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.

We derive precise late-time asymptotics for solutions to the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes including as special cases the Schwarzschild and Reissner-Nordström families of black holes. We also obtain late-time asymptotics for the time derivatives of all orders and for the radiation field along null infinity. We show that the leading-order term in the asymptotic expansion is related to the existence of the conserved Newman-Penrose quantities on null infinity. As a corollary we obtain a characterization of all solutions which satisfy Price's polynomial law τ −3 as a lower bound. Our analysis relies on physical space techniques and uses the vector field approach for almost-sharp decay estimates introduced in our companion paper. In the black hole case, our estimates hold in the domain of outer communications up to and including the event horizon. Our work is motivated by the stability problem for black hole exteriors and strong cosmic censorship for black hole interiors.

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We study the Cauchy problem for the wave equation 2 g ψ = 0 on extreme Kerr backgrounds. Specifically, we consider regular axisymmetric initial data prescribed on a Cauchy hypersurface Σ 0 which connects the future event horizon with spacelike or null infinity, and we solve the linear wave equation on the domain of dependence of Σ 0 . We show that the spacetime integral of an energy-type density is bounded by the initial conserved flux corresponding to the stationary Killing field T , and we derive boundedness of the non-degenerate energy flux corresponding to a globally timelike vector field N . Finally, we prove uniform pointwise boundedness and power-law decay for ψ up to and including the event horizon H + . Published by Elsevier Inc.

We present a new vector field approach to almost-sharp decay for the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes. Specifically, we derive a new hierarchy of higher-order weighted energy estimates by employing appropriate commutator vector fields. In cases where an integrated local energy decay estimate holds, like in the case of sub-extremal Reissner-Nordström black holes, this hierarchy leads to almost-sharp global energy and pointwise time-decay estimates with decay rates that go beyond those obtained by the traditional vector field method. Our estimates play a fundamental role in our companion paper where precise late-time asymptotics are obtained for linear scalar fields on such backgrounds. States,

In this paper, we prove precise late-time asymptotics for solutions to the wave equation supported on angular frequencies greater or equal to on the domain of outer communications of subextremal Reissner-Nordström spacetimes up to and including the event horizon. Our asymptotics yield, in particular, sharp upper and lower decay rates which are consistent with Price's law on such backgrounds. We present a theory for inverting the time operator and derive an explicit representation of the leadingorder asymptotic coefficient in terms of the Newman-Penrose charges at null infinity associated with the time integrals. Our method is based on purely physical space techniques. For each angular frequency we establish a sharp hierarchy of r-weighted radially commuted estimates with length 2 + 5. We complement this hierarchy with a novel hierarchy of weighted elliptic estimates of length + 1.