Dynamic and thermodynamic stability of relativistic, perfect fluid stars

We consider perfect fluid bodies ('stars') in general relativity, with the local state of the fluid specified by its 4-velocity, u(a), its ` particle number density', n, and its ` entropy per particle', s. A star is said to be in dynamic equilibrium if it is a stationary, axisymmetric solution to the Einstein-fluid equations with circular flow. A star is said to be in thermodynamic equilibrium if it is in dynamic equilibrium and its total entropy, S, is an extremum for all variations of initial data that satisfy the Einstein constraint equations and have fixed total mass, M, particle number, N, and angular momentum, J. We prove that for a star in dynamic equilibrium, the necessary and sufficient condition for thermodynamic equilibrium is constancy of angular velocity, Omega, redshifted temperature, T, and redshifted chemical potential, mu. A star in dynamic equilibrium is said to be linearly dynamically stable if all physical, gauge invariant quantities associated with linear perturbations of the star remain bounded in time; it is said to be mode stable if there are no exponentially growing solutions that are not pure gauge. A star in thermodynamic equilibrium is said to be linearly thermodynamically stable if delta S-2 < 0 for all variations at fixed M, N, and J; equivalently, a star in thermodynamic equilibrium is linearly thermodynamically stable if delta M-2 -T delta S-2 -mu delta N-2 -Omega delta(2)J > 0 for all variations that, to first order, satisfy dM = dN = dJ = 0 (and, hence, delta S = 0). Friedman previously identified positivity of canonical energy, E, as a criterion for dynamic stability and argued that all rotating stars are dynamically unstable to sufficiently non-axisymmetric perturbations (the CFS instability), so our main focus is on axisymmetric stability (although we develop our formalism and prove many results for non-axisymmetric perturbations as well). We show that for a star in dynamic equilibrium, mode stability holds with respect to all axisymmetric perturbations if E is positive on a certain subspace, V, of axisymmetric Lagrangian perturbations that, in particular, have vanishing Lagrangian change in angular momentum density. Conversely, if E fails to be positive on V, then there exist perturbations that cannot become asymptotically stationary at late times. We further show that for a star in thermodynamic equilibrium, for all Lagrangian perturbations, we have Er = delta M-2-Omega delta(2)J, where Er denotes the ` canonical energy in the rotating frame', so positivity of Er for perturbations with delta J = 0 is a necessary condition for thermodynamic stability. For axisymmetric perturbations, we have E = Er, so a necessary condition for thermodynamic stability with respect to axisymmetric perturbations is positivity of E on all perturbations with delta J = 0, not merely on the perturbations in V. Many of our results are in close parallel with the results of Hollands and Wald for the theory of black holes.