  Core collapse supernovae are a promising source of detectable gravitational
waves. Most of the existing (multidimensional) numerical simulations of core
collapse in general relativity have been done using approximations of the
Einstein field equations. As recently shown by Dimmelmeier et al (2002a,b), one
of the most interesting such approximation is the so-called conformal flatness
condition (CFC) of Isenberg, Wilson and Mathews. Building on this previous work
we present here new results from numerical simulations of relativistic
rotational core collapse in axisymmetry, aiming at improving the dynamics and
the gravitational waveforms. The computer code used for these simulations
evolves the coupled system of metric and fluid equations using the 3+1
formalism, specialized to a new framework for the gravitational field equations
which we call CFC+. In this approach we add new degrees of freedom to the
original CFC equations, which extend them by terms of second post-Newtonian
order. The corrections for CFC+ are computed solving a system of elliptic
linear equations. The new formalism is assessed with time evolutions of both
rotating neutron stars in equilibrium and gravitational core collapse of
rotating polytropes. Gravitational wave signals for a comprehensive sample of
collapse models are extracted using either the quadrupole formula or directly
from the metric. We discuss our results on the dynamics and the gravitational
wave emission through a detailed comparison between CFC and CFC+ simulations.
The main conclusion is that, for the neutron star spacetimes analyzed in the
present work, no significant differences are found among CFC, CFC+, and full
general relativity, which highlights the suitability of the former.
