Abstract
We study analytic properties of the dispersion relations in classical hydrody- namics by treating them as Puiseux series in complex momentum. The radii of convergence of the series are determined by the critical points of the associated complex spectral curves. For theories that admit a dual gravitational description through holography, the critical points correspond to level-crossings in the quasinormal spectrum of the dual black hole. We illustrate these methods in N = 4 supersymmetric Yang-Mills theory in 3+1 dimensions, in a holographic model with broken translation symmetry in 2+1 dimensions, and in con- formal field theory in 1+1 dimensions. We comment on the pole-skipping phenomenon in thermal correlation functions, and show that it is not specific to energy density correlations.
Highlights
Hydrodynamic variables: local temperature, fluid velocity, and charge density [1]
We introduced spectral curves as a useful tool for investigating analytic properties of gapless collective excitations in classical hydrodynamics
We showed that the dispersion relations of hydrodynamic modes, such as shear and sound modes, are generically given by Puiseux series expansions in rational powers of the spatial momentum squared
Summary
We start with a brief review of how the hydrodynamic dispersion relations are derived. It is straightforward to write down the linearised constitutive relations at any order, by noting that under the spatial SO(ds), the stress fluctuation δT ij is a rank-two tensor, momentum density δT 0i is a vector, and the energy density δT 00 is a scalar. Substituting the constitutive relations (2.4) into the conservation equations (2.1), we find a system of ds+1 linear equations for ds+1 hydrodynamic variables This system has non-trivial solutions provided the determinant of the corresponding matrix vanishes. (2.9), (2.10) give simple explicit expressions for ωshear(q2) and ωsound(q2) in terms of three scalar functions γη(q2), γs(q2) and H(q2), whose small-q limits are given by eq (2.8) In this way of implementing the derivative expansion, the hydrodynamic dispersion relations are the only solutions to (2.9), (2.10). We will assume that the functions γη(q2, ω), γs(q2, ω), H(q2, ω) are defined by the exact response functions of the T μν operator
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