Mixmaster chaos in an AdS black hole interior

Abstract

We derive gravitational backgrounds that are asymptotically Anti-de Sitter, have a regular black hole horizon and which deep in the interior exhibit mixmaster chaotic dynamics. The solutions are obtained by coupling gravity with a negative cosmological constant to three massive vector fields, within an Ansatz that reduces to ordinary differential equations. At late interior times the equations are identical to those analysed in depth by Misner and by Belinskii-Khalatnikov-Lifshitz fifty years ago. We review and extend known classical and semiclassical results on the interior chaos, formulated as both a dynamical system of `Kasner eras' and as a hyperbolic billiards problem. The volume of the universe collapses doubly-exponentially over each Kasner era. A remarkable feature is the emergence of a conserved energy, and hence a `time-independent' Hamiltonian, at asymptotically late interior times. A quantisation of this Hamiltonian exhibits arithmetic chaos associated with the principal congruence subgroup $\Gamma(2)$ of the modular group. We compute a large number of eigenvalues numerically to obtain the spectral form factor. While the spectral statistics is anomalous for a chaotic system, the eigenfunctions themselves display random matrix behaviour.