BTZ dynamics and chaos

We find an effective action for gravitational interactions with scalars inAdS₃to first order inG_Nat 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 $$ {\lambda}_L=\frac{2\pi }{\beta } $$λL=2πβ for OTOC computed first in arXiv:1412.6087 for a Schwarzchild black hole inAdS₃. Applying the same procedure to rotating BTZ we find the Lyapunov index to be $$ {\lambda}_L=\frac{2\pi }{\beta +}>\frac{2\pi }{\beta } $$λL=2πβ+>2πβ whereβ₊=β(1− μ_L), with $$ {\mu}_L=\frac{r_{-}}{r_{+}} $$μL=r−r+ being the chemical potential for angular momentum. We thus comment on a possible generalization to a part of the proof given in arXiv:1503.01409 to accommodate this result.