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Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I

Abstract

We study the problem of stability and instability of extreme Reissner-Nordström spacetimes for linear scalar perturbations. Specifically, we consider solutions to the linear wave equation \({\square_{g}\psi=0}\) on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface Σ0 crossing the future event horizon \({\mathcal{H}^{+}}\) . We obtain boundedness, decay and non-decay results. Our estimates hold up to and including the horizon \({\mathcal{H}^{+}}\) . The fundamental new aspect of this problem is the degeneracy of the redshift on \({\mathcal{H}^{+}}\) . Several new analytical features of degenerate horizons are also presented.

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Correspondence to Stefanos Aretakis.

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Communicated by P.T. Chruściel

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Aretakis, S. Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I. Commun. Math. Phys. 307, 17 (2011). https://doi.org/10.1007/s00220-011-1254-5

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Keywords

  • Black Hole
  • Wave Equation
  • Event Horizon
  • Linear Stability
  • Extreme Black Hole