On the Chaos Bound in Rotating Black Holes

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.