Axially Symmetric Perturbations of Kerr Black Holes I: A gauge-invariant construction of ADM Energy

TL;DR

This paper develops a gauge-invariant, positive-definite energy functional for linear axially symmetric perturbations of Kerr black holes, using Hamiltonian reduction and Einstein-wave map systems, aiding stability analysis.

Contribution

It introduces a novel gauge-invariant energy functional for Kerr perturbations derived from Hamiltonian reduction and Einstein-wave map formulation.

Findings

Constructed a positive-definite, gauge-invariant energy functional.

Demonstrated the energy functional acts as a Hamiltonian for perturbation evolution.

Applicable to the exterior of Kerr black holes for stability studies.

Abstract

Based on the Hamiltonian dimensional reduction of $3+1$ axially symmetric, Ricci-flat Lorentzian spacetimes to a $2+1$ Einstein-wave map system with the (negatively curved) hyperbolic 2-plane target, we construct a positive-definite, (spacetime) gauge-invariant energy functional for linear axially symmetric perturbations in the exterior of Kerr black holes, in a manner that is also gauge-independent on the target manifold. We also show that the positive-definite energy functional serves as a Hamiltonian for the constrained evolution of the linear perturbations