On Extended Thermodynamics: From Classical to the Relativistic Regime

TL;DR

This paper reviews recent advances in the thermodynamics of relativistic continuous media, emphasizing observational breakthroughs and their implications for the theory of relativistic dissipative fluids and irreversible thermodynamics.

Contribution

It provides a comprehensive summary of the progress in non-equilibrium thermodynamics of relativistic media over recent decades.

Findings

Relativistic dissipative fluids are crucial in astrophysics and subnuclear physics.

Recent observational data motivate the development of relativistic thermodynamics.

The paper connects classical thermodynamics with modern relativistic theories.

Abstract

The recent monumental detection of gravitational waves by LIGO, the subsequent detection by the LIGO/VIRGO observatories of a binary neutron star merger seen in the gravitational wave signal $GW170817$,the first photo of the event horizon of the supermassive black hole at the center of the $M87$ galaxy released by the EHT telescope and the ongoing experiments on Relativistic Heavy Ion Collisions at the BNL and at the CERN, demonstrate that we are witnessing the second golden era of observational relativistic gravity. These new observational breakthroughs, although in the long run would influence our views regarding this Kosmos, in the short run, they suggest that relativistic dissipative fluids (or magnetofluids) and relativistic continuous media play an important role on astrophysical-and also subnuclear-scales. This realization brings into the frontiers of current research theories of irreversible thermodynamics of relativistic continuous media. Motivated by these considerations, in this paper, we summarize the progress that has been made in the last few decades in the field of non equilibrium thermodynamics of relativistic continuous media. For coherence and completeness purposes, we begin with a brief description of the balance laws for classical (Newtonian) continuous media and introduce the classical irreversible thermodynamics (CIT) and the role of the local-equilibrium postulate within this theory. Tangentially, we touch the program of rational thermodynamics (RT), the Clausius-Duhem inequality, the theory of constitutive relations and the emergence of the entropy principle and its role in the description of continuous media.

Equations

$$u^{\mu }\mapsto {\hat{u}}^{\mu }=u^{\mu }+{\delta }^{\mu },{\delta }^{\mu }\le O_1.$$

$$Q^{\mu }(u)=\frac{1}{2}{u}^{\mu }[{\beta }_0{\pi }^2+{\beta }_1{q}^{\nu }{q}_{\nu }+{\beta }_2{\pi }^{\lambda \nu }{\pi }_{\lambda \nu }]-{\alpha }_0\pi q^{\mu }-{\alpha }_1{\pi }^{\mu \nu }{q}_{\nu }+R^{\mu }(u),$$

$${\varphi }_t:U\to {\varphi }_t(U):=U_t\subset R^3:\vec{X}\to {\varphi }_t(\vec{X}):=\varphi (t,\vec{X}),$$

$$\vec{V}(\vec{X},t)=\frac{\partial \varphi (\vec{X},t)}{\partial t},\vec{g}(\vec{X},t)=\frac{{\partial }^2\varphi (\vec{X},t)}{\partial t^2}=\frac{\partial \vec{V}(\vec{X},t)}{\partial t}.$$

$$\vec{u}(\vec{x},t)=\vec{V}(\vec{X},t)\Big{|}_{\vec{X}=\phi^{-1}(t,x)},$$

$$\frac{\partial \vec{V}(\vec{X},t)}{\partial t}=\frac{\partial \vec{u}(\vec{x},t)}{\partial t}+\vec{u}\cdot \nabla \vec{u},$$

$$\vec{g}(\vec{X},t)=\frac{\partial \vec{u}(\vec{x},t)}{\partial t}+\vec{u}\cdot \nabla \vec{u}.$$

$$q(\vec{x},t)=Q(\vec{X},t)\Big{|}_{X=\phi^{-1}(t,x)},$$

$$\frac{\partial Q(\vec{X},t)}{\partial t}=\frac{Dq}{dt}:=\frac{\partial q(\vec{x},t)}{\partial t}+\vec{u}\cdot \nabla q(\vec{x},t),$$

$$\frac{D}{dt}=\frac{\partial }{\partial t}+\vec{u}\cdot \nabla ,$$

$$\dot{f}:=\frac{Df}{dt}=\frac{\partial f}{\partial t}+\vec{u}\cdot \nabla f.$$

$$h(\vec{x},t,\vec{n})=\vec{q}(\vec{x},t)\cdot \vec{n}.$$

$$\frac{d}{dt}{\int }_{{\varphi }_t(\hat{U})}R(\vec{x},t)dV(t)={\int }_{{\varphi }_t(\hat{U})}\left[\frac{\partial R}{\partial t}+\nabla \cdot (R\vec{u})\right]dV(t).$$

$$\frac{d}{dt}{\int }_{{\varphi }_t(\hat{U})}\rho (\vec{x},t)dV(t)=0.$$

$$\frac{d}{dt}{\int }_{{\varphi }_t(\hat{U})}\vec{x}\times \rho \vec{u}d\vec{x}={\int }_{{\partial {\varphi }_t(\hat{U})}_t}\vec{x}\times \vec{t}(\vec{x},t,\vec{n})da+{\int }_{{\varphi }_t(\hat{U})}\vec{x}\times \rho \vec{b}(\vec{x},t)d\vec{x}$$

$$\frac{d}{dt}{\int }_{{\varphi }_t(\hat{U})}\rho (\vec{x},t)\vec{u}(\vec{x},t)d\vec{x}={\int }_{\partial {\varphi }_t(\hat{U})}\vec{t}(\vec{x},t,\vec{n})da+{\int }_{{\varphi }_t(\hat{U})}\rho (\vec{x},t)\vec{b}(\vec{x},t)d\vec{x},$$

$$t^a(\vec{x},t,\vec{n})={\sigma }^{ab}(\vec{x},t)g_{bc}{n}^c={\sigma }^a{}_c{n}^c,a,b,c\in (1,2,3),$$

$$\rho \dot{\vec{u}}(\vec{x},t)=\nabla \cdot \sigma +\rho \vec{b},(\nabla \cdot \sigma {)}^i=\frac{\partial {\sigma }^{ij}}{\partial x^j},i,j\in (1,2,3).$$

$$\rho \left(\frac{\partial u^i}{\partial t}+u^j\frac{\partial u^i}{\partial x^j}\right)=\frac{\partial {\sigma }^{ij}}{\partial x^j}+\rho b^i,i,j\in (1,2,3),$$

$$T(t)={\int }_{{\varphi }_t(\hat{U})}\frac{u^2}{2}dm,U(t)={\int }_{{\varphi }_t(\hat{U})}edm,dm=\rho dV(t),$$

$$\frac{d}{dt}{\int }_{{\varphi }_t(\hat{U})}\left(\frac{u^2}{2}+e\right)dm={\int }_{{\varphi }_t(\hat{U})}(\vec{u}\cdot \vec{b}+Q)dm+{\int }_{\partial {\varphi }_t(\hat{U})}(\vec{u}\cdot \vec{t}+h)da.$$

$$\rho \frac{D}{dt}\left(\frac{u^2}{2}+e\right)=\rho (\vec{u}\cdot \vec{b}+Q)+\nabla \cdot (\sigma \cdot \vec{u}-\vec{q}),\nabla \cdot (\sigma \cdot \vec{u}):=\frac{\partial ({\sigma }^i{}_j{u}^j)}{\partial x^i}.$$

$$\rho \frac{De}{dt}=\rho Q-\nabla \cdot \vec{q}+\sigma :\nabla \vec{u},\sigma :\nabla \vec{u}={\sigma }^{ij}{u}_{i,j}$$

$${\sigma }^{ij}(\vec{x},t)=-P(\vec{x},t)g^{ij},i,j\in (1,2,3).$$

$$\rho \dot{\vec{u}}(\vec{x},t)=-\nabla P(\vec{x},t)+\rho \vec{b}(\vec{x},t),$$

$${\sigma }^{ij}=-P{g}^{ij}-{\hat{\sigma }}^{ij},{\hat{\sigma }}^{ij}=P^v{g}^{ij}+{\sigma }_{(v)}{}^{ij},{\sigma }_{(v)}{}^{ij}{g}_{ij}=0,$$

$$\rho \dot{e}=\rho Q-{\nabla }_{\mu }{q}^{\mu }-(P+P^v){\nabla }_{\mu }{u}^{\mu }-{\sigma }_{(v)}^{ij}{u}_{(t)ij},$$

$$u_{(i,j)}=\left(\frac{1}{3}{\nabla }^a{u}_a\right)g_{ij}+\frac{1}{2}\left(u_{i,j}+u_{j,i}-\left[\frac{2}{3}({\nabla }^a{u}_a)g_{ij}\right]\right)=\left(\frac{1}{3}{\nabla }^a{u}_a\right)g_{ij}+u_{(t)}{}_{ij}.$$

$$\rho \dot{v}={\nabla }_{\mu }{u}^{\mu },$$

$$\frac{d}{dt}{\int }_{{\varphi }_t(\hat{U})}s(\vec{x},t)dm,$$

Full text

Motivated by these considerations, in this paper, we summarize the progress that has been made in the last few decades in the field of non equilibrium thermodynamics of relativistic continuous media. For coherence and completeness purposes, we begin with a brief description of the balance laws for classical (Newtonian) continuous media and introduce the classical irreversible thermodynamics (CIT) and the role of the local-equilibrium postulate within this theory. Tangentially, we touch the program of rational thermodynamics (RT), the Clausius-Duhem inequality, the theory of constitutive relations and the emergence of the entropy principle and its role in the description of continuous media. We discuss at some length, the theories of non equilibrium thermodynamics that sprang out of a fundamental paper written by Müller in $1967$, with emphasis on the principles of extended irreversible thermodynamics (EIT) and the field of rational extended irreversible thermodynamics (REIT).

Subsequently, after a brief introduction to relativistic fluids, we discuss the Israel-Stewart transient (or causal) thermodynamics and discuss its main features. Moreover, we introduce the Liu-Müller-Ruggeri theory of irreversible thermodynamics of relativistic fluids and analyze and compare this theory to the transient thermodynamics. We also discuss the class of dissipative relativistic fluids of divergence type developed in the late $1990$ by Pennisi, Geroch and Lindblom and the emergence of symmetric-hyperbolic (or causal) system of dynamical equations in the description of the such fluid states.

As we shall see in this review, the current efforts aiming to develop viable theories of irreversible thermodynamics of continuous media is focused on theories whose dynamical equations constitute a symmetric-hyperbolic and preferably causal set of dynamical equations. By design, this class of theories eliminates propagation of disturbances with unbounded speed, a necessary condition for the viability of the underlying theory.

Although it is fair to state that substantial progress has been made in the field of non equilibrium thermodynamics of classical media and many predictions of the extended theories (in the form of (EIT) or (REIT)) have been placed under experimental scrutiny, at the relativistic level the situation is different. Even though the time spent aiming to the development of a sensible theory (or theories) of non equilibrium thermodynamics of relativistic fluids (or continuous media) is relatively short, enormous steps in the right direction have been taken. Still however, as we shall see in this review, a successful theory of relativistic dissipation is lacking.

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PACS numbers:

1 Introduction

The date $14th$ of September of $2015$ will remain in the annals of gravitational physics as a landmark date. At that date, the upgraded Laser Interferometer Gravitational Wave Observatory (LIGO) captured gravitational waves from two black holes circling each other, the first detection of gravitational waves ever recorded in the human history. The event announced in a press conference on $11th$ of February in $2016$ and the reader is referred to the web page of the LIGO observatory [1] for a thorough analysis of this detection.

Two years later, specifically on $17th$ of August $2017$, the LIGO/VIRGO gravitational wave observatory network recorded a gravitational wave signal, refereed as GW170817, which is consistent with a binary neutron star inspiral and merger. For an analysis of this signal, its interpretation and consequences the reader is refereed to Refs. \refciteNS1,NS2.

On the $10th$ of April $2019$, the Event Horizon Telescope (EHT) collaboration, released the first ”photograph of the event horizon” of the supermassive black hole at the center of the $M87$ galaxy (for details, visit Ref. \refciteEHT). This photograph establishes the existence of an accretion disk around the black hole and here is worth recalling that modeling of such a disk presuppose viscous accreting matter.

In another exciting development, ongoing experiments at the Relativistic Heavy Ion Collider at BNL and at Large Hadron Collider at CERN, show that in the relativistic heavy ion collision the quark-gluon plasma formed in these terrestrial “mini bing-bangs” is reliably described by a viscous relativistic fluid. For evidences supporting this unexpected connection the reader is referred to recent reviews in Refs. \refciteDKK,FHS,RR_1.

These latest observational breakthroughs would have far reaching consequences upon our understanding of this Kosmos. At the fundamental level, once again, we reassured that Einstein’s General Theory of Relativity, in the words of Penrose is a superb theory111We borrow the term ”superb theory” from the classification of fundamental theories proposed by Penrose in Ref. \refcitePen1, (see discussion on pages $(225,261)$ of that ref.). where its minute predictions one after the other are observationally confirmed. Moreover, in an era where the LIGO/VIRGO observatories are operational and in the near future the advanced KAGRA observatory is expected to be operational, accretion of matter into black holes and other compact objects, plasmas in the early universe, supernovae explosions and core collapse, are going to be placed under observational scrutiny and in these scenarios dissipative fluids or more generally relativistic continuous media play an important role. In this connection, it is worth mentioning that the current modeling of neutron stars suggest that viscosity and thermal conductivity play an important role in their stability while results from relativistic heavy ion collisions suggest that viscosity is also relevant at a subnuclear domain.

These latest developments challenges relativists and high energy physicists alike to develop reliable theories of irreversible thermodynamics of relativistic continuous media in order to confront the new observational realities.

Interestingly, for relativistic fluids, the conventional first order theories of irreversible relativistic thermodynamics of Eckart [9] and Landau-Lifshitz [10], predict instantaneous propagation of thermal and viscous effects which to use the words of Israel and Stewart [11] “is an offense to the intuition, which expects propagation at about the mean molecular speed; in a consistent relativistic theory it ought to be completely prohibited”. Beyond this deficiency, these first order theories, suffer from other drawbacks: in Ref.\refciteHis2 it was proved that small-amplitude disturbances around equilibrium states diverge exponentially on a very short time scale. These features of the conventional first order theories has been a source of concern and many researchers are skeptical whether these theories could model accurately process characterized by rapid spatial and temporal variations of heat fluxes and viscous stresses222A referee kindly pointed out to us that actually the situation is even worst: They are (i.e. first order theories) ill-posed. The exponential modes grow without bound as the frequency increases. Thus, any high frequency perturbation, no matter how small we take it, can become as big as we want at any finite time just by increasing its frequency. Thus, these theories, do not have predictive power. We thank the referee for this comment.. Fortunately however, in the last few decades, there has been a progress into the field of non equilibrium thermodynamics of continuous media both at the classical and the general relativistic domain. Theories of extended irreversible thermodynamics have been developed which at least for states near equilibrium, lead to symmetric-hyperbolic system of equations predicting causal propagation of thermal and stress disturbances (see for instance Refs. \refciteIsr2,Isr1,His1,Mul1,Mul2,JVL,Mul4,Mul6,Ger1,Ger2,Ger3). Motivated by these developments, this article introduces theories of extended irreversible thermodynamics of continuous media (mostly Newtonian or relativistic fluids). Our emphasis is on the structure of theories refereed as “hyperbolic theories” and in order to provide a self contained introduction to their origin and the necessity of introducing these theories, we begin by first offering a brief account of the historical development of the subject.

Thermodynamics is an empirical science which has been developed after persistent studies of the behavior of matter under external stimuli and these studies culminated in the formulation of the four laws of thermodynamics (for an enjoyable reading of this historical development see Ref. \refciteMul3). Gradually, and beginning with the work of Onsager [28, 29], states near equilibrium began to be incorporated into the field and this lead to the development of the Classical Irreversible Thermodynamics abbreviated here after as: (CIT). The cornerstone that underlies this theory is the local thermodynamical equilibrium (often abbreviated as (LTE)) postulate which (when suitably applied) assigns an entropy to non equilibrium states and moreover implies that its evolution is deduced from the Gibbs relation combined with the balance laws. For a simple, heat conducting, viscous fluid, (CIT) leads to the standard Fourier-Navier-Stokes theory and this success led to considerable amount of scientific work. Moreover, (CIT) had some experimental confirmation and the reader is refereed to Refs.\refciteJVL,Mei,Gro,Gya,Gla for an overview regarding these confirmations.

Despite these successes, (CIT) suffers from some serious drawbacks. The (LTE) postulate implies that the entropy of non equilibrium states depends upon the same variables as the entropy of equilibrium states. It is conceivable however, that other variables may influence the thermodynamical behavior of non equilibrium states. Moreover, (CIT) predicts parabolic set of dynamical equations which yield infinite speed of propagation of thermal and viscous signals. Although these predictions are not really in contradiction with the classical (Newtonian) framework, as we have already mentioned, they are in contradiction with the spirit of the relativistic framework (and our intuition). The problem of the infinite propagation of thermal signals remained open until $1967$, when Müller in an influential paper [17] has shown that the paradox of Fourier’s heat conduction (propagation of temperature disturbance with infinity velocity) is a consequence of an insufficient description of the off-equilibrium thermodynamical state. For a simple fluid, and for states near equilibrium, he proposed a generalized entropy that gets quadratic contributions from the heat flux and stresses, a hypothesis that is in a glaring contradiction to the spirit of the (LTE) postulate as applied within the (CIT). Via this bold hypothesis, Müller arrived at a theory which removes the paradox of infinite propagation of heat conduction at least for fluids with an appropriate equation of state.

