### 13.2 Instabilities of rotating perfect fluid stars

The importance of the canonical energy stems from the fact that it can be used to test the stability of the system. In particular:
• Dynamical instabilities are only possible for motions such that . This makes intuitive sense since the amplitude of a mode for which vanishes can grow without bounds and still obey the conservation laws.
• If the system is coupled to radiation (e.g. gravitational waves) which carries positive energy away from the system (which should be taken to mean that ) then any initial data for which will lead to an unstable evolution.

Consider a real frequency normal-mode solution to the perturbation equations, a solution of form . One can readily show that the associated canonical energy becomes

where the expression in the bracket is real. For the canonical angular momentum we get
Combining Equation (250) and Equation (251) we see that, for real frequency modes, we will have
where is the pattern speed of the mode.

Now notice that Equation (251) can be rewritten as

Using cylindrical coordinates, and , one can show that
But
and we must have (for uniform rotation)

Equation (256) forms a key part of the proof that rotating perfect fluid stars are generically unstable in the presence of radiation [45]. The argument goes as follows: Consider modes with finite frequency in the limit. Then Equation (256) implies that co-rotating modes (with ) must have , while counter-rotating modes (for which ) will have . In both cases , which means that both classes of modes are stable. Now consider a small region near a point where (at a finite rotation rate). Typically, this corresponds to a point where the initially counter-rotating mode becomes co-rotating. In this region . However, will change sign at the point where (or, equivalently, the frequency ) vanishes. Since the mode was stable in the non-rotating limit this change of sign indicates the onset of instability at a critical rate of rotation.