  We study the dynamical stability against bar-mode deformation of rapidly
spinning neutron stars with differential rotation. We perform fully
relativistic 3D simulations of compact stars with $M/R \geq 0.1$, where $M$ is
the total gravitational mass and $R$ the equatorial circumferential radius. We
adopt an adiabatic equation of state with adiabatic index $\Gamma=2$. As in
Newtonian theory, we find that stars above a critical value of $\beta \equiv
T/W$ (where $T$ is the rotational kinetic energy and $W$ the gravitational
binding energy) are dynamically unstable to bar formation. For our adopted
choices of stellar compaction and rotation profile, the critical value of
$\beta = \beta_{dGR}$ is $\sim 0.24-0.25$, only slightly smaller than the
well-known Newtonian value $\sim 0.27$ for incompressible Maclaurin spheroids.
The critical value depends only very weakly on the degree of differential
rotation for the moderate range we surveyed. All unstable stars form bars on a
dynamical timescale. Models with sufficiently large $\beta$ subsequently form
spiral arms and eject mass, driving the remnant to a dynamically stable state.
Models with moderately large $\beta \gtrsim \beta_{dGR}$ do not develop spiral
arms or eject mass but adjust to form dynamically stable ellipsoidal-like
configurations. If the bar-mode instability is triggered in supernovae collapse
or binary neutron star mergers, it could be a strong and observable source of
gravitational waves. We determine characteristic wave amplitudes and
frequencies.

Astrophysics;General Relativity and Quantum Cosmology