Post-Newtonian hydrodynamics and post-Newtonian gravitational wave generation for numerical relativity

We present an approximate formalism for the equations of evolution of a moderately relativistic self-gravitating fluid which takes into account the dominant (quantitative and qualitative) new effects entailed by Einstein's theory. This (first plus second-and-a-half) post-Newtonian scheme is equivalent to known results of the literature, but casts them in an explicitly ‘quasi-Newtonian’ form, so that it becomes as easy to implement on a computer as a Newtonian-gravity code for non-relativistic hydrodynamics (in particular, all the relativistic non-localities are reduced to Poisson equations with compact-supported sources). We complete this scheme by a correspondingly accurate post-Newtonian gravitational waveform extraction formalism which goes beyond the ‘standard quadrupole equation’ in including the first relativistic corrections to the emission of gravitational radiation in a form which can be easily implemented numerically. We view our formalisms as simple and robust tools allowing one, with only a minimal computer investment, to study the most important quantitative and qualitative characteristics of the three-dimensional gravitational dynamics, and wave generation, of a wide range of semi-relativistic sources, such as the collapse of a star to the neutron star stage, or the coalescence of neutron star binaries.