Relativistic thermodynamics of perfect fluids

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Relativistic thermodynamics of perfect ﬂuids

Sylvain D. Brechet, Marin C. A. Girard

Institute of Physics, Station 3, Ecole Polytechnique F´ed´erale de Lausanne - EPFL, CH-1015

Lausanne, Switzerland

Abstract

The relativistic continuity equations for the extensive thermodynamic quantities

are derived based on the divergence theorem in Minkowski space outlined by

St¨uckelberg. This covariant approach leads to a relativistic formulation of the

ﬁrst and second laws of thermodynamics. The internal energy density and the

pressure of a relativistic perfect ﬂuid carry inertia, which leads to a relativistic

coupling between heat and work. The relativistic continuity equation for the

relativistic inertia is derived. The relativistic corrections in the Euler equation

and in the continuity equations for the energy and momentum are identiﬁed.

This relativistic theoretical framework allows a rigorous derivation of the rela-

tivistic transformation laws for the temperature, the pressure and the chemical

potential based on the relativistic transformation laws for the energy density,

the entropy density, the mass density and the number density.

Contents

1 Introduction 2

2 Relativistic continuity equations 6

3 Relativistic second law of thermodynamics 14

4 Relativistic ﬁrst law of thermodynamics 16

5 Relativistic orbital angular momentum continuity equation 18

Preprint submitted to Journal of L

X Templates October 11, 2022

arXiv:2210.04282v1 [cond-mat.stat-mech] 9 Oct 2022

6 Relativistic matter continuity equation 19

7 Dynamics of a relativistic dust 21

8 Stress-energy-momentum tensor of a relativistic dust 23

9 Dynamics of a relativistic perfect ﬂuid 26

10 Stress-energy-momentum tensor of a relativistic perfect ﬂuid 27

11 Relativistic temperature 31

12 Relativistic chemical potential 34

13 Relativistic pressure 35

14 Relativistic force density 37

15 Relativistic energy continuity equation 38

16 Relativistic momentum continuity equation 40

17 Relativistic inertia continuity equation 41

18 Relativistic Euler equation 44

19 Conclusion 45

Appendix A Relativistic kinematics 49

Appendix B Relativistic dynamics 55

1. Introduction

In his seminal paper entitled “On the electrodynamics of moving bodies” [1] and

published in the “Annalen der Physik” in 1905 during the “annus mirabilis”,

Albert Einstein laid the foundation of special relativity by rooting ﬁrmly his

theory on the transformation laws for the electromagnetic ﬁelds derived inde-

pendently in 1899 by Hendrik A. Lorentz [2] and in 1900 by Henri Poincar´e. [3]

Shortly thereafter, an important eﬀort was made to establish relativistic theo-

ries for all branches of physics, notably by Albert Einstein himself [4] in 1907.

Max Planck [5] derived in 1908 the ﬁrst relativistic transformation law for the

temperature,

T=γ−1˜

T(Einstein and Planck) (1)

where ˜

Tis the temperature in the rest frame ˜

Rand Tis the temperature in an

inertial frame Rand γ > 0 is the Lorentz factor. This transformation law was

at the heart of the reigning paradigm of relativistic thermodynamics for over

50 years adopted notably by Louis de Broglie [6], Olivier C. de Beauregard [7],

Max von Laue [8], Richard C. Tolman [9] and Wolfgang Pauli [10]. Heinrich

Ott called into question this paradigm. [11] Taking into account the relativistic

transformation law for the mass, i.e. M=γ˜

M, where ˜

Mis the mass in the rest

frame ˜

Rand Mis the mass in an inertial frame R, Ott showed in 1963 that the

relativistic transformation law for the temperature becomes, [11]

T=γ˜

T(Ott and Arzelies) (2)

and Henri Arzelies reached independently the same conclusion in 1965. [12]

The diﬀerence between the relativistic transformation laws (1) and (2) for the

temperature is referred to unfortunately as the Einstein and Ott controversy.

