Fast scrambling of mutual information in Kerr-${\mathrm{AdS}}_4$ spacetime

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-${\lambda }_L^{(\min )}$ at late times and find that it is bounded by $\kappa =\frac{2\pi T_H}{(1-\mu \mathcal{L})}$, where $\mu$ 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 $\kappa$ with $\kappa t_{*}=\log \mathcal{S}$ for large black holes with a large entropy $\mathcal{S}$. We also find a term that increases the scrambling time by $\log (1-\mu \mathcal{L}{)}^{-1}$ but which does not scale with the entropy of the Kerr geometry.

Figure 1

Figure depicting the extremal surface (red) at $t=0$ in extended Kerr spacetime, which is analyzed by dividing into three segments denoted as I, II, and III. The shift $V$ coordinate at $U=0$ is $\alpha$. Note, the lengths II and III are the same and so are the lengths along the red curve on the either sides of $U=0$ and $V=\alpha /2$.

Figure 2

Plots for various values of $\mathcal{L}=s(\frac{r_{-}}{r_{+}}){\mu }^{-1}$ with $r_{+}=0.57$ and $\ell =1$. The blue, orange, and green curves correspond to $s=\{0,\frac{2}{3},\frac{9}{10}\}$, respectively. These curves approach zero as $r_{-}\to r_{+}$. The red curve corresponds to $s=1$ and approaches a constant since $\mathcal{L}\to {\mu }^{-1}$ as $r_{-}\to r_{+}$.

Figure 3

Plots for various values of $\mathcal{L}=s(\frac{r_{-}}{r_{+}}){\mu }^{-1}$ with $r_{+}=0.57$ and $\ell =1$. The blue, orange, and green curves are for ${\lambda }_L^{(\min )}$ corresponding to $s=\{0,\frac{9}{10},1\}$, respectively, while the red, violet, and brown curves correspond to the value of $\kappa$ for similar respective values of $s$. The red curve ${\kappa }_0$, i.e., the temperature of the geometry, is greater than only the ${\lambda }_L^{(\min )}$ for $\mathcal{L}=0(s=0)$; the values of ${\lambda }_L^{(\min )}$ for $\mathcal{L}>0$ can be greater than this curve.

Figure 4

Plot of $\kappa /{\lambda }^{(\min {)}_L}$ against $r_{-}/r_{+}$ with $\ell =1$. This plot is independent of the shock waves specific angular momentum $\mathcal{L}$.