Müller’s hypothesis, received well within the scientific community and his thesis extended in a variety of ways. One popular extension lead to the development of a theory of extended irreversible thermodynamics of classical continuous media, designated here after by the acronym: (EIT). This theory assigns a generalized entropy to arbitrary non equilibrium states which depends upon dissipative fluxes (i.e. quantities that appear in the balance laws such as Cauchy’s stress, heat flux, etc.). This hypothesis, combined with a generalized Gibbs relation and the imposition of the second law leads to phenomenological equations that describe the temporal and spatial variations of the fluxes and these equations generalize the Maxwell-Cattaneo[34, 35] system. The theory leads to a successful treatment of heat transport in micro and nano systems, shock structure of waves propagating on hydrodynamical systems, etc. and for an overview of successes (and failures) of (EIT), the reader is refereed to the latest edition of the book[19], the $1984$ Barcelona conference proceedings [36], and in Refs. \refciteGC1,GC2,Sie1.

Müller’s original hypothesis even though supported by the kinetic theory of dilute gases, it was not entirely free of mathematical problems. For a heat conducting, viscous fluid the resulting system of equations is generally non-hyperbolic and the characteristic speeds of disturbances are real provided some of the thermodynamical variables are suitably bounded. This property is however unsatisfactory since even if these bounds hold by the initial data, they may fail to the future of the initial surface (for a constructive criticism of Müller’s original approach consult Ref. \refciteRug1). The quest for mathematical rigor lead Liu, Müller, Ruggeri and collaborators to develop of a new theory, referred as Rational Extended Irreversible Thermodynamics -abbreviated here after as (REIT)-which is characterized by the following properties:

a) the dynamical equations of the theory are of balance type,

b) the set of constitutive equations are local in space and time,

c) the dynamical equations constitute a symmetric-hyperbolic system of field equations.

The tractable features of (REIT) is the restoration of finite propagation of the heat and stress disturbances, and the symmetric-hyperbolic nature of the underlying field equations (for an introduction to this theory consult Refs. \refciteMul4, \refciteMul5, \refciteMulW).

The symmetric-hyperbolic nature of the underlying field equations of (REIT), and the ensuing prediction of the finite propagation of the heat and stress disturbances made relativists dreaming for an extension of (REIT) into the relativistic regime. However, formulating a theory describing the irreversible thermodynamics of relativistic continuous media and obeying requirements $a)-c)$ it is not an easy task.

Besides the mathematical difficulties of extending conditions a) to c) to the relativistic regime, one confronts the problem related to the absence of preferable rest frames333In more simple terms this problem is associated with the inability within theories describing the thermodynamics of relativistic fluids to uniquely single out the fluid’s four velocity a notion that is the landmark of the non relativistic fluid dynamics. We will discuss that issue further ahead (see coment $e)$ on page $(27)$). for off-equilibrium states444We are assuming that within each theory, one would be able to define the class of equilibrium states. . In general, off-equilibrium states of fluids, or of elastic media, may single out more than one preferable rest frames. The Eckart frame, the Landau-Lifshitz frame, the entropy frame555The energy frame and particle frame will be defined further ahead. The entropy frame is introduced by Carter in Ref. \refciteCar0. etc., are examples of such frames and in general it is not clear which one, if any of them should be employed to express the thermodynamical laws666One may argue for the development of a theory (or theories) that is (are) ”frame-independent”. In fact, as we shall see further ahead, the formulation of the Liu-Müller-Ruggeri theory, the class of relativistic fluid theories of divergence type, etc do not require a rest frame (or the fluid’s four velocity) for their formulation. However, at some stage, their interpretation becomes accessible to intuition whenever some rest frame (or four velocity) in invoked. All so far experience of the Fourier-Navier-Stokes fluids, where gradients of the velocity field enter into the description of stresses, shows that frames add to our intuition, and this trend also holds at the relativistic regime.. This subtle issue, has been settled by Israel who for long period of time has been stressing that as long as considerations are restricted to states near equilibrium, a theory of small deviations from equilibrium can be constructed which is manifestly invariant under first order changes of the rest frame777 This important property has been pointed out long ago in a M.Sc thesis written by Aitken [41], then student of Israel.. For a simple fluid, it is shown, in the Appendix A, that on the tangent space of any event within the region occupied by the fluid, an invariant “cone” of opening pseudo-angle888The magnitude of the three velocity $\vec{v}$ that appears in the definition of $\epsilon=\frac{v}{c}=O_{1}$, stands for the relative velocity of the Eckart frame relative to the Landau frame (the symbols $O_{1},O_{2}....$ will be introduced further ahead). $\epsilon=\frac{v}{c}$ can be defined that has the following property: Any four velocity $u$ that falls within this ”cone” can be used as a potential rest frame (or as a potential four velocity of the fluid) so that observers at rest relative to this frame deduce thermodynamical variables for the non equilibrium state under consideration. Even though, these thermodynamical variables are frame dependent and despite the plurality of such rest frames, as long as $\epsilon=\frac{v}{c}=O_{1}$ a consistent thermodynamical theory can be developed that is manifestly invariant under first-order changes of the rest-frame $u$, i.e. change999 Under such transformations, many thermodynamical variables transform in a well defined manner and these transformation laws are discussed in Appendix D. in the rest frame described by

[TABLE]

Motivated by these considerations and results from relativistic kinetic theory of gases, Israel in Ref.[13] proposed that for states near equilibrium of a simple fluid, there exist a linear relationship between the primary variables consisting of the energy momentum tensor $T^{\mu\nu}$ the particle current $J^{\mu}$ and the entropy four vector $S^{\mu}$ which however is modified by a term $Q^{\mu}$ quadratic in the deviations $\delta J^{\mu},\delta T^{\mu\nu}$ from the state of (local) thermodynamical equilibrium. Based on this hypothesis, and independently of Müller’s considerations, he formulated in $1976$ a theory of irreversible thermodynamics for fluid states near equilibrium which elaborated further in Israel and Stewart [11] and nowadays is known as transient (or causal) thermodynamics (of relativistic fluids). In Ref. \refciteIsr2, the fundamental relation between the primary variables and the $Q^{\mu}$ term has been obtained in an elegant manner based on the covariant form of the Gibbs relation and by invoking “the release of variation assumption”, a term that will be introduced in section $9$. For a simple fluid, and within the hydrodynamical approximation advocated in [11] the $Q^{\mu}(u)$ term has the form:

[TABLE]

where $\beta_{i}$, $i\in(0,1,2)$, $\alpha_{j}$, $j\in(0,1)$ are undetermined functions, $q^{\mu}$ is a frame invariant heat flux101010This quantity is defined precisely further ahead (see eq. (119)). It is frame invariant in the sense that under a change of the velocity of the frame changes by an $O_{2}$ term., $(\pi,\pi^{\mu\nu})$ are stresses deduced by an observer with four velocity $u$ and $R^{\mu}(u)$ is a term whose specific nature will be discussed further ahead. The phenomenological laws resulting from the above choice of $Q^{\mu}(u)$ will be derived in section $(12)$ and as we shall see the functions $(\alpha_{j}$,$\beta_{i})$ are left unspecified. By appealing to Boltzmann equation for a relativistic gas, and within the Grad’s approximation [42], Israel and Stewart[11], have evaluated the coefficients $(\alpha_{j},\beta_{i})$ and have shown that their theory, at least for these values, predicts that the characteristic velocities of perturbations about equilibrium states are finite and sub-luminal i.e, they are real and less than the speed of light. The same conclusion has been also reached by Hiscock and Lindblom in [14] who have shown within the framework of transient thermodynamics a close connection between causality and stability of equilibrium states.

The results quoted above, show that transient thermodynamics at least for the values of the coefficients $(\alpha_{i},\beta_{i})$ predicted by the relativistic Boltzmann equation eliminates deficiencies that plagues the theories of Eckart and Landau-Lifshitz (or more generally the class of first order theories). Therefore transient thermodynamics, at least when applied to fluid states near equilibrium, seems to be a tractable theory with encouraging properties. However, the fact that the resulting phenomenological equations contain unspecified functions and the absence of a rigorous proof demonstrating that the dynamical equations constitute a symmetric-hyperbolic system, lead to the development of alternative theories. The common feature of these alternative relativistic fluid theories is the symmetric-hyperbolic nature of their dynamical equations whose evolution respects causality.

One such theory, describing states of relativistic fluids, proposed by Liu, Müller and Ruggeri in Ref. \refciteMul6, and this theory extends the principles of Rational Extended Irreversible Thermodynamics (REIT) to the relativistic regime. The Liu-Müller-Ruggeri theory uses the particle current $J^{\mu}$ and the conserved symmetric stress tensor $T^{\mu\nu}$ as the basic dynamical variables, but the dynamical equations $\nabla_{\mu}J^{\mu}=\nabla_{\mu}T^{\mu\nu}=0$ are enlarged by an additional set of equations. These additional equations contain the divergence of a third rank totally symmetric tensor field $A^{\mu\nu\lambda}$ and a second rank traceless tensor $I^{\nu\lambda}$ refereed as the dissipation tensor. Liu, Müller and Ruggeri[22] assumed that the tensors $A^{\mu\nu\lambda}$, $I^{\nu\lambda}$ as well as an entropy vector $S^{\mu}$ are constitutive functions i.e. are functions of the dynamical variables $J^{\mu}$ and $T^{\mu\nu}$. The functional form of these constitutive relations are restricted by appealing to the following three fundamental principles:

a) Entropy principle,

b) Relativity principle,

c) Symmetric-hyperbolic nature of the dynamical equations.

In section 12, we shall discuss the implications of these principles and shall see that for states near equilibrium, the theory leads to phenomenological eqs. for the heat flux and shear stresses that contain three unknown functions.

The approach of Liu-Müller-Ruggeri has been extended further by Pennisi [23] and by Geroch and Lindblom [24], and these extensions lead to the development of a class of relativistic fluid theories having the mathematically tractable property that their dynamical equations are determined from the knowledge of a scalar generating function depending however upon $14$ suitably defined variables and from the knowledge of the traceless dissipation tensor $I^{\nu\lambda}$. The advantage of this class of theories lies in the flexible structure they possesses. It has been demonstrated that there exist families of generating functions that lead to causal dissipative fluid theories and in section 13, we shall discuss these dissipative fluid theories in a more details.

This review is organized as follows: In sections (2-4), we remind the reader of a few properties of classical (Newtonian) continuous media with emphasis on the structure of the balance laws for mass, linear momentum and total energy. Since these laws in general, fail to yield a closed system of equations, we provide a brief introduction to the theory of constitutive relations and point out the importance of the second law of thermodynamics in the description of continuous media. In particularly, within the the program of Rational Thermodynamics (RT), we discuss the implementation of the second law via the Clausius-Duhem inequality and thus set the scene for the introduction of the entropy principle and its role as a selection rule for specifying appropriate constitutive relations.

In section 5, we introduce (CIT) i.e. the irreversible thermodynamics of classical continuous media and discuss the local-equilibrium hypothesis and the implementation of the second law within this theory. In section 6, we introduce theories of extended irreversible thermodynamics and define in particularly two classes of such theories: the first one is the Extended Irreversible Thermodynamics refereed as (EIT) and the second one as Rational Extended Irreversible thermodynamics indicated as (REIT). Section 7 discusses continuous media within the relativistic regime and introduces primary and auxiliary variables describing arbitrary (non equilibrium) states of such media. From section 8 onward, we restrict attention mostly to the description of relativistic fluids. As a precursor to the development of transient thermodynamics, we identify a class of fluid states in global (or local) thermodynamical equilibrium and derive the covariant form of the Gibbs relation for this class of states. In section 9, we introduce the Israel-Stewart transient thermodynamics. First order theories and the phenomenological equations for the theory of Eckart and Landau-Lifshitz are derived as ”limits” of the transient thermodynamics, and these theories are discussed in section 10. In section 11, the phenomenological equations for the second order theories describing fluid states near equilibrium are derived. In section $12$, we discuss the Liu-Müller-Ruggeri theory describing relativistic fluids and we compare the predictions of this theory to transient thermodynamics. In section $13$, we discuss the thermodynamics of the class of relativistic fluid theories of divergence type introduced by Pennisi and Geroch-Lindblom. In the conclusion section, we discuss the current state of irreversible thermodynamics of relativistic fluids and open problems related to the thermodynamics of such media.

The paper contains four Appendixes. In the Appendix A, and within transient thermodynamics, we discuss in a coordinate free manner the identification of fluid states near equilibrium. Appendix B, deals with some mathematical aspects regarding the implementation of the entropy principle and provides an introduction to Liu’s procedure and to an alternative procedure introduced by Boillat, Ruggeri and coworkers for implementing the entropy principle. In Appendix C, we remind the reader of the first, second law and the identification of equilibrium states for relativistic continuous media. In Appendix D, intermediate calculations regarding the transformations properties of thermodynamical variables under change of the rest frame within the context of transient thermodynamics, are outlined.

2 Balance Laws for Continuous Classical Media

As a prelude, in the next three sections, we discuss a few relevant properties of classical continuous media. For our purposes, it is sufficient to consider an electrically neutral, continuous medium which at $t=0$ occupies a smooth bounded region $U$ of Euclidean $\mathbb{R}^{3}$ with smooth boundary $\partial U$. The kinematics of the medium is described by one parameter family of orientation preserving diffeomorphisms111111We assume the motion to be smooth enough so that all mathematical operations defined in the next sections are well defined. In Ref. \refciteMH, $C^{r}$ regular motions are defined, but in this work we do not enter in such fine mathematical details. $\phi_{t},t\geq 0$ defined according to:

[TABLE]

so that the family of the images $\phi_{t}(U):=U_{t},t>0$ describe the evolution of the initial distribution $U:=\phi_{0}(U)$. Local coordinates $\vec{X}:=(X^{1},X^{2},X^{3})$ over $U$ serve as Lagrangian labels for the elements of the medium. Thus, for a fixed $\vec{X}$, the set: $\{{\phi(t,\vec{X}),t\geq 0}\}$ describes a smooth trajectory of the element of the medium labelled by $\vec{X}$. For this fluid element, its velocity $\vec{V}(\vec{X},t)$ and acceleration $g(\vec{X},t)$ are defined by:

[TABLE]

The Eulerian121212An $\vec{x}\in\mathbb{R}^{3}$ is the Eulerian coordinate of a point on the trajectory $\{{\phi(\vec{X},t),t\geq 0}\}$ provided $\vec{x}=\phi(\vec{X},t)$ for some $t\geq 0$. velocity field $\vec{u}(\vec{x},t)$ is defined via

[TABLE]

and by the chain rule it follows that

[TABLE]

which means that the “Lagrangian and Eulerian accelerations” are related via:

[TABLE]

For any function $Q(\vec{X},t)$, we associate its Eulerian counterpart $q(\vec{x},t)$ via

[TABLE]

and again the chain rule implies

[TABLE]

where the Lagrangian coordinates $(\vec{X},t)$ and Eulerians $(\vec{x},t)$ are related via $(\vec{x},t)=(\phi(\vec{X},t),t)$. The operator

[TABLE]

acting upon scalars, differentiates along the flow lines of the velocity field and often it is denoted by on overdot, for instance for a sufficiently smooth $f$, we write;

[TABLE]

We recall that for continuous media, the notion of the stress describes the internal forces generated by the medium itself and according to the Stress principle of Cauchy131313This principle and its history is discussed in more details on page $132$ of Ref. \refciteMH. for any oriented surface element at $\vec{x}$ with normal vector $\vec{n}$ that finds itself within the medium, the force $\vec{t}(\vec{x},t,\vec{n})$ that the medium generates at $(t,\vec{x})$ depends upon $t$, the spatial point $\vec{x}=\phi(\vec{X},t)$ and the normal vector $\vec{n}$. The so defined ”vector field” $\vec{t}(\vec{x},t,\vec{n})$ is the Cauchy stress vector field and plays an important role in the structure of the dynamical laws describing the evolution of the medium.

Another important attribute of continuous media is their property of permiting the phenomenon of the heat conduction to take place. This phenomenon, is described by the heat flux function $h(\vec{x},t,\vec{n})$ which determines the rate of heat conduction at time $t$ across any oriented surface element at $\vec{x}$ with unit normal $\vec{n}$. Further ahead, we shall see, that the first law of thermodynamics requires the existence of a vector field $\vec{q}(\vec{x},t)$, refereed as the heat flux vector, related to the heat flux function $h(\vec{x},t,\vec{n})$ via

[TABLE]

With the introduction of the Cauchy stress and the heat flux function, we now turn our attencion to the description of the balance laws141414For the development of the balance laws, the transport theorem is very helpful: If $R(\vec{x},t)$ is any sufficiently smooth scalar function and $\phi_{t}(\hat{U})$ is any smooth region of the fluid at time $t$ transported from a subregion $\hat{U}$ of the initial $U$, then the following identity holds:

$\frac{d}{dt}\int_{\phi_{t}(\hat{U})}R(\vec{x},t)dV(t)=\int_{\phi_{t}(\hat{U})}\left[\frac{\partial R}{\partial t}+\nabla\cdot(R\vec{u})\right]dV(t).$

(12)

The derivation of this identity can be found in [43] (or in any text of advanced calculus).. In the classical framework, these laws incorporate the following principles:

a) mass is neither created nor destroyed.

b) Newton’s second law is valid in the sense that the rate of change of linear momentum of any part of the medium equals to the total external and internal forces acting on this part of the medium.

c) Energy neither is created nor is destroyed.

d) Entropy never decreases in the forward time direction.

