Petrov type I Condition and Dual Fluid Dynamics

TL;DR

This paper investigates the Petrov type I condition's role in deriving dual fluid dynamics from vacuum Einstein gravity, revealing limitations at higher orders and proposing a method to extract second-order transport coefficients.

Contribution

It demonstrates that the Petrov type I condition alone is insufficient for non-relativistic fluids at higher orders and introduces a way to derive higher-order hydrodynamics with some transport coefficients.

Findings

Non-relativistic fluid does not satisfy Petrov type I at next order without extra constraints.

The inverse procedure can derive higher-order hydrodynamics and extract some transport coefficients.

The Petrov type I dual fluid does not match the vacuum Einstein gravity dual in the non-relativistic limit.

Abstract

Recently Lysov and Strominger [arXiv:1104.5502] showed that imposing 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 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 of vacuum Einstein gravity in the 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.