### 4.3 Stability

The stability of radial oscillations for non-rotating stars in general relativity is well understood.
Especially, the stability of static spherically symmetric stars can be determined by examining the
mass-radius relation for a sequence of equilibrium stellar models, see for example Chapter 24 in [150]. The
radial perturbations are described by a Sturm–Liouville second order equation with the frequency of the
mode being the eigenvalue , then for real the modes will be stable while for imaginary they
will be unstable [52], see also Chapter 17.2 in [188].
The stability of the non-radially pulsating stars (Newtonian or relativistic) is determined by the
Schwarzschild discriminant

where is the star’s adiabatic index. This can be understood if we define the local buoyancy force
per unit volume acting on a fluid element displaced a small radial distance to be
where is the local acceleration of gravity. When is negative in some region the buoyancy force is
positive and the star is unstable against convection, while when is positive the buoyancy force is
restoring and the star is stable against convection. Another way of discussing the stability is through the
so-called Brunt–Väisälä frequency which is the characteristic frequency of the local fluid
oscillations. Following earlier discussions when is positive, the fluid element undergoes oscillations,
while when is negative the fluid is locally unstable. In other words, in Newtonian theory
stability to non-radial oscillations can be guaranteed only if everywhere within the
star [65]. In general relativity [78], this is a sufficient condition, and so if the quasi-normal
modes are stable. For an extensive discussion of stellar instabilities for both non-rotating and
rotating stars (which are actually more interesting for the gravitational wave astronomy) refer
to [177, 140, 192].
For completeness the same applies as outlined at the end of Section 3.3. A model calculation of Price
and Husain [168], however indicated that the nearly Newtonian quasi-normal modes might be a
basis for the fluid perturbations. Further mathematical investigation is needed to clarify this
issue.