On the Chaos Bound in Rotating Black Holes

TL;DR

This work analyzes chaos in rotating BTZ black holes by two complementary methods: gravity-based elastic eikonal computations and a Chern-Simons formulation of AdS3 gravity. Both approaches reveal two competing chaotic channels associated with left- and right-moving modes, each governed by a Lyapunov exponent λ_L^{±} tied to effective temperatures β_{±} = β(1 ∓ ℓΩ). The CS analysis derives a Schwarzian-like boundary action and, via analytic continuation of Euclidean correlators, reproduces the same dichotomy of chaotic growth and shows frame-dependent butterfly velocities. The findings reconcile apparent violations of the chaos bound by interpreting the bound separately for each moving sector, with implications for holographic descriptions of rotating systems and their microscopic duals.

Abstract

We study out-of-time-order correlators (OTOCs) of rotating BTZ black holes using two different approaches: the elastic eikonal gravity approximation, and the Chern-Simons formulations of 3-dimensional gravity. Within both methods the OTOC is given as a sum of two contributions, corresponding to left and right moving modes. The contributions have different Lyapunov exponents, $λ_L^{\pm}=\frac{2π}β\frac{1}{1\mp \ell Ω}$, where $Ω$ is the angular velocity and $\ell$ is the AdS radius. Since $λ_L^{-} \leq \frac{2π}β \leq λ_L^{+}$, there is an apparent contradiction with the chaos bound. We discuss how the result can be made consistent with the chaos bound if one views $β_{\pm}=β(1\mp \ell Ω)$ as the effective inverse temperatures of the left and right moving modes.

Figures (2)

• Figure 1: Penrose diagram for the rotating BTZ black hole. The figure only shows the region of interest. The same pattern repeats itself indefinitely above and below Banados:1992gq. The regions $1_{++}$ and $1_{+-}$ denote the right and left exterior regions, while the regions $2_{+-}$ and $2_{++}$ denote the past and future interiors. The coordinates $(\tilde{U},\tilde{V})$ are defined as in Fidkowski:2003nf: $V=e^{\frac{\pi}{2}}\text{tan}\left( \frac{\tilde{V}}{2}\right)$ and $U=e^{\frac{\pi}{2}}\text{tan}\left( \frac{\tilde{U}}{2}\right)$, with $\tilde{U},\tilde{V} \in [-\pi,\pi]$.
• Figure 2: Left: the 'in' state ${\color{red}{V_{\phi_3}(t_3)}} {\color{blue}{W_{\phi_4}(t_4)}}| \text{TFD}\rangle$ represented in a bulk spatial slice that touches the right boundary at time $t_3$. Right: the 'out' state ${\color{blue}{W_{\phi_2}(t_2)^{\dagger}}} {\color{red}{V_{\phi_1}(t_1)^{\dagger}}}| \text{TFD}\rangle$ represented in a bulk slice that touches the right boundary at time $t_2$.