Chaos and pole-skipping in rotating black holes

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 ω∗ = i2πT 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.

Fast Scrambling of Mutual Information in Kerr-AdS$_4$

“…Static black branes in AdS have been shown to posses this phenomena of pole-skipping with λ L = 2πT H . This peculiar phenomena of pole-skipping has also been justified as arising from an effective theory of hydrodynamics of maximally chaotic systems [15,16]. It was also shown to be a generic feature of planar holographic black branes in AdS [17].…”

“…κ 0 , for non-rotating shockwaves. The in-falling null coordinates used in [16] are precisely non-rotating and-according to the general arguments presented in section 2, must see a blueshift given by 2πT H . Our results further suggest that adapting the analysis of [16] to rotating coordinates provided in section 2 one must be able see pole-skipping at points ω * = iκ = 2πiT H /(1 − µ L) 7 .…”

“…For the case of rotating BTZ this was first analysed in [18] where pole skipping points implied 2 possible Lyapunov indices corresponding to left and right temperatures of the dual CFT 2 as found in [8,9]. The case for Kerr-AdS 4 was recently analysed [16] and the Lyapunov index and the butterfly velocity were inferred by analysing the ingoing solutions to metric perturbations along null directions at the horizon. Here the Lyapunov index was found to be λ L = 2πT H = κ 0 i.e.…”

“…We can in principle solve for f (θ) to find the how the backreaction spreads away from the equator; a similar analysis can also be done for a point null particle perturbation at the equator. This must indeed reveal interesting dynamics and butterfly velocity associated with such perturbations to the rotating geometry; see [16] for a analysis of the butterfly velocity in a slowly rotating Kerr AdS 4 . However for the case at hand we need not be concerning ourselves with dynamics away from the equator and choose f (θ = π/2) = 1.…”

Fast Scrambling due to Rotating Shockwaves in BTZ

“…It would be interesting to understand a similar computation for scrambling of in fallen information in Kerr-AdS which is a work we leave for the near future. It is worth pointing out the investigations of pole skipping in Kerr-AdS in [32]. The JT model arising in the near horizon region of near extremal Kerr in 4 and 5 dimensions were investigated in [33] and later in [34,35] and were found to have many interesting non-universal features.…”