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# Petrov type I condition and dual fluid dynamics

## Abstracts

 Recently Lysov and Strominger [arXiv:1104.5502] showed that imposing Petrov type I condition on a (p+1)-dimensional time like 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 in compressible 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 their rotational 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 of vacuum Einstein gravity in the non-relativistic limit. We discuss the procedure both on the finite cut off surface via the non-relativistic hydrodynamic expansion and on the highly accelerated surface via the near horizon expansion.
Journal:JOURNAL OF HIGH ENERGY PHYSICS
Recommended references： Cai, Rong-Gen,Li, Li,Yang, Qing,Zhang, Yun-Long.(2016).Petrov type I condition and dual fluid dynamics.JOURNAL OF HIGH ENERGY PHYSICS.[ChinaXiv:201605.01165] (Click&Copy)
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