Quantum Field Theory of Black Hole Perturbations with Backreaction II. Spherically symmetric 2nd order Einstein sector

TL;DR

This paper develops a gauge-invariant Hamiltonian formalism for second-order spherically symmetric black hole perturbations in pure gravity, confirming its consistency with established linear results and enabling higher-order analysis.

Contribution

It introduces a second-order Hamiltonian formalism for black hole perturbations that matches known linear results and can be extended to higher orders.

Findings

Exact agreement with Moncrief's Hamiltonian for linear perturbations.

Validation of the gauge-invariant formalism through comparison with established gauges.

Framework ready for higher-order perturbation analysis.

Abstract

In this second paper of our series we focus on the classical pure gravity sector of spherically symmetric black hole perturbations and expand the reduced Hamiltonian to second order. To compare our manifestly gauge invariant formalism with established results in the literature we have to translate our results derived in Gullstrand-Painlev\'e gauge to the gauges used in those works. After several canonical transformations we expectedly find exact agreement with the Hamiltonian given by Moncrief which generates the Regge-Wheeler and Zerilli equations of motion for the linear axial (often denoted odd) and polar (often denoted even) perturbations respectively. This confirms the validity of our method which immediately generalises to higher orders.