Rotating massive stars in general relativity

Abstract

Equilibrium models of uniformly rotating massive stars are investigated, using a weak field, slow rotation approximation, which is shown to be adequate for all cases of interest. The fate of radial perturbations about these equilibrium configurations is investigated using a linear­ized stability analysis to determine the oscillation frequency σ in a peturbation ∝ e iσ t . An eigenvalue equation for σ 2 is obtained which can be made self adjoint with respect to the spatial metric, and a variational principle to determine σ 2 is derived. Numerical determinations of σ 2 have been carried out for a variety of masses, radii and rotational velocities, and these results are incorporated in a simple formula that gives the dependence of σ 2 on these quantities. The condition for instability, σ 2 negative, is determined, and it is found that for large masses and maximum rotation velocity, so that when centrifugal force balances gravity at the surface, a massive star becomes unstable when its radius is 208 times the Schwarzschild radius 2 GM/c 2 .