  The special relativistic hydrodynamics with weak gravity is hitherto unknown
in the literature. Whether such an asymmetric combination is possible was
unclear. Here, the hydrodynamic equations with Poisson-type gravity considering
fully relativistic velocity and pressure under the weak gravity and the
action-at-a-distance limit are consistently derived from Einstein's general
relativity. Analysis is made in the maximal slicing where the Poisson's
equation becomes much simpler than our previous study in the zero-shear gauge.
Also presented is the hydrodynamic equations in the first post-Newtonian
approximation, now under the {\it general} hypersurface condition. Our
formulation includes the anisotropic stress.
