# From Navier-Stokes to Einstein

 Title: From Navier-Stokes to Einstein Author: Bredberg, Irene; Keeler, Cynthia; Lysov, Vyacheslav; Strominger, Andrew E. Note: Order does not necessarily reflect citation order of authors. Citation: Bredberg, Irene, Cynthia Keeler, Vyacheslav Lysov, and Andrew E. Strominger. 2012. From Navier-Stokes to Einstein. Journal of High Energy Physics 2012(7): 146. Full Text & Related Files: From Navier-Stokes to Einstein.pdf (629.7Kb; PDF) 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 $$\sum_c$$ whose extrinsic curvature is given by the stress tensor of the Navier-Stokes fluid. We consider a “near-horizon” limit in which $$\sum_c$$ 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. Published Version: doi:10.1007/JHEP07(2012)146 Other Sources: http://arxiv.org/abs/1101.2451 Terms of Use: This article is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAP Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:10919793 Downloads of this work: