Canonical Energy and Hertz Potentials for Perturbations of Schwarzschild Spacetime

TL;DR

This paper proves the positivity of canonical energy for metric perturbations of Schwarzschild spacetime generated by Hertz potentials, aiding the understanding of black hole stability and potentially extending to Kerr black holes.

Contribution

It demonstrates that canonical energy for Hertz potential-generated perturbations of Schwarzschild is positive, simplifying stability analysis and linking to previous linear stability proofs.

Findings

Canonical energy is positive for Hertz potential-generated perturbations.

Relates the DHR stability energy to canonical energy of Hertz potential perturbations.

Connects Regge-Wheeler variables to Hertz potential formalism.

Abstract

Canonical energy is a valuable tool for analyzing the linear stability of black hole spacetimes; positivity of canonical energy for all perturbations implies mode stability, whereas the failure of positivity for any perturbation implies instability. Nevertheless, even in the case of $4$-dimensional Schwarzschild spacetime --- which is known to be stable --- manifest positivity of the canonical energy is difficult to establish, due to the presence of constraints on the initial data as well as the gauge dependence of the canonical energy integrand. Consideration of perturbations generated by a Hertz potential would appear to be a promising way to improve this situation, since the constraints and gauge dependence are eliminated when the canonical energy is expressed in terms of the Hertz potential. We prove that the canonical energy of a metric perturbation of Schwarzschild that is generated by a Hertz potential is positive. We relate the energy quantity arising in the linear stability proof of Dafermos, Holzegel and Rodnianski (DHR) to the canonical energy of an associated metric perturbation generated by a Hertz potential. We also relate the Regge-Wheeler variable of DHR to the ordinary Regge-Wheeler and twist potential variables of the associated perturbation. Since the Hertz potential formalism can be generalized to a Kerr black hole, our results may be useful for the analysis of the linear stability of Kerr.