From Navier-Stokes to Einstein

Irene Bredberg, Cynthia Keeler, Vyacheslav Lysov, Andrew Strominger

Research output: Contribution to journalArticlepeer-review

100 Scopus citations


We show by explicit construction that for every solution of the incompressible Navier-Stokes equation in p + 1 dimensions, there is a uniquely associated "dual" solution of the vacuum Einstein equations in p + 2 dimensions. The dual geometry has an intrinsically flat timelike boundary segment Σ c whose extrinsic curvature is given by the stress tensor of the Navier-Stokes fluid. We consider a "near-horizon" limit in which Σ c becomes highly accelerated. The near-horizon expansion in gravity is shown to be mathematically equivalent to the hydrodynamic expansion in fluid dynamics, and the Einstein equation reduces to the incompressible Navier-Stokes equation. For p = 2, we show that the full dual geometry is algebraically special Petrov type II. The construction is a mathematically precise realization of suggestions of a holographic duality relating fluids and horizons which began with the membrane paradigm in the 70's and resurfaced recently in studies of the AdS/CFT correspondence.

Original languageEnglish (US)
Article number146
JournalJournal of High Energy Physics
Issue number7
StatePublished - 2012
Externally publishedYes


  • Black holes
  • Classical theories of gravity
  • Holography and condensed matter physics (AdS/CMT)

ASJC Scopus subject areas

  • Nuclear and High Energy Physics


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