Title: Stability and instability of the sub-extremal Reissner-Nordström black hole interior for the Einstein-Maxwell-Klein-Gordon equations in spherical symmetry,
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 at infinity- in presence of a
massive and charged scalar field, motivated by the strong cosmic censorship
conjecture in that setting :
1. 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 spacetime 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.
2. Instability : We prove that for the class of space-times considered in the
stability part, whose scalar field in addition obeys a polynomial averaged-L^2
(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 C^2 strong cosmic censorship conjecture for
one-ended asymptotically initial data.