  We study by computational means the dynamical stability against bar-mode
deformation of rapidly and differentially rotating stars in a post-Newtonian
approximation of general relativity. We vary the compaction of the star $M/R$
(where $M$ is the gravitational mass and $R$ the equatorial circumferential
radius) between 0.01 and 0.05 to isolate the influence of relativistic
gravitation on the instability. For compactions in this moderate range, the
critical value of $\beta = T/W$ for the onset of the dynamical instability
(where $T$ is the rotational kinetic energy and W the gravitational binding
energy) slightly decreases from $\sim$ 0.26 to $\sim$ 0.25 with increasing
compaction for our choice of the differential rotational law. Combined with our
earlier findings based on simulations in full general relativity for stars with
higher compaction, we conclude that relativistic gravitation enhances the
dynamical bar-mode instability, i.e. the onset of instability sets in for
smaller values of $\beta$ in relativistic gravity than in Newtonian gravity. We
also find that once a triaxial structure forms after the bar-mode perturbation
saturates in dynamically unstable stars, the triaxial shape is maintained, at
least for several rotational periods. To check the reliability of our numerical
integrations, we verify that the general relativistic Kelvin-Helmholtz
circulation is well-conserved, in addition to rest-mass and total mass-energy,
linear and angular momentum. Conservation of circulation indicates that our
code is not seriously affected by numerical viscosity. We determine the
amplitude and frequency of the quasi-periodic gravitational waves emitted
during the bar formation process using the quadrupole formula.
