  Viewing gravitational energy-momentum as equal by observation, but different
in essence from inertial energy-momentum naturally leads to the gauge theory of
volume-preserving diffeormorphisms of a four-dimensional in- ner space. To
analyse scattering in this theory the gauge field is coupled to two Dirac
fields with different masses. Based on a generalized LSZ reduction formula the
S-matrix element for scattering of two Dirac particles in the gravitational
limit and the corresponding scattering cross-section are calculated to leading
order in perturbation theory. Taking the non-relativistic limit for one of the
initial particles in the rest frame of the other the Rutherford-like
cross-section of a non-relativistic particle scattering off an infinitely heavy
scatterer calculated quantum mechanically in Newtonian gravity is recovered.
This provides a non-trivial test of the gauge field theory of volume-preserving
diffeomorphisms as a quantum theory of gravity
