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From Petrov-Einstein to Navier-Stokes · Around

2011

From Petrov-Einstein to Navier-Stokes

Lysov, Vyacheslav, Strominger, Andrew

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We consider a p+1-dimensional timelike hypersurface \Sigma_c embedded with a flat induced metric in a p+2-dimensional Einstein geometry.

It is shown that imposing a Petrov type I condition on the geometry reduces the degrees of freedom in the extrinsic curvature of \Sigma_c to those of a fluid in \Sigma_c.
Moreover, expanding around a limit in which the mean curvature of the embedding diverges, the leading-order Einstein constraint equations on \Sigma_c are shown to reduce to the non-linear incompressible Navier-Stokes equation for a fluid moving in \Sigma_c.

Built on

I. Bredberg, C. Keeler, V. Lysov, and A. Strominger, “Wilsonian Approach to Fluid/Gravity Duality,” arXiv:1006.1902 [hep-th]
Original
1902
Earlier work this paper cites.
Pergamon Press, 1969
A. Petrov, Einstein Spaces · 1969
Earlier work this paper cites.
Pergamon Press, first ed., 1975
L. Landau and E. Lifshitz, Fluid Mechanics · 1975
Earlier work this paper cites.

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Cambridge Univ. Pr., 2003
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I. Bredberg, C. Keeler, V. Lysov, and A. Strominger, “From Navier-Stokes To Einstein,” arXiv:1101.2451 [hep-th]
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G. Compere, P. McFadden, K. Skenderis, and M. Taylor, “The holographic fluid dual to vacuum Einstein gravity,” arXiv:1103.3022 [hep-th]
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I. Bredberg and A. Strominger
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Then

E. Gourgoulhon and J. L. Jaramillo, “A 3+1 perspective on null hypersurfaces and isolated horizons,” Phys. Rept
2006
Later among the works it cites.
A. Coley, “Classification of the Weyl Tensor in Higher Dimensions and Applications,” Class. Quant. Grav
Original
2008
Later among the works it cites.

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