The coexistence of null and spacelike singularities inside spherically symmetric black holes

Abstract: In our previous work [Van de Moortel, The breakdown of weak null singularities, Duke Mathematical Journal 172 (15), 2957-3012, 2023], we showed that dynamical black holes formed in charged spherical collapse generically feature both a null weakly singular Cauchy horizon and a stronger (presumably spacelike) singularity, confirming a longstanding conjecture in the physics literature. However, this previous result, based on a contradiction argument, did not provide quantitative estimates on the stronger singularity. In this study, we adopt a new approach by analyzing local initial data inside the black hole that are consistent with a breakdown of the Cauchy horizon. We prove that the remaining portion is spacelike and obtain sharp spacetime estimates near the null-spacelike transition. Notably, we show that the Kasner exponents of the spacelike portion are positive, in contrast to the well-known Oppenheimer-Snyder model of gravitational collapse. Moreover, these exponents degenerate to (1,0,0) towards the null-spacelike transition. Our result provides the first quantitative instances of a null-spacelike singularity transition inside a black hole. In our companion paper, we moreover apply our analysis to carry out the construction of a large class of asymptotically flat one or two-ended black holes featuring coexisting null and spacelike singularities.