Radiation Driven Instability of Rapidly Rotating Relativistic Stars: Criterion and Evolution Equations Via Multipolar Expansion of Gravitational Waves

I suggest a novel approach for deriving evolution equations for rapidly rotating relativistic stars affected by radiation-driven Chandrasekhar-Friedman-Schutz instability. This approach is based on the multipolar expansion of gravitational wave emission and appeals to the global physical properties of the star (energy, angular momentum, and thermal state), but not to canonical energy and angular momentum, which is traditional. It leads to simple derivation of the Chandrasekhar-Friedman-Schutz instability criterion for normal modes and the evolution equations for a star, affected by this instability. The approach also gives a precise form to simple explanation of the Chandrasekhar-Friedman-Schutz instability; it occurs when two conditions are met: (a) gravitational wave emission removes angular momentum from the rotating star (thus releasing the rotation energy) and (b) gravitational waves carry less energy, than the released amount of the rotation energy. To illustrate the results, I take the r-mode instability in slowly rotating Newtonian stellar models as an example. It leads to evolution equations, where the emission of gravitational waves directly affects the spin frequency, being in apparent contradiction with widely accepted equations. According to the latter, effective spin frequency decrease is coupled with dissipation of unstable mode, but not with the instability as it is. This problem is shown to be superficial, and arises as a result of specific definition of the effective spin frequency applied previously. Namely, it is shown, that if this definition is taken into account properly, the evolution equations coincide with obtained here in the leading order in mode amplitude. I also argue that the next-to-leading order terms in evolution equations were not yet derived accurately and thus it would be more self-consistent to omit them.