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The theory of small, adiabatic, non-radial perturbations of a star away from hydrostatic equilibrium is developed within the framework of general relativity. The unperturbed equilibrium confignration is an arbitrary, non-rotating, general-relativistic stellar model. The departures from equilibrium are analyzed into tensorial spherical harmonics and then into complex normal modes with various mixtures of incoming and outgoing gravitational waves. A discussion is given of the expansion of real, physical pulsations with purely outgoing gravitational radiation in terms of the complex normal modes. Criteria are developed for stability against non-radial pulsations; and methods are devised for computing numerically the pulsation frequencies, eigenfunctions, and gravitational-radiation damping times of the stable, real quasi-normal modes of pulsation.
Thorne et al. (Fri,) studied this question.