## 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) |
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Article number | 146 |

Journal | Journal of High Energy Physics |

Volume | 2012 |

Issue number | 7 |

DOIs | |

State | Published - Aug 6 2012 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- Nuclear and High Energy Physics