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Gravity & Hydrodynamics: Lectures on the fluid-gravity correspondence

Mukund Rangamani

TL;DR

The work outlines the gravity–fluid correspondence, showing how long-wavelength, strongly coupled field theory dynamics can be encoded in dynamical black holes in AdS via a boundary derivative expansion. By matching bulk solutions to boundary stress tensors, it provides a systematic method to derive relativistic (and conformal) hydrodynamics and extract transport coefficients from gravity, including higher-order corrections and Weyl covariance. The framework also extends to include matter fields and non-relativistic limits, enabling holographic descriptions of forced fluids, charged plasmas, and Schrödinger-invariant systems, with explicit results for quantities like η/s and higher-order transport coefficients. It further connects bulk horizon dynamics to boundary entropy currents, and discusses generalizations and conceptual links to the membrane paradigm, highlighting the approach’s significance for studying strong coupling phenomena such as the quark–gluon plasma. Overall, the fluid-gravity correspondence provides a powerful, universal holographic toolkit for probing the hydrodynamic regime of strongly interacting quantum systems and their gravity duals.

Abstract

We discuss recent developments in the hydrodynamic description of strongly coupled conformal field theories using the AdS/CFT correspondence. In particular, we review aspects of the fluid-gravity correspondence which provides a map between a class of inhomogeneous, dynamical, black hole solutions in asymptotically AdS spacetimes and arbitrary fluid flows in the strongly interacting boundary field theory. We explain how the geometric duals to the fluid dynamics are constructed in a boundary derivative expansion and use the construction to extract the hydrodynamic transport coefficients. In addition we also describe the recent developments extending the correspondence to incorporate matter fields and to non-relativistic systems. Based on lectures given at the CERN Winter School on Supergravity, Strings and Gauge Theories, Geneva, Switzerland (February 2009).

Gravity & Hydrodynamics: Lectures on the fluid-gravity correspondence

TL;DR

The work outlines the gravity–fluid correspondence, showing how long-wavelength, strongly coupled field theory dynamics can be encoded in dynamical black holes in AdS via a boundary derivative expansion. By matching bulk solutions to boundary stress tensors, it provides a systematic method to derive relativistic (and conformal) hydrodynamics and extract transport coefficients from gravity, including higher-order corrections and Weyl covariance. The framework also extends to include matter fields and non-relativistic limits, enabling holographic descriptions of forced fluids, charged plasmas, and Schrödinger-invariant systems, with explicit results for quantities like η/s and higher-order transport coefficients. It further connects bulk horizon dynamics to boundary entropy currents, and discusses generalizations and conceptual links to the membrane paradigm, highlighting the approach’s significance for studying strong coupling phenomena such as the quark–gluon plasma. Overall, the fluid-gravity correspondence provides a powerful, universal holographic toolkit for probing the hydrodynamic regime of strongly interacting quantum systems and their gravity duals.

Abstract

We discuss recent developments in the hydrodynamic description of strongly coupled conformal field theories using the AdS/CFT correspondence. In particular, we review aspects of the fluid-gravity correspondence which provides a map between a class of inhomogeneous, dynamical, black hole solutions in asymptotically AdS spacetimes and arbitrary fluid flows in the strongly interacting boundary field theory. We explain how the geometric duals to the fluid dynamics are constructed in a boundary derivative expansion and use the construction to extract the hydrodynamic transport coefficients. In addition we also describe the recent developments extending the correspondence to incorporate matter fields and to non-relativistic systems. Based on lectures given at the CERN Winter School on Supergravity, Strings and Gauge Theories, Geneva, Switzerland (February 2009).

Paper Structure

This paper contains 44 sections, 138 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Fluid dynamics:
  3. Gravitational solutions:
  4. Relevance to real world physics:
  5. Summary of the lectures:
  6. Outline of the lectures:
  7. Elements of fluid dynamics
  8. Ideal fluids
  9. Dissipative fluids
  10. Defining the velocity field:
  11. Defining the velocity field:
  12. Enumerating independent operators:
  13. Decomposing velocity gradients:
  14. Causality issues in relativistic viscous fluids
  15. Conformal fluids
  16. ...and 29 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).