BTZ dynamics and chaos

Abstract

Abstract 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 λ L = 2 π β $$ {\lambda}_L=\frac{2\pi }{\beta } $$ 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 λ L = 2 π β + > 2 π β $$ {\lambda}_L=\frac{2\pi }{\beta +}>\frac{2\pi }{\beta } $$ where β + = β(1 − μ L ), with μ L = r − r + $$ {\mu}_L=\frac{r_{-}}{r_{+}} $$ 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.

Keywords

• AdS-CFT Correspondence
• Black Holes in String Theory