### Abstract

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 language | English (US) |
---|---|

Article number | 146 |

Journal | Journal of High Energy Physics |

Volume | 2012 |

Issue number | 7 |

DOIs | |

State | Published - Aug 6 2012 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Nuclear and High Energy Physics

### Cite this

*Journal of High Energy Physics*,

*2012*(7), [146]. https://doi.org/10.1007/JHEP07(2012)146

**From Navier-Stokes to Einstein.** / Bredberg, Irene; Keeler, Cynthia; Lysov, Vyacheslav; Strominger, Andrew.

Research output: Contribution to journal › Article

*Journal of High Energy Physics*, vol. 2012, no. 7, 146. https://doi.org/10.1007/JHEP07(2012)146

}

TY - JOUR

T1 - From Navier-Stokes to Einstein

AU - Bredberg, Irene

AU - Keeler, Cynthia

AU - Lysov, Vyacheslav

AU - Strominger, Andrew

PY - 2012/8/6

Y1 - 2012/8/6

N2 - 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.

AB - 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.

KW - Black holes

KW - Classical theories of gravity

KW - Holography and condensed matter physics (AdS/CMT)

UR - http://www.scopus.com/inward/record.url?scp=84864447901&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864447901&partnerID=8YFLogxK

U2 - 10.1007/JHEP07(2012)146

DO - 10.1007/JHEP07(2012)146

M3 - Article

VL - 2012

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

SN - 1029-8479

IS - 7

M1 - 146

ER -