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Boundedness and growth for the massive wave equation on asymptotically anti-de Sitter black holes. (English) Zbl 1292.83033

Summary: We study the global dynamics of free massive scalar fields on general, globally stationary, asymptotically AdS black hole backgrounds with Dirichlet, Neumann or Robin boundary conditions imposed on \({\psi}\) at infinity. This class includes the regular Kerr-AdS black holes satisfying the Hawking-Reall bound \(r_+^2 > |a|l\). We establish a suitable criterion for linear stability (in the sense of uniform boundedness) of \({\psi}\) and demonstrate how the issue of stability can depend on the boundary condition prescribed. In particular, in the slowly rotating Kerr-AdS case, we obtain the existence of linear scalar hair (i.e. non-trivial stationary solutions) for suitably chosen Robin boundary conditions.

MSC:

83C57 Black holes
35L05 Wave equation
83C15 Exact solutions to problems in general relativity and gravitational theory
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
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