Gravitational radiation from radial infall of a particle into a Schwarzschild black hole. A numerical study of the spectra, quasi-normal modes and power-law tails

TL;DR

This paper presents a highly accurate numerical analysis of gravitational waves emitted by a particle falling into a Schwarzschild black hole, detailing waveforms, spectra, quasi-normal modes, and power-law tails.

Contribution

It introduces an improved numerical method for computing gravitational radiation, achieving higher accuracy in waveforms, spectra, and mode frequencies compared to previous studies.

Findings

Accurate frequencies of black hole quasi-normal modes obtained.

Detected power-law tails with amplitudes much smaller than the main wave.

Enhanced numerical precision improves understanding of gravitational wave features.

Abstract

The computation of the gravitational radiation emitted by a particle falling into a Schwarzschild black hole is a classic problem studied already in the 1970s. Here we present a detailed numerical analysis of the case of radial infall starting at infinity with no initial velocity. We compute the radiated waveforms, spectra and energies for multipoles up to l = 6, improving significantly on the numerical accuracy of existing results. This is done by integrating the Zerilli equation in the frequency domain using the Green's function method. The resulting wave exhibits a "ring-down" phase whose dominant contribution is a superposition of the quasi-normal modes of the black hole. The numerical accuracy allows us to recover the frequencies of these modes through a fit of that part of the wave. Comparing with direct computations of the quasi-normal modes we reach a \sim 10^{-4} to \sim 10^{-2} accuracy for the first two overtones of each multipole. Our numerical accuracy also allows us to display the power-law tail that the wave develops after the ring-down has been exponentially cut-off. The amplitude of this contribution is \sim 10^2 to \sim 10^3 times smaller than the typical scale of the wave.