Mixmaster chaos in an AdS black hole interior
Abstract
We derive gravitational backgrounds that are asymptotically Antide 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 BelinskiiKhalatnikovLifshitz 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 doublyexponentially over each Kasner era. A remarkable feature is the emergence of a conserved energy, and hence a `timeindependent' 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2023
 DOI:
 10.48550/arXiv.2312.11622
 arXiv:
 arXiv:2312.11622
 Bibcode:
 2023arXiv231211622D
 Keywords:

 High Energy Physics  Theory;
 General Relativity and Quantum Cosmology
 EPrint:
 74 pages including references and appendices. 18 figures. v2: added reference and a minor clarification. v3: added reference, details on numerics and comment on unitarity bound. v4: minor clarifications and references added