Homogeneous spacelike singularities inside spherical black holes

Recent numerical simulations have found that the Cauchy horizon inside spherical charged black holes, when perturbed nonlinearly by a self-gravitating, minimally-coupled, massless, spherically-symmetric scalar field, turns into a null weak singularity which focuses monotonically to $r=0$ at late times, where the singularity becomes spacelike. Our main objective is to study this spacelike singularity. We study analytically the spherically-symmetric Einstein-Maxwell-scalar equations asymptotically near the singularity. We obtain a series-expansion solution for the metric functions and for the scalar field near $r=0$ under the simplifying assumption of homogeneity. Namely, we neglect spatial derivatives and keep only temporal derivatives. We find that there indeed exists a generic spacelike singularity solution for these equations (in the sense that the solution depends on enough free parameters), with similar properties to those found in the numerical simulations. This singularity is strong in the Tipler sense, namely, every extended object would inevitably be crushed to zero volume. In this sense this is a similar singularity to the spacelike singularity inside uncharged spherical black holes. On the other hand, there are some important differences between the two cases. Our model can also be extended to the more general inhomogeneous case. The question of whether the same kind of singularity evolves in more realistic models (of a spinning black hole coupled to gravitational perturbations) is still an open question.