Thermodynamic equilibrium and relativity: Killing vectors and Lie derivatives

07/11/2017
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The main concepts of general relativistic thermodynamics and general relativistic statistical mechanics are reviewed in a quantum framework. The main building block of the proper relativistic extension of classical thermodynamics laws is the four-temperature vector.
The general relativistic thermodynamic equilibrium condition demands to be a Killing vector eld. A remarkable consequence of this condition is that all Lie derivatives of all physical observables along the four-temperature 
ow vanish.

Thermodynamic equilibrium and relativity: Killing vectors and Lie derivatives

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application/pdf Thermodynamic equilibrium and relativity: Killing vectors and Lie derivatives Francesco Becattini
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contenu protégé  Document accessible sous conditions - vous devez vous connecter ou vous enregistrer pour accéder à ou acquérir ce document.
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The main concepts of general relativistic thermodynamics and general relativistic statistical mechanics are reviewed in a quantum framework. The main building block of the proper relativistic extension of classical thermodynamics laws is the four-temperature vector.
The general relativistic thermodynamic equilibrium condition demands to be a Killing vector eld. A remarkable consequence of this condition is that all Lie derivatives of all physical observables along the four-temperature 
ow vanish.
Thermodynamic equilibrium and relativity: Killing vectors and Lie derivatives

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Thermodynamic equilibrium and relativity: Killing vectors and Lie derivatives F. Becattini1 University of Florence, Via G. Sansone 1, I-50019, Sesto F.no (Firenze) Italy, becattini@fi.infn.it Abstract. The main concepts of general relativistic thermodynamics and general relativistic statistical mechanics are reviewed in a quantum framework. The main building block of the proper relativistic exten- sion of classical thermodynamics laws is the four-temperature vector β. The general relativistic thermodynamic equilibrium condition demands β to be a Killing vector field. A remarkable consequence of this con- dition is that all Lie derivatives of all physical observables along the four-temperature flow vanish. Keywords: Relativistic thermodynamics, General relativity, Thermo- dynamic equilibrium 1 Introduction Relativistic thermodynamics and relativistic statistical mechanics are widespreadly used in advanced research topics: high energy astrophysics, cosmology, and rel- ativistic nuclear collisions. The standard cosmological model views the primor- dial Universe as a curved manifold with matter content at local thermodynamic equilibrium. Similarly, the matter produced in high energy nuclear collisions is assumed to reach and maintain local thermodynamic equilibrium for a large fraction of its lifetime. In this paper, we review the basic concepts of thermodynamic equilibrium in a quantum relativistic framework including general relativity. We will follow a rather informal approach leaving mathematical rigour aside for the ease of read- ing and to better illustrate the reasoning and the methods. We will see that the key role in the extension of thermodynamics to the quantum relativistic realm is played by the inverse temperature or, simply, four-temperature, vector β. This vector field has a precise physical meaning in terms of ideal thermometers, its magnitude being the inverse temperature marked by an ideal thermometer mov- ing along the flow. At global thermodynamic equilibrium, the four-temperature β must be a Killing vector and we will show that, as a consequence, the Lie derivatives of all physical quantities vanish. Notation In this paper we use the natural units, with ~ = c