Petrov type I Spacetime and Dual Relativistic Fluids

The Petrov type I condition for the solutions of vacuum Einstein equations in both of the non-relativistic and relativistic hydrodynamic expansions is checked. We show that it holds up to the third order of the non-relativistic hydrodynamic expansion parameter, but it is violated at the fourth order even if we choose a general frame. On the other hand, it is found that the condition holds at least up to the second order of the derivative expansion parameter. Turn the logic around, through imposing the Petrov type I condition and Hamiltonian constraint on a finite cutoff surface, we show that the stress tensor of the relativistic fluid can be recovered with correct first order and second order transport coefficients dual to the solutions of vacuum Einstein equations.