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Strong cosmic censorship in the presence of matter: the decisive effect of horizon oscillations on the black hole interior geometry

Christoph Kehle and Maxime Van de Moortel

Vol. 17 (2024), No. 5, 1501–1592

Motivated by the strong cosmic censorship conjecture in the presence of matter, we study the Einstein equations coupled with a charged/massive scalar field with spherically symmetric characteristic data relaxing to a Reissner–Nordström event horizon. Contrary to the vacuum case, the relaxation rate is conjectured to be slow (nonintegrable), opening the possibility that the matter fields and the metric coefficients blow up in amplitude at the Cauchy horizon, not just in energy. We show that whether this blow-up in amplitude occurs or not depends on a novel oscillation condition on the event horizon which determines whether or not a resonance is excited dynamically:

  • If the oscillation condition is satisfied, then the resonance is not excited and we show boundedness and continuous extendibility of the matter fields and the metric across the Cauchy horizon.

  • If the oscillation condition is violated, then by the combined effect of slow decay and the resonance being excited, we show that the massive uncharged scalar field blows up in amplitude.

    In a companion paper, we will show that in that case a novel null contraction singularity forms at the Cauchy horizon, across which the metric is not continuously extendible in the usual sense.

Heuristic arguments in the physics literature indicate that the oscillation condition should be satisfied generically on the event horizon. If these heuristics are true, then our result falsifies the C0-formulation of strong cosmic censorship by means of oscillation.

strong cosmic censorship, black hole, Einstein equations, scattering, charged scalar field
Mathematical Subject Classification
Primary: 35Q75, 35Q76, 83C05, 83C57, 83C75
Received: 31 May 2021
Revised: 7 April 2022
Accepted: 21 December 2022
Published: 20 June 2024
Christoph Kehle
Institute for Theoretical Studies
ETH Zürich
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States
Maxime Van de Moortel
Department of Mathematics
Princeton University
Princeton, NJ
United States

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