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Dynamic and Thermodynamic Stability of Superconducting-superfluid Stars

Delong Kong, Yu Tian, Hongbao Zhang

Abstract

We give a comprehensive analysis of the dynamic and thermodynamic stability of neutron stars composed of superconducting-superfluid mixtures within the Iyer-Wald formalism. We derive the first law of thermodynamics and the necessary and sufficient condition under which dynamic equilibrium implies thermodynamic equilibrium. By constructing the phase space and canonical energy, we show that the dynamic stability for perturbations, restricted in symplectic complement of trivial perturbations with the ADM 3-momentum unchanged, is equivalent to the non-negativity of the canonical energy. Furthermore, dynamic stability against restricted axisymmetric perturbations guarantees the dynamic stability against all axisymmetric perturbations. We also prove that the positivity of canonical energy on all axisymmetric perturbations within the Lagrangian displacement framework with fixed angular momentum is necessary for thermodynamic stability. In particular, the equivalence of dynamic and thermodynamic stability for spherically symmetric perturbations of static, spherically symmetric isentropic configurations is established.

Dynamic and Thermodynamic Stability of Superconducting-superfluid Stars

Abstract

We give a comprehensive analysis of the dynamic and thermodynamic stability of neutron stars composed of superconducting-superfluid mixtures within the Iyer-Wald formalism. We derive the first law of thermodynamics and the necessary and sufficient condition under which dynamic equilibrium implies thermodynamic equilibrium. By constructing the phase space and canonical energy, we show that the dynamic stability for perturbations, restricted in symplectic complement of trivial perturbations with the ADM 3-momentum unchanged, is equivalent to the non-negativity of the canonical energy. Furthermore, dynamic stability against restricted axisymmetric perturbations guarantees the dynamic stability against all axisymmetric perturbations. We also prove that the positivity of canonical energy on all axisymmetric perturbations within the Lagrangian displacement framework with fixed angular momentum is necessary for thermodynamic stability. In particular, the equivalence of dynamic and thermodynamic stability for spherically symmetric perturbations of static, spherically symmetric isentropic configurations is established.

Paper Structure

This paper contains 18 sections, 8 theorems, 234 equations.

Table of Contents

  1. Introduction
  2. Relativistic model of superconducting-superfluid mixtures
  3. Master function
  4. Dynamical fields, variations, and Lagrangian displacements
  5. Lagrangian of superconducting-superfluid mixtures
  6. Lagrangian framework for diffeomorphism covariant theories
  7. First law of thermodynamics and thermodynamic equilibrium
  8. First law of thermodynamics
  9. Existence of desired solutions to the linearized constraints
  10. Thermodynamic equilibrium
  11. Lagrangian formulation of superconducting-superfluid mixtures: symplectic structure, phase space, and trivial displacements
  12. Lagrangian and symplectic form
  13. Speed of sound and causal behavior
  14. Phase space
  15. Trivial displacements
  16. ...and 3 more sections

Key Result

Lemma 1

Let $\delta\mathcal{N}_{\text{X}}$, $\delta\mathcal{S}$, and $\delta\mathcal{J}_{\text{X}}$ be specified arbitrary smooth, axisymmetric functions with compact support, such that $\delta\mathcal{J}_{\text{X}}/\varphi^{a}\varphi_{a}$ also is smooth. Then we can choose the remaining initial data $\left

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • Lemma 5
  • proof
  • Theorem 6
  • ...and 2 more