Perturbations of the almost Killing equation and their implications

TL;DR

This paper investigates how the almost Killing equation behaves under perturbations of symmetric spacetimes, revealing conditions under which the perturbed equation remains well-behaved and exploring related thermodynamical implications.

Contribution

It analyzes the perturbations of the almost Killing equation in symmetric spacetimes, providing conditions for stability and extending understanding of approximate symmetries in astrophysical contexts.

Findings

Perturbed almost Killing equation avoids unbounded Hamiltonian issues for nonradiative, non-trace metric perturbations.

Second-order perturbations behave similarly for traceless perturbations with some caveats.

Thermodynamical implications of these perturbations are discussed.

Abstract

Killing vectors play a crucial role in characterizing the symmetries of a given spacetime. However, realistic astrophysical systems are in most cases only approximately symmetric. Even in the case of an astrophysical black hole, one might expect Killing symmetries to exist only in an approximate sense due to perturbations from external matter fields. In this work, we consider the generalized notion of Killing vectors provided by the almost Killing equation, and study the perturbations induced by a perturbation of a background spacetime satisfying exact Killing symmetry. To first order, we demonstrate that for nonradiative metric perturbations (that is, metric perturbations with nonvanishing trace) of symmetric vacuum spacetimes, the perturbed almost Killing equation avoids the problem of an unbounded Hamiltonian for hyperbolic parameter choices. For traceless metric perturbations, we obtain similar results for the second-order perturbation of the almost Killing equation, with some additional caveats. Thermodynamical implications are also explored.