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Communications in Mathematical Physics

, Volume 350, Issue 2, pp 639–697 | Cite as

Unstable Mode Solutions to the Klein–Gordon Equation in Kerr-anti-de Sitter Spacetimes

  • Dominic DoldEmail author
Open Access
Article

Abstract

For any cosmological constant \({\Lambda = -3/\ell^{2} < 0}\) and any \({\alpha < 9/4}\), we find a Kerr-AdS spacetime \({({\mathcal{M}}, g_{{\rm KAdS}})}\), in which the Klein–Gordon equation \({\Box_{g_{{\rm KAdS}}}\psi + \alpha/\ell^{2}\psi = 0}\) has an exponentially growing mode solution satisfying a Dirichlet boundary condition at infinity. The spacetime violates the Hawking–Reall bound \({r_{+}^{2} > |a|\ell}\). We obtain an analogous result for Neumann boundary conditions if \({5/4 < \alpha < 9/4}\). Moreover, in the Dirichlet case, one can prove that, for any Kerr-AdS spacetime violating the Hawking–Reall bound, there exists an open family of masses \({\alpha}\) such that the corresponding Klein–Gordon equation permits exponentially growing mode solutions. Our result adopts methods of Shlapentokh-Rothman developed in (Commun. Math. Phys. 329:859–891, 2014) and provides the first rigorous construction of a superradiant instability for negative cosmological constant.

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Cambridge Centre for Analysis, Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUnited Kingdom

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