Stationary Solution to Charged Hairy Black Hole in AdS4: Kasner Interior, Rotating Shock Waves, and Fast Scrambling

TL;DR

This work constructs and analyzes a stationary charged hairy $AdS_4$ black hole with scalar hair coupled to a Maxwell field, revealing a Kasner interior and no inner horizon. By injecting rotating and charged gravitational shock waves, the authors compute holographic mutual information via the Ryu–Takayanagi prescription to study scrambling, showing the scrambling time scales as $t_* \sim \frac{1}{\lambda_L}\log S$ and that chaos is shaped by boundary deformation, rotation, and charge. The key findings are that boundary deformation lowers the normalized Lyapunov exponent while increasing shock angular momentum raises it, and charge-induced scrambling-time delays grow with charge density and angular momentum but are suppressed by the scalar boundary source. These results illuminate how interior Kasner geometry and exterior boundary data jointly govern chaotic dynamics, with potential implications for holographic superconductors and deeper interior transitions.

Abstract

We consider a stationary solution of a charged black hole with scalar hair in AdS$_4$, where the scalar field is coupled to a $U(1)$ Maxwell gauge field. Near the singularity, the spacetime transitions into a more general Kasner geometry. The black hole is then injected with rotating and charged gravitational shock waves in the Dray-'t Hooft solution. These shock waves lengthen the wormhole connecting the two asymptotic boundaries, thereby disrupting the correlations between them. The correlation, quantified by the quantum mutual information between subregions on the left and right boundaries, vanishes at a characteristic timescale known as the scrambling time, which depends logarithmically on the black hole entropy. The mutual information is computed holographically using the Ryu-Takayanagi prescription for entanglement entropy. We investigate how the rotation and charge of both the black hole and the shock waves affect chaotic properties such as the scrambling time delay and the Lyapunov exponent. The interaction between the charges of the black hole and the shock waves introduces a delay in the scrambling process. We find that as the strength of the boundary deformation increases, both the Lyapunov exponent and the scrambling time delay decrease monotonically. Furthermore, the angular momentum of the shock waves enhances both the Lyapunov exponent and the scrambling time delay.

Figures (16)

• Figure 1: Numerical solutions at $\phi_0/T=6.48714$. In this plot, we use $q=0.10, N_{H1}=-A_{xH1}=-1/2, \mathcal{L}=0.1,$ and $\rho=2$. The horizon is located at $r_H=1$, while the boundary solutions are represented by $r\rightarrow0$.
• Figure 2: Collapse of the Einstein-Rosen bridge when the rotation parameter $N_{H1}$ is varied. Here, we use $q=0.10,\rho=2$, and $\phi_0/T=6.48714$. The dashed lines represent the solutions when $\phi=0$. The "would-be" inner horizons are located at $r\approx2.72$ for $N_{H1}=0$, $r\approx2.86$ for $N_{H1}=-1/3$, and $r\approx2.95$ for $N_{H1}=-1/2$.
• Figure 3: Collapse of the Einstein-Rosen bridge when the coupling $q$ is varied. Here, we use $N_{H1}=-1/2, \rho=2$, and $\phi_0/T=5.59021$. The "would-be" inner horizon when $\phi=0$ is located at $r\approx2.95$.
• Figure 4: Plot of the metric component $g_{tt}$ as a function of $r$ in the exterior region. The solid lines correspond to the solution with $\phi_0/T = 238.799$ and $q = 0.1$, $\rho = 2$, while the dashed lines represent the case with $\phi = 0$. Blue lines indicate $N_{H1} = -1/2$, and red lines indicate $N_{H1} = -3/4$.
• Figure 5: Plot of the metric component $g_{tt}$ as a function of $r$ in the exterior region. The solid lines correspond to the solution with $\phi_0/T=242.795$ and $N_{H1}=-1/2$, $\rho=2$, while the gray dashed line represents the case with $\phi=0$. The red line indicates $q=0.1$, the blue line indicates $q=0.2$, and the orange line indicates $q=0.3$.