The construction of a relativistic thermodynamics theory is still controversial after more than 110 years. To the date there is no agreement on which set of relativistic transformations of thermodynamic quantities is the correct one, or if the problem even has a solution. Starting from Planck and Einstein, several authors have proposed their own reasoning, concluding that a moving body could appear cooler, hotter or at the same temperature as measured by a local observer. In this article we present a review of the main theories of relativistic thermodynamics, with an special emphasis on the physical assumptions adopted by each one. We also present a set of relativistic transformations that we have derived by assuming the laws of Thermodynamics to be covariant. We found that under such assumptions a moving body appears to be hotter. Since relativistic thermodynamics is a topic that can be treated as part of an undergraduate course of classical thermodynamics or modern physics, the review and our own derivations presented here aim to encourage undergraduate physics students to open a discussion on the fundamental assumptions in thermodynamics and to engage in research activities early in their scientific career.

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What is the temperature of a moving body

Calcul des relations de dispersion dans un systeme faisceau plasma dans le cas ou les temperatures du faisceau et du plasma, et la vitesse du faisceau sont relativistes. Quand il n'y a aucune contribution des particules resonnantes a la dispersion des modes, il apparait un nouveau terme dans la relation de dispersion, plus important que celui de Bohm-Gross. Quand il y a contribution des particules resonnantes, les corrections relativistes dues a la temperature deplacent la vitesse de phase vers zero

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022377800015117

Plasma waves in hot relativistic beam-plasma systems. Part 1. Dispersion relations

The isothermal compression of an ideal gas is analysed using a relativistic thermodynamics formalism based on the principle of inertia of energy (Einstein's equation) and the asynchronous formulation (Cavalleri and Salgarelli 1969 Nuovo Cimento 42 722–54), which is similar to the formalism developed by van Kampen (1968 Phys. Rev. 173 295–301) and Hamity (1969 Phys. Rev. 187 1745–52). In this 4-vector Minkowski formalism mechanical and thermodynamical processes are described by the first law of thermodynamics expressed as ΔUμ = Wμ + Qμ, in a Lorentz covariant way. This exercise is considered useful for undergraduate physics students interested in foundations of physics, with the only prerequisites in first courses in thermodynamics and special relativity.

An undergraduate exercise in the first law of relativistic thermodynamics

Classical optical experiments that confirm the validity of special relativity are considered. Transformations of spatial coordinates and time that were proposed at different times for the passage from one inertial reference frame (IRF) to another and that differ from the classical Lorentz transformations are critically analyzed. It is shown that, although some of these transformations are capable of explaining the results of single classical optical experiments, in particular, the Michelson-Morley experiments, neither of them, except for the Tangherlini transformations, can explain the results of the entire set of these experiments. The discrepancy between the predictions of incorrect transformations and the results of the well-known experiments is caused by the absence of a clearly formulated procedure for synchronizing spaced clocks in a rest IRF (where the observer is located) and a moving IRF, which should be consistent with the transformation of time. A number of relativistic and quantum effects are indicated, which have been predicted but not yet detected, to a search for which efforts of physicists are directed, and which are convenient to describe with the help of the formalism of the Tangherlini transformations.

Classical optical experiments and special relativity: A review

It is shown that there are no energy currents perpendicular to the forces on an object. Many reasons for the erroneous nature of the von Laue energy current are given. This energy current is of no help in formulating equilibrium conditions in the conventional synchronous formulation, where additionalad hoc momenta due to the nonconservation of simultaneity have to be added. Equilibrium maintained by continuous time-dependent forces cannot be defined by the simultaneous vanishing of the forces. On the other hand, in the asynchronous formulation of Cavalleri and Salgarelli, equilibriumis given in terms of only the forces and torques. The relevant torque tensor ℐΜΝ is defined. This tensor is the sum of quantities evaluated at different times, corresponding to the Lorentz transformation of simultaneous measurements in the proper inertial frameS. Consequently, if ℐΜΝ= 0, then ℐΜΝ1, the torque in another inertial frameS′, also vanishes. Numerous objections are fully discussed. It is shown that there is no paradox in the Trouton-Noble experiment. Finally, it is shown why it is impossible to formulate the usual Hamiltonian for interacting particles in special relativity in terms of the co-ordinate timet; instead, one must use individual proper timesΤ.

Equilibrium in special relativity.

For point bodies with variable rest mass it is shown that the definition of forcem
0dU
 i
 /dτ=F
i is better than the usual d(m
0
U
i))/dτ=F
i and the explicit equation of motion is given bym
0dU
i/dτ=F

 ext
 +(z
−1
U

 i
 B
 −U
i)dm
0/dτ, where the external forceF

 ext
 has been separated from the reaction force due to the ejected mass,U

 i
 B
 is the four-velocity of the mass centre of the ejected mass andz=U

 s
 B
 U2c−2. For extended bodies it is shown that the von Laue mechanical-energy flux (in the usual «synchronous» formulation) is not plausible. Moreover a new formulation, called «asynchronous» is given. By this formulation many problems receive an immediate solution. The resultant «asynchronous» force density is always orthogonal to the four-velocity, and the equations of motion for continuous media turn out to be formally equal to the classical equations. Expanding the new equations to the first order when the pressure is a regular function of space and time, one obtains the usual «synchronous» equations. In the asynchronous formulation, all quantities relevant to extended bodies transform in a covariant way. In particular, momentum and energy turn out to have the expression proposed by Rohrlich. The above concepts are applied to relativistic thermodynamics. It is shown that, for point bodies, heat transforms as in the recent Ott formulation. For extended bodies, in the usual «synchronous» formulation, heat transforms as pointed out by von Laue. In the «asynchronous» formulation we always haveQ=γQ
0. TemperatureT is considered invariant and the relation dQ=TdS valid in the rest system only.

G. Salgarelli

Papers

Revision of the relativistic dynamics with variable rest mass and application to relativistic thermodynamics