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 sp inning neutron stars with differential rotation. We perform fully relativis tic 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 adiab atic index Gamma = 2. As in Newtonian theory, we find that stars above a cr itical 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 c ritical value of beta = beta (dGR) is similar to0.24-0.25, only slightly sm aller than the well-known Newtonian value similar to0.27 for incompressible Maclaurin spheroids. The critical value depends only very weakly on the de gree of differential rotation for the moderate range we surveyed. All unsta ble stars form bars on a dynamical timescale. Models with sufficiently larg e 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 adju st to form dynamically stable ellipsoidal-like configurations. If the bar-m ode instability is triggered in supernova collapse or binary neutron star m ergers, it could be a strong and observable source of gravitational waves. We determine characteristic wave amplitudes and frequencies.