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 [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.
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