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Slow Decay of Waves in Gravitational Solitons

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Abstract

We consider a family of globally stationary (horizonless), asymptotically flat solutions of five-dimensional supergravity. We prove that massless linear scalar waves in such soliton spacetimes cannot have a uniform decay rate faster than inverse logarithmically in time. This slow decay can be attributed to the stable trapping of null geodesics. Our proof uses the construction of quasimodes which are time periodic approximate solutions to the wave equation. The proof is based on previous work to prove an analogous result in \(\text{ Kerr-AdS}_4\) black holes (Holzegel and Smulevici in Anal PDE 7(5):1057–1090, 2014). We remark that this slow decay is suggestive of an instability at the nonlinear level.

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Notes

  1. Alternate proofs for this nonlinear stability result have been obtained in [15, 16].

  2. Supersymmetric spacetimes admit Killing spinors, i.e., non-trivial spinor fields with are covariantly constant with respect to an appropriate connection.

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Acknowledgements

This work was supported by NSERC Discovery Grant RGPIN-2018-04887. We are thankful to Stefanos Aretakis for helpful comments on a preliminary version of this paper and for discussions during the 2017 Atlantic General Relativity workshop at Memorial University. We also acknowledge helpful communications with Joe Keir on the derivation of certain estimates in Sect.  7. We thank Graham Cox for clarifying various aspects of PDEs and Ivan Booth and James Lucietti for useful discussions.

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  1. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C5S7, Canada

    Sharmila Gunasekaran & Hari K. Kunduri

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  1. Sharmila Gunasekaran
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Gunasekaran, S., Kunduri, H.K. Slow Decay of Waves in Gravitational Solitons. Ann. Henri Poincaré 22, 821–872 (2021). https://doi.org/10.1007/s00023-020-01010-3

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