Brickwall model for hyperbolic black holes and chaos

TL;DR

The paper investigates quantum chaos in black holes using the brickwall model for probe scalar fields in ($d+1$)-dimensional hyperbolic AdS spacetimes. By retaining angular momentum $J$ and imposing Gaussian boundary conditions on the stretched horizon, it computes normal modes and analyzes level-spacing statistics, spectral form factors, and Krylov complexity to diagnose chaos across dimensions. The study finds that chaos signatures persist in moderate higher dimensions but transition from a logarithmic to a power-law spectral form as $d$ grows, and that in the asymptotically large-$d$ limit the spectrum degenerates, diminishing chaotic behavior. This work clarifies the universality and limitations of the brickwall approach, linking black hole microphysics to random-matrix theory while highlighting dimensional sensitivity and potential connections to holographic chaos diagnostics.

Abstract

We study the quantum chaotic behavior of black holes within the brickwall model, focusing on probe scalar fields in ($d+1$)-dimensional hyperbolic AdS black holes. The brickwall model has captured the normal modes of BTZ black holes ($d=2$) with Gaussian-distributed boundary conditions on the stretched horizon and their connection to quantum chaos signatures of random matrix theory. Here, we extend this framework to higher-dimensional AdS black holes ($d>2$), exploring how black hole normal modes encode chaotic dynamics across dimensions and examining the universality of this approach. We show that quantum chaos is prominent in lower dimensions and persists in higher dimensions. Specifically, deviations from the logarithmic spectrum in $d=2$ evolves into a power-law spectrum at higher $d$, highlighting the sensitivity of black hole normal modes to dimensionality, while retaining signatures of quantum chaos despite the spectral deformation. Our results are supported by conventional diagnostics, including level spacing distributions and spectral form factors, as well as modern tools like Krylov complexity. Finally, we discuss the limitations of the brickwall model in capturing chaotic behavior in the parametrically large dimension limit, where the spectrum becomes constant leading to degeneracy and a non-chaotic regime, while emphasizing its effectiveness as a tool for studying quantum aspects of black holes in moderate higher dimensions.

Figures (9)

• Figure 1: Normal mode spectrum for $d=3$ (AdS$_4$) with $\sigma = 0.0022$ (left) and $0.2$ (right).
• Figure 2: Normal mode spectrum for $\sigma = 0.0022$ with $d=2, 100, 500, 1000$ (green, orange, yellow, purple). The black dashed line is the analytic result from \ref{['ANDLIMIT']} for $d=3000$, giving $\omega = 0.000157193$.
• Figure 3: Level spacing distributions of scalar fields for $d=2$ (a-c), $d=100$ (d-f), and $d=500$ (g-i). The solid curves represent the Wigner surmise from random matrix theory \ref{['WS']}, corresponding to the GSE ($\beta=4$), GUE ($\beta=2$), and GOE ($\beta=1$).
• Figure 4: Level spacing distributions of scalar fields for $d=100$ with $\sigma=0$ (left) and $0.2$ (right), where the black solid line is the Poisson distribution \ref{['PS']}.
• Figure 5: Dependence of the level spacing distribution parameter $\beta$ on $\sigma$ for $d=100$: Wigner surmise \ref{['WS']} (left) and Brody distribution \ref{['BD']} (right).