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 .