Newtonian hydrodynamics with general relativistic pressure

We present the general relativistic pressure correction terms in Newtonian hydrodynamic equations to the nonlinear order: these are equations (1.1)-(1.3). 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 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.