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Local Fluid Dynamical Entropy from Gravity

Sayantani Bhattacharyya, Veronika E Hubeny, R. Loganayagam, Gautam Mandal, Shiraz Minwalla, Takeshi Morita, Mukund Rangamani, Harvey S. Reall

TL;DR

The paper analyzes the holographic duals of arbitrarily flowing conformal fluids in the long-wavelength limit and proves that the corresponding bulk spacetimes possess regular event horizons whose radial location $r_H(x)$ is locally determined by boundary fluid data via a derivative expansion. It constructs a boundary entropy current by pulling back the horizon area $(d-1)$-form using a natural horizon-to-boundary map defined by ingoing null geodesics, and shows its divergence is non-negative due to the horizon area increase theorem. Up to second order in derivatives, explicit gravity-derived expressions for the entropy current $J_S^{\mu}$ are obtained for the dual fluid, and its Weyl-covariant structure is analyzed to identify constraints on admissible currents; the analysis reveals a five-parameter space of Weyl-invariant currents with non-negative divergence, of which the gravitational construction selects a two-parameter subfamily. The results provide a direct link between horizon dynamics and boundary hydrodynamics, supporting a membrane-like interpretation of horizon behavior and offering a framework to explore higher-derivative corrections and more general horizons in holographic fluids.

Abstract

Spacetime geometries dual to arbitrary fluid flows in strongly coupled N=4 super Yang Mills theory have recently been constructed perturbatively in the long wavelength limit. We demonstrate that these geometries all have regular event horizons, and determine the location of the horizon order by order in a boundary derivative expansion. Intriguingly, the derivative expansion allows us to determine the location of the event horizon in the bulk as a local function of the fluid dynamical variables. We define a natural map from the boundary to the horizon using ingoing null geodesics. The area-form on spatial sections of the horizon can then be pulled back to the boundary to define a local entropy current for the dual field theory in the hydrodynamic limit. The area theorem of general relativity guarantees the positivity of the divergence of the entropy current thus constructed.

Local Fluid Dynamical Entropy from Gravity

TL;DR

The paper analyzes the holographic duals of arbitrarily flowing conformal fluids in the long-wavelength limit and proves that the corresponding bulk spacetimes possess regular event horizons whose radial location is locally determined by boundary fluid data via a derivative expansion. It constructs a boundary entropy current by pulling back the horizon area -form using a natural horizon-to-boundary map defined by ingoing null geodesics, and shows its divergence is non-negative due to the horizon area increase theorem. Up to second order in derivatives, explicit gravity-derived expressions for the entropy current are obtained for the dual fluid, and its Weyl-covariant structure is analyzed to identify constraints on admissible currents; the analysis reveals a five-parameter space of Weyl-invariant currents with non-negative divergence, of which the gravitational construction selects a two-parameter subfamily. The results provide a direct link between horizon dynamics and boundary hydrodynamics, supporting a membrane-like interpretation of horizon behavior and offering a framework to explore higher-derivative corrections and more general horizons in holographic fluids.

Abstract

Spacetime geometries dual to arbitrary fluid flows in strongly coupled N=4 super Yang Mills theory have recently been constructed perturbatively in the long wavelength limit. We demonstrate that these geometries all have regular event horizons, and determine the location of the horizon order by order in a boundary derivative expansion. Intriguingly, the derivative expansion allows us to determine the location of the event horizon in the bulk as a local function of the fluid dynamical variables. We define a natural map from the boundary to the horizon using ingoing null geodesics. The area-form on spatial sections of the horizon can then be pulled back to the boundary to define a local entropy current for the dual field theory in the hydrodynamic limit. The area theorem of general relativity guarantees the positivity of the divergence of the entropy current thus constructed.

Paper Structure

This paper contains 38 sections, 109 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Local Event Horizon
  3. Coordinates adapted to a null geodesic congruence
  4. Spacetime dual to hydrodynamics
  5. The event horizon in the derivative expansion
  6. The event horizon at second order in derivatives
  7. The Local Entropy Current
  8. Abstract construction of the area $(d-1)$-form
  9. Entropy $(d-1)$-form in global coordinates
  10. Properties of the area-form and its dual current
  11. Non-negative divergence:
  12. Lorentz invariance:
  13. The Horizon to Boundary Map
  14. Classification of ingoing null geodesics near the boundary
  15. Our choice of $t^\mu(x)$
  16. ...and 23 more sections

Figures (2)

  • Figure 1: Penrose diagram of the uniform black brane and the causal structure of the spacetimes dual to fluid mechanics illustrating the tube structure. The dashed line in the second figure denotes the future event horizon, while the shaded tube indicates the region of spacetime over which the solution is well approximated by a tube of the uniform black brane.
  • Figure 2: The event horizon $r=r_H(x^\mu)$ sketched as a function of the time $t$ and one of the spatial coordinates $x$ (the other two spatial coordinates are suppressed).