Black hole non-modal linear stability: The Schwarzschild (A)dS cases

Abstract

The non-modal linear stability of the Schwarzschild black hole established in Dotti (2014 Phys. Rev. Lett. 112 191101) is generalized to the case of a non-negative cosmological constant Λ. Two gauge invariant combinations G± of perturbed scalars made out of the Weyl tensor and its first covariant derivative are found such that the map with domain the set of equivalent classes under gauge transformations of solutions of the linearized Einstein's equation, is invertible. The way to reconstruct a representative of in terms of is given. It is proved that, for an arbitrary perturbation consistent with the background asymptote, and are bounded in the the outer static region. At large times, the perturbation decays leaving a linearized Kerr black hole around the Schwarzschild or Schwarschild de Sitter background solution. For negative cosmological constant it is shown that there are choices of boundary conditions at the time-like boundary under which the Schwarzschild anti de Sitter black hole is unstable. The root of Chandrasekhar's duality relating odd and even modes is exhibited, and some technicalities related to this duality and omitted in the original proof of the case are explained in detail.