Numerical solution of the 2+1 Teukolsky equation on a hyperboloidal and horizon penetrating foliation of Kerr and application to late-time decays

TL;DR

This paper develops a stable numerical method for solving the Teukolsky equation on Kerr black holes using a hyperboloidal foliation, enabling detailed analysis of late-time decay behaviors of perturbations, including extremal cases.

Contribution

It introduces a spin-dependent rescaling for stable, long-term evolutions of the Teukolsky equation on Kerr spacetime with hyperboloidal slicing, and applies it to study late-time decay phenomena.

Findings

Verified power-law decay rates for perturbations

Observed prolonged quasi-normal modes in rapidly rotating black holes

Identified oscillatory decay and amplification in extremal Kerr cases

Abstract

In this work we present a formulation of the Teukolsky equation for generic spin perturbations on the hyperboloidal and horizon penetrating foliation of Kerr recently proposed by Racz and Toth. An additional, spin-dependent rescaling of the field variable can be used to achieve stable, long-term, and accurate time-domain evolutions of generic spin perturbations. As an application (and a severe numerical test), we investigate the late-time decays of electromagnetic and gravitational perturbations at the horizon and future null infinity by means of 2+1 evolutions. As initial data we consider four combinations of (non-)stationary and (non-)compact-support initial data with a pure spin-weighted spherical harmonic profile. We present an extensive study of late time decays of axisymmetric perturbations. We verify the power-law decay rates predicted analytically, together with a certain "splitting" behaviour of the power-law exponent. We also present results for non-axisymmetric perturbations. In particular, our approach allows to study the behaviour of the late time decays of gravitational fields for nearly extremal and extremal black holes. For rapid rotation we observe a very prolonged, weakly damped, quasi-normal-mode phase. For extremal rotation the field at future null infinity shows an oscillatory behaviour decaying as the inverse power of time, while at the horizon it is amplified by several orders of magnitude over long time scales. This behaviour can be understood in terms of the superradiance cavity argument.