CFT Hydrodynamics: Symmetries, Exact Solutions and Gravity

TL;DR

This work shows that relativistic conformal field theory hydrodynamics at finite temperature, in its slow, incompressible limit, reproduces the incompressible Navier–Stokes framework and exhibits rich symmetry structures inherited from conformal invariance. It constructs exact one-dimensional ideal-CFT solutions with finite-time singularities, demonstrates that their higher-dimensional images under special conformal transformations lack a non-trivial slow-motion limit, and analyzes non-equilibrium shocks, including viscous smoothing and Knudsen-number scaling. The paper then draws a precise parallel between CFT hydrodynamics and gravitational dual descriptions, deriving domain-wall (shock) gravity solutions from hydrodynamic data and showing how non-relativistic NS flows enable globally valid gravity solutions. Overall, the results illuminate how fluid dynamics of strongly coupled CFTs relates to gravity through AdS/CFT, with shocks and viscous effects providing concrete bridges between field theory and bulk spacetime structure.

Abstract

We consider the hydrodynamics of relativistic conformal field theories at finite temperature and its slow motions limit, where it reduces to the incompressible Navier-Stokes equations. The symmetries of the equations and their solutions are analyzed. We construct exact solutions with finite time singularities of one-dimensional relativistic conformal hydrodynamic motions, and use them to generate multi-dimensional solutions via special conformal transformations. These solutions, however, are shown to have no non-trivial slow motions limit. A simple non-equilibrium steady state in the form of a shock solution is constructed, and its inner structure is analyzed. We demonstrate that the derivation of the gravitational dual description of conformal hydrodynamics is analogous to the derivation of hydrodynamics equations from the Boltzmann equation. The shock solution is shown to correspond to a domain-wall solution in gravity. We show that the solutions to the non-relativistic incompressible Navier-Stokes equations play a special role in the construction of global solutions to gravity.