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Conformal Nonlinear Fluid Dynamics from Gravity in Arbitrary Dimensions

Sayantani Bhattacharyya, R. Loganayagam, Ipsita Mandal, Shiraz Minwalla, Ankit Sharma

TL;DR

This work establishes a comprehensive map between conformal fluid dynamics in d dimensions and long-wavelength, asymptotically AdS_{d+1} gravity. By employing a Weyl-covariant derivative framework and a derivative expansion, the authors construct explicit bulk metrics and boundary stress tensors accurate to second order, valid on curved boundaries, and confirm consistency with exact AdS Kerr black holes. They derive a covariant entropy current from the bulk horizon, proving a local second-law-like monotonicity, and demonstrate that AdS Kerr solutions naturally fit the fluid-dynamical description. The results reveal universal structures in the second-order transport data and provide a robust platform for exploring holographic hydrodynamics in arbitrary dimensions, with potential extensions to charged or non-conformal setups.

Abstract

We generalize recent work to construct a map from the conformal Navier Stokes equations with holographically determined transport coefficients, in d spacetime dimensions, to the set of asymptotically locally AdS_{d+1} long wavelength solutions of Einstein's equations with a negative cosmological constant, for all d>2. We find simple explicit expressions for the stress tensor (slightly generalizing the recent result by Haack and Yarom (arXiv:0806.4602)), the full dual bulk metric and an entropy current of this strongly coupled conformal fluid, to second order in the derivative expansion, for arbitrary d>2. We also rewrite the well known exact solutions for rotating black holes in AdS_{d+1} space in a manifestly fluid dynamical form, generalizing earlier work in d=4. To second order in the derivative expansion, this metric agrees with our general construction of the metric dual to fluid flows.

Conformal Nonlinear Fluid Dynamics from Gravity in Arbitrary Dimensions

TL;DR

This work establishes a comprehensive map between conformal fluid dynamics in d dimensions and long-wavelength, asymptotically AdS_{d+1} gravity. By employing a Weyl-covariant derivative framework and a derivative expansion, the authors construct explicit bulk metrics and boundary stress tensors accurate to second order, valid on curved boundaries, and confirm consistency with exact AdS Kerr black holes. They derive a covariant entropy current from the bulk horizon, proving a local second-law-like monotonicity, and demonstrate that AdS Kerr solutions naturally fit the fluid-dynamical description. The results reveal universal structures in the second-order transport data and provide a robust platform for exploring holographic hydrodynamics in arbitrary dimensions, with potential extensions to charged or non-conformal setups.

Abstract

We generalize recent work to construct a map from the conformal Navier Stokes equations with holographically determined transport coefficients, in d spacetime dimensions, to the set of asymptotically locally AdS_{d+1} long wavelength solutions of Einstein's equations with a negative cosmological constant, for all d>2. We find simple explicit expressions for the stress tensor (slightly generalizing the recent result by Haack and Yarom (arXiv:0806.4602)), the full dual bulk metric and an entropy current of this strongly coupled conformal fluid, to second order in the derivative expansion, for arbitrary d>2. We also rewrite the well known exact solutions for rotating black holes in AdS_{d+1} space in a manifestly fluid dynamical form, generalizing earlier work in d=4. To second order in the derivative expansion, this metric agrees with our general construction of the metric dual to fluid flows.

Paper Structure

This paper contains 23 sections, 98 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Manifest Weyl Covariance
  3. Weyl covariant notation for conformal hydrodynamics
  4. Classification of fluid dynamical Weyl covariant Tensors to second order
  5. Perturbative Construction of Solutions
  6. Equations of motion and uniform brane solutions
  7. Slow variation and bulk tubes and our zero order ansatz
  8. Perturbation theory at long wavelengths
  9. Weyl Covariance
  10. The bulk metric and boundary stress tensor to second order
  11. The metric dual to hydrodynamics
  12. Energy momentum tensor of fluids with metric duals
  13. Causal structure and the local entropy current
  14. The event horizon of our solutions
  15. Entropy current as the pullback of Area form
  16. ...and 8 more sections

Figures (1)

  • Figure 1: Penrose diagram of the uniform black brane illustrating the causal Eddington-Finkelstein tubes running along ingoing null geodesics . The tubes with $x^\mu_{\text{Schwarzschild}}$=constant are also shown. Note that we have suppressed the other regions of the penrose diagram not germane to the discussion in this paper.