Canonical energy and Hertz potentials for perturbations of Schwarzschild spacetime
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 4D 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 ReggeWheeler variable of DHR to the ordinary ReggeWheeler 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.
 Publication:

Classical and Quantum Gravity
 Pub Date:
 December 2018
 DOI:
 10.1088/13616382/aae9ae
 arXiv:
 arXiv:1807.09883
 Bibcode:
 2018CQGra..35w5004P
 Keywords:

 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 v2: Eqs.(3.39a) and (3.41) corrected and other minor revisions made. v1: 30 pages