The Zeroth Law of Thermodynamics in Special Relativity
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The Zeroth Law of Thermodynamics in Special Relativity L. Gavassino1 Received: 29 August 2020 / Accepted: 9 October 2020 / Published online: 5 November 2020 © The Author(s) 2020
Abstract We critically revisit the definition of thermal equilibrium, in its operational formulation, provided by standard thermodynamics. We show that it refers to experimental conditions which break the covariance of the theory at a fundamental level and that, therefore, it cannot be applied to the case of moving bodies. We propose an extension of this definition which is manifestly covariant and can be applied to the study of isolated systems in special relativity. The zeroth law of thermodynamics is, then, proven to establish an equivalence relation among bodies which have not only the same temperature, but also the same center of mass four-velocity. Keywords Thermodynamics · Special relativity
1 Introduction The modern covariant formulation of the second law of thermodynamics relies on the assumption that it is possible to define an entropy four-current s𝜈 with non-negative divergence, ∇𝜈 s𝜈 ≥ 0 [19, 20]. The validity of this assumption is strongly supported by relativistic kinetic theory, both classical [9] and quantum [10, 13], and by the hydrodynamics of locally isotropic fluids [15], including perfect fluids [42] and chemically reacting fluids [6]. This approach finds application in every branch of the relativistic hydrodynamics, including the problem of relativistic dissipation [21, 31] and multifluid hydrodynamics [1, 7, 8]. In the absence of dissipation ( ∇𝜈 s𝜈 = 0 ), it is possible to define the total entropy of a system as the flux of the entropy current through an arbitrary spacelike hypersurface 𝛴 crossing the system,
S=−
∫
s𝜈 d𝛴𝜈 .
𝛴
* L. Gavassino [email protected] 1
Polish Academy of Sciences: Polska Akademia Nauk, Warsaw, Poland
1Vol:.(1234567890) 3
(1)
Foundations of Physics (2020) 50:1554–1586
1555
Since the result of the integral does not depend on the choice of 𝛴 , the entropy is a Lorentz scalar, in agreement with [41] and with the microscopic statistical interpretations of the entropy [18, 22, 23, 40, 45]. This enables us to define, starting from a theory which is necessarily local (to ensure causality), the equilibrium thermodynamics of an isolated macroscopic body, which allows to make a contact with statistical mechanics [17]. The long-lasting debate on the definition of the temperature of moving bodies, traditionally called Planck–Ott imbroglio [32], originates at this point. At first [28, 35, 41, 44], the discussion was oriented in the direction of defining a transformation law of the temperature under Lorentz boosts and work in this direction is still ongoing (see [12] for a recent review). Supporters of both Planck’s and Ott’s views agree on the fact that the transformation should involve a Lorentz factor, however the opinions diverge on its position (at the denominator according to Planck [41], at the numerator according to Ott [35]). Nowadays this approach has become a quest for
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