Linear Stability of Schwarzschild-Anti-de Sitter spacetimes I: The system of gravitational perturbations

TL;DR

This paper proves the linear stability of Schwarzschild-Anti-de Sitter black holes under gravitational perturbations, showing solutions remain bounded and decay logarithmically, using a hierarchical approach and results from the Teukolsky system.

Contribution

It establishes the first part of a series demonstrating linear stability of Schwarzschild-AdS black holes with new decay estimates and a hierarchical analysis of linearised gravity equations.

Findings

Solutions remain globally bounded on the black hole exterior.

Solutions decay logarithmically in time to a linearised Kerr-AdS metric.

No residual gauge solution is needed for decay of curvature and Ricci coefficients.

Abstract

This is the main paper of a series establishing the linear stability of Schwarzschild-Anti-de Sitter (AdS) black holes to gravitational perturbations. Specifically, we prove that solutions to the linearisation of the Einstein equations $\textrm{Ric}(g) = \Lambda g$ with $\Lambda<0$ around a Schwarzschild-AdS metric arising from regular initial data and with standard Dirichlet-type boundary conditions imposed at the conformal boundary (inherited from fixing the conformal class of the non-linear metric) remain globally uniformly bounded on the black hole exterior and in fact decay inverse logarithmically in time to a linearised Kerr-AdS metric. The proof exploits a hierarchical structure of the equations of linearised gravity in double null gauge and crucially relies on boundedness and logarithmic decay results for the Teukolsky system, which are independent results proven in Part II of the series. Contrary to the asymptotically flat case, addition of a residual pure gauge solution to the original solution is not required to prove decay of all linearised null curvature and Ricci coefficients. One may however normalise the solution at the conformal boundary to be in standard AdS-form by adding such a pure gauge solution, which is constructed dynamically from the trace of the original solution at the conformal boundary and quantitatively controlled by initial data.