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Stability and instability of extreme Reissner-Nordström black hole spacetimes for linear scalar perturbations. I. (English) Zbl 1229.85002
Summary: We study the problem of stability and instability of extreme Reissner-Nordström space-times 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 space-time, arising from regular initial data prescribed on a Cauchy hypersurface \(\Sigma_0\) 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.

MSC:
85A05 Galactic and stellar dynamics
83C57 Black holes
35L05 Wave equation
83C22 Einstein-Maxwell equations
83C50 Electromagnetic fields in general relativity and gravitational theory
83C75 Space-time singularities, cosmic censorship, etc.
83C25 Approximation procedures, weak fields in general relativity and gravitational theory
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
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