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 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.

Linear Waves in the Interior of Extremal Black Holes I

“…In fact, Aretakis showed in [6,7] that solutions to (1.1) on extremal ReissnerNordström or their transversal derivatives along the event horizon do not decay as a consequence of the existence of conserved quantities that do not vanish for generic data, the Aretakis constants. In [4] he showed moreover that, under the assumption of pointwise decay of solutions and their tangential derivatives along the event horizon in affine time v, second-order transversal derivatives even blow up as v → ∞.…”

“…In view of the decay results along H + in [6,7], it is sufficient to prove the results of Theorem 1, starting from appropriate characteristic initial data.…”

“…6 That is to say,Ũ is constant along outgoing null hypersurfaces andṼ is constant along ingoing null hypersurfaces that are preserved under isometries of spherical symmetry. We can therefore express U = M − r | v=v 0 (Ũ ) and use this expression to extend U as a function on the entire manifold M. Then U ∈ (0, ∞).…”

“…Specifically, we consider (1.1) in extremal Reissner-Nordström (|e| = M). Aretakis proved decay in time for solutions φ and their tangential derivatives along the event horizon in [6,7], starting from Cauchy data on a spacelike hypersurface intersecting the event horizon. We first show that, under the assumption of the decay rates from [6,7], φ is continuously extendible beyond the Cauchy horizon (the inner horizon).…”

Linear Waves in the Interior of Extremal Black Holes II

“…For results (B), (C) and (D) we imposed stronger decay estimates in affine time on φ and its tangential derivatives along the event horizon than those that had previously been established in [6,7] for φ arising from Cauchy data. The required decay estimates have been obtained in [3] for suitable Cauchy data.…”