Dynamic and Thermodynamic Stability of Relativistic, Perfect Fluid Stars

TL;DR

This paper investigates the conditions for dynamic and thermodynamic stability of relativistic perfect fluid stars, establishing criteria based on canonical energy and analyzing the implications for axisymmetric perturbations.

Contribution

It provides a rigorous analysis linking dynamic and thermodynamic stability criteria for relativistic stars, extending previous work by explicitly relating canonical energy to stability conditions.

Findings

Positivity of canonical energy E indicates axisymmetric stability.

Thermodynamic equilibrium requires constant angular velocity, temperature, and chemical potential.

Positivity of E_r is necessary for thermodynamic stability with respect to perturbations.

Abstract

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.