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Bounded Energy Waves on the Black Hole Interior of Reissner–Nordström–de Sitter

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Abstract

Motivated by the strong cosmic censorship conjecture, in the presence of a cosmological constant, we consider solutions of the scalar wave equation \(\Box _g\phi =0\) on fixed subextremal Reissner–Nordström–de Sitter backgrounds \(({\mathcal M}, g)\), without imposing symmetry assumptions on \(\phi \). We provide a sufficient condition, in terms of surface gravities and a parameter for an exponential decaying Price Law, for a local energy of the waves to remain bounded up to the Cauchy horizon. The energy we consider controls, in particular, regular transverse derivatives at the Cauchy horizon; this allows us to extend the solutions with bounded energy, to the Cauchy horizon, as functions in \(C^0\cap H^1_\mathrm{loc}\). Our results correspond to another manifestation of the potential breakdown of strong cosmic censorship in the positive cosmological constant setting.

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Authors and Affiliations

  1. Instituto Universitário de Lisboa (ISCTE-IUL), Lisbon, Portugal

    João L. Costa

  2. Center for Mathematical Analysis, Geometry and Dynamical Systems Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Anne T. Franzen

Authors
  1. João L. Costa
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  2. Anne T. Franzen
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Correspondence to João L. Costa.

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Communicated by James A. Isenberg.

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Costa, J.L., Franzen, A.T. Bounded Energy Waves on the Black Hole Interior of Reissner–Nordström–de Sitter. Ann. Henri Poincaré 18, 3371–3398 (2017). https://doi.org/10.1007/s00023-017-0592-z

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  • DOI: https://doi.org/10.1007/s00023-017-0592-z

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