  We present the generalization of a recently introduced modified gravitational
potential for self-gravitating fluids. The use of this potential allows for an
accurate approximation of general relativistic effects in an otherwise
Newtonian hydrodynamics code also in cases of rapid rotation. We test this
approach in numerical simulations of astrophysical scenarios related to compact
stars, like supernova core collapse with both a simplified and detailed
microphysical description of matter, and rotating neutron stars in equilibrium.
We assess the quality of the new potential, and demonstrate that it provides a
significant improvement compared to previous formulations for such potentials.
Newtonian simulations of compact objects employing such an effective
relativistic potential predict inaccurate pulsation frequencies despite the
excellent agreement of the collapse dynamics and structure of the compact
objects with general relativistic results. We analyze and discuss the reason
for this behavior.