A private letter sent by Einstein to von Laue in 1952 shows that Einstein’s

point of view clearly changed over time. [13] The transformation law for the

temperature derived in this letter is identical to the transformation law (2)

derived later by Ott. So the controversy should be renamed adequately the

Planck and Einstein controversy. A third relativistic transformation law of the

temperature was postulated by Peter T. Landsberg [14] and Nicolaas G. van

Kampen [15] who argued independently in 1967 and 1968 that the temperature

should be a Lorentz invariant quantity,

T=˜

T(Landsberg and van Kampen) (3)

All these authors agree on the frame independence of the entropy that should

be a Lorentz invariant quantity as advocated initially by Albert Einstein [4] and

Max Planck, [5]

S=˜

S(Einstein, Planck et al.) (4)

where ˜

Sis the entropy in the rest frame ˜

Rand Sis the entropy in an inertial

frame R. Since the entropy is Lorentz invariant and the temperature is the

intensive conjugate variable of the entropy, the relativistic transformation law

for the internal energy has to be the same as the relativistic transformation

law for the temperature. The relativistic transformation law for the pressure

is another essential result for a relativistic theory of thermodynamics. Albert

Einstein [4], Max Planck [5] and Arzelies [12] reached the conclusion that the

pressure is Lorentz invariant,

p= ˜p(Einstein, Planck and Arzelies) (5)

where ˜pis the pressure in the rest frame ˜

Rand pis the pressure in an inertial

frame R. These authors used the mechanical deﬁnition of the pressure as the

ratio of the normal force and the surface area. [16] Sutcliﬀe derived another

transformation law using the thermodynamic deﬁnition of the pressure as the

intensive conjugate variable of the volume in 1965, [17]

p=γ2˜p(Sutcliﬀe) (6)

Our aim is to establish a rigorous theoretical framework for the relativistic

thermodynamics of a perfect ﬂuid based on the Lorentz invariance between an

inertial frame Rand a local rest frame ˜

R. Our main objective is to settle once

for all the debate on the relativistic transformations laws of thermodynamic

quantities by using a consistent and systematic approach. Another important

purpose is to determine the structure of the relativistic continuity equations

for extensive thermodynamic scalar and vector quantities. Such an approach

is expected to lead to a clear identiﬁcation of the relativistic corrections to

the dynamics of a perfect ﬂuid. Finally, our synthetic approach allows us to

obtain elegant axiomatic statements of the relativistic ﬁrst and second law of

thermodynamics.

This publication is structured as follows : the structure of the relativistic con-

tinuity equations for scalar and vector quantities is established in Sec. 2 by

following the approach outlined by St¨uckelberg. [18] The relativistic statements

of the second and ﬁrst law are presented in Sec. 3 and 4 together with the rela-

tivistic continuity equations for the entropy and energy-momentum. In Sec. 5,

we establish the relativistic continuity equation for the orbital angular momen-

tum, which implies the symmetry of the stress-energy-momentum tensor. The

relativistic matter continuity equation is derived in Sec. 6. As a prelude to the

study of the dynamics and thermodynamics of a perfect ﬂuid, we ﬁrst examine

the dynamics of a relativistic dust in Sec. 7 and derive the corresponding stress-

energy-momentum tensor in Sec. 8. Taking into account the pressure and the

internal energy density, we generalise our analysis to the study of the dynamics

of a perfect ﬂuid in Sec. 9 and derive the corresponding stress-energy-momentum

tensor in Sec. 10. This leads to the introduction of a relativistic inertia density

that diﬀers from the mass density. Using the thermodynamic deﬁnition of the

temperature, the pressure and the chemical potential, we establish rigorously the

relativistic transformation laws for these three intensive quantities in Sec. 11, 12

and 13 respectively. The relativistic force density is established in Sec. 14. The

relativistic continuity equations for the energy and the momentum are derived

from the relativistic continuity equation for the energy-momentum in Sec. 15

and 16, and the relativistic corrections are highlighted. In Sec. 17, we derive a

relativistic continuity equation for the inertia and show how and why it diﬀers

from the continuity equation for the mass that is usually assumed to hold. We

derive the relativistic Euler equation from the relativistic continuity equations

for the momentum and the inertia in Sec. 18 and highlight the relativistic cor-

rections. Finally, we conclude our analysis of the relativistic thermodynamics

of a perfect ﬂuid in Sec. 19. In Appendices Appendix A and Appendix B, we

present the fundamental concepts of relativistic kinematics and dynamics of a

particle, which are the cornerstone of a relativistic theory.

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摘要：

RelativisticthermodynamicsofperfectuidsSylvainD.Brechet,MarinC.A.GirardInstituteofPhysics,Station3,EcolePolytechniqueFederaledeLausanne-EPFL,CH-1015Lausanne,SwitzerlandAbstractTherelativisticcontinuityequationsfortheextensivethermodynamicquantitiesarederivedbasedonthedivergencetheoreminMinkowskisp...

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Relativistic thermodynamics of perfect fluids

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