Abstract
We reveal an \mathfrak{iso}(2,1)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)(2,1), which allows to describe the evolution of the geometry inside the black hole in terms of geodesics and horocycles of AdS{}_22. 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)BMS3=Diff(S1)⋉Vect(S1) group, where S^1S1 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.
Highlights
We reveal an iso(2, 1) Poincaré algebra of conserved charges associated with the dynamics of the interior of black holes
Concerning point ii), we will see that there is a subtle sense in which a boundary has to be considered. This is because non-closed homogeneous cosmological models, like FLRW or the black hole interior which we study, require an IR cutoff in order for the spatial integrals to be well-defined and to obtain for the symmetry-reduced action a mechanical model depending only on time
The generalized Kantowski–Sachs line element (2.2) is very similar to the homogenous metric of FLRW cosmology, and we might expect a similar sl(2, ) algebraic structure as that identified in [37,38,39,40]. We show that this structure is extended to a Poincaré iso(2, 1) Lie algebra encoding the dynamics of the between the trapped (BH) interior
Summary
Symmetries play a fundamental role in modern physics. They give rise to conservation laws via Noether’s theorem [1], control the structure of classical solutions of a system, and organize its quantum states via representation theory. Concerning point ii), we will see that there is a subtle sense in which a boundary has to be considered This is because non-closed homogeneous cosmological models, like FLRW or the black hole interior which we study, require an IR cutoff in order for the spatial integrals to be well-defined and to obtain for the symmetry-reduced action (and symplectic structure and Hamiltonian) a mechanical model depending only on time. This cutoff, which we will introduce as a fiducial length L0 restricting the radial integrals, plays a subtle role and appears in particular as a shift in the Hamiltonian constraint. In appendix A we summarize some of the notations which are used throughout the paper
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