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Quasimodes and a lower
bound on the uniform energy decay rate for Kerr–AdS
spacetimes
Gustav Holzegel and Jacques Smulevici |
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Vol. 7 (2014), No. 5, 1057–1090
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Abstract |
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We construct quasimodes for the Klein–Gordon equation on the black hole exterior of Kerr–AdS (anti- de Sitter) spacetimes. Such quasimodes are associated with time-periodic approximate solutions of the Klein–Gordon equation and provide natural candidates to probe the decay of solutions on these backgrounds. They are constructed as the solutions of a semiclassical nonlinear eigenvalue problem arising after separation of variables, with the (inverse of the) angular momentum playing the role of the semiclassical parameter. Our construction results in exponentially small errors in the semiclassical parameter. This implies that general solutions to the Klein Gordon equation on Kerr–AdS cannot decay faster than logarithmically. The latter result completes previous work by the authors, where a logarithmic decay rate was established as an upper bound. |
Keywords
wave equation, black holes, decay estimates, Kerr – anti-de
Sitter
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Mathematical Subject Classification 2010
Primary: 58J50
Secondary: 83C57
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Milestones
Received: 3 June 2013
Revised: 24 January 2014
Accepted: 20 April 2014
Published: 27 September 2014
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Authors |
| Gustav Holzegel | |
| Department of Mathematics Imperial College London South Kensington Campus London SW7 2AZ United Kingdom |
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| Jacques Smulevici | |
| Laboratoire de Mathématiques Université Paris Sud 11 bât. 425 91405 Orsay France |
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