The study of the linear (small-amplitude) pulsation of stars is the principal way to investigate their stability. Stars that evolve to an unstable state may explode, collapse, undergo large-amplitude pulsation, or otherwise greatly change their structure. Linear stability therefore delimits the range of stars that one expects to find in nature and points toward the kinds of stars that might become, for example, supernovae or black holes.
Stability theory is also very interesting mathematically, having to do with functional analysis and the spectral theory of differential operators. The eigenvalues of the spectrum are characteristic frequencies of pulsation, and the associated pulsation solutions are called normal modes of the star. The theory of the pulsations of non-rotating stars in Newtonian gravity has a lot in common with spectral theory in quantum mechanics, which by the time I began my research (1969) was well understood. But even in Newtonian gravity, the pulsation theory of rotating stars was more challenging and less well developed. Moreover, the theory of linear pulsations was significantly more complicated in general relativity, partly because of the complexity of the theory, but mainly because pulsations emit gravitational radiation, which means that their characteristic frequencies must typically be complex rather than real.
In my early work, done for my PhD, I tried to develop the formal basis for studying pulsations, which included a variational principle for fluid systems in general relativity and sufficient criteria for the stability of rotating stars in both Newtonian gravity and in general relativity. I went on to work with John Friedman to develop the full mathematical theory of the pulsations of rotating stars in general relativity. With Rafael Sorkin I discovered that the Lagrangian fluid displacement variables that Friedman and I (and most of the community) were using had a gauge group on their initial data. Pure-gauge perturbations, called the trivials, had to be understood before using stability criteria. Sorkin and I also developed the relationship between conservation laws and variational principles in general relativity (the Noether theorem) and showed that coordinate-dependent pseudotensors could be replaced by a coordinate-invariant Noether operator.
Friedman and I incorporated the trivials into pulsation theory and showed for the first time that there was a general class of perturbations of rotating stars that is unstable in general relativity, destabilized by the emission of gravitational waves. Chandrasekhar had discovered the first instance of this class a few years earlier by studying unrealistic Maclaurin spheroids, but we showed that it was generic. This instability, now called the Chandrasekhar-Friedman-Schutz (CFS) instability, may power gravitational wave emission from certain kinds of known stars, called low-mass X-ray binaries.
Further research in the 1970s and 1980s, mainly with graduate students and postdocs in my group, explored the properties of rotating stellar pulsation in various circumstances: CFS modes in Maclaurin spheroids, the continuous spectrum in differentially rotating stars, the spectrum of modes in flat rotating discs (amenable to semi-analytic treatment), the way instability sets in along a sequence of stellar models, and the development of instabilities in shearing fluids with self-gravity.
The subject remains a current interest of mine. The discovery by Nils Andersson that Rossby modes (r-modes) of rotating stars are CFS unstable has produced an explosion of research into the possibility that these modes radiate detectable gravitational waves, and I have contributed to this debate. I have returned also to the interesting subject of shear instabilities in rotating discs.