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
Now notice that Equation (251) can be rewritten as
Equation (256) forms a key part of the proof that rotating perfect fluid stars are generically unstable in the presence of radiation . 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.
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