From Petrov-Einstein to Navier-Stokes
Abstract
We consider a p+1-dimensional timelike hypersurface 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 c to those of a fluid in c. Moreover, expanding around a limit in which the mean curvature of the embedding diverges, the leading-order Einstein constraint equations on c are shown to reduce to the non-linear incompressible Navier-Stokes equation for a fluid moving in c.
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