  We revisit the analysis made by Hwang and Noh [JCAP 1310 (2013)] aiming the
construction of a Newtonian set of equations incorporating pressure effects
typical of the General Relativity theory. We explicitly derive the Hwang-Noh
equations, comparing them with similar computations found in the literature.
Then, we investigate $i)$ the cosmological expansion, $ii)$ linear cosmological
perturbations theory and $iii)$ stellar equilibrium by using the new set of
equations and comparing the results with those coming from the usual Newtonian
theory, from the Neo-Newtonian theory and from the General Relativity theory.
We show that the predictions for the background evolution of the Universe are
deeply changed with respect to the General Relativity theory: the acceleration
of the Universe is achieved with positive pressure. On the other hand, the
behaviour of small cosmological perturbations reproduces the one found in the
relativistic context, even if only at small scales. We argue that this last
result may open new possibilities for numerical simulations for structure
formation in the Universe. Finally, the properties of neutron stars are
qualitatively reproduced by Hwang-Noh equations, but the upper mass limit is at
least one order of magnitude higher than the one obtained in General
Relativity.
