Late-time singularity inside nonspherical black holes

It was long believed that the singularity inside a realistic, rotating black hole must be spacelike. However, studies of the internal geometry of black holes indicate a more complicated structure is typical. While it seems likely that an observer falling into a black hole with the collapsing star encounters a crushing spacelike singularity, an observer falling in at late times generally reaches a null singularity which is vastly different in character to the standard Belinsky, Khalatnikov, and Lifschitz (BKL) spacelike singularity [V. A. Belinsky, I. M. Khalatnikov, and E. M. Lifshitz, Sov. Phys. JETP 32, 169 (1970)]. In the spirit of the classic work of BKL we present an asymptotic analysis of the null singularity inside a realistic black hole. Motivated by current understanding of spherical models, we argue that the Einstein equations reduce to a simple form in the neighborhood of the null singularity. The main results arising from this approach are demonstrated using an almost plane symmetric model. The analysis shows that the null singularity results from the blueshift of the late-time gravitational wave tail; the amplitude of these gravitational waves is taken to decay as an inverse power of advanced time as suggested by perturbation theory. The divergence of the Weyl curvature at the null singularity is dominated by the propagating modes of the gravitational field, that is, $C_{\alpha \beta \gamma \delta }{C}^{\alpha \beta \gamma \delta }\sim {\Psi }_0{\Psi }_4\sim v^{-\mathrm{}(2l+3)\mathrm{}}{e}^{2\kappa v},$ as $\overrightarrow{v}\infty$ at the Cauchy horizon. Here, ${\Psi }_0$ and ${\Psi }_4$ are the Newman-Penrose Weyl scalars, and $l>~2$ is the multipole order of the perturbations crossing the event horizon. The null singularity is weak in the sense that tidal distortion remains bounded along timelike geodesics crossing the Cauchy horizon. These results are in agreement with previous analyses of black hole interiors. We briefly discuss some outstanding problems which must be resolved before the picture of the generic black hole interior is complete.