Bounded energy waves on the black hole interior of Reissner–Nordström–de Sitter

Abstract

Motivated by the strong cosmic censorship conjecture, in the presence of a cosmological constant, we consider solutions of the scalar wave equation \Box _g\phi =0 on fixed subextremal Reissner–Nordström–de Sitter backgrounds ({\mathcal M}, g), without imposing symmetry assumptions on \phi . We provide a sufficient condition, in terms of surface gravities and a parameter for an exponential decaying Price Law, for a local energy of the waves to remain bounded up to the Cauchy horizon. The energy we consider controls, in particular, regular transverse derivatives at the Cauchy horizon; this allows us to extend the solutions with bounded energy, to the Cauchy horizon, as functions in C^0\cap H^1_\mathrm{loc}. Our results correspond to another manifestation of the potential breakdown of strong cosmic censorship in the positive cosmological constant setting.