Black hole nonmodal linear stability: the Schwarzschild (A)dS cases

Abstract

The nonmodal linear stability of the Schwarzschild black hole established in Phys. Rev. Lett. 112 (2014) 191101 is generalized to the case of a nonnegative cosmological constant $\Lambda$. Two gauge invariant combinations $G_{\pm }$ of perturbed scalars made out of the Weyl tensor and its first covariant derivative are found such that the map $[h_{\alpha \beta }]\to \left(G_{-}\left([h_{\alpha \beta }]\right),G_{+}\left([h_{\alpha \beta }]\right)\right)$ with domain the set of equivalent classes $[h_{\alpha \beta }]$ under gauge transformations of solutions of the linearized Einstein's equation, is invertible. The way to reconstruct a representative of $[h_{\alpha \beta }]$ in terms of $(G_{-},G_{+})$ is given. It is proved that, for an arbitrary perturbation consistent with the background asymptote, $G_{+}$ and $G_{-}$ 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 is a choice 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 $\Lambda =0$ case are explained in detail.

Typos corrected, changes in the Introduction (including example of nonmodal instability)

References in corpus (4)

• Instability of the negative mass Schwarzschild naked singularity
• Improved outer boundary conditions for Einstein's field equations
• Linear perturbations of self-gravitating spherically symmetric configurations
• Petrov type of linearly perturbed type D spacetimes

Cited by in corpus (9)

• Symmetry operators and decoupled equations for linear fields on black hole spacetimes
• Open problems in mathematical physics
• Non-modal effects in black hole perturbation theory: Transient Superradiance
• Instability of asymptotically anti de Sitter black holes under Robin conditions at the timelike boundary
• Black hole nonmodal linear stability: odd perturbations of Reissner-Nordström
• Linear Stability of Black Holes and Naked Singularities
• Gauge invariant description of weak gravitational field on a spherically symmetric background with cosmological constant
• Generalized wave operators, weighted Killing fields, and perturbations of higher dimensional spacetimes
• Black hole nonmodal linear stability: even perturbations in the Reissner-Nordström case