Pole-Skipping in Rotating BTZ Black Holes

Abstract

Motivated by the connection between pole-skipping phenomena of two point functions and four point out-of-time-order correlators, we study the pole-skipping phenomena for rotating BTZ black holes. In particular, we investigate the effect of rotations on the pole-skipping point for various fields with spin $s = 1/2, 1, 2/3$, extending the previous research for $s=0, 2$. We derive an analytic full tower of the pole-skipping points of fermionic ($s=1/2$) and vector ($s=1$) fields by the exact holographic Green's functions. For the \textit{non-extremal} black hole, the leading pole-skipping frequency is $\omega_{\text{leading}}=2\pi i T_h {(s-1+\nu \Omega)}/{(1-\Omega^2)}$ where $T_h$ is the temperature, $\Omega$ the rotation, and $\nu:=(\Delta_+ - \Delta_-)/2$, the difference of conformal dimensions ($\Delta_{\pm}$). These are confirmed by another independent method: the near-horizon analysis. For the \textit{extremal} black hole, we find that the leading pole-skipping frequency can occur at $\omega_{\text{leading}}^{\text{extremal}}=-2\pi i T_R {(s+1)}$ only when $\nu = s+1$, where $T_R$ is the temperature of the right moving mode. It is non-trivial because it cannot be achieved by simply taking the extreme limit ($T_h\rightarrow 0\,, \Omega\rightarrow 1$) of the non-extremal black hole result.