Applying the turning point theorem provided by Sorkin , Friedman Ipser and Sorkin  shows that, in the case of rotating stars, the secular axisymmetric instability sets in when the mass becomes maximum along a sequence of constant angular momentum. An equivalent criterion is provided by Cook et al. : The secular axisymmetric instability sets in when the angular momentum becomes minimum along a sequence of constant rest mass.
The instability develops on a timescale that is limited by the time required for viscosity to redistribute the star's angular momentum. This timescale is long compared to the dynamical timescale and comparable to the spin-up time following a pulsar glitch. When it becomes secularly unstable, a star evolves in a quasi-stationary fashion until it encounters the dynamical instability and collapses to a black hole. Thus, the onset of the secular instability to axisymmetric perturbations separates stable neutron stars from neutron stars that will collapse to a black hole.
Goussard et al.  extend the stability criterion to hot protoneutron stars with nonzero total entropy. In this case, the loss of stability is marked by the configuration with minimum angular momentum along a sequence of both constant rest mass and total entropy.
In the nonrotating limit, Gondek et al.  compute frequencies and eigenfunctions of axisymmetric pulsations of hot proto-neutron stars and verify that the secular instability sets in at the maximum mass turning point, as is the case for cold neutron stars.
|Rotating Stars in Relativity
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