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I am confused as to what the Chandrasekhar-Friedman-Schutz (CFS) instability is, exactly. It seems to refer to this paper by Chandrasekhar, but I do not think this paper covers the full instability. In the first part of the paper, Chandrasekhar shows that a Jacobi ellipsoid radiates off angular momentum in gravitational waves to become a Maclauren ellipsoid. I do not yet understand the second part of the paper.

This webpage says that in a rotating star, a non-axially symmetric perturbation rotating slower than the star will radiate off angular momentum, thus increasing the size of the perturbation. Apparently, any rotating star is unstable in general relativity because of this mechanism.

Questions: I am very confused as to what the exact nature of the instability is. Do quickly rotating stars collapse into black holes, or do they just radiate off angular momentum and settle down into slowly rotating stars? Have we detected this instability experimentally in the real world? Moreover: damping effects (like friction) in a real star work against the instability in slowly rotating stars. Can they fight the instability also for fast rotating stars?

Also, is there a good source that gives an introduction to the instability? Papers by Friedman and Shutz (for example, this one) seem to focus heavily on obtuse mathematics (i.e. "trivial" Lagrangian perturbations) and don't present a clear physical picture of what the instability actually is.

Quillo
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user1379857
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1 Answers1

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The Chandrasekhar-Friedman-Schutz (CFS) instability is a gravitational-wave-driven instability that affects rotating stars in general relativity. A non-axisymmetric perturbation can extract angular momentum from the star if it rotates slower than the background fluid as seen in an inertial frame. Since the gravitational waves carry away negative angular momentum, the perturbation grows rather than decays, leading to an instability.

In Chandrasekhar’s original paper (1970), he studied how a Jacobi ellipsoid—a triaxial rotating fluid body—radiates angular momentum via gravitational waves and transitions into a Maclaurin spheroid. The second part of his paper then considers small non-axisymmetric perturbations in a general relativistic star and finds that they can be destabilized by gravitational-wave emission. Friedman and Schutz (1978) extended this to show that in general relativity, any rotating star is, in principle, unstable to this mechanism because there are always modes that satisfy the necessary condition for instability.

Regarding your questions:

  1. Does the instability cause collapse or just spin-down?
    The instability does not directly cause black hole formation; rather, it extracts angular momentum until the star reaches a stable configuration where the instability ceases. For realistic neutron stars, this typically means spinning down to a lower rotation rate.

  2. Has this instability been detected?
    So far, there is no direct observational evidence of the CFS instability operating in real neutron stars. This is likely because realistic damping mechanisms (like viscosity and magnetic fields) suppress the instability in all but the fastest-spinning stars.

  3. Can damping mechanisms suppress the instability in fast-rotating stars?
    Yes, but only if they are strong enough. In slowly rotating stars, viscosity (especially bulk and shear viscosity) can completely suppress the instability. In fast-rotating stars, bulk viscosity from weak interaction processes is much less effective, but shear viscosity and magnetic field effects can still significantly slow down the growth of the instability. Superfluid effects in neutron star interiors may also play a crucial role.

For further reading, Andersson (2003) "Gravitational waves from instabilities in relativistic stars" (arXiv:astro-ph/0211057) or the review by Friedman & Stergioulas Rotating Relativistic Stars are excellent.

Suraj Gavhale
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