Second order perturbations of a Schwarzschild black hole: inclusion of odd parity perturbations

TL;DR

This paper develops second-order perturbation equations for Schwarzschild black holes, including odd parity modes, providing explicit formulas for evolution, energy, and waveforms relevant for black hole collision modeling.

Contribution

It introduces a second-order perturbation framework for Schwarzschild black holes that includes odd parity perturbations, extending previous first-order analyses.

Findings

Explicit second-order evolution equations derived

Formulas for radiated energies and waveforms provided

Applicable to black hole collision scenarios in close limit approximation

Abstract

We consider perturbations of a Schwarzschild black hole that can be of both even and odd parity, keeping terms up to second order in perturbation theory, for the $\ell=2$ axisymmetric case. We develop explicit formulae for the evolution equations and radiated energies and waveforms using the Regge-Wheeler-Zerilli approach. This formulation is useful, for instance, for the treatment in the ``close limit approximation'' of the collision of counterrotating black holes.