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.
Highlights
To studying a two-particle scattering process near the horizon of a black hole in the dual gravitational description [11]
In the last section we studied the computation of the ordered correlation functions (OTOCs) in the CFT dual to the KerrAdS black hole, and in particular we showed that the angular profile of the gravitational shock waves relevant for the OTOC are governed by the shock wave equation (3.9)
In this paper we have explored the connection between many-body quantum chaos and energy dynamics in the holographic theory dual to the Kerr-AdS black hole
Summary
As explained in the Introduction, our goal in this paper is to investigate the connection between many-body quantum chaos and energy dynamics in the context of the holographic theory dual to the Kerr-AdS black hole. In the coordinate frame {t, r, x, φ}, the angular velocity of the outer horizon ΩH is given by aΞ ΩH = r02 + a2. The temperature of the CFT is given by (2.3) and there is a chemical potential Ω for rotation about the North pole given by the angular velocity of the horizon relative to the conformal boundary [62, 63, 69]. The singularities in the metric at the outer horizon can be removed by introducing “ingoing” (v) and “outgoing” (u) co-ordinates defined by (r2 + a2). In these co-ordinates, the metric can be written as ds2 = − ∆r Σ r02 + a2x2 2α(r02 + a2).
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