A quasi-radial stability criterion for rotating relativistic stars

The stability properties of relativistic stars against gravitational collapse to black holes is a classical problem in general relativity. In 1988, a sufficient criterion for secular instability was established by Friedman, Ipser & Sorkin, who proved that a sequence of uniformly rotating barotropic stars are secularly unstable on one side of a turning point and then argued that a stronger result should hold: that the sequence should be stable on the opposite side, with the turning point marking the onset of secular instability. We show here that this expectation is not met. By computing in full general relativity the F-mode frequency for a large number of rotating stars, we show that the neutral-stability point, that is, where the frequency becomes zero, differs from the turning point for rotating stars. Using numerical simulations, we validate that the new criterion can be used to assess the dynamical stability of relativistic rotating stars.