The first principle expresses mass conservation. For any continuous medium is assigned the mass density function $\rho(\vec{x},t)$ with the property that the integral $M(V)=\int_{V}\rho d\vec{x}$ is the mass contained within $V$ (assuming here that both $\rho(\vec{x},t)$ and $V$ are sufficiently smooth for the integral to exist). This $\rho(\vec{x},t)$ incorporates the conservation of mass principle, provided151515Here after and in order to avoid repetitions, the symbol $\phi_{t}(\hat{U})$ that appears in integrals stands for the evolution under $\phi_{t}$ of any arbitrary open subset $\hat{U}$ of U.

[TABLE]

This condition, coupled with the transport theorem leads to the following easily verifiable Lemma:

Lemma 2.1**.**

*Let $\phi_{t},t>0$ a motion and $\rho(\vec{x},t)$ a smooth density function, then the following are equivalent:

*1) Conservation of mass holds,

*2) $\rho(\vec{x},t)J(\vec{X},t)=\rho_{0}(\vec{X})$, where $\vec{x}=\phi(\vec{X},t)$,

*3) the equation of continuity holds: $\frac{D\rho}{Dt}+\rho\nabla\cdot\vec{u}=0\quad\Longleftrightarrow\quad\frac{\partial\rho(t,x)}{\partial t}+\nabla\cdot(\rho\vec{u})=0$,

where $\rho_{0}(\vec{X})$ is the density of mass at the $t=0$ and $J(\vec{X},t)=det(\frac{\partial\phi^{i}}{\partial X^{j}})$ is the Jacobian of the family of maps: $\phi(\vec{X},t)=(\phi^{1}(\vec{X},t),\phi^{2}(\vec{X},t),\phi^{3}(\vec{X},t))$.

The following definition expresses the balance of linear momentum:

Definition 2.2**.**

*For any motion $\phi_{t},t\geq 0$ of a continuous medium characterized by a mass density $\rho(\vec{x},t)$, Cauchy’s stress vector field $\vec{t}(\vec{x},t,n)$ and finds itself in an external force field $\vec{b}(\vec{x},t)$, the balance of linear momentum161616 Just for completeness we also mention the balance of angular momentum.

Definition 2.3**.**

Under the same assumptions as in the Definition $1$, we are saying that the balance of angular momentum is satisfied provided

$\frac{d}{dt}\int_{\phi_{t}(\hat{U})}{\vec{x}}\times\rho\vec{u}d{\vec{x}}=\int_{{\partial\phi_{t}(\hat{U})}_{t}}{\vec{x}}\times\vec{t}(\vec{x},t,\vec{n})da+\int_{\phi_{t}(\hat{U})}{\vec{x}}\times\rho\vec{b}(\vec{x},t)d{\vec{x}}$

(14)

where $\vec{x}$ stands for the position vector relative to same origin and $\vec{x}\times\vec{u}$ is the operation of the standard cross product of the Euclidean $R^{3}$. is satisfied provided for every $\phi_{t}(\hat{U})$ the following relation holds:*

[TABLE]

where ${\partial\phi_{t}(\hat{U})}$ stands for the boundary of $\phi_{t}(\hat{U})$ and $da$ is the surface element of ${\partial\phi_{t}(\hat{U})}$.

This integral relation expresses linear momentum balance, but as it stands, it does not lead to a local conservation law of linear momentum. However, as long as (15) holds, and under some mild restrictions upon the smoothness of $\phi_{t},t\geq 0$ and the Cauchy’s stress vector field $\vec{t}(\vec{x},t,\vec{n})$, it can be shown that there exist a unique second rank tensor field171717 For the derivation of this important property see Theorem $2.2$ on page $134$ in Ref. \refciteMH. $\sigma=\sigma(\vec{x},t)$ referred as the Cauchy’s stress tensor, so that the components of $\vec{t}(\vec{x},t,\vec{n})$ can be written in the form:

[TABLE]

where $\sigma^{ab}(\vec{x},t)$ are the components of the Cauchy’s stress tensor, $g_{ab}$ are the components of the Euclidean metric of $\mathbb{R}^{3}$ and as always $n^{a}$ stand for the components of the outward pointing normal vector field of the ${{\partial\phi_{t}(\hat{U})}}$. Using this representation of $\vec{t}(\vec{x},t,\vec{n})$ in (15), and assuming mass conservation, then the divergence theorem yields the local law181818Similar analysis holds for the balance law of the angular momentum. Here a local conservation law for angular momentum requires the Cauchy’s stress tensor to be symmetric. For a derivation of the local angular momentum conservation law see Theorem $2.10$ on page $138$ of Ref. \refciteMH.:

[TABLE]

This law in Cartesian coordinates takes the form191919It is should be mentioned, that both the balance of linear momentum and angular momentum discussed here, use the linear structure of the underlying Euclidean $\mathbb{R}^{3}$ and in particularly the global existence of Cartesian coordinate systems. Therefore special care is required when these laws are required to be written down relative to arbitrary coordinates systems.

[TABLE]

and as derived here, these equations have a formal character. They acquire a well defined meaning provided first the dependence of $\sigma(\vec{x},t)$ upon the motion of the medium itself is spelled out. We will come back to this point further ahead.

3 The First Law for Continuous Classical Media

So far, the thermodynamical state of the medium is described by the mass density $\rho(\vec{x},t)$, the stress tensor $\sigma(\vec{x},t)$, the external force field/unit mass $\vec{b}(\vec{x},t)$. However these variables alone do not specify the medium and they are enlarged by the inclusion of:

a) the internal energy/unit mass $e(\vec{x},t)$,

b) the heat flux vector $\vec{q}(\vec{x},t)$ so that $h(\vec{x},t,\vec{n})=-\vec{q}(\vec{x},t)\cdot\vec{n}=-g_{ab}q^{a}(\vec{x},t)n^{b}$,

c) the heat supply/unit mass $Q(\vec{x},t)$.

Also for any $t>0$ and any $\phi_{t}(\hat{U})$, we introduce the total kinetic energy $T(t)$ and total internal energy $U(t)$ given by

[TABLE]

and in terms of these variables we have the following definition:

Definition 3.1**.**

For a motion $\phi_{t},t>0$, a state of the medium defined by $\rho(\vec{x},t)$, $\vec{u}(\vec{x},t)$, $e(\vec{x},t)$, $h(\vec{x},t,\vec{n})$, $Q(\vec{x},t)$ obeys the balance of energy principle provided:

[TABLE]

This energy principle is a restatement of the first law of thermodynamics as applied to continuous media. It asserts that within any $\phi_{t}(\hat{U})$, the rate of change of the total energy (= kinetic energy + internal energy (including potential energy)) is due to the work done by the combined external force $\vec{b}$ and the stress force $\vec{t}$ on this $\phi_{t}(\hat{U})$ augmented by the amount of the total heat crossing $\partial\phi_{t}(\hat{U})$ or to the total heat supplied to $\phi_{t}(\hat{U})$ by an external agency.

As for the case of linear momentum, this integral law yields a local conservation law provided the heat flux function $h(\vec{x},t,\vec{n})$ and the Cauchy’s stress field $\vec{t}(\vec{x},t,\vec{n})$ are written in terms of the heat vector202020The proof that the local law of the energy conservation, then requires the representation $h(\vec{x},t,\vec{n})=-\vec{q}(\vec{x},t)\cdot\vec{n}$ see proposition $(3.2)$ in Ref.[43]. $\vec{q}(\vec{x},t)$ and Cauchy’s stress tensor $\sigma(\vec{x},t)$. Under these conditions and by applying the divergence theorem to (20) one obtains a local law:

[TABLE]

Combining this law with the local law in (17) expressing linear momentum conservation and some algebra, yields:

[TABLE]

which describes the manner that the internal energy $e$ varies as it transported along along the flow lines of the fluid’s velocity field.

To get insights into the structure of the so far derived balance laws, let us assume that for a particular medium the Cauchy tensor $\sigma(t,\vec{x})$ has the special form:

[TABLE]

For such medium, the law of linear momentum (17) yields

[TABLE]

which is recognized as the Euler’s equation with $P(\vec{x},t)$ interpreted as the (thermodynamical) pressure for an ideal fluid.

However, for an arbitrary medium the Cauchy’s stress tensor $\sigma(\vec{x},t)$ may exhibit a much more complicated structure than the one exhibited in (24).

In order to describe this more general case, it customarily to decomposed it according to:

[TABLE]

where $P$ is the thermodynamical pressure, $P^{v}$ is the bulk viscous stress, while the traceless part $\sigma_{(v)}^{ij}$ is the shear viscous stress212121This terminology is motivated by the theory of Navier-Stokes fluids and in such theory often $\sigma^{ij}$ are also refereed as a the components of the pressure tensor.. In view of this splitting, the evolution of the internal energy $e$ in (22) takes the equivalent form:

[TABLE]

where $u_{(t)ij}$ stands for the symmetric trace free part of $u_{i,j}$ defined via the decomposition: $u_{{i,j}}=\frac{1}{2}(u_{i,j}+u_{j,i})+\frac{1}{2}(u_{i,j}-u_{j,i})=u_{(ij)}+u_{[ij]}$ so that

[TABLE]

For latter use, we introduce here the specific density $v(\vec{x},t)=\rho(\vec{x},t)^{-1}$. This $v(\vec{x},t)$ and as a consequence of the continuity equation satisfies

[TABLE]

a formula that will be helpful further ahead.

The balance laws derived in the last two sections apply to a large class of (classical) continuous media. It is a peculiarity of the nature of continuous media that these laws alone fail to yield a closed system of equations. Closure is accomplished by the specifications of constitutive relations. These relations amount to specifying the functional dependance of the Cauchy’s stress tensor $\sigma(\vec{x},t)$, the heat flux vector $\vec{q}(\vec{x},t)$ and internal energy $e(\vec{x},t)$ upon suitable variables describing the medium and often these variables are refereed as basic variables. For instance, for a heat conducting, viscous fluid a set of basic variables consists of the density, the velocity field, temperature (and may be their derivatives). For the specification of the constitutive222222For an update on issues regarding the theory of constitutive relations, the reader is refereed to Refs. \refciteMul4, \refciteMH, \refciteTrues1, \refciteTrues2, \refciteTrues3, \refciteRug4, for further details. relations describing a particular medium, certain selection rules apply and at this stage the entropy and the second law play a central role. In the next section, we shall discuss the interplay between balance laws, constitutive relations and thermodynamic and it should be mentioned that the focus of the rest of the paper is to delineate the frontiers where hydrodynamics ends and thermodynamics begins.

4 The Second Law and Continuous Media-The Entropy Principle

Although the derivation of the balance laws discussed in the last two sections were free of ambiguities, maters complicate considerably once we pass to the subtle task of assigning entropy to arbitrary states of a continuous media. In fact, it is fair to claim that except for states describing thermodynamical equilibrium, there exist no universally accepted recipes to assign a meaningful entropy to non equilibrium states of continuous media. The possible choices of this non equilibrium entropy, the implementation of the second law and its consequences will be the central themes for the rest of this paper. In the next section, we shall introduce classical irreversible thermodynamics (CIT) where the local thermodynamical equilibrium postulate determines the entropy of off-equilibrium states. In section$-6$, we introduce the Extended Irreversible Thermodynamics (EIT), where Müller’s postulate determines the entropy of states near equilibrium, while within the Rational Extended Irreversible Thermodynamics (REIT), constitutive relations determines the entropy of off-equilibrium states.

Below and motivated by reasons of presentations, we do not follow the historical development, but start discussing first the issue of assigning an entropy to off-equilibrium states and the implementation of the second law within the framework of Rational Thermodynamics232323 For an introduction to the principles of Rational Thermodynamics, the reader is refereed to Refs. \refciteTrues1,Trues2,Trues3, see also discussion in Refs. \refciteJVL,MH. abbreviated here after as (RT).

Rational thermodynamics, leaves the balance laws for continuous media derived in the previous sections intact, but introduces two additional functions:

the entropy/unit mass $s(\vec{x},t)$,

the local temperature $T(\vec{x},t)$.

The local temperature $T(\vec{x},t)$ is considered to be an absolute element and the theory implements the second law of thermodynamics via the Clausius-Duhem inequality according to the definition:

Definition 4.1**.**

For any motion $\phi_{t},t>0$, a state of a continuous media described by $\rho(\vec{x},t)$, $e(\vec{x},t)$, $h(\vec{x},t,\vec{n})$, $Q(\vec{x},t)$, $s(\vec{x},t)$, $T(\vec{x},t)$ satisfies the second law of thermodynamics, if the rate of entropy production within any $\phi_{t}(\hat{U})$, i.e.

[TABLE]

satisfies the Clausius-Duhem integral inequality:

[TABLE]

Thus within (RT) the rate of entropy increase within $\phi_{t}(\hat{U})$, is greater (or at best equal) to the entropy generated by the heat supplied reversibly to $\phi_{t}(\hat{U})$ and the entropy generated by heat flux through the boundary ${\partial\phi_{t}}(\hat{U})$. For states subject to: $Q(\vec{x},t)=h(\vec{x},t)=0$, the inequality implies

[TABLE]

i.e. the entropy cannot decrease in the forward time direction.

The Clausius-Duhem inequality can be expressed in a local form provided that one employs the heat flux vector $\vec{q}(\vec{x},t)$ via $h(\vec{x},t,\vec{n})=-\vec{q}(\vec{x},t)\cdot\vec{n}=-g_{ab}q^{a}(\vec{x},t)n^{b}$ where as before $\vec{n}$ is the outward pointing normal vector of $\partial\phi_{t}(\hat{U})$. In that event, (30) yields the point wise form of the Clausius-Duhem inequality:

[TABLE]

which is the local formulation of the second law within the theory. Within (RT), the entropy density $s$ is considered to be a constitutive function i.e. $s$ dependents upon suitably defined set of basic variables describing the state of the medium and the explicit dependance of this $s$ upon the basic variables is a thorny issue.

Coleman and Noll[47] in the $1960s$, were the first to suggest a limited version of what nowadays is refereed as the entropy principle. They suggested that the dependance of $s(\vec{x},t)$ upon the basic variables must be assigned in such a manner so that the second law of thermodynamics is satisfied for any arbitrary thermodynamic processes242424In the language of thermodynamists, a thermodynamic process is any solution of the balance equations.. Müller[48, 49] in the $1970s$ refined the Coleman-Noll approach and this refinement cemented the modern version of the entropy principle. Specifically Müller at first postulated that for any medium, there exists a scalar additive quantity the entropy density $s$, and an entropy flux vector $\vec{J}$, such that in the absence of any heat supply $Q$ these $s$, and $\vec{J}$, satisfy the entropy inequality:

[TABLE]

Moreover, he postulated that both $s$ and $\vec{J}$ are considered to be constitutive functions and this introduces an additional degree of flexibility in the formulation of the second law252525 Compare (32) to the form of the entropy inequality in (33). The latter, does not make any reference to the heat flux nor makes no reference to the notion of temperature. Away from equilibrium states, the notion of temperature is a subtle concept.. In this modern version, the entropy principle asserts that the dependance of both i.e $s$ and $\vec{J}$ (or more generally of any constitutive function) upon the basic variables should be chosen in such a manner so that every solution of the balance laws satisfy the above form of the entropy inequality.

Nowadays, the entropy inequality and entropy principle as refined by Müller, lie at the center of the thermodynamics of classical continuous media (and as we shall see further ahead they remain so in the theory of relativistic continuous media). However implementing the entropy principle is a delicate matter. Does there exists an algorithmic procedure to pick up the right form of the constitutive relations so that the entropy principle holds?

Since the late $1960$ until recently, there have been invented various procedures to implement the entropy principle. The Coleman-Noll procedure (see Refs. \refciteColN1, \refciteColN2), the Müller procedure (see Refs. \refciteMulN1, \refciteMulN2), the Liu-procedure (see Ref.\refciteLiu) and the procedure developed by Boillat, Ruggeri and coworkers (see Refs. \refciteRugF1, \refciteRugF2, \refciteRugF3, \refciteBoi, \refciteRugStr). In order to facilitate matters, in the Appendix $B$, we outline the basic features of the Liu’s and Boillat-Ruggeri’s procedures and the manner that these procedures implement the principle.

We finish this section by mentioning that the entropy inequality (33) can be cast in the equivalent form

[TABLE]

where the scalar $\sigma$ is interpreted as the the density of entropy production per unit volume and unit time and the restriction $\sigma\geq 0$ implements the second law. The entropy inequality in the form shown in (34) will be used further ahead.

