The imposition of Cauchy data to the Teukolsky equation II: Numerical comparison with the Zerilli-Moncrief approach to black hole perturbations

TL;DR

This paper compares numerical evolutions of black hole perturbations using the Teukolsky and Zerilli approaches, demonstrating their agreement and validating the Teukolsky code for nonrotating black holes.

Contribution

It establishes a relation between the Teukolsky function and Moncrief waveform, enabling direct comparison of two perturbation methods for Schwarzschild black holes.

Findings

Complete agreement of waveforms from both methods within numerical errors

Validation of the Teukolsky evolution code for nonrotating black holes

Explicit relation between Psi and hypersurface quantities

Abstract

We revisit the question of the imposition of initial data representing astrophysical gravitational perturbations of black holes. We study their dynamics for the case of nonrotating black holes by numerically evolving the Teukolsky equation in the time domain. In order to express the Teukolsky function Psi explicitly in terms of hypersurface quantities, we relate it to the Moncrief waveform phi_M through a Chandrasekhar transformation in the case of a nonrotating black hole. This relation between Psi and phi_M holds for any constant time hypersurface and allows us to compare the computation of the evolution of Schwarzschild perturbations by the Teukolsky and by the Zerilli and Regge-Wheeler equations. We explicitly perform this comparison for the Misner initial data in the close limit approach. We evolve numerically both, the Teukolsky (with the recent code of Ref. [1]) and the Zerilli equations, finding complete agreement in resulting waveforms within numerical error. The consistency of these results further supports the correctness of the numerical code for evolving the Teukolsky equation as well as the analytic expressions for Psi in terms only of the three-metric and the extrinsic curvature.