Note on the Petrov-like boundary condition at finite cutoff surface in gravity/fluid duality

Previously it was shown that imposing a Petrov-like boundary condition on a hypersurface may reduce the Einstein equation to the incompressible Navier-Stokes equation, but all these correspondences are established in the near-horizon limit. In this paper, we demonstrate that this strategy can be extended to an arbitrary finite cutoff surface which is spatially flat, and the Navier-Stokes equation is obtained by employing a nonrelativistic long-wavelength limit.