On secular instabilities of rigidly rotating stars in general relativity - I. Theoretical formalism

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Abstract
This is the first of two papers on secular instabilities in rigidly rotating, axisymmetric, stationary incompressible stars. The equations which give the e-folding times of the instabilities caused by an important class of nonaxisymmetric perturbations of the stars are derived. The stars are modelled as Maclaurin spheroids. The instabilities are due to gravitational radiation reaction and viscosity in the star and they cause its angular momentum to decrease. The method used is a generalization of the multipole approach of Roberts & Stewartson. Using these equations, we show in a second paper that all incompressible, inviscid stars are secularly unstable due to gravitational radiation reaction. Generally both viscosity and gravitational radiation reaction cause secular instabilities to occur, however we find that in some cases these two dissipative effects counter-balance each other and actually increase the secular stability of the stars. The e-folding times for solar density stars typically exceed the present age of the universe, but for white dwarfs and more especially neutron stars, these times are of astrophysical interest.

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