From Navier-Stokes to Einstein

  • Irene Bredberg
  • Cynthia Keeler
  • Vyacheslav Lysov
  • Andrew Strominger


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 Σc whose extrinsic curvature is given by the stress tensor of the Navier-Stokes fluid. We consider a “near-horizon” limit in which Σc 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.


Classical Theories of Gravity Black Holes Holography and condensed matter physics (AdS/CMT) 


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Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • Irene Bredberg
    • 1
  • Cynthia Keeler
    • 1
  • Vyacheslav Lysov
    • 1
  • Andrew Strominger
    • 1
  1. 1.Center for the Fundamental Laws of NatureHarvard UniversityCambridgeU.S.A

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