behavior at null infinity.

Showed the method's effectiveness in studying signal teleportation and luminosities in extreme-mass-ratio binaries.

Abstract

In the context of blackhole perturbation theory, we describe both exact evaluation of an asymptotic waveform from a time series recorded at a finite radial location and its numerical approximation. From the user's standpoint our technique is easy to implement, affords high accuracy, and works for both axial (Regge-Wheeler) and polar (Zerilli) sectors. Our focus is on the ease of implementation with publicly available numerical tables, either as part of an existing evolution code or a post-processing step. Nevertheless, we also present a thorough theoretical discussion of asymptotic waveform evaluation and radiation boundary conditions, which need not be understood by a user of our methods. In particular, we identify (both in the time and frequency domains) analytical asymptotic waveform evaluation kernels, and describe their approximation by techniques developed by Alpert, Greengard, and Hagstrom. This paper also presents new results on the evaluation of far-field signals for the ordinary (acoustic) wave equation. We apply our method to study late-time decay tails at null-infinity, "teleportation" of a signal between two finite radial values, and luminosities from extreme-mass-ratio binaries. Through numerical simulations with the outer boundary as close in as r = 30M, we compute asymptotic waveforms with late-time t^{-4} decay (l = 2 perturbations), and also luminosities from circular and eccentric particle-orbits that respectively match frequency domain results to relative errors of better than 10^{-12} and 10^{-9}. Furthermore, we find that asymptotic waveforms are especially prone to contamination by spurious junk radiation.