  The stability properties of relativistic stars against gravitational collapse
to black hole is a classical problem in general relativity. A sufficient
criterion for secular instability was established by Friedman, Ipser and Sorkin
(1988), who proved that a sequence of uniformly rotating barotropic stars is
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, i.e., 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.