5 Classical Irreversible Thermodynamics

In this section, we introduce the first theory of irreversible thermodynamics describing classical continuous media. The theory initiated long ago by Onsager262626Besides Onsager, the works by Eckart, Meixner, Progogine, are also associated with the developments of (CIT). For a historical account consult Ref.\refciteJVL, and also the critical review in Ref.\refciteMulW and will be abbreviated here after as: (CIT). The theory leaves intact the balance laws for mass, linear momentum and total energy introduced earlier on but uses the local thermodynamical equilibrium postulate (local-equilibrium in short) to assign an entropy to non equilibrium states.

According to (CIT) a medium finds itself in a state of a local-equilibrium, if at any point $(\vec{x},t)$ a sufficiently small cell can be introduced so that the cell by itself is considered to be a thermodynamical subsystem satisfying the following property: within this cell, the state variables are well defined (i.e. do not exhibit wild fluctuations) and obey the same thermodynamical relations as if this subsystem was in a state of a global thermodynamical equilibrium. Accepting this hypothesis, then it is relatively straihtforward to assign the equilibrium entropy $s(t,\vec{x})$ at that cell272727 This can be done for instance by appealing to Clausious, Caratheodory’s or to Gibbs axiomatic approach to the equilibrium entropy. Notice however, that (CIT) postulates something more: it postulates that the functional form of the this local equilibrium entropy can be taken as describing the physical entropy of the underlying state.. It is worthwhile to briefly describe states of a simple, heat conducting, viscous fluid within the framework of (CIT).

Under the hypothesis that the fluid finds itself in a state of local-equilibrium, at any $(\vec{x},t)$ the entropy density $s(\vec{x},t)$ depends only upon $e(\vec{x},t)$ and specific volume $v(\vec{x},t)=\rho^{-1}(\vec{x},t)$ and moreover $s(\vec{x},t),e(\vec{x},t)$ and $v(\vec{x},t)=\rho^{-1}(\vec{x},t)$ satisfy :

[TABLE]

which is the familiar form of the Gibbs equilibrium relation (except from the crucial dependance of the thermodynamical variables upon $(\vec{x},t)$).

It follows from this relation that the evolution of $s(\vec{x},t)$ along the flow lines obeys:

[TABLE]

while for any two nearby fluids element at $(\vec{x},t)$ and $(\vec{x}+d\vec{x},t)$, the density $s(\vec{x},t)$ satisfy

[TABLE]

where $\nabla$ stands for the gradient operator on the Euclidean $\mathbb{R}^{3}$.

Now (36) implies

[TABLE]

and using to (26-28), yields

[TABLE]

which can be cast in the form:

[TABLE]

[TABLE]

where $\vec{J}_{E}$ is the entropy flux while $\sigma$ is the entropy production per unit volume and unit time282828It is a common notation to refer to $\vec{q},P^{v},\sigma_{v}^{ij}$ in (41), as dissipative fluxes while the corresponding

$\frac{\nabla T}{T^{2}},\quad\frac{\nabla_{a}u^{a}}{T}\quad\text{and}\quad\frac{u_{(t)ij}}{T}$

as thermodynamical forces..

Whenever $Q=0$, then (40-41) show that the time evolution of fluid states characterized by $\vec{q}=P^{v}=\sigma_{v}^{ij}=0$ do not generate entropy. For this case, the balance laws show that we are dealing with Eulerian hydrodynamics and these Eulerian states may be viewed as the equilibrium states within the space of all states describing heat conducting, viscous fluids.

However, for states characterized by non vanishing heat flux or (and) Cauchy stress, formulas (41) shows that their evolution generates entropy. A look at (40) shows that the second law is satisfied provided $\sigma$ in (41) is semi-positive definite and this holds provided the following linear constitutive relations hold:

[TABLE]

[TABLE]

[TABLE]

where the coefficients $\mu_{1},\mu_{0},\mu_{2}$ are in general temperature dependent and subject to the restrictions: $\mu_{1}\geq 0$, $\mu_{0}\geq 0$ and $\mu_{2}\geq 0$. By introducing the coefficients of the thermal conductivity $k$, bulk viscosity $\zeta$ and shear viscosity $\eta$ via

[TABLE]

then (42- 44) take the form

[TABLE]

which are the standard forms of the Fourier and Navier-Stokes linear constitutive relations known long time ago 292929It is worth noting that the constitutive relations in (46) have deduced by implementing the second law within (CIT)..

The Fourier-Navier-Stokes theory of heat conducting, viscous fluid is the standard theory describing laboratory and astrophysical fluids and it is reliable303030Although in this section, for illustration purposes, we applied the principles of (CIT) to states of a heat conducting, viscous fluid, nevertheless, (CIT) can be employed to describe states for a fluid mixture or states of other physical systems and the reader is referred to the Refs. \refciteJVL,Mei,Gro,Gya,Gla for further applications. as long as restrictions as restricted to length scales greater than certain microscopic distances (for instance mean free path for the case of a gas).

On the other hand, it was recognized long ago, that as a consequence of the structure of the constitutive relations in (46), the Fourier-Navier-Stokes system predicts that thermal and viscous signals propagate with unbounded speed and this unsatisfactory property motivated the search for alternative theories describing the irreversible thermodynamics of a heat conducting, viscous fluid. A large number of such alternative theories have been put forward after Müller’s[17] in $1967$ introduced a key postulate regarding the entropy of non equilibrium states and in the next section we discuss this postulate in details.

6 Extended Irreversible Thermodynamics

Müller[17] in $1967$, suggested that for a simple, heat conducting, viscous fluid, the entropy of non equilibrium states differs drastically from the entropy resulting by invoking the local equilibrium postulate within the framework of (CIT). For states near equilibrium, Müller[17] postulated that their entropy receives quadratic contributions313131According to a historical account on Ref \refciteJou1, it appears that other workers before Müller’s $1967$ paper were contemplating the enlargement of the entropy function by the inclusion of dissipative fluxes, but it seems that it was Müller’s paper[17] that trigger the development of new theories. from the fluxes $(\vec{q},P^{v},\sigma_{(v)}{}^{ij})$ that appear in the balance laws. Specifically, he assigned to these near equilibrium states a generalized entropy $s_{ge}(\vec{x},t)$ that has the form:

[TABLE]

where $s_{lep}(e,v)$ is the entropy density assigned by the local equilibrium postulate within (CIT), and $(a,b,c)$ are smooth functions of $(e,v)$. Remarkably, Müller’s hypothesis yields a theory that cures the problems of the Fourier-Navier-Stokes mentioned in the last section. By appealing to (47) and imposing the second law, a new set of constitutive relations are derived which are structurally different than those predicted by the (CIT). In turn, these new constitutive relations are the crucial elements in arriving at a set of dynamical equations for the Fourier-Navier-Stokes system predicting finite propagation for the heat and shear waves.

Due to the close connection of Müller’s postulate to the development of transient thermodynamics of relativistic fluids, below, we treat briefly non equilibrium states of a simple, heat conducting, viscous fluid within the Extended Irreversible Thermodynamics323232We remind again the reader, that in this paper, the term (EIT), stands for the theory that abandons the recipe of assigning entropy to non equilibrium states as within (CIT) by appealing local equilibrium postulate. The (EIT) assigns an entropy to non equilibrium states that depends explicitly upon fluxes appearing in the balance laws and whose evolutions are obtained by imposing the second law combined with an extended version of the Gibbs relation. (EIT) is discussed at length in the book of Ref.\refciteJVL, see also Ref.\refcitelast. However often in the literature, the acronym (EIT) stands for alternative theories of non equilibrium thermodynamics of continuous media which in one or another way are based on a form of non equilibrium entropy which is motivated largely by Müller’s original idea. For an overview of such theories, consult for instance Refs. \refciteRug2,Rug3. (abbreviated as (EIT)) which is a theory that adopts Müller’s $1967$ postulate. According to this theory, for an arbitrary non equilibrium state of this fluid, one assigns a generalized entropy $s_{ge}(\vec{x},t)$ which is a smooth function of $(e,v,\vec{q},P^{v},\sigma_{(v)}{}^{ij})$ i.e. $s_{ge}=s_{ge}(e,v,\vec{q},P^{v},\sigma_{(v)}{}^{ij})$. For states near equilibrium, a series expansion of this $s_{ge}(e,v,\vec{q},P^{v},\sigma_{(v)}{}^{ij})$ around $(e,v,0,0,0)$ yields (47) and the resulting theory leads to Müller’s conclusions while for states away from equilibrium, this $s_{ge}(e,v,\vec{q},P^{v},\sigma_{(v)}{}^{ij})$ may be expanded around a background non equilibrium steady state (for the definition and treatment of such states, see Ref.\refcitelast section $9$).

Within this theory, let $(e,v,\vec{q},P^{v},\sigma_{(v)}{}^{ij})$ a background non equilibrium steady state and let $(e+de,v+dv,\vec{q}+d\vec{q},P^{v}+dP^{v},\sigma_{(v)}{}^{ij}+d\sigma_{(v)}{}^{ij}$ a ”nearby non equilibrium state”. The difference $ds_{ge}$ in the generalized entropies of these two states at least formally, can be written in the form

[TABLE]

where for typographical convenience we write $s$ intead of $s_{ge}(e,v,\vec{q},P^{v},\sigma_{(v)}{}^{ij})$, summation over the indices $(i,j)$ is understood and the partial derivative of $s$ with respect to $e$ is taken by keeping $(v,\vec{q}$, $P^{v}$, $\sigma_{(v)}^{ij})$ fixed (similar restrictions for the other derivatives as well). This relation is interpreted within (EIT) as a “generalized Gibbs relation” and introduces formally a non-equilibrium absolute temperature $\Theta$ via

[TABLE]

and a non-equilibrium thermodynamical pressure $\pi$ via

[TABLE]

The remaining partial derivatives in (48) are written333333For a discussion regarding the physical significance and observability of the non equilibrium temperature $\Theta$ defined in (49)- not to be confused with the local-equilibrium temperature $T^{-1}=\frac{\partial s(e,v)}{\partial e}$-as well as the significance of the non equilibrium pressure $\pi$ defined in (50), the reader is refereed to chapters $(2,3)$ of Ref. \refciteJVL, see also Refs. \refcitelast, \refcitenew1. in the form:

[TABLE]

[TABLE]

[TABLE]

where the scalar functions $(a_{10},a_{00},a_{21})$ depend in general upon $(e,v,\vec{q},P^{v},\sigma_{(v)}{}^{ij})$. With this notation, the generalized Gibbs relation (48) takes the form

[TABLE]

which implies that the evolution of the generalized entropy along the flow lines is governed by

[TABLE]

Multiplying this equation by $\rho$ and using (26-28), one finds343434Notice that equation (55) in combination to (26-28) imply that a term $\frac{\pi-P}{\Theta}\nabla_{a}u^{a}$ ought to be included in the right hand side of (56) and here we see the first conceptual problems of the (EIT) to present themselves. It is assumed here that the thermodynamical pressure $P$ and the non-equilibrium thermodynamical pressure $\pi$ are related and in fact that they are equal. For states near to local equilibrium that may be the case but for states away from equilibrium it is far from clear whether this is the case.

[TABLE]

where $u_{(t)ij}$ stands for the symmetric traceless part of $u_{i,j}$ and as before $\sigma^{ij}_{(v)}$ is the symmetric traceless part of the stress tensor $\sigma^{ij}$ (see the decomposition in (25)).

Ignoring for the moment conceptual problems related to the physical significance of the non-equilibrium absolute temperature $\Theta$, the non-equilibrium thermodynamical pressure $\pi$ as well as the meaning of the other partial derivatives in the generalized Gibbs relation, the evolution equation $\dot{s}$ in (56) can be cast in the form: $\rho\dot{s}+\nabla\cdot\vec{J}=\sigma$, where the restriction $\sigma\geq 0$ enforces the second law. The entropy flux $\vec{J}$ is considered to be a constitutive function (see comments following eq. (33)) and for isotropic states353535For more details regarding the choice of (57) and the results of this section consult sections $(2.3,2.4)$ of Ref.\refciteJVL., $\vec{J}$ is postulated to have the structure:

[TABLE]

where $\beta^{\prime}$ and $\beta^{\prime\prime}$ are unspecified coefficients depending upon $e$ and $v$. For this choice of $\vec{J}$, the entropy production $\sigma$ takes the form

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

where $(\partial_{i}q_{j})_{st}$ stands for the symmetric and traceless part of the tensor $\partial_{j}q_{i}$.

It follows from (58), that a simple manner to implement the second law (i.e. to enforce a non negative $\sigma$) is via the choices:

[TABLE]

where $\mu_{1}\geq 0,\mu_{0}\geq 0,\mu_{2}\geq 0$, are new phenomenological coefficients that may depend upon $(e,v)$. By combining (62) with (59-61) one obtains a set of constitutive relations for this theory that are structurally different to those constructed within the framework of (CIT) (compare (62) to (46)).

To get insights into these relations, one neglects in (62) quadratic terms in the fluxes and products of fluxes as well as time gradients of $e$ and $u$. Under these simplifications, (62) combined with (59-61) yields:

[TABLE]

[TABLE]

[TABLE]

For stationary and spatially homogeneous states, the spatial and temporal gradients of the fluxes are zero and thus

[TABLE]

Comparing these relations with the standard Navier-Stokes formulas in (46) fixes the parameters $\mu_{0},\mu_{1},\mu_{2}$ to the values

[TABLE]

where $(k,\zeta,\eta)$ stand for the coefficients of thermal conductivity, bulk viscosity and shear viscosity, and we identified $\Theta$ with the local equilibrium temperature $T$.

Leaving aside intermediate details (see Ref. \refciteJVL) and introducing relaxation times $(\tau_{0},\tau_{1},\tau_{2})$ via

[TABLE]

[TABLE]

[TABLE]

the linearized evolution equations (63-65) take the form:

[TABLE]

[TABLE]

[TABLE]

and these equations in the limit where $\beta^{\prime}$ and $\beta^{\prime\prime}$ are vanishing reduce to the Maxwell-Cattaneo laws [19, 34, 35]

[TABLE]

[TABLE]

[TABLE]

The time evolution equations for the fluxes $(\vec{q},P^{v},\sigma_{(v)}{}^{ij})$ shown in $(\ref{G1}-\ref{G3})$ are one of the striking implications of (EIT) when this theory is applied to spatially homogeneous states. These evolution equations combined with the balance yields a closed system of equations which are very different than those predicted by (CIT).

One of the issues concerning this new system is whether it predicts finite propagation for heat and viscous disturbances. According to Ref. \refciteJVL page (85), it appears that the (EIT) passes that test provided the generalized entropy $s_{ge}=s_{ge}(e,v,\vec{q},P^{v},\sigma_{(v)}{}^{ij})$ is chosen to be a concave function of its arguments i.e. the second variation $\delta^{2}s_{ge}$ of $s_{ge}$ evaluated on the background state is negative definite. Moreover, it is stated in [19], that as long as the generalized entropy $s_{ge}$ is chosen to be a concave function of its arguments, then this property of $s_{ge}$ is equivalent to the symmetric-hyperbolic363636Since in this theory the background state is arbitrary, it would be nice if this property reexamined further. The structure of the dynamical equations needed to have (or transformed into) the form of conservation laws. nature of the underlying equation which in turn ensures that the characteristic propagation speeds are real and finite and in the same reference document these assertions through specific examples.

We conclude this brief introduction of (EIT) by mentioning that this theory has been developed enormously in the last few decades and the reader is referred to the vast literature (see for instance [19, 20, 36, 37, 38, 39]) for further discussion and open problems within (EIT).

Müller’s original hypothesis, regarding the notion of generalized entropy $s_{ge}(\vec{x},t)$ discussed at the beginning of this section, as well as many of the theories arising from his thesis, have been placed under intense scrutiny. A comparison of the predictions of these extended theories to the standard theory of Fourier-Navier-Stokes fluids reveal signs of concern. In Refs. \refciteAnil1,Maj1 the shock structure of Müller’s original theory was investigated and compared to the structure predicted by the Fourier-Navier-Stokes theory. It is found that regular shock structure exist only for sufficiently low Mach numbers and this prediction lead to questioning the concept of an extended entropy.

This crisis, as well as the quest to have a satisfactory resolution of the velocity propagation of thermal and shear waves, lead Liu, Müller, Ruggeri and collaborators to develop a new theory dealing with classical fluids, refereed as Rational Extended Irreversible Thermodynamics (abbreviated here after as (REIT)). This theory takes for granted the moments of the non relativistic Boltzmann equation for a monoatomic classical dilute gas truncated at some large integer $N$. Using these moments, and employing Grads $13$-moment distribution function, Müller and Liu in Ref. \refciteMul5 derived equations for the heat flux $q^{i}$ and the components of Cauchy stress $\sigma^{ij}$ for a classical dilute monoatomic gas. They noticed that these equations take the form of equations of balance form, i.e. equations of the form

[TABLE]

where

[TABLE]

[TABLE]

and $T$ signifies transpose (the structure of such balance laws and some of their basic properties are discussed in the Appendix 2).

