Fast scrambling of mutual information in Kerr- AdS4 spacetime

Abstract

We compute the disruption of mutual information between the hemispherical subsystems on the left and right conformal field theories of a thermofield double state described by a Kerr geometry in ${\mathrm{AdS}}_{4}$ due to shock waves along the equatorial plane. The shock waves and the subsystems considered respect the axisymmetry of the geometry. At late times, the disruption of the mutual information is given by the lengthening of the Hubeny-Rangamani-Takayanagi surface connecting the two subsystems; we compute the minimum value of the Lyapunov index-${\ensuremath{\lambda}}_{L}^{(\mathrm{min})}$ at late times and find that it is bounded by $\ensuremath{\kappa}=\frac{2\ensuremath{\pi}{T}_{H}}{(1\ensuremath{-}\ensuremath{\mu}\mathcal{L})}$, where $\ensuremath{\mu}$ is the horizon velocity, and $\mathcal{L}$ is the angular momentum per unit energy of the shock wave. At very late times, we find the scrambling time for such a system is governed by $\ensuremath{\kappa}$ with $\ensuremath{\kappa}{t}_{*}=\mathrm{log}\mathcal{S}$ for large black holes with a large entropy $\mathcal{S}$. We also find a term that increases the scrambling time by $\mathrm{log}(1\ensuremath{-}\ensuremath{\mu}\mathcal{L}{)}^{\ensuremath{-}1}$ but which does not scale with the entropy of the Kerr geometry.