Canonical Energy and Hertz Potentials for Perturbations of Schwarzschild Spacetime

Understand

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.