We find an effective action for gravitational interactions with scalars in $AdS_3$ to first order in $G_N$ at the conformal boundary. This action can be understood as an action for the Brown-Henneaux modes and is given by the square-root product of right and left moving Schwarzian derivatives for conformal transformations of the boundary. We thus reproduce the result $\lambda_L=\frac{ 2\pi}{ \beta}$ for OTOC computed first in arXiv:1412.6087 for a Schwarzchild black hole in $AdS_3$. Applying the same procedure to rotating BTZ we find the Lyapunov index to be $\lambda_L=\frac{ 2\pi}{ \beta_+}>\frac{ 2\pi}{ \beta}$ where $\beta_+=\beta(1-\mu_L)$, with $\mu_L=\frac{ r_-}{ r_+}$ being the chemical potential for angular momentum. We thus comment on a possible modification to a part of the proof given in arXiv:1503.01409 to accommodate this result.
Categories: hep-th
