Generic instability of rotating relativistic stars

Abstract

All rotating perfect fluid configurations having two-parameter equations of state are shown to be dynamically unstable to nonaxisymmetric perturbations in the framework of general relativity. Perturbations of an equilibrium fluid are described by means of a Lagrangian displacement, and an action for the linearized field equations is obtained, in terms of which the symplectic product and canonical energy of the system can be expressed. Previous criteria governing stability were based on the sign of the canonical energy, but this functional fails to be invariant under the gauge freedom associated with a class of trivial Lagrangian displacements, whose existence was first pointed out by Schutz and Sorkin [12]. In order to regain a stability criterion, one must eliminate the trivials, and this is accomplished by restricting consideration to a class of “canonical” displacements, orthogonal to the trivials with respect to the symplectic product. There nevertheless remain perturbations having angular dependencee^imφ (φ the azimuthal angle) which, for sufficiently largem, make the canonical energy negative; consequently, even slowly rotating stars are unstable to short wavelength perturbations. To show strict instability, it is necessary to assume that time-dependent nonaxisymmetric perturbations radiate energy to null infinity. As a byproduct of the work, the relativistic generalization of Ertel's theorem (conservation of vorticity in constant entropy surfaces) is obtained and shown to be Noetherrelated to the symmetry associated with the trivial displacements.