A framework for perturbations and stability of differentially rotating stars

The paper provides a new framework for the description of linearized adiabatic Lagrangian perturbations and stability of differentially rotating Newtonian stars. It overcomes problems in a previous framework by Dyson & Schutz and provides the basis of a rigorous analysis of the stability of such stars. For this, the governing equations of the oscillations are written as a first-order system in time. From that system, the formal generator of time evolution is read off and a Hilbert space is given, in which generates a strongly continuous group. As a consequence, the governing linearized equations have a well-posed initial-value problem. The spectrum of the (in general non-normal) generator relevant for stability considerations is shown to coincide with the spectrum of an operator polynomial whose coefficients can be read off from the governing equations. Finally, we give for the first time sufficient criteria for stability in the form of inequalities for the coefficients of the polynomial. These show that a negative canonical energy of the star does not necessarily indicate instability. It is still unclear whether these criteria are strong enough to prove stability for realistic stars. However, their usefulness has already been demonstrated in another paper, where they lead to a new result in the discussion of the stability of rotating (Kerr) black holes. That stability is a classical open problem in general relativity.