Thermodynamic Equilibrium in General Relativity

Abstract

The thermodynamic equilibrium condition for a static self-gravitating fluid in the Einstein theory is defined by the Tolman-Ehrenfest temperature law, $T\sqrt{g_{00}(x^i)}=constant$, according to which the proper temperature depends explicitly on the position within the medium through the metric coefficient $g_{00}(x^i)$. By assuming the validity of Tolman-Ehrenfest "pocket temperature", Klein also proved a similar relation for the chemical potential, namely, $\mu \sqrt{g_{00}(x^i)}=constant$. In this letter we prove that a more general relation uniting both quantities holds regardless of the equation of state satisfied by the medium, and that the original Tolman-Ehrenfest law form is valid only if the chemical potential vanishes identically. In the general case of equilibrium, the temperature and the chemical potential are intertwined in such a way that only a definite (position dependent) relation uniting both quantities is obeyed. As an illustration of these results, the temperature expressions for an isothermal gas (finite spherical distribution) and a neutron star are also determined.

arXiv comment7 pages, 2 figures, PRD accepted

References in corpus (14)

• Neutron Star Structure and the Equation of State
• Composition and Structure of Protoneutron Stars
• Constraining neutron star matter with QCD
• The outer crust of non-accreting cold neutron stars
• Thermal Relaxation in Young Neutron Stars
• Thermal time and the Tolman-Ehrenfest effect: temperature as the "speed of time"
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• Temperature Evolution Law of Imperfect Relativistic Fluids
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• Relativistic Gravothermal Instabilities
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• Heat conduction in general relativity
• Tolman-Ehrenfest-Klein Law in non-Riemannian geometries
• Tolman-Ehrenfest's criterion of thermal equilibrium extended to conformally static spacetimes
• On thermodynamics of compact objects
• Determining the minimal mass of a proto-neutron star with chirally constrained nuclear equations of state
• Gravito-thermal transports, Onsager reciprocal relation and gravitational Wiedemann-Franz law
• How the Schwarzschild-de Sitter horizons remain in thermal equilibrium at vastly different temperatures
• Josephson's effect in the Schwarzschild background
• Temperature upper bound of an ideal gas
• Linear response in a charged gas in curved spacetime and covariant heat equation
• Thermodynamic parameters of fluids on conformally connected spacetimes