Proof of linear instability of the Reissner–Nordström Cauchy horizon under scalar perturbations

Abstract

Abstract. It has long been suggested that solutions to linear scalar wave equation g φ = 0 on a fixed subextremal Reissner-Nordström spacetime with non-vanishing charge are generically singular at the Cauchy horizon. We prove that generic smooth and compactly supported initial data on a Cauchy hypersurface indeed give rise to solutions with infinite nondegenerate energy near the Cauchy horizon in the interior of the black hole. In particular, the solution generically does not belong to W 1,2 loc . This instability is related to the celebrated blue shift effect in the interior of the black hole. The problem is motivated by the strong cosmic censorship conjecture and it is expected that for the full nonlinear Einstein-Maxwell system, this instability leads to a singular Cauchy horizon for generic small perturbations of Reissner-Nordström spacetime. Moreover, in addition to the instability result, we also show as a consequence of the proof that Price's law decay is generically sharp along the event horizon.

Linear Waves in the Interior of Extremal Black Holes I

“…Rigorous results pertaining to the upper bound in (1.3) have been obtained in [21,24,25,41,49]. Moreover, Luk-Oh showed in [37] that ∂ v φ cannot decay with a polynomial rate faster than v −4 along the event horizon of subextremal Reissner-Nordström, for generic, compactly supported data. This fact is crucially used in the proof of Theorem B.…”

“…Theorem B (Luk-Oh [37]). Let φ be a solution to (1.1) in subextremal ReissnerNordström arising from generic, smooth, compactly supported data on a spacelike hypersurface that is asymptotically flat at two ends.…”

“…The methods in [21,24,25,37,41,49] break down in extremal Reissner-Nordström, in view of the absence of the local red-shift effect. Heuristics and numerics regarding latetime tails for extremal Reissner-Nordström in [33,44,47] suggest an extremal variant of "Price's law" that in particular predicts:…”

“…The required decay along the event horizon can be shown to hold for solutions arising from suitable Cauchy data; see upcoming work with Angelopoulos and Aretakis [1]. The finiteness of the local energy of φ at the Cauchy horizon stands in sharp contrast to the interior region of subextremal Reissner-Nordström, where the corresponding energy of φ has been shown to blow up for generic data [37]. This distinctive property of the extremal Reissner-Nordström interior was first observed by Murata-Reall-Tanahashi in a numerical setting [42].…”

Linear Waves in the Interior of Extremal Black Holes II

“…The analogue of (A) has recently been obtained for the wave equation in subextremal Reissner-Nordström (e 2 < M 2 ) [21] and subextremal Kerr (a 2 < M 2 ) [20] by Franzen (see also the results of Hintz [25] in the very slowly rotating setting, where a 2 M 2 ), whereas the analogue of (B) has been shown to fail in subextremal Reissner-Nordström for generic Cauchy data [29] by Luk-Oh. See also related results concerning instabilities in subextremal Kerr [19,30].…”