Stability and Instability of Extreme Reissner–Nordström Black Hole Spacetimes for Linear Scalar Perturbations II

Abstract

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.

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 this paper, we impose Cauchy data for (1.1) on a spacelike hypersurface in extremal Reissner-Nordström. It is appropriate to consider a hypersurface that is asymptotically flat at one end and intersects the black hole interior; see the discussion in [5]. Due to the geometry of the interior, must necessarily be incomplete.…”

“…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.…”

“…The requirement (1.2) follows from improved decay results in the exterior of extremal Reissner-Nordström for solutions arising from suitable Cauchy data on , compared to the decay results in [6,7]. See upcoming work with Angelopoulos and Aretakis [1].…”

“…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).…”