These results of Müller and Liu in Ref. \refciteMul5 lead to the foundation of (REIT). Within (REIT), it is postulated that the moments of the Boltzmann equation ought to be treated as phenomenological equations describing heat conducting, viscous fluids so that the heat flux $q^{i}$ and the components of Cauchy stress $\sigma^{ij}$ for such fluids satisfy equations analogous to (77). This is the central hypothesis underlying (REIT) and thus at a first side it appears that this theory is dealing with dilute gases. However, that is not the case. Ought to be kept in mind that the problem of the closure of the $N$ moments within (REIT) is dealt via the entropy principle and other methods of classical continuous media. In particular an entropy law is incorporated as an additional balance law in (77) and under appropriate restrictions a symmetric - hyperbolic system of field equations is emerging (for a brief discussion on that issue see Appendix B). The symmetric- hyperbolic nature of the dynamical equations within (REIT) is to be compared with the parabolic nature of the dynamical equations characterizing for instance the Fourier-Navier-Stokes system. Thus in summary, Müller’s original idea of an extended entropy lead eventually to the emergence of a symmetric- hyperbolic system of field equations as the dynamical equations describing states of classical fluids. Due to space limitations, we shall not discuss any further the principles of (REIT). For a detailed introduction to this theory, the reader is refereed to Ref. \refciteMul5, the monograph entitled: Rational Extended Thermodynamics, Ref. \refciteMul4 and the relatively recent review in Ref. \refciteMulW.

The current status of (REIT) is best described by quoting a passage373737There is a comments regarding the title of the book of Ref. \refciteMul4. The term rational thermodynamics should not confused with the theory of rational thermodynamics (RT) introduced in Refs. \refciteTrues1,Trues2,Trues3 and mentioned briefly in section $4$. The authors of Ref. \refciteMul4 explain the title of their book in the introduction section as follows: *……Rational Extended Thermodynamics. The literature is full of papers referring to extended thermodynamics which, however, are devoid of rational methodology and mathematical cohesion. The epithet rational in the present title is chosen so as to emphasize the systematic procedure which the book espouses, a procedure typical for a deductive science.

from the introduction section of this monograph:

*…. the shock wave structure calculated in extended thermodynamics….is worse than the shock wave structure in ordinary thermodynamics; and again: many moments are needed to put things right….. When enough moments are used to describe the state, (REIT) leads to perfect agreement of theory and experiment.

*We are not going to pursue any further the analysis of irreversible thermodynamics of classical (Newtonian) media either in the form of (EIT) or (REIT). We hope that this brief exposition highlights the potentialities (and challenging open problems) of non equilibrium thermodynamics of continuous media.

The rest of the paper is focused on aspects of the thermodynamics of relativistic continuous media and confronts the subtleties arising from the blending of thermodynamics with the principle of general covariance.

7 Continuous media in a relativistic setting

From this section onward, we discuss thermodynamical aspects of continuous media propagating on an arbitrary smooth four dim. spacetime $(M,g)$. Thermodynamical properties of such media has been the subject of many past investigations, see for instance [9, 10, 61, 62, 63, 64, 65, 66, 67, 68], and these investigations cover the nature of relativistic equilibrium, the formulation of the second law, aspects of irreversible thermodynamics of relativistic fluids, elastic solids etc. However, the aim of the following sections is to discuss the progress that has been made in the last few decades regarding the development of theories of irreversible thermodynamics of relativistic media that predict finite propagation of disturbances and admitting stable equilibrium states. To put matters in mathematically precise terms, the aim is to introduce theories of irreversible thermodynamics of relativistic media that are described by dynamical equations that constitute a symmetric-hyperbolic and causal system of equations. Such systems of equations guarantee the well posedness of the Cauchy problem and moreover general perturbations of a background solution propagate within the light cone.

As for the case of theories of non equilibrium thermodynamics of classical (Newtonian) media outlined in the previous sections, similarly the field of irreversible thermodynamics of relativistic media has evolved. As early as $1940$ Eckart [9] and later in $1950s$ Landau and Lifshitz [10], introduced the first theories of dissipative relativistic fluids that now days are refereed as conventional theories. In the late $1970$, Israel[13] and Israel and Steward[11] introduced the theory of transient thermodynamics. In the late $1980$’s Liu, Müller and Ruggeri, [22] (see also [21]) developed a version of irreversible thermodynamics for relativistic fluids that extends the principles of Rational Extended Irreversible Thermodynamics (REIT) to the general relativistic regime. Motivated by the approach of Liu, Müller and Ruggeri, in the early $1990$, Pennisi[23], Geroch and Lindblom[24, 25], developed the theory of relativistic dissipative fluids of divergence type. In the following sections, we discuss these theories and whenever convenient, we illustrate their principles by considering states of either simple fluids or fluid mixtures. In order to pave the way towards to these developments, in the Appendix C, we remind the reader of a few basic aspects of the thermodynamics of relativistic media.

We begin by setting the scene for the development of Israel-Stewart’s transient thermodynamics383838The theories of Eckart [9] Landau and Lifshitz [9] will emerge as a special limit of the transient thermodynamics. and shall illustrate its principles by specializing the theory for the moment to arbitrary fluid states393939As it will become clear further ahead transient thermodynamics describes states near equilibrium, however for this initial setting up of the theory, this point is irrelevant. Also, part of the discussion that follows remains valid if instead of fluids, more general continuous media are considered, like relativistic elastic media, polarized media etc. For the description of off-equilibrium states of such media, see for instance Refs. \refciteKr1, \refciteKr2, \refciteIsr3.. Within this theory, states of a simple fluid propagating on a smooth spacetime $(M,g)$, are described by a set of primary variables404040Besides these primary variables, a complete specification of arbitrary fluid states requires the specification of additional auxiliary variables and these variables will be introduced further ahead. consisting of the conserved and symmetric414141It will be assumed here after that the fluid is isolated and interacts only with a background gravitational field. energy momentum tensor $T$, a conserved timelike particle current $J$ and the entropy four vector $S$ obeying:

[TABLE]

where the inequality satisfied by $S^{\mu}$ is dictated by the second law and at this point we do not impose any restriction upon the dependance of $S^{\mu}$ upon other basic variables, this dependance will enter the scene gradually.

For a fluid mixture, an arbitrary state involves $n-$particle currents described by $n-$timelike vector fields $J_{A}$ with $A\in(1,2,.......n)$. In the absence of chemical or nuclear reactions, the primary variables $(T,S,J_{A})$ for this fluid mixture, satisfy

[TABLE]

while in the presence of chemical reactions the $n-$particle currents $J_{A}$ satisfy:

[TABLE]

here $k$ is the number of reactions that involve the species of type $A$, $r_{i}$ is the $i_{th}$ reaction rate and $C^{i}_{A}$ are the stoichiometric coefficients (for properties of these coefficients see for instance Refs. \refciteIsr1,Gro).

For classical fluids, it is common to assume that the energy momentum tensor satisfies the weak energy condition i.e. $T_{\mu\nu}u^{\mu}u^{\nu}\geq 0$ for all future directed timelike vectors $u$ and thus by Synge’s theorem Ref.\refciteSyn, $T$ admits a unique timelike eigenvector $u_{E}$, $g(u_{E},u_{E})=-1$ that defines the Landau-Lifshitz or energy frame.

On the other hand, every particle current $J_{A}$ defines a unique timelike future directed vector field $u_{A}$ via $J_{A}{}^{\mu}=n_{A}{u_{A}}^{\mu}$ with $g(u_{A},u_{A})=-1$ and each one of these $n$-fields $u_{A}$ define their own rest frame. For the case of a simple fluid, the unique $u_{N}$ parallel to $J$ defines the Eckart or particle frame. Thus the primary variables assigned to an arbitray state of a relativistic fluid (away from the case of relativistic perfect fluids), offers the possibility to introduce more than one rest frame and in general no fundamental reason exists to choose one versus the other424242To view matters from different perspective: the four velocity associated with states of a dissipative fluid ceases to be a well defined element. For instance, within the context of a simple dissipative fluid, should one identify the fluids four velocity with the flux of energy i.e. assign the fluid’s for velocity to $u_{E}$ or identify the fluid’s four velocity with $u_{N}$ i.e. with the flux of particles or may be take the fluid’s velocity to be a combination of the two? There is not a satisfactory resolutions of this dilemma. In this work, this dilemma appear as an issue regarding the specification of suitable classes of fields of rest frames. Transient thermodynamics deals satisfactory with the absence of what we are intuitively accustomed i.e. the fluids four velocity by building a framework where the possible choice of fluid’s four velocity is restricted within a suitable cone specified further below..

It is a common practice amongst relativists to express the thermodynamical properties of the fluid either relative to the Eckart frame or relative the Landau-Lifshitz frame. This tendency gives the impression that the laws of irreversible thermodynamics of relativistic fluids are tied to a particular frame, even though the primary variables do not single out such a frame. As we have mentioned in the introduction, a suggestion by Israel asserts that as long as considerations are restricted to states close to thermal equilibrium, there is some sort of ”gauge freedom” regarding the choice of the rest frame (or according to the comment $e$ of the previous page a ”gauge freedom” in the choice of the fluid’s four velocity). A consistent thermodynamical theory can be developed that is manifestly invariant under changes of the rest-frame $u$, as long as $u$ remains within the “cone” of opening angle $\epsilon$ defined by $u_{E}$ and $u_{N}$ (for further discussion on this cone see Appendix A) and this theory is the Israel-Stewart transient thermodynamics that we are aiming to develop in the following sections.

8 Global thermodynamical equilibrium

Of importance for the development of transient thermodynamics, is the identification of a class of fluid states refereed as equilibrium states. The primary variables for such class of states define a unique rest frame (and thus a unique fluid four velocity) and in this case naturally the thermodynamical laws are expressed relative to this special rest frame434343We ought to be aware however, that there exist systems that do not admit a rest frame, in the sense that the frame moves with the speed of light. This for instance occurs for the Hawking radiation field on the event horizon of a black hole. (for more discussion on this point see Appendix C or Ref. \refciteMTW). As we shall show with more details further ahead, transient thermodynamics deals with the description of states that are considered to be perturbations of equilibrium states and thus this later family of states needs to be specified precisely444444 Hiscock and Lindblom in Ref.\refciteHis1, identified the class of equilibrium states within transient thermodynamics by a different root than the one that we are going to follow. They first postulated a particular functional form for the entropy $S^{\mu}$ for fluid states and subsequently required that equilibrium states are those subject to: $\nabla_{\mu}S^{\mu}=0$. They arrived at the same conditions as those postulated by Israel in Ref.\refciteIsr1 and conditions $1)-5)$ that we are discussing in this section..

For a fluid mixture, the primary variables describing equilibrium states within transient thermodynamics satisfy the following five conditions (for a detailed discussion regarding the choice of these conditions, consult Ref.\refciteIsr1):

$1$) The entropy production vanishes i.e.

[TABLE]

$2$) There exists a unique hydrodynamical $4$ velocity $u$, $g(u,u)=-1$ such that the primary variables $(T^{\mu\nu},J_{A}^{\mu},S^{\mu})$ for all $A\in(1,2,....n)$, take the form:

[TABLE]

where $(\rho,P,n_{A})$ stand for the energy density, thermodynamical pressure and particle densities measured by an observer comoving with the flow defined by $u$, while $s=-S^{\mu}u_{\mu}$ is the entropy density perceived by that observer.

$3$) There exist an equation of state of the form $s=s(\rho,n_{A})$ from which the equilibrium pressure $P(\rho,n_{A})$ can be derived from the relation

[TABLE]

with the temperature $T$ and the thermal potentials $\Theta_{A}$, $A\in(1,..,n)$ defined from the Gibbs equilibrium relation

[TABLE]

(the origin of the last two fundamental relations are discussed in the Appendix C).

$4$) The thermal potentials $\Theta_{A}$ and the reaction rates $r_{i}$ obey

[TABLE]

$5$) The motion is rigid in the sense of Born i.e. satisfies

[TABLE]

(for properties of this type of fluid motions, see for instance discussion in Ref. \refciteSyn).

When these five conditions hold simultaneously, then states defined by the primary variables $(T^{\mu\nu},J_{A}^{\mu},S^{\mu})$ in (82), describes global thermodynamical equilibrium. To get insights into these states, let us define the inverse temperature vector $b^{\mu}$ via

[TABLE]

and note that this definition, in combination to $\Theta_{A}=\text{const.}$ and (86), implies that $b^{\mu}$ is a timelike Killing field454545We may add here that conformally invariant fluid theories (see discussion and references at the end of section $(13)$) have equilibrium states where $b^{\mu}$ is a conformal Killing vector field. We expect that this is due to the radiation equation of state $P=\frac{1}{3}\rho$ which is dictated by conformal invariance, although this claim needs to be checked in details. i.e. obeys

[TABLE]

This equation implies

[TABLE]

as well as the Tolman-Klein law:

[TABLE]

Thus states in global thermodynamical equilibrium are special. Besides the vanishing of the entropy production $\nabla_{\mu}S^{\mu}=0$, and the restrictive nature of the primary variables shown in (82), they require stationarity of the background spacetime $(M,g)$ and in addition the four velocity $u^{\mu}$ should be parallel to the timelike Killing field $b^{\mu}$ and these conditions are very restrictive.

On the other hand, states that obey conditions $1)$ through $3)$ describe states in a local thermodynamical equilibrium (as opposed to states describing global thermodynamical equilibrium). These states are characterized by the vanishing of entropy production i.e. $\nabla_{\mu}S^{\mu}=0$ and this property follows as a consequence of (82) coupled to (83) (see Appendix A for details regarding this point). Moreover, for these states, the fluid flow it is not any longer required to be rigid since condition $5)$ it is not required to hold and thus the background $(M,g)$ it is not required to be stationary464646There exist plenty of systems admitting states in local equilibrium. For instance, any state describing a simple perfect fluid propagating in an arbitrary $(M,g)$ satisfies conditions $1)$ to $3)$ but not in general conditions $4)$ to $5)$. Relativistic kinetic theory offers other states in local thermodynamical equilibrium. For a simple relativistic gas, a local Maxwellian distribution makes the collision integral in the relativistic Boltzmann equation to vanish and thus locally the entropy production vanishes. Moreover a local Maxwellian distribution, introduces a natural rest frame and the corresponding energy momentum tensor and particle currents have the form as in (82). Small deviations from a local Maxwellian distribution satisfy relation (100) derived further ahead. The interest reader is referred to Refs. \refciteEhl, \refciteO1,O2,O3,An1 for an introduction to this theory while for a relation of kinetic theory to phenomenology see sections $(3-7)$ of Ref. \refciteIsr2.. States in local (or global) thermodynamical equilibrium can be thought as comprising a $(n+4)$ dimensional space $\hat{E}$ parametrized by the $n-$thermal potentials $\Theta_{A}$, $A\in(1,...,n)$ and an inverse temperature vector $b^{\mu}$ defined in (87) and this observation will be useful further ahead.

In the above formulation of conditions $1)-5)$, the rest frame $u$ or the unique hydrodynamical four velocity defined in (82), played a prominent role. However, this prominence appears to subside by passing to a covariant form of the equilibrium Gibbs relation. To derive this version, note that the standard form of the Gibbs relation (84) in combination to (83) imply the following identities derived first by Israel[13]

[TABLE]

[TABLE]

where $X$ stands for an arbitrary function. Choosing $X=1$ in (92) yields (84) while for a simple fluid the choice $X=V:=\frac{1}{n}$ yields

[TABLE]

where $\hat{s}=sn^{-1}$ is the entropy per particle and $e$ is the internal energy per particle defined according to $\rho=n(m+e)$ (we are employing units so that $k=c=1$). However, the most important relation hidden in (92) arises by choosing: $X=u^{\mu}$. Remembering that we are dealing with states obeying $1)$ to $3)$ (or possibly $1)$ to $5)$), then for the choice $X=u^{\mu}$, the identity (92) implies474747Notice that in the following three relations, we denoted the primary variables that describe states in (local or global) equilibrium by $(T_{0}^{\mu\nu},J_{0A}^{\mu},S_{(0)}^{\mu})$ and the reason for introducing this notation will become clear shortly.

[TABLE]

while (83) yields

[TABLE]

Moreover, multiplying (91) by $u^{\mu}$ and after some algebra, we obtain the useful relation

[TABLE]

Relation (94) is the covariant version of (the equilibrium) Gibbs relation which involves only covariant objects and thus eliminates any quantity defined relative to a specific frame. This covariant version describes reversible transformations from the (global or local) equilibrium state parametrized by $(\Theta_{A},b^{\mu})$ to a nearby (global or local) equilibrium state484848Within the mathematical framework introduced in the Appendix C, the $d$-variations that appear in (94) are variations along a fiber $Y$ over a point on $(M,g)$ but leave us within the equilibrium manifold $\hat{E}$, see Appendix C. parametrized by $(\Theta_{A}+d\Theta_{A},b^{\mu}+db^{\mu})$. It is an aesthetically pleasing formula and as we shall see in the next section, plays an important role in the formulation of the transient thermodynamics.

9 The Principles of Transient Thermodynamics

In section 6, we have seen that the deficiencies in the Fourier-Navier-Stokes system appear to be cured by abandoning the formula for the entropy of off-equilibrium states arising by appealing to (CIT) and introducing instead the notion of the generalized entropy that receives quadratic contributions from dissipative fluxes.

