Stability of Schwarzschild-AdS for the Spherically Symmetric Einstein-Klein-Gordon System

Abstract

In this paper, we study the global behavior of solutions to the spherically symmetric coupled Einstein-Klein-Gordon (EKG) system in the presence of a negative cosmological constant. For the Klein-Gordon mass-squared satisfying a ≥ −1 (the Breitenlohner-Freedman bound being a > −9/8), we prove that the Schwarzschild-AdS spacetimes are asymptotically stable: Small perturbations of Schwarzschild-AdS initial data again lead to regular black holes, with the metric on the black hole exterior approaching, at an exponential rate, a Schwarzschild-AdS spacetime. The main difficulties in the proof arise from the lack of monotonicity for the Hawking mass and the asymptotically AdS boundary conditions, which render even (part of) the orbital stability intricate. These issues are resolved in a bootstrap argument on the black hole exterior, with the redshift effect and weighted Hardy inequalities playing the fundamental role in the analysis. Both integrated decay and pointwise decay estimates are obtained. As a corollary of our estimates on the Klein-Gordon field, one obtains in particular exponential decay in time of spherically-symmetric solutions to the linear Klein-Gordon equation on Schwarzschild-AdS.

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Correspondence to Jacques Smulevici.

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Communicated by P. T. Chruściel

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Holzegel, G., Smulevici, J. Stability of Schwarzschild-AdS for the Spherically Symmetric Einstein-Klein-Gordon System. Commun. Math. Phys. 317, 205–251 (2013). https://doi.org/10.1007/s00220-012-1572-2

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Keywords

  • Black Hole
  • Wave Equation
  • Boundary Term
  • Hardy Inequality
  • Hairy Black Hole