# BTZ dynamics and chaos

## TL;DR

This paper derives an effective boundary action for gravitational interactions in AdS3, reproduces known chaos bounds, and explores how rotation affects Lyapunov exponents in BTZ black holes.

## Contribution

It introduces a boundary action for AdS3 gravity with scalars, extending the Schwarzian framework to rotating BTZ black holes and analyzing chaos bounds.

## Key findings

- Reproduces the chaos bound $rac{2\pi}{eta}$ for non-rotating black holes.
- Finds increased Lyapunov exponent $rac{2\pi}{eta_+}$ for rotating BTZ.
- Suggests modifications to existing proofs of chaos bounds for rotating cases.

## 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 $\lambda_L=\frac{2\pi}{\beta}$ 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.