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 g ψ = 0 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 obtain boundedness, decay and non-decay results. Our estimates hold up to and including the horizon H + . The fundamental new aspect of this problem is the degeneracy of the redshift on H + . Several new analytical features of degenerate horizons are also presented.

Abstract. This paper contains the second part of a two-part series on the stability and instability of extreme Reissner-Nordström spacetimes for linear scalar perturbations. We continue our study of solutions to the linear wave equation g ψ = 0 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.

We show that axisymmetric extremal horizons are unstable under scalar perturbations. Specifically, we show that translation invariant derivatives of generic solutions to the wave equation do not decay along such horizons as advanced time tends to infinity, and in fact, higher order derivatives blow up. This result holds in particular for extremal Kerr-Newman and Majumdar-Papapetrou spacetimes and is in stark contrast with the subextremal case for which decay is known for all derivatives along the event horizon.