(ed.). Gravitational waves (IOP, 2001)(422s).

74 Source calculationsthat the loss of energy to gravitational waves would damp out any perturbations,and indeed this is normally the case. However, it was a remarkable discovery ofChandrasekhar [27] that the opposite sometimes happens.A rotating star is idealized as an axially symmetric perfect-fluid system.In the Newtonian theory the pulsations of a perturbed fluid can be describedby normal modes which are the solutions of perturbed Euler and gravitationalfield equations. If the star is stable, the eigenfrequencies σ of the normal modesare real; if the star is unstable, there is at least one pair of complex-conjugatefrequencies, one of which represents an exponentially growing mode and the othera decaying mode. (We take the convention that the time-dependence of a mode isexp(iσ t).)In general relativity, the situation is, in principle, the same, except that thereis a boundary condition on the perturbation equations that insists that gravitationalwaves far away be outgoing, i.e. that the star loses energy to gravitational waves.This condition forces all eigenfrequencies to be complex. The sign of theimaginary part of the frequency determines stability or instability.The loss of energy to gravitional radiation can destabilize a star that wouldotherwise (i.e. in Newtonian theory) be stable. This is because it opens a pathwayto lower-energy configurations that might not be accessible to the Newtonianstar. This normally happens because gravitational radiation also carries awayangular momentum, a quantity that is conserved in the Newtonian evolution ofa perturbation.The sign of the angular momentum lost by the star is a critical diagnosticfor the instability. A wave that moves in the positive angular direction around astar will radiate positive angular momentum to infinity. A wave that moves in theopposite direction, as seen by an observer at rest far away, will radiate negativeangular momentum. In a spherical star, both actions result in the damping ofthe perturbation because, for example, the positive-going wave has intrinsicallypositive angular momentum, so when it radiates its angular momentum decreasesand so its amplitude decreases. Similarly, the negative-going wave has negativeangular momentum, so when it radiates negative angular momentum its amplitudedecreases.The situation can be different in a rotating star, as first pointed out byFriedman and Schutz [28]. The angular momentum carried by a wave dependson its pattern angular velocity relative to the star’s angular velocity, not relativeto an observer far away. If a wave pattern travels backwards relative to the star,it represents a small effective slowing down of the star and therefore carriesnegative angular momentum. This can lead to an anomalous situation: if a wavetravels backwards relative to the star, but forwards relative to an inertial observer(because its angular velocity relative to the star is smaller than the star’s angularvelocity), then it will have negative angular momentum but it will radiate positiveangular momentum. The result will be that its intrinsic angular momentum willget more negative, and its amplitude will grow.This is the mechanism of the Chandrasekar–Friedman–Schutz (CFS)