Canonical energy and linear stability of Schwarzschild

Consider linearised perturbations of the Schwarzschild black hole in 4 dimensions. Using the linearised Newman-Penrose curvature component, which satisfies the Teukolsky equation, as a Hertz potential we generate a 'new' metric perturbation satisfying the linearised Einstein equation. We show that the canonical energy, given by Hollands and Wald, of the 'new' metric perturbation is the conserved Regge-Wheeler-like energy used by Dafermos, Holzegel and Rodnianski to prove linear stability and decay of perturbations of Schwarzschild. We comment on a generalisation of this strategy to prove the linear stability of the Kerr black hole.