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Off-shell hydrodynamics from holography

Michael Crossley, Paolo Glorioso, Hong Liu, Yifan Wang

TL;DR

This work develops a holographic Wilsonian RG framework to obtain an action principle for hydrodynamics from a gravity dual, focusing on non-dissipative horizons at leading order. The gapless hydrodynamic DOF are the relative boundary–horizon embeddings $X^a(\sigma)$, and the zeroth-order UV action depends only on $H=g^{-1}ar{h}$; in the horizon limit it reduces to the conformal ideal-fluid action of Dubovsky et al., with the horizon geometry giving rise to the boundary entropy current $j^ u = s u^ u$ where $s= rac{ig({ m Sdet}Hig)^{1/2}}{4G_N}$. At higher orders, non-dissipative formulations encounter divergences, indicating that dissipation is intrinsic to a consistent action principle; the full Schwinger-Keldysh approach requires incorporating the IR sector $ig( ext{Ψ}_{ m IR}ig)$ to capture dissipative dynamics. The framework reveals a geometric origin for hydrodynamic DOF and entropy transport and suggests natural extensions to charged fluids, anomalies, and higher-derivative gravity.

Abstract

We outline a program for obtaining an action principle for dissipative fluid dynamics by considering the holographic Wilsonian renormalization group applied to systems with a gravity dual. As a first step, in this paper we restrict to systems with a non-dissipative horizon. By integrating out gapped degrees of freedom in the bulk gravitational system between an asymptotic boundary and a horizon, we are led to a formulation of hydrodynamics where the dynamical variables are not standard velocity and temperature fields, but the relative embedding of the boundary and horizon hypersurfaces. At zeroth order, this action reduces to that proposed by Dubovsky et al. as an off-shell formulation of ideal fluid dynamics.

Off-shell hydrodynamics from holography

TL;DR

This work develops a holographic Wilsonian RG framework to obtain an action principle for hydrodynamics from a gravity dual, focusing on non-dissipative horizons at leading order. The gapless hydrodynamic DOF are the relative boundary–horizon embeddings , and the zeroth-order UV action depends only on ; in the horizon limit it reduces to the conformal ideal-fluid action of Dubovsky et al., with the horizon geometry giving rise to the boundary entropy current where . At higher orders, non-dissipative formulations encounter divergences, indicating that dissipation is intrinsic to a consistent action principle; the full Schwinger-Keldysh approach requires incorporating the IR sector to capture dissipative dynamics. The framework reveals a geometric origin for hydrodynamic DOF and entropy transport and suggests natural extensions to charged fluids, anomalies, and higher-derivative gravity.

Abstract

We outline a program for obtaining an action principle for dissipative fluid dynamics by considering the holographic Wilsonian renormalization group applied to systems with a gravity dual. As a first step, in this paper we restrict to systems with a non-dissipative horizon. By integrating out gapped degrees of freedom in the bulk gravitational system between an asymptotic boundary and a horizon, we are led to a formulation of hydrodynamics where the dynamical variables are not standard velocity and temperature fields, but the relative embedding of the boundary and horizon hypersurfaces. At zeroth order, this action reduces to that proposed by Dubovsky et al. as an off-shell formulation of ideal fluid dynamics.

Paper Structure

This paper contains 21 sections, 140 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setup
  3. Isolating hydrodynamical degrees of freedom
  4. Saddle point evaluation
  5. Einstein gravity
  6. Action for an ideal fluid
  7. Solving the dynamical equations
  8. Solving the dynamical equations
  9. Effective action for $\tau$ and $X^a$
  10. Horizon limit
  11. Entropy current
  12. Hydrodynamical action and volume-preserving diffeomorphism
  13. More on the off-shell gravity solution
  14. Generalization to higher orders
  15. Structure of derivative expansions to general orders
  16. ...and 6 more sections

Figures (2)

  • Figure 1: (a) Complex bulk manifold ${{\mathcal{M}}}_c$ consisting of two copies of asymptotic AdS spacetimes patched together at a horizon hypersurface. Also labeled are stretched horizons $\Sigma_1, \Sigma_2$ discussed around \ref{['1.2']}. (b) A boundary theory Schwinger-Keldysh contour used to describe non-equilibrium physics. The two AdS regions map to the two horizontal legs of the Schwinger-Keldysh contour, while the analytic continuation around the horizon corresponds to the vertical leg which defines the initial thermal density matrix.
  • Figure 2: The gapless degrees of freedom in the path integrals \ref{['uvint']} are the relative embedding coordinates $X^a_1 (\sigma^\mu)$ between the horizon and the boundary. $X^a_1$ can be understood geometrically as follows: start with $\sigma^\mu$ at $\Sigma_1$, shooting a congruence of geodesics orthogonal to $\Sigma_1$ toward the boundary, the intersections of these geodesics with the boundary define $X^a_1$.