The bar-mode instability in differentially rotating neutron stars: Simulations in full general relativity

We study the dynamical stability against bar-mode deformation of rapidly spinning neutron stars with differential rotation. We perform fully relativistic three-dimensional simulations of compact stars with MIR greater than or equal to 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 = 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 similar to0.24-0.25, only slightly smaller than the well-known Newtonian value similar to0.27 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 greater than or similar to 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 supernova collapse or binary neutron star mergers, it could be a strong and observable source of gravitational waves. We determine characteristic wave amplitudes and frequencies.