  We present the general relativistic pressure correction terms in Newtonian
hydrodynamic equations to the nonlinear order: these are equations
(\ref{mass-conservation-Mink})-(\ref{Poisson-eq-Mink}). The derivation is made
in the zero-shear gauge based on the fully nonlinear formulation of
cosmological perturbation in Einstein's gravity. The correction terms {\it
differ} from many of the previously suggested forms in the literature based on
hand-waving manners. We confirm our results by comparing with (i) the nonlinear
perturbation theory, (ii) the first order post-Newtonian approximation, and
(iii) the special relativistic limit, and by checking (iv) the consistency with
full Einstein's equation.

Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Theory