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It has been known for several decades that Einstein's field equations, when projected onto a null surface, exhibits a structure very similar to non-relativistic Navier-Stokes equation.
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As an aside, we may mention that the condition of for a 2 × 2 2\times 2 symmetric tensor S A B S^{AB} (with 3 independent components) to be expressible as a viscous stress-tensor of a 2-dimensional velocity field (with 2 independent components) with viscosity coefficients η , ξ \eta,\xi is ∂ A ∂ B S A B = ( 1 / 2 ) [ 1 + ( η / ξ ) ] S \partial_{A}\partial_{B}S^{AB}=(1/2)[1+(\eta/\xi)]S . Using this result and the (1+3) decomposition of of Einstein’s equations, one can show that ( η / ξ ) = − 1 (\eta/\xi)=-1 when the fluid interpretation is possible for Einstein’s equations
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