Rong-Gen Cai, Li Li, Qing Yang, Yun-Long Zhang
Recently Lysov and Strominger [arXiv:1104.5502] showed that imposing a Petrov type $I$ condition on a $(p+1)$-dimensional timelike hypersurface embedded in a $(p+2)$-dimensional vacuum Einstein gravity reduces the degrees of freedom in the extrinsic curvature of the hypersurface to that of a fluid on the hypersurface, and that the leading-order Einstein constraint equations in terms of the mean curvature of the embedding give the non-linear incompressible Navier-Stokes equations of the dual fluid. In this paper we show that the non-relativistic fluid dual to vacuum Einstein gravity does not satisfy the Petrov type $I$ condition at next order, unless additional constraint such as the irrotational condition is added. In addition, we show that this procedure can be inversed to derive the non-relativistic hydrodynamics with higher order corrections through imposing the Petrov type $I$ condition, and that some second order transport coefficients can be extracted, but the dual "Petrov type $I$ fluid" does not match the dual fluid constructed from the geometry in non-relativistic limit. We discuss the procedure both on the finite cutoff surface via the non-relativistic hydrodynamic expansion and on the highly accelerated surface via the near horizon expansion.
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http://arxiv.org/abs/1302.2016
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