# On the Chaos Bound in Rotating Black Holes

## TL;DR

This paper investigates chaos bounds in rotating BTZ black holes by analyzing out-of-time-order correlators, revealing mode-dependent Lyapunov exponents that can be reconciled with the chaos bound through effective temperatures.

## Contribution

It introduces a dual-approach analysis of OTOCs in rotating BTZ black holes, showing mode-specific Lyapunov exponents and resolving apparent contradictions with the chaos bound.

## Key findings

- Lyapunov exponents differ for left and right moving modes.
- Effective inverse temperatures explain the mode-dependent chaos bounds.
- The results reconcile chaos bounds with rotating black hole dynamics.

## 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, $\lambda_L^{\pm}=\frac{2\pi}{\beta}\frac{1}{1\mp \ell \Omega}$, where $\Omega$ is the angular velocity and $\ell$ is the AdS radius. Since $\lambda_L^{-} \leq \frac{2\pi}{\beta} \leq \lambda_L^{+}$, 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 $\beta_{\pm}=\beta(1\mp \ell \Omega)$ as the effective inverse temperatures of the left and right moving modes.