Chandrasekhar–Friedman–Schutz instability

Remarks

A wave that moves forward in the inertial frame radiates positive angular momentum through gravitational waves. A wave that moves backward relative to the fluid carries negative angular momentum in the rotating star. Losing positive angular momentum then makes the negative angular momentum of the mode larger in size, so the amplitude grows. In the rotating frame a backward f mode of order $m$ has a frequency $\sigma_m$ that grows roughly like $\sqrt{m}$, so its pattern speed $\sigma_m / m$ tends to $0$. Its pattern speed in the inertial frame is $\Omega - \sigma_m / m$, which lies in $(0, \Omega)$ for $m$ large; this gives the second clause. The growth time increases fast with $m$, so in real stars viscosity damps all but the lowest modes; without the hypothesis of zero viscosity the second clause fails. Chandrasekhar found the effect for the Maclaurin spheroids, and Friedman and Schutz showed that it is generic.

Going deeper

For a student

The instability is a radiation-driven secular instability, similar to the one that drives a viscous Maclaurin spheroid away from axial symmetry. It is a candidate for limiting the spin of young neutron stars and for producing a source of gravitational waves. Modes that are often discussed are the f modes for fast rotation and the r modes, which are governed by the Coriolis force. Whether the instability matters in a given star depends on the competition between the growth time and the damping time due to viscosity.