Cai, RG (reprint author), Chinese Acad Sci, Inst Theoret Phys, State Key Lab Theoret Phys, Beijing 100190, Peoples R China.
; Yang, Qing
; Zhang, Yun-Long] Chinese Acad Sci, Inst Theoret Phys, State Key Lab Theoret Phys, Beijing 100190, Peoples R China
; [Zhang, Yun-Long] Univ Southampton, Math Sci & STAG Res Ctr, Southampton SO17 1BJ, Hants, England
The Petrov type I condition for the solutions of vacuum Einstein equations in both of the nonrelativistic and relativistic hydrodynamic expansions is checked. We show that it holds up to the third order of the nonrelativistic 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.