  We consider perfect fluid bodies (stars) in general relativity, characterized
by particle number density and entropy per particle. A star is said to be in
dynamic equilibrium if it is a stationary, axisymmetric solution to the
Einstein-fluid equations with circular flow. We prove that for a star in
dynamic equilibrium, the necessary and sufficient condition for thermodynamic
equilibrium (extremum of total entropy S) is constancy of angular velocity
(\Omega), redshifted temperature, and redshifted chemical potential. 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 nonaxisymmetric perturbations (the CFS instability), so our
main focus is on axisymmetric stability. 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 no Lagrangian change in
angular momentum density. Conversely, if E fails to be positive on V, then
there exist perturbations that can't become asymptotically stationary at late
times. We further show that for a star in thermodynamic equilibrium, the
canonical energy in the rotating frame, E_r, is related to second order changes
in ADM mass, M, and angular momentum, J, by E_r = \delta^2 M - \Omega \delta^2
J. Thus, positivity of E_r for perturbations with \delta J = 0 is a necessary
condition for thermodynamic stability (local maximum of S). For axisymmetric
perturbations we have E = E_r, 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 to results of Hollands and Wald for the
theory of black holes.
