Stability of Anti-de Sitter spacetime for Robin boundary conditions

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Written by Dominika Hunik

We consider the four-dimensional spherically symmetric Einstein-Klein-Gordon equations with cosmological constant Λ and mass square m²=2/3Λ. These equations are well-behaved at the conformal boundary $\scri$ which makes them a good toy model for studying how the stability properties of anti-de Sitter spacetime depend on the boundary conditions at $\scri$. We find that for the Dirichlet and Neumann boundary conditions the dynamics is similar to the massless case, i.e. for small perturbations of AdS of size ε we observe instability against black hole formation on the time scale 1/ε². However, for the Robin boundary conditions sufficiently small perturbations do not grow which indicates that there is a threshold for instability. We conjecture that the existence of the threshold is due to the fact that the Robin spectrum of linear perturbations of AdS, in contrast to the Dirichlet and Neumann spectra, is not fully resonant.