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Linear Stability of Schwarzschild-Anti-de Sitter spacetimes I: The system of gravitational perturbations
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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.
Forward citations
Cited by 5 Pith papers
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Stationary gravitational modes on Kerr-anti-de Sitter spacetimes
For Kerr-AdS black holes crossing the Hawking-Reall bound, stationary gravitational modes exist for all sufficiently large azimuthal numbers, and the parameters accumulate on the threshold as the azimuthal number grows.
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Near-Boundary Asymptotics and Unique Continuation for the AdS--Einstein--Maxwell System
Given identical boundary data near infinity, two solutions of the AdS-Einstein-Maxwell equations must agree near that boundary, provided the boundary region satisfies a null-convexity condition.
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A scattering construction for nonlinear wave equations on Kerr--Anti-de Sitter spacetimes
For a broad class of nonlinear wave equations on the Kerr-AdS exterior, exponentially decaying event horizon data is shown to yield future-global smooth solutions with exponential decay, even beyond the Hawking-Reall bound.
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On a fixed spacetime with stable trapping, the cubic wave equation produces a forward angular-mode cascade and growth of higher-order derivatives, behavior absent for linear waves.
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Tails from the Bulk: Gravitational Decay in AdS$_5$
Gravitational perturbations of Schwarzschild-AdS5 with analytic SO(3)-symmetric data decay at late times as v^{-2α/C}, with subleading oscillations periodic in log v with period C(y+).
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