The development of the transient (or causal) thermodynamics parallels similar route. Israel in Ref.\refciteIsr1, motivated from the relativistic kinetic theory of gases, put forward the hypothesis that an arbitrary non equilibrium fluid state, beyond the primary variables $T^{\mu\nu},J_{A}^{\mu}$ and $S^{\mu}$ is characterized by an additional (perhaps an infinite) set of auxiliary variables denoted collectively by $X_{(i)}^{\mu\nu\lambda...}$ where $i\in(1,2,....).$ Furthermore, he suggested that an equation of state (EOS) should exist of the form

[TABLE]

having the following properties:

a) For any equilibrium state (where all $X_{(i)}^{\mu\nu\lambda...}$ vanish) this (EOS) reduces to the linear relation between $(T_{(0)}{}^{\mu\nu}$, $J{{}_{A}}_{(0)}^{\mu}$, $S_{(0)}^{\mu})$ shown in (95).

b) Away from equilibrium states, and as long as consideration are restricted to states near equilibrium, then $S^{\mu}$ in (97), can be expanded in a Taylor series around a background equilibrium state and in such expansion contributions higher than quadratic, at a first instance, can be neglected.

The assumption that an (EOS) of the form (97) exists that obeys conditions a) and b) constitutes the backbone of transient thermodynamics. Assuming validity of a) and b), let $(\Theta_{A},b^{\mu})$ be an arbitrary point on the space of equilibrium states $\hat{E}$ so that $(T_{(0)}{}^{\mu\nu},J{{}_{A}}_{(0)}^{\mu},S_{(0)}^{\mu})$ stand for the primary variables describing this particular equilibrium state. Let now $(dS^{\mu},dT^{\mu\nu},dJ_{A}{}^{\mu})$ be an infinitesimal perturbations of this equilibrium state so that

[TABLE]

define a new state that remains “infinitesimally near” to $(T_{(0)}{}^{\mu\nu},J{{}_{A}}_{(0)}^{\mu},S_{(0)}^{\mu})$ but lies494949In the Appendix C, a mathematical framework is outlined where the notion of states close to the equilibrium manifold $\hat{E}$ acquires a well defined mathematical meaning. off-$\hat{E}$.

“The release of variation postulate” introduced in [13, 11], states that the infinitesimal perturbations $(dS^{\mu},dT^{\mu\nu},dJ_{A}{}^{\mu})$ in (98) are not independent but satisfy a (non equilibrium) covariant Gibbs relation of the form

[TABLE]

i.e. a relation that has the same functional form as the equilibrium Gibbs relation in (94) except that presently the perturbations505050Notice that the $d$-variations that appears in (99) are considered to be fiber variations i.e. variations that move us along the fiber $Y$ over a basis point and ”away” from the equilibrium manifold $\hat{E}$. These variations are distinct to the $d$-variations appearing in the equilibrium Gibbs relation (94). The underlying mathematical framework is briefly outlined in Appendix C. $(dS^{\mu},dT^{\mu\nu},dJ_{A}{}^{\mu})$ lead us off -$\hat{E}$. Recalling that $(T_{(0)}{}^{\mu\nu},J{{}_{A}}_{(0)}^{\mu},S_{(0)}^{\mu})$ obey (95), addition of (95) and (99) yields the following fundamental relation between the primary variables $(S^{\mu},T^{\lambda\mu},J_{A}{}^{\mu})$ and the parameters $(\Theta_{A},b^{\mu})$ describing the background equilibrium state

[TABLE]

where the term $Q^{\mu}$ takes care of the quadratic and higher order contributions in the Taylor series expansion of $S^{\mu}$ around the equilibrium state. In this formula, $P(\Theta_{A},b)$ is the thermodynamical pressure of the background equilibrium state and the perturbations $\delta J_{A}^{\mu}:=J^{\mu}-J_{0}{}^{\mu}$, $\delta T^{\mu\nu}:=T^{\mu\nu}-T_{0}{}^{\mu\nu}$ describe deviations from the equilibrium state.

Formula (100) has similar structure as the generalized entropy $s_{ge}(\vec{x},t)$ introduced in the development of (EIT) in section $(6)$. Depending upon the structure of the $Q^{\mu}$ term, the entropy vector $S^{\mu}$ may receive contributions from terms describing deviations from the background equilibrium state. As we shall see further ahead, the Eckart [9] theory and Landau-Lifshitz [10] theory are generated by taking $Q^{\mu}:=0$, while second order theories515151The terms “first order theories”, “second order theories” seem to have been coined by Hiscock and Lindblom in Ref.[14]. postulate that $Q^{\mu}\neq 0$. However before we discuss properties of the resulting theories, we first point out a few implications of the fundamental relation in (100).

As it stands, this relation combines the primary variables describing the non equilibrium state and the variables of a reference background equilibrium state and this mixing make difficult to extract out of (100) the relevant physics. However, one point worth recognizing is the non uniqueness property of the reference equilibrium state parametrized by $(\Theta_{A},b^{\mu})$. Israel and Stewart observe that if the parameters $(\Theta_{A},b^{\mu})$ that specify an equilibrium state $(T_{(0)}{}^{\mu\nu},J{{}_{A}}_{(0)}^{\mu},S_{(0)}^{\mu})$ which is near to $(T^{\mu\nu},J_{A}^{\mu},S^{\mu},X_{i}^{\mu\nu....})$, are displaced to nearby values $(\Theta^{\prime}_{A}=\Theta_{A}+\delta\Theta_{A},b^{\prime}_{\mu}=b_{\mu}+\delta b_{\mu})$, then (100) can be written in the equivalent form:

[TABLE]

[TABLE]

where terms $(\delta\Theta_{A})^{c}$ and $(\delta b_{\mu})^{c}$, with $c\geq 2$ are neglected. In turn, (101) implies that if terms quadratic in $(\delta\Theta_{A},\delta b_{\mu})$, are neglected, then both $(\Theta_{A},b_{\mu})$ and $(\Theta^{\prime}_{A}=\Theta_{A}+\delta\Theta_{A},b^{\prime}_{\mu}=b_{\mu}+\delta b_{\mu})$ can serve as a reference background equilibrium state for $(T^{\mu\nu},J_{A}^{\mu},S^{\mu},X_{i}^{\mu\nu....})$.

This freedom in the choice of the reference equilibrium state simplifies matter considerably. For a non equilibrium state525252In order to avoid technicalities, in the following analysis, we restrict attention to the case of a simple fluid. The case of fluid mixture can be treated similarly and details can be found in ref.\refciteIsr2. specified by $(T^{\mu\nu},J^{\mu},S^{\mu},X_{i}^{\mu\nu....}),$ Israel and Stewart assign a reference equilibrium state in the following manner: First they choose a four velocity $u^{\mu}$ within the cone of opening angle $\epsilon$ defined by $u_{E}$ and $u_{N}$ (see Appendix A for the definition of this cone). Once a choice of this $u^{\mu}$ has been made, the primary variables $(T_{(0)}{}^{\mu\nu},J{}_{(0)}^{\mu})$ identifying a reference equilibrium state that is near to $(T^{\mu\nu},J^{\mu},S^{\mu},X_{i}^{\mu\nu....}),$ are chosen so that the following fitting conditions hold:

[TABLE]

while the rest of the thermodynamical variables $s(u),T(u)$ and $P(u)$ for this reference state are constructed by appealing to the equilibrium equation of state and the equilibrium Gibbs relation (84). For the so defined reference equilibrium state, the fundamental relation (100) in view of (95) yields

[TABLE]

[TABLE]

which implies that the entropy density $s(x)=-u_{\mu}S^{\mu}$ of the actual state and the entropy density $s_{0}(x)=-u_{\mu}S_{0}^{\mu}$ of the reference equilibrium state as measured by the $u$-observer, obey

[TABLE]

i.e. the two densities agree to first order deviations from the equilibrium and differences between these densities appear in second order deviations. Moreover (104) shows that under the assumption that the fitting conditions (103) hold, then among all states with the same $(\rho(u),n(u))$ then $s(x)=-u_{\mu}S^{\mu}$ attains its maximum value at equilibrium, if and only if $Q^{\mu}$ is timelike and future directed.

Relation (104) has another important consequence: the thermodynamical pressure535353Notice that in addition, there exist two inter-related issues that needed to be clarified. First how one identifies the thermodynamical pressure $P(u)$ and secondly how one differentiates $P(u)$ from the bulk pressure $\pi(u)$ that appears in the formula (106) bellow. The analysis that follows addresses also these two issues. $P{(u)}$ for the off-equilibrium state as measured by the $u$ observer and the equilibrium pressure $P(\Theta,b)$ that appears in (100) satisfy

[TABLE]

i.e. the two pressures agree to first order in deviations from equilibrium and this property can be seen as follows: For the off-equilibrium state specified by $(T^{\mu\nu},J^{\mu},S^{\mu},X_{i}^{\mu\nu....})$, the thermodynamical pressure $P(u)$ and the bulk pressure $\pi(u)$ as perceived by the $u$ observer satisfy

[TABLE]

where $\Delta(u)^{\mu}{}_{\nu}$ is the projection tensor associated to the particular four velocity $u$ (relations (106) will become clear further below). Relative to the rest frame specified by $u$ and for a simple fluid, the thermodynamical pressure $P(u)$ can be defined either by appealing to the relation $s(u)=b\left(\rho(u)+P(u)\right)-\Theta(u)n(u)$ see equacion (289) in the Appendix C or by appealing to the non equilibrium Gibbs relation, see equation (286) in the same Appendix. The latter equation implies:

[TABLE]

and thus for any process that maintains the entropy particle $\hat{s}:=sn^{-1}$ fixed, obeys:

[TABLE]

which identifies the thermodynamic pressure $P(u)$ according to

[TABLE]

This formula is util to establish validity of (105). Indeed we find

[TABLE]

where the partial derivatives are computed at fixed entropy per particle and particle density and we passed to the second equality by appealing to relations (103) and (104). Therefore the thermodynamical pressure $P(u)$ in (106) and the equilibrium pressure $P(\Theta,b)$ appearing in (100), agree to first order deviations from equilibrium. Since we are interested only in the dynamics of first order deviations from equilibrium, here after we do not differentiate between the two pressures. Parenthetically, notice that (108) offers the operational means of differentiating545454Although the derivation of (108) presupposes a simple fluid, a similar formula holds for the case of a fluid mixture. Details of this can be found in Ref.[75]. between $P(u)$ and $\pi(u)$.

We now consider some implications arising from the non-uniqueness of the four velocity $u$ that enters into the theory through the fitting conditions (103). Within the framework of the transient thermodynamics, the four velocity $u$ that enters into these fitting conditions is arbitrary except that it is restricted to lie within the cone of opening angle $\epsilon$ specified by $u_{E}$ and $u_{N}$. Once however, a choice of $u$ has been made, the primary variables $T^{\mu\nu}$ and $J^{\mu}$ can be decomposed555555In this and the remaining sections, we write $\rho(u),P(u),h(u),\tau(u)^{\mu}{}_{\nu}$ etc. in order to remind the reader that these quantities are measured by the $u-$observer, and ought to keep in mind that in general the decompositions in (110), (111), (112) and (113) depend upon the chosen $u$ and thus are frame dependent. according to

[TABLE]

[TABLE]

where $h(u)^{\mu},n(u)^{\mu}$ is the energy flow and particle “drift” relative to the $u$-frame, $\Delta^{\mu\nu}(u)$ is the projection tensor565656In order to avoid proliferation of new symbols, the four velocity $u$ in (110) should not be confused with the four velocity $u$ that defines the unique rest frame in equilibrium states, (see (82)). and the spatial symmetric pressure tensor $\tau(u)^{\mu\nu}$ is decomposed according to

[TABLE]

where $\pi(u)$ and $\pi(u)^{\mu\nu}$ stand for the bulk and shear stresses (as measured by the $u$-observer). Combining these decompositions with the fitting conditions (103) and the relation $T(u)s(u)={\rho(u)+P(u)}-T(u)\Theta(u)n(u)$ which implies: $P(u)b^{\mu}=s(u)u^{\mu}-\rho(u)b^{\mu}+\Theta(u)[J^{\mu}-n(u)^{\mu}]$, we find that the fundamental relation (100) takes the equivalent form:

[TABLE]

The dependance of the right handside of this relation upon the arbitrarily chosen four velocity $u$ raises delicate questions regarding the interpretation of the theory and the implementation of the second law.

For states near equilibrium, the variables $\tau^{\mu\nu}(u),\rho(u),n(u),$ $P(u),s(u),T(u),\Theta(u)$ in (110-113), except the term $Q^{\mu}$, are all invariant to first order deviation from equilibrium and thus they are considered as been “frame independent”. To define this later term more precisely, let $Z(u)$ stands for an arbitrary thermodynamical variable measured by the observer $u$, and let a change of the rest frame which is implemented by a change: $u^{\mu}\to{\hat{u}}^{\mu}$. Under such change, the variation $\delta Z:=Z(\hat{u})-Z(u)$ can be cast in the form:

[TABLE]

where $(a_{0},a_{1},a_{2}....)$ are well defined functions and $O_{1},O_{2},...$ stand for terms of first, second order….. deviations from equilibrium575757Notice that once a first order change in the frame is implemented via $u^{\mu}\to\hat{u}^{\mu}=u^{\mu}+\hat{\epsilon}^{\mu}$ then besides the term $\epsilon:=O_{1}$ is also introduced the smallness parameter $\hat{\epsilon}$ (see the formulation of the following lemma) and the terms $(a_{0},a_{1},a_{2}....)$ may contain this new smallness parameters $\hat{\epsilon}$.. Variable characterized by the property that $a_{1}:=0$ are considered to be frame-invariant and the following two lemmas describe the transformations properties of several thermodynamical variables under a change of the rest frame (Israel [13]).

Lemma 9.1**.**

Let ($u$, $\hat{u}$) be two arbitrary (future directed) timelike unit vectors within the cone of opening angle $\epsilon$ spanned by $u_{E}$ and $u_{N}$ and let

[TABLE]

with $\delta^{\mu}$ subject to: $\delta^{\mu}=\hat{\epsilon}^{\mu}:=\hat{\epsilon}\leq O_{1}$ so that $\hat{u}^{\mu}-u^{\mu}=\hat{\epsilon}\leq O_{1}$. Let the primary variables $T^{\mu\nu}$ and $J^{\mu}$ are decomposed relative to the $u$-frame according to (110) and (111) while relative to the to the $\hat{u}$-frame these decompositions are written in the form:

[TABLE]

Then under the transformation under first-order changes of the rest-frame $u^{\mu}$ described by

[TABLE]

*that follows (114), the following relations hold:

[TABLE]

[TABLE]

[TABLE]

Lemma 9.2**.**

Under the same assumptions as in the previous lemma, the combination

[TABLE]

is frame invariant i.e.

[TABLE]

The proofs of these two Lemmas are discussed in the Appendix D (see also derivation in Ref[75]). For completeness, we also mention that the the variation $\delta$ of the other thermodynamical variables like $(P(u),n(u),s(u),T(u),\Theta(u))$ under a first order change in the frame $u^{\mu}\to\hat{u}^{\mu}=u^{\mu}+\hat{\epsilon}^{\mu}$ described above, are all of order $\hat{\epsilon}O_{1}$ with exception the variation $\delta n^{\mu}$ in the particle drift $n^{\mu}(u)$. Like the energy flux $h(u)^{\mu}$, its variation is of order $\hat{\epsilon}$.

As a first consequence of these Lemmas, it is worth noticing that the spatial vector $q^{\mu}(u)$ in (119) can be interpreted as defining the frame independent heat flux vector, since relative to the Eckart frame this $q^{\mu}(u)$ reduces to the energy flux585858For the case of fluid mixture, the invariant heat flux $q^{\mu}(u)$ is replaced by $q_{A}{}^{\mu}(u)=h(u)^{\mu}-\frac{\rho(u)+P(u)}{n_{A}(u)}n_{A}(u)^{\mu}$ which has the property that if $u$ is identified by the one of the $u_{A}$ satisfying $J_{A}^{\mu}=n_{A}u_{A}^{\mu}$ then $q_{A}{}^{\mu}(u_{A})=h(u_{A})^{\mu}$. Notice however, that for a fluid mixture, the drifts of the particles taken relative to the energy frame are more convenient variables. $h^{\mu}(u_{N})$, while relative to the energy frame specified by $u_{E}$, this $q(u_{E})^{\mu}$ is proportional to the particle drift $\nu^{\mu}$.

In addition these lemmas, permit us to eliminate the explicit dependance of the various thermodynamical variables upon the arbitrarily chosen $u^{\mu}$. For instance, if in (113), we eliminate the energy flux $h^{\mu}(u)$ in favor of the heat flux $q^{\mu}(u)$, then ($S^{\mu},J^{\mu})$ reduce to

[TABLE]

and if for the moment we ignore the $Q^{\mu}$ contribution in $S^{\mu}$, then the resulting $S^{\mu}$ is manifestly frame invariant under frame change as long as $O_{2},O_{3}....$ deviations from equilibrium are neglected. For this reason in the entropy flux $S^{\mu}$ in (121), we write $s,n$.. instead of $s(u),n(u)$ etc and this convention will used further ahead.

