Newtonian hydrodynamic equations with relativistic pressure and velocity
  We present a new approximation to include fully general relativistic pressure
and velocity in Newtonian hydrodynamics. The energy conservation, momentum
conservation and two Poisson's equations are consistently derived from
Einstein's gravity in the zero-shear gauge assuming weak gravity and
action-at-a-distance limit. The equations show proper special relativity limit
in the absence of gravity. Our approximation is complementary to the
post-Newtonian approximation and the equations are valid in fully nonlinear
situations.
