Symmetries of the Black Hole Interior and Singularity Regularization

We reveal an $\mathfrak{iso}(2,1)$ Poincar\'e algebra of conserved charges associated with the dynamics of the interior of black holes. The action of these Noether charges integrates to a symmetry of the gravitational system under the Poincar\'e group ISO$(2,1)$, which allows to describe the evolution of the geometry inside the black hole in terms of geodesics and horocycles of AdS${}_2$. At the Lagrangian level, this symmetry corresponds to M\"obius transformations of the proper time together with translations. Remarkably, this is a physical symmetry changing the state of the system, which also naturally forms a subgroup of the much larger $\textrm{BMS}_{3}=\textrm{Diff}(S^1)\ltimes\textrm{Vect}(S^1)$ group, where $S^1$ is the compactified time axis. It is intriguing to discover this structure for the black hole interior, and this hints at a fundamental role of BMS symmetry for black hole physics. The existence of this symmetry provides a powerful criterion to discriminate between different regularization and quantization schemes. Following loop quantum cosmology, we identify a regularized set of variables and Hamiltonian for the black hole interior, which allows to resolve the singularity in a black-to-white hole transition while preserving the Poincar\'e symmetry on phase space. This unravels new aspects of symmetry for black holes, and opens the way towards a rigorous group quantization of the interior.