Abstract
We show non-linear stability and instability results in spherical symmetry for the interior of a charged black hole—approaching a sub-extremal Reissner–Nordström background fast enough—in presence of a massive and charged scalar field, motivated by the strong cosmic censorship conjecture in that setting:Stability We prove that spherically symmetric characteristic initial data to the Einstein–Maxwell–Klein–Gordon equations approaching a Reissner–Nordström background with a sufficiently decaying polynomial decay rate on the event horizon gives rise to a space–time possessing a Cauchy horizon in a neighbourhood of time-like infinity. Moreover, if the decay is even stronger, we prove that the space–time metric admits a continuous extension to the Cauchy horizon. This generalizes the celebrated stability result of Dafermos for Einstein–Maxwell-real-scalar-field in spherical symmetry.Instability We prove that for the class of space–times considered in the stability part, whose scalar field in addition obeys a polynomial averaged-L2 (consistent) lower bound on the event horizon, the scalar field obeys an integrated lower bound transversally to the Cauchy horizon. As a consequence we prove that the non-degenerate energy is infinite on any null surface crossing the Cauchy horizon and the curvature of a geodesic vector field blows up at the Cauchy horizon near time-like infinity. This generalizes an instability result due to Luk and Oh for Einstein–Maxwell-real-scalar-field in spherical symmetry.This instability of the black hole interior can also be viewed as a step towards the resolution of the C2 strong cosmic censorship conjecture for one-ended asymptotically flat initial data.
Highlights
This instability of the black hole interior can be viewed as a step towards the resolution of the C2 strong cosmic censorship conjecture for one-ended asymptotically flat initial data
We prove that assuming an upper and lower bound on the scalar field φ on the event horizon of the black hole, the Cauchy horizon exhibits both stability and instability features, namely: 1. Stability the perturbed black hole still admits a Cauchy horizon—near time-like infinity—like the original unperturbed Reissner–Nordström black hole, and in some cases we can even extend the metric continuously beyond this Cauchy horizon
In this paper we prove that the expected asymptotic decay of the scalar field on the event horizon—known as generalised Price’s law—8 implies some stability and instability features for a more realistic and richer generalization of the charged space–time model of Dafermos in spherical symmetry
Summary
If we assume the following point-wise lower bound on the scalar field for some > 0: v−3s+ |∂vφ||H+ v−s , the modified mass blows up as one approaches the Cauchy horizon:. The statement that the tail of the scalar field decays polynomially—for all models— is called generalized Price’s law This conjecture is still an open problem for the charged and massive model of the present paper and requires a stability study of the black hole exterior. The result of Luk and Oh is more quantitative but on the other hand it relies crucially on the hypothesis s > 1 giving an initial integrable decay of Ω2 − ΩR2N, r − rRN and φ This is why— the method can be adapted in the presence of a charged and massive field—the proof fails for s ≤ 1 which is the expectation in many interesting cases as claimed by Price’s law of Conjecture 1.6. In the “Appendix A”, we use our stability framework to “localise” in coordinates the part of the apparent horizon that is close to time-like infinity
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