Numerical investigation of the late-time tails of the solutions of the Fackerell-Ipser equation

TL;DR

This paper numerically investigates the late-time decay behavior of solutions to the Fackerell-Ipser equation on Kerr spacetime, revealing power-law decay and quasinormal ringing, and proposes a Price's law for this context.

Contribution

It introduces a numerical analysis of the Fackerell-Ipser equation's late-time tails on Kerr spacetime, including decay exponents and a proposed Price's law specific to this equation.

Findings

Solutions decay to static solutions or zero at late times

Solutions exhibit quasinormal ringing before decay

Decay exponents are extracted for various initial data parameters

Abstract

The late-time behaviour of the solutions of the Fackerell-Ipser equation (which is a wave equation for the spin-zero component of the electromagnetic field strength tensor) on the closure of the domain of outer communication of sub-extremal Kerr spacetime is studied numerically. Within the Kerr family, the case of Schwarzschild background is also considered. Horizon-penetrating compactified hyperboloidal coordinates are used, which allow the behaviour of the solutions to be observed at the event horizon and at future null infinity as well. For the initial data, pure multipole configurations that have compact support and are either stationary or non-stationary are taken. It is found that with such initial data the solutions of the Fackerell-Ipser equation converge at late times either to a known static solution (up to a constant factor) or to zero. As the limit is approached, the solutions exhibit a quasinormal ringdown and finally a power-law decay. The exponents characterizing the power-law decay of the spherical harmonic components of the field variable are extracted from the numerical data for various values of the parameters of the initial data, and based on the results a proposal for a Price's law relevant to the Fackerell-Ipser equation is made. Certain conserved energy and angular momentum currents are used to verify the numerical implementation of the underlying mathematical model. In the construction of these currents a discrete symmetry of the Fackerell-Ipser equation, which is the product of an equatorial reflection and a complex conjugation, is also taken into account.