Abstract
We explore a new class of general properties of thermal holographic Green’s functions that can be deduced from the near-horizon behaviour of classical perturbations in asymptotically anti-de Sitter spacetimes. We show that at negative imaginary Matsubara frequencies and appropriate complex values of the wavenumber the retarded Green’s functions of generic operators are not uniquely defined, due to the lack of a unique ingoing solution for the bulk perturbations. From a boundary perspective these ‘pole-skipping’ points correspond to locations in the complex frequency and momentum planes at which a line of poles of the retarded Green’s function intersects with a line of zeroes. As a consequence the dispersion relations of collective modes in the boundary theory at energy scales ω ∼ T are directly constrained by the bulk dynamics near the black-brane horizon. For the case of conserved U (1) current and energy-momentum tensor operators we give examples where the dispersion relations of hydrodynamic modes pass through a succession of pole- skipping points as real wavenumber is increased. We discuss implications of our results for transport, hydrodynamics and quantum chaos in holographic systems.
Highlights
The real-time formulation of holography initiated in [1] relates the Fourier space retarded Green’s functions GR(ω, k) of boundary operators to the solutions of classical bulk equations for perturbations obeying ingoing boundary conditions at the black hole horizon
Unlike the case described in [14, 18,19,20], we believe it is unlikely that the pole-skipping phenomena that we describe in this paper are related in a straightforward way to the underlying quantum chaotic properties of holographic systems
In this paper we have shown that a simple analysis of the near-horizon properties of classical perturbations leads to a series of non-trivial constraints on the properties of holographic Green’s functions at frequencies ω ∼ T
Summary
At frequencies ωn = −i2πT n and certain complex values of momentum kn, the imposition of the ingoing boundary condition at the horizon is not sufficient to uniquely specify φ (up to an overall normalisation constant) At these special points in complex Fourier space any solution to (2.7) is regular at the horizon in ingoing coordinates. The locations of these special points can and systematically be determined by expanding (2.7) near the horizon of the black hole, and can be used to obtain highly non-trivial information about the boundary Green’s function GROO(ω, k). As such we will show how to directly obtain non-trivial information about the dispersion relations of poles and zeroes of GROO(ω, k) from a simple analysis of perturbations near the black hole horizon, and will demonstrate this explicitly in several examples
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