Chaos and pole-skipping in rotating black holes

TL;DR

This paper explores the relationship between quantum chaos and energy dynamics in rotating black holes, revealing how shock wave profiles relate to pole-skipping phenomena and energy fluctuations in the dual field theory.

Contribution

It introduces a PDE for shock wave angular profiles in Kerr-AdS black holes and links these profiles to energy response and pole-skipping in the boundary theory.

Findings

Shock wave profiles satisfy a PDE related to chaos diagnostics.

Pole-skipping occurs at complex frequency $rac{i 2  T}$ with specific angular conditions.

Explicit OTOC computation constrains collective mode dispersion relations.

Abstract

We study the connection between many-body quantum chaos and energy dynamics for the holographic theory dual to the Kerr-AdS black hole. In particular, we determine a partial differential equation governing the angular profile of gravitational shock waves that are relevant for the computation of out-of-time ordered correlation functions (OTOCs). Further we show that this shock wave profile is directly related to the behaviour of energy fluctuations in the boundary theory. In particular, we demonstrate using the Teukolsky formalism that at complex frequency $\omega_* = i 2 \pi T$ there exists an extra ingoing solution to the linearised Einstein equations whenever the angular profile of metric perturbations near the horizon satisfies this shock wave equation. As a result, for metric perturbations with such temporal and angular profiles we find that the energy density response of the boundary theory exhibit the signatures of "pole-skipping" - namely, it is undefined, but exhibits a collective mode upon a parametrically small deformation of the profile. Additionally, we provide an explicit computation of the OTOC in the equatorial plane for slowly rotating large black holes, and show that its form can be used to obtain constraints on the dispersion relations of collective modes in the dual CFT.