We now consider the implementation of the second law i.e. impose the inequality $\nabla_{\mu}S^{\mu}\geq 0$. To do so, we form the covariant divergence of (100), which in view of the conservation laws satisfied by the primary variables, yields:

[TABLE]

Since (96) implies

[TABLE]

therefore (122) takes the form

[TABLE]

By inserting the decompositions of $T^{\mu\nu}$ and $J^{\mu}$ in (110) in view of the special forms of $T_{0}{}^{\mu\nu}$ and $J_{0}{}^{\mu}$, we find

[TABLE]

[TABLE]

where in above and here after $a_{\mu}=u^{\nu}\nabla_{\nu}u_{\mu}$ stands for the four acceleration595959Note a misprint in equation $(2.28)$ of Ref.[11]. The term proportional to the four acceleration in that equation should appear with a negative sign. It is however quoted correctly in [13], see equations $(72,73)$ in that reference. of the velocity field606060For the case of a fluid mixture, the term $n(u)^{\mu}\nabla_{\mu}\Theta(u)$ in $W$ should be replaced by $\sum\limits_{A}n_{A}(u)^{\mu}\nabla_{\mu}\Theta_{A}(u)$. $u$.

It follows from the last two formulas that in order to impose and work out consequences of the second law, the term $Q^{\mu}$ needs to be specified. This amounts to the specification of a particular theory and for completeness, we first treat the class of first order theories and these theories are generated by assuming that $Q^{\mu}:=0$ in the fundamental relation (100) (or equivalently in (121)).

10 The Eckart and Landau-Lifshitz first order theories

The choice $Q^{\mu}=0$ in (121), implies that the entropy vector $S^{\mu}$ reduces to

[TABLE]

which is the entropy vector due to the convected motion (first term) followed by an irreversible contribution generated by the heat flux $q^{\mu}$ (second term). Even though this $S^{\mu}$ has a simple form, we should not loose side of the fact that we are dealing with fluid states admitting dissipation and heat conduction and in the right hand side of (127) is hidden the choice of the rest frame $u$. Depending upon the choice of the rest frame, these class of theories include as particular case the Landau-Lifshitz and Eckart theories. Below we shall analyze only these latter theories (a general treatment of first order theories can be found in Ref. \refciteHis2).

For the Landau-Lifshitz theory (for a detailed account of this theory, see for instance [10]), one chooses the rest frame $u$ to be the unique timelike eigenvector $u_{E}$ of the energy momentum tensor $T_{\mu\nu}$. For this choice, the decompositions616161In the formulas (128) through (131), for simplicity of the presentation we write $u$ instead of $u_{E}$. in (110-111) imply:

[TABLE]

and upon substituting these decompositions in (124) with $Q^{\mu}=0$, we get

[TABLE]

where in above and here after $\langle\nabla_{\mu}u_{\nu}\rangle$ signifies the symmetric traceless part of $\nabla_{\mu}u_{\nu}$. Returning to the definition of the heat flux vector $q^{\mu}$ defined in (119) and taking $h^{\mu}=0$, we write 129 in the form:

[TABLE]

Implementation of the second law leads to the phenomenological relations:

[TABLE]

where in a standard notation, $\kappa$ is the coefficient of thermal conductivity and $(\zeta_{v}$, $\zeta)$ are the coefficients of bulk and shear viscosity.

The Eckart theory (for an introduction to this theory, see for instance Ref. \refciteEck), is generated by choosing the rest frame to be the four velocity $u_{N}$ obeying $J^{\mu}=nu_{N}^{\mu}$. For this choice, we have626262Here again in the formulas (132-134) that follow, $u$ stands for $u_{N}$.

[TABLE]

and upon substituting these decompositions in (124) with $Q^{\mu}=0$, a short calculus shows that

[TABLE]

Relative to this frame, formula (119) shows that the heat flux $q^{\mu}$ is identical to energy flux i.e. $h^{\mu}=q^{\mu}$ and by similar analysis as for the case Landau-Lifshitz case, we deduce the following phenomenological relations:

[TABLE]

where the coefficients $(\kappa$, $\zeta_{v}$, $\zeta)$ have the same meaning as for the Landau-Lifshitz theory.

Although the theories of Landau-Lifshitz and Eckart are simple theories, nevertheless as we mentioned in the introduction, they are pathological theories. They exhibit infinite propagation of disturbances, instabilities of the equilibrium states and for a critical analysis of the problems plugging them see for instance Refs. \refciteHis2,His1.

11 Second Order Theories: The Hydrodynamical Approximation

In this section, we consider second order theories i.e. theories generated by choosing a non vanishing term $Q^{\mu}$ in the fundamental relation (100), and the choice that we discuss below generates the Israel-Stewart transient thermodynamics.

Israel and Stewart, motivated from relativistic kinetic theory of diluted gases, proposed that the entropy flux vector $S^{\mu}$ (and thus $Q^{\mu}$), should be independent of the gradients of $J^{\mu}$ and $T^{\mu\nu}$ and should be quadratic in the deviations from the state of local equilibrium. To describe these deviations in a practical and intuitive manner, they first choose an admissible four velocity $u$ and make use of the expansions in (110-112) that define variables $\pi(u),\pi(u)^{\mu\nu},h(u)^{\mu}$ and also use the invariant heat flux $q(u)^{\mu}$ defined in (119). Moreover Israel and Stewart postulate that of the infinite number of the auxiliary variables $X_{(i)}^{\mu\nu\lambda...}$ with $i\in(1,2,....)$, that would appear in the exact equation of state shown in (97), none of them will appear explicitly in the $Q^{\mu}(u)$ term.

Denoting here after by $Q^{\mu}(u)$ the term $Q^{\mu}$ evaluated relative to the $u$-frame, Israel-Stewart proposed that for a simple fluid $Q^{\mu}(u)$ should have the form:

[TABLE]

where $R^{\mu}(u)$ stands for

[TABLE]

and $\alpha_{j}$, $j\in(0,1)$, $\beta_{i}$, $i\in(0,1,2)$, are undetermined coefficients (notice that the $R^{\mu}(u)$ term is free of arbitrary functions636363An $R^{\mu}(u)$ term of the form $T(\rho+P)R^{\mu}(u)=[\frac{1}{2}\gamma_{1}u^{\mu}h^{\nu}h_{\nu}+\gamma_{2}\tau^{\mu\nu}h_{\nu}+\gamma_{3}\pi h^{\mu}]$ combined with the requirement that the entire $Q^{\mu}(u)$ should be frame-independent to order $O_{2}$, demands $\gamma_{1}=\gamma_{2}=1$ and $\gamma_{3}=0$. For these values of the $\gamma_{i}$, $i\in(1,2,3)$ the term $Q^{\mu}$ in (135) becomes frame independent and for this reason the explicit dependance of $T$, $\pi$, $\pi^{\mu\nu}$,…etc upon $u$ has been omitted. For further discussion regarding the structure of the $Q^{\mu}(u)$ term in (135), the reader is referred to original article of Israel [13] and Israel and Stewart Ref.[11] (see also Ref.\refciteF1).).

For a fluid mixture, it is more convenient to express $Q^{\mu}(u)$, in terms of the $n$-particle drifts

[TABLE]

where $u_{E}{}^{\mu}$ specifies the energy frame. In terms of these drifts and in the notation of ref. Ref.\refciteIsr1, the term $Q^{\mu}(u)$ reads:

[TABLE]

where the summation over the indice $(A,B)$ extends over all $n$-particle species, and the coefficients $(a_{\ 0}^{A},a_{\ 1}^{A},\beta_{0},\beta_{1}^{AB},\beta_{2})$ with $A,B\in(1,2,...n)$ are undetermined functions, while $R^{\mu}$ in (137) has the form as in (136).

As for the case of first order theories, the phenomenological equations for the Israel-Stewart transient thermodynamics follow by imposing the second law and for generality purposes, we analyze the implications of the second law for the choice of $Q^{\mu}(u)$ shown in (137).

A direct substitution of this $Q^{\mu}(u)$ in (125) yields a long expression. To minimize algebra, we use the W term shown in (126) and the form of $R$ in (136) and compute $W-\nabla_{\mu}R^{\mu}$:

[TABLE]

where $a_{\mu}$ is the four acceleration of the four velocity $u$. Moreover eliminating the drifts ${n}^{\mu}_{A}(u)$ in favor of the the $n$ particle drifts $\nu_{A}^{\mu}$ relative to the energy frame using:

[TABLE]

and by appealing to the identity (91), written in the form

[TABLE]

one concludes that:

[TABLE]

Taking into account that

[TABLE]

and by eliminating the $h^{\mu}a_{\mu}$ from (141) we obtain finally

[TABLE]

With this simplification, we now have

[TABLE]

Following the approach[11], we neglect the gradients of the coefficients $(a_{\ 0}^{A},a_{\ 1}^{A},\beta_{0},\beta_{1}^{AB},\beta_{2})$ and thus the right hand side of (144) can be written in the form:

[TABLE]

where $f^{A}_{\ \mu},f$ and $f_{\mu\nu}$ are functions to be determined. Carrying out the differentiation in the right hand side of (144) and grouping terms we arrive at

[TABLE]

where in above and here after an overdot signifies differentiation along $u^{\mu}$ i.e. $\dot{\pi}=u^{\mu}\nabla_{\mu}\pi$, etc. Using the decomposition $\tau^{\mu\nu}=\pi\Delta^{\mu\nu}+\pi^{\mu\nu}$, the right hand side of (146) yields:

[TABLE]

Since $h_{\lambda}u^{\lambda}=0$ and $u^{\mu}u_{\mu}=-1$, the following identity holds:

[TABLE]

and this identity transforms (147) into the form:

[TABLE]

By inspection of the right hand side, we can write down the phenomenological relations between $(\pi,\pi^{\mu\nu},\nu^{\mu}_{A}),A\in(1,2,....n)$ that enforce the second law. For this, it is sufficient to assume that $\pi$, $\pi^{\mu\nu}$ and $\nu^{\ \mu}_{A}$ depend linearly on the “strains” i.e. $\nabla_{\mu}u^{\mu}$, $\langle\nabla_{\mu}u_{\nu}\rangle$, $\nabla_{\mu}\Theta_{A}$ i.e. to set

[TABLE]

[TABLE]

[TABLE]

where $k_{AB}$ $A,B,\in(1,2,...,n)$, is an $(n\times n)$ a semi-positive matrix with real entries and the angular bracket in the last equation signifies:

[TABLE]

Equations (150, 151, 152) are the phenomenological relations that follow by imposing the second law $\nabla_{\mu}S^{\mu}\geq 0$ for the choice $Q^{\mu}$ shown in (137). Originally, they have been derived by Israel in Ref.\refciteIsr1, although in this original article Israel analysis includes the case where there inter-particle reactions in a fluid mixture.

As they are derived here, they are valid for any rest frame specified by a four velocity $u$ as long as this $u$ lies between $(0,min(\epsilon_{1},\epsilon_{2},....\epsilon_{n}))$ where $\epsilon_{i}$ is the opening angle defined by $u_{E}$ and the four velocity $u^{\mu}{}_{I}$ parallel to the corresponding $I-$type particle $J^{\mu}_{I}$ (see comments in the Appendix A). It might be worth mentioning that even though in the derived equaciones (150-152) appear explicitly the $n$-particle drifts $\nu^{\mu}_{\ A}:=n_{A}{}^{\mu}(u_{E}),A\in(1,...,n)$ eventually these drifts can be eliminated by appealing to (139) which shows that (150-152) hold relative to an ”arbitrary” fram $u$.

They simplify slightly whenever they are expressed relative to a particular frame like the energy frame or the particle frame646464For a fluid mixture the particle frame is not well defined, or more precisely there exist $n$ such frames specified by the corresponding $u_{A}$. Relative to each of these frames the corresponding species of type $A$ is at rest even though the other species exhibit a drift relative to this particular frame. Often, for a fluid mixture, the material four velocity $u_{M}$ is introduce via $u_{M}{}^{\mu}=\sum\limits_{A=1}^{n}\frac{w_{A}J_{A}^{\mu}}{w_{A}n_{A}}$ where $w_{A}$ are arbitrary weights and $n_{A}=-u^{A}_{\mu}J_{A}^{\mu}$. Relative to this ”material frame” a heat flux $q^{\mu}$ is defined via $q^{\mu}=h^{\mu}(u_{M})$. We shall not write the phenomenological equations relative to this material frame. They can be derived by a straightforward generalization of the approach of this section.. For a simple fluid and relative to the energy frame, the phenomenological equations are obtained from (150, 151, 152) by setting $h^{\mu}=0$ everywhere, removing the summation symbols and the indices $(A,B)$. They reduce to the form:

[TABLE]

[TABLE]

[TABLE]

where in these equations $u^{\mu}$ stands for $u_{E}{}^{\mu}$, $\Delta^{\mu\nu}=\Delta^{\mu\nu}(u_{E})$ etc.

For a simple fluid, it is often convenient to eliminate the particle drift $\nu^{\mu}$ from (154, 155, 156) in favor of the frame invariant heat flux vector $q^{\mu}$ defined in (119). Evaluating $q^{\mu}$ in the energy frame, yields

[TABLE]

Returning to the form of $Q^{\mu}$ in (137), removing the summation symbols, eliminating $\nu^{\mu}$ in favor of $q^{\mu}$ and setting $h^{\mu}=0$, we obtain a “new” $Q^{\mu}$ term for a simple fluid. After some algebra the phenomenological equations resulting from the modified expression $\nabla_{\mu}S^{\mu}\geq 0$ in (144) are:

[TABLE]

[TABLE]

[TABLE]

and these are the equations given in Israel [13] and Israel and Stewart [11]. It is understood that in (157-159) $u$ stands for $u_{E}$.

The structure of the phenomenological equations for a simple fluid relative to the Eckart frame, can be obtained using the term $Q^{\mu}$ shown in (135, 136) and recalling that for this frame $W-\nabla_{\mu}R^{\mu}$ reduces to:

[TABLE]

The resulting phenomenological equations have the form

[TABLE]

[TABLE]

[TABLE]

where in these equations $u$ stands for $u_{N}$.

The coefficients $\alpha_{0},\alpha_{1},\beta_{1}$ in (157,158,159) and their barred versions in (161,162,163) are related via

[TABLE]

and these relations are important in proving that the phenomenological equations (157-159) and (161-163) are equivalent to first order deviations from equilibrium. In order to establish that property, we should keep in mind that the four velocities $u_{E}$ and $u_{N}$ are related via

[TABLE]

as well as the following relation:

[TABLE]

discussed in Ref.\refciteIsr1.

The sets of the phenomenological equations656565As derived above, these equations are equivalent to an $O_{1}$ accuracy in the change of frame. However the stability properties of these equations need to be accessed. In particularly the gauge freedom associated with different choices of a rest frame calls for a thorough investigation of their stability properties. It is conceivable that different choices of the four velocity $u$ may lead to different stability properties. For an illuminating discussion of that issue consult Refs.\refciteRP1,RP2. derived above yield the evolution equations for the heat flux, bulk and shear stresses and these equations combined to the equations arising from the conservation laws yield a closed system of equations. Their solutions describe the behavior of near equilibrium states whose evolution is compatible with the second law. Although the implementation of the second law is welcomed, by itself, it is not a sufficient reason for the physical acceptance of the theory. As it is clear from the so far analysis, of central importance is the issue whether the set of dynamical equations constitutes a hyperbolic system of equations or the stronger restriction whether they constitute a causal set of dynamical equations at least for states near equilibrium. For the moment within the context of tarnsient thermodynamics it is not known whether the resulting equations satisfy that constraint . However there are strong evidences that the theory rests on strong foundations.

Israel and Stewart in Ref.\refciteIsr2 went outside the realm of phenomenology and derived the equations of transient thermodynamics from microphysical considerations. Starting from the relativistic Boltzmann equation for a dilute gas and within Grads approximation [42] they were able in Ref. \refciteIsr2 to derive the equations of transient thermodynamics by evaluating the first three moments of the Boltzmann equation666666Specifically in [11] they have shown that the first three moments of the distribution function $f(x,p)$ i.e. $\hat{J}^{\mu}(x)=\int f(x,p)p^{\mu}d\Omega$, $\hat{T}^{\mu\nu}(x)=\int f(x,p)p^{\mu}p^{\nu}d\Omega$, $\hat{A}^{\mu\nu\lambda}(x)=\int f(x,p)p^{\mu}p^{\nu}p^{\lambda}d\Omega$, as a consequence of the Boltzmann equation satisfy: $\nabla_{\mu}\hat{J}^{\mu}=\nabla_{\mu}\hat{T}^{\mu\nu}=0$, $\nabla_{\mu}\hat{A}^{\mu\nu\lambda}=\hat{I}^{\nu\lambda}$ with $\hat{I}^{\nu\lambda}=\int C[f]p^{\nu}p^{\lambda}d\Omega$ is the second moment of the collision term $C[f]$ for the Boltzmann equation. . Moreover they were able to evaluate the five coefficients $(\alpha_{i},\beta_{i})$ in (161-163) and they have shown that these coefficients are purely thermodynamical functions i.e. independent of the cross section that enters in the collision term in Boltzmann’s equation. Subsequently they evaluated the wave fronts speeds about the equilibrium state and they found that all of them are finite and causal. These conclusions shows that transient thermodynamics at least for the coefficients $(\alpha_{i},\beta_{i})$ predicted by the relativistic Boltzmann equation avoids infinite propagation of disturbances.

The same conclusion has been also reached by Hiscock and Lindblom in [14] who have shown causality for transient thermodynamics holds for a wider range of circumstances. Moreover Hiscock and Lindblom in [14] addressed the stability property of these equilibrium states by examining the behavior of small perturbations about equilibrium states. They demonstrated a striking connection between causality and stability: at least within the framework of transient thermodynamics, these notions are equivalent in the following sense. If the theory possesses stable equilibrium states, then linear perturbations propagate causally and conversely i.e. if the characteristic velocities are subluminal (and the perturbation equations form a hyperbolic system) then the stability of equilibrium states is guaranteed676767Since the theory admits a gauge freedom regarding the choice of the rest frame, issues related to the stability properties under change of the frame are needed to be clarified. For a discussion regarding that point and further subtleties see Refs: \refciteRP1,RP2 and further references therein.. The stability of equilibrium states implies that all solutions of the perturbations equations about the equilibrium state with regular initial data are bounded functions of time and this property has been established by constructing an appropriate energy functional quadratic in the perturbations. These results suggest that transient thermodynamics rectifies the predictions of Eckart and Landau-Lifshitz first order theories.

Even though all so far evidences regarding transient thermodynamics are encouraging, signaling a robust theory, nevertheless more work is needed. As we have already mentioned, for instance, it would of interest to investigate whether the dynamical equations constitute a symmetric-hyperbolic system of equations, their causality etc.

We finish the treatment of transient thermodynamics by mentioning that various aspects of the theory can be found in Refs: \refciteHis3, \refciteHis4, \refciteBel, \refciteRMa1, \refciteF1, \refciteCMon, \refciteJS.

12 Relativistic (REIT)

In this section, we introduce an alternative theory of non equilibrium thermodynamics describing states of relativistic fluids and this theory developed by Liu, Müller and Ruggeri in Ref. \refciteMul6 (and is discussed in more detail in chapters $(6)$ in Ref. \refciteMul4). This theory can be considered as a relativistic extension of the classical rational extended thermodynamics (REIT) that we briefly introduced at the end of section 6 and for this reason often in the sequel we refer to it as relativistic (REIT).

Motivated by the desire to place irreversible thermodynamics of fluids on a solid mathematical foundation, the authors in Ref. \refciteMul6 introduced a new theory that deals with states of a relativistic simple fluid. The distinguishing feature of this theory is that the dynamical fluid equations, by appropriate restrictions, constitute a symmetric hyperbolic, causal system and thus finite propagation of disturbances is automatically built in.

In this theory, the $10$ components of the energy momentum tensor $T^{\mu\nu}=T^{\nu\mu}$ and the $4$ components of the particle current $J^{\mu}$ are consider to be the dynamical variables686868In the terminology of the theory of constitutive relations these variables are refereed as the basic variables. describing the fluid. They satisfy the equations696969 The structure of eqs. (167,168), the trace free property of $I^{\mu\nu}$ and the relation $A^{\mu\nu}{}_{\nu}=m^{2}c^{2}J^{\mu}$ are motivated by relativistic kinetic theory of a simple gas, see the comment 66 in the previous page.:

[TABLE]

[TABLE]

where $A^{\mu\nu\lambda}$ is a completely symmetric tensor field while $I^{\mu\nu}$ is symmetric and traceless $I^{\mu}{}_{\mu}=0$ and $A^{\mu\nu}{}_{\nu}=m^{2}c^{2}J^{\mu}$ so that $\nabla_{\mu}A^{\mu\nu}{}_{\nu}=0$. Because of these symmetries, the system in (167, 168) involves $14$ equations, which is equal to the number of the unknown components in $J^{\mu}$ and $T^{\mu\nu}$. Thus (167 -168) could serve as the dynamical equations for the $14$ unknown fields, provided that

[TABLE]

i.e. the fields $A^{\mu\nu\lambda}$ and $I^{\mu\nu}$ are viewed as a set of constitutive relations.

In addition to the balance laws (167, 168), the authors complete the system by including an entropy vector $S^{\mu}$ which is also a constitutive function707070It is of interest to point out here the different philosophy that underlies the present theory and the transient theory developed in the previous sections. Whereas for the transient theory the entropy vector $S^{\mu}$ (see eq. (97)), depends upon the primary variables and perhaps an infinite set of auxiliary variables $X_{(i)}^{\mu\nu\lambda...}$ with $i\in(1,2,....)$, for the present theory $S^{\mu}$ depends only upon the primary variables i.e. $S^{\mu}=S^{\mu}(J^{\mu},T^{\mu\nu})$. i.e.

[TABLE]

and demand that for any solution717171 A solution $(J^{\mu},T^{\mu\nu})$ of the field equations (167,168) is refereed in [22, 21] as a thermodynamical property. $(J^{\mu},T^{\mu\nu})$ of (167,168) the entropy four vector $S^{\mu}$ obeys:

[TABLE]

The authors place severe constraints upon the form of the constitutive functions $A^{\mu\nu\lambda}(J^{\mu},T^{\mu\nu})$, $I^{\mu\nu}=I^{\mu\nu}(J^{\mu},T^{\mu\nu})$ and $S^{\mu}=S^{\mu}(J^{\mu},T^{\mu\nu})$ by appealing to the following three principles:

• a)

Entropy Principle

• b)

Principle of Relativity

• c)

Hyperbolicity.

Since the applications of these principles as means to pin down the structure of the constitutive functions involve long computations, in this section, we only highlight the essential steps and frequently refer the reader to the original article of Ref \refciteMul6 and to chapter727272Actually, chapter $6$ in Ref. \refciteMul4 provides a detail and systematic treatment of the theory and the interested reader is recommended to go through this chapter. $(6)$ in the monograph of Ref.\refciteMul4 for further details.

The authors begin by breaking the frame invariance of the theory and employ the Eckart737373In principle one could however decompose $J^{\mu}$ and $T^{\mu\nu}$ relative to any arbitrary four velocity $\hat{u}^{\mu}$ and carry out the analysis of this section relative to this new frame $\hat{u}^{\mu}$. This freedom in the choice of the rest frame raise the question whether the resulting theories are physically equivalent. As long as one restricts attention to states near equilibrium ( a term that needs to be defined precisely within the theory), one expects identical situation as for the case of the transient thermodynamics, although these expectations ought to be checked in details.

frame to perform the calculations. Accordingly and relative to this frame, they decompose the energy momentum tensor $T^{\mu\nu}$ and the particle current $J^{\mu}$ according to:

[TABLE]

where as before the quantities747474 Since all the subsequent computations are done relative to the Eckart frame and thus there is no danger of confusion, in this section, we drop the dependance of $(\rho,P,\pi,q^{\mu},\pi^{\mu\nu})$ upon $u$ (this dependance introduced in the decompositions (110-111) as a reminder that they are frame dependent). $(\rho,P,\pi,q^{\mu},\pi^{\mu\nu},n)$ have the same interpretation as those appearing in the expansions shown in (110-111). Notice that due to the signature employed in Ref.\refciteMul6, the components of the projection tensor $\Delta^{\mu\nu}$ are given by $\Delta^{\mu\nu}=\frac{1}{c^{2}}u^{\mu}u^{\nu}-g^{\mu\nu}$, while the four velocity $u$ is normalized according to $g(u,u)=c^{2}$. Moreover for the present section, the energy flow $h^{\mu}$ in (110) is denoted by $q^{\mu}$ and is refereed as heat flux while relative to the Eckart frame, the particle drift $n^{\mu}$ defined in (111) is vanishing identically.

For latter use, notice that $\pi^{\alpha\beta}=\left(\Delta^{\alpha}_{\ \mu}\Delta^{\beta}_{\ \nu}-\frac{1}{3}\Delta^{\alpha\beta}\Delta_{\mu\nu}\right)T^{\mu\nu}$ while the thermodynamical pressure $P$, bulk pressure $\pi$, energy density $\rho$, heat flux vector $q^{\mu}$ and particle density $n$ are determined by

[TABLE]

In the treatment that follows, the $14$ components of $(J^{\mu},T^{\mu\nu})$ are traded757575As it will become clear further ahead, this trade off make the physical content of the theory more accessible to intuition, although as we have already mentioned the freedom in the choice of the frame needs to be treated with care. in favor of the $14$ fields $(\rho,P+\pi,q^{\mu},\pi^{\mu\nu},n,u^{\mu})$ defined in (172, 173).

The principle of relativity, when applied to the constitutive relations $A^{\mu\nu\lambda}(J^{\mu},T^{\mu\nu})$, $I^{\mu\nu}=I^{\mu\nu}(J^{\mu},T^{\mu\nu})$ and $S^{\mu}=S^{\mu}(J^{\mu},T^{\mu\nu})$, dictates that these objects should remains form invariant under arbitrary coordinate transformations which means that they behave as isotropic tensor-functions with respect to such transformations. Although one could study the structure of such tensor fields, the authors avoid that root. Instead, by employing the components of $(u^{\mu},q^{\mu},g^{\mu\nu},\pi^{\mu\nu})$ and scalars, they constructed the most general form of the tensors $I^{\alpha\beta}$ and $A^{\alpha\beta\mu}$ that are linear in $(\pi$, $\pi^{\alpha\beta},q^{\alpha})$. They have the form:

[TABLE]

[TABLE]

where $(B^{\pi}_{\ 1},B_{3},B_{4})$ respectively $(C^{0}_{\ 1},C^{\pi}_{\ 1},C_{3},C_{5})$ are functions of $(n,\rho)$ to be determined and $m$ is a mass scale. A similar construction shows that the entropy current $S^{\alpha}$ quadratic in $(\pi,\pi^{\alpha\beta},q^{\alpha})$ reads:

[TABLE]

where again $(A^{\pi}_{\ 1},A^{\pi^{2}}_{\ 1},A^{q}_{\ 1},A^{t}_{\ 1},A^{0}_{\ 2},A^{\pi}_{\ 2},A^{0}_{\ 3})$ are functions to be determined while $(n,s)$ are the particle number density and entropy per particle as measured relative to the Eckart frame. The determination of the $14$ coefficients $(A,...B,...C,...)$ that appear in (174-176) is a major undertaking and in this task, the entropy principle plays an important role.

A clever device to take a maximal advantage of the entropy principle was put forward by Liu in Ref. \refciteLiu. In the Appendix B, we provide a brief introduction to Liu’s procedure for the implementation of the entropy principle and also discuss a variance of this procedure developed by Ruggeri and coworkers see [53]. Both of these procedures use Lagrange multipliers and these multipliers arise via the following considerations.

With reference to the system (167,168) coupled to the entropy inequality (171), let the fields $(\zeta,\zeta_{a},\zeta_{ab})$ with $\zeta_{ab}=\zeta_{ba}$ and $g^{ab}\zeta_{ab}=0$ acting as Lagrange multipliers in the following sense: the inequality

[TABLE]

is required to hold for arbitrary smooth configuration $(T^{\mu\nu},J^{\mu})$ (and not only for solutions of (167,168)). The existence of the field of the Lagrange multipliers $(\zeta,\zeta_{a},\zeta_{ab})$ that satisfy this inequality has been addressed in Ref. \refciteLiu (see also Appendix B). Introducing a new field $S^{\prime\alpha}$ via

[TABLE]

then (177) can be rewritten in the form

[TABLE]

Since $(J^{\mu},T^{\mu\nu})$ and $(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ have the same number of components, the latter can be traded for the former through a non singular transformation of the form767676This transformation and in particularly, its global nature, is subtle. In the section $(3)$ of Ref.\refciteMul4 and for the case of non relativistic fluids, the authors discuss this point in some details. They show that global invertibility is ensured provide the entropy density is a concave function of the basic variables. For the relativistic case matters are not that simple. However as we shall see further ahead one expects local invertibility to hold (see also discussion in Appendix B).

[TABLE]

Based in an analysis of Ref.\refciteLiu, the authors in Refs. \refciteMul6, \refciteMul4 proved that this transformation can be generated from the vector function $S^{\prime\alpha}$ defined in (178) considered now as a function of $(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$. Performing the differentiation in $\nabla_{\mu}S^{\prime\mu}$, then (179) yields

[TABLE]

and since this inequality must hold for arbitrary $(\zeta,\zeta_{\beta},\zeta_{\beta\alpha})$, it implies:

[TABLE]

where in above, as before, the angular bracket signifies taking the symmetric and trace free part.

Introducing the entropy production $\sigma$ via $\sigma:=-\zeta_{\beta\mu}I^{\beta\mu}$, then validity of (181) now demands:

[TABLE]

It follows from (182) that the components $(J^{\mu},T^{\mu\nu},A^{\mu\nu\lambda})$ can be considered as depending upon the multipliers $(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ and moreover these components can be generated777777 As we shall seen in the next section as long as $T^{\mu\nu}=T^{\nu\mu}$, actually the components of $(J^{\mu},T^{\mu\nu},A^{\mu\nu\lambda})$ can be obtained by differentiating a scalar function and this observation simplifies mater considerably. We shall discuss this property in the next section. by differentiating the ”vector potential” $S^{\prime\alpha}=S^{\prime\alpha}(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$.

Due to the significance of this ”vector potential”, in Refs, \refciteMul6,\refciteMul4 construct at first an explicit representation of $S^{\prime\alpha}$ as function of $(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ by appealing to the principle of relativity. As for the case of the tensors $I^{\alpha\beta},A^{\alpha\beta\mu}$ and $S^{\alpha}$ treated above, this principle requires that $S^{\prime\alpha}$ to behave as an isotropic vector function under arbitrary coordinate transformations. Since however $S^{\prime\alpha}=S^{\prime\alpha}(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ and the only vectors that one can construct out of the Lagrange multipliers that are quadratic in $\zeta_{\mu\nu}$ are $\zeta_{\mu},\zeta_{\mu\nu}\zeta^{\nu},\zeta_{\mu\nu}^{2}\zeta^{\nu}$, it follows that $S^{\prime\alpha}$ must be a linear combination of these vectors with suitable scalar coefficients functions of the Lagrange multipliers. Introducing the scalars

[TABLE]

then it was shown in [22, 21] that the most general form $S^{\prime\alpha}$ quadratic in $\zeta_{\mu\nu}$ must have the form787878 The representation of $S^{\prime}_{\alpha}$ in (185) is one of the central formulas and the reader is refereed to Refs. \refciteMul4, section $(2.3)$ chapter $6$ for more details leading to its derivation.:

[TABLE]

where $(\Gamma_{0},\Gamma_{1},\Gamma_{2})$ are functions of $\zeta$ and $G_{0}$. In arriving at the above formula, the authors took into considerations the symmetries of the fields $(T^{\mu\nu},A^{\mu\nu\lambda})$ and the requirement that $(J^{\mu},T^{\mu\nu},A^{\mu\nu\lambda})$ should be linear functions of $\zeta_{\mu\nu}$. These conditions fix the functions $(\Gamma_{1},\Gamma_{2})$ in terms of the $\Gamma_{0}(\zeta,G_{0})$ and two arbitrary function of $\zeta$. Therefore, although in the following formulas the functions $\Gamma_{1},\Gamma_{2}$ appear explicitly, in fact they are considered as determined by $\Gamma_{0}(\zeta,G_{0})$ and two arbitrary functions of $\zeta$.

Using $S^{\prime}_{\alpha}$ shown in (185) combined with (182), it follows that the components of $J^{\alpha}$, $T^{\alpha\beta}$ and $A^{\beta\mu\alpha}$ to linear order in $\zeta_{\alpha\beta}$ have the form:

[TABLE]

[TABLE]

[TABLE]

while the entropy vector $S^{\alpha}$ has the form:

[TABLE]

The above representations of $(J^{\alpha},T^{\alpha\beta},A^{\beta\mu\alpha},S^{\alpha})$ are formal since they depend upon the Lagrange multipliers $(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ whose physical significance is not yet clear. So the second step in the approach of Refs.\refciteMul4,Mul6 is to express the multipliers in terms of the physically relevant quantities $(\rho,P+\pi,q^{a},\pi^{\mu\nu},n,u^{\mu})$ that appear in (172,173). Combining (172,173). with (186), (187) one gets:

[TABLE]

[TABLE]

These relations are considered as a system of $14$ equations relating the Lagrange multipliers $(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ to the $14$ fields $(n,\rho,\pi^{\mu\nu},\text{etc})$ appearing in (172,173). However, due to their non linear nature, resolving these equations for the multipliers is not a trivial task. In \refciteMul6,\refciteMul4 settled that problem ”perturbativelly”: they first established the relationship between $(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ and the physical variables $(n,\rho,\pi^{\mu\nu},\text{etc})$ for the particular family of fluid states namely the family of equilibrium states. The identification of the equilibrium states within their theory needs some considerations. Such states are characterized by the vanishing of the entropy production $\sigma$ (see (183)) and this condition requires $I^{\mu\nu}=0$. However the additional requirement that $\sigma$ should be a local minimum in equilibrium demands that equilibrium states are characterized by the property: $\zeta_{\mu\nu}=0$ (for details leading to this conclusion see discussion in Refs. \refciteMul6,\refciteMul4).

Accepting this property, and using the subscript $E$ to denote equilibrium values, one finds from the expression for $S^{\alpha}$ and (190, 191) the following formulas valid only for equilibrium sates:

[TABLE]

[TABLE]

[TABLE]

where $S_{E}$ is the equilibrium entropy density as measured by the $u^{\mu}$ observer i.e. $S_{E}=S^{\mu}{}_{E}u_{\mu}$.

Recalling that $\Gamