Stability and Instability of Extreme Reissner-Nordstr?m Black Hole Spacetimes for Linear Scalar Perturbations II

摘要

This paper contains the second part of a two-part series on the stability and instability of extreme Reissner-Nordstr?m spacetimes for linear scalar perturbations. We continue our study of solutions to the linear wave equation □_gψ on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface Σ_0 crossing the future event horizon H~+. We here obtain definitive energy and pointwise decay, non-decay and blow-up results. Our estimates hold up to and including the horizon H~+. A hierarchy of conservations laws on degenerate horizons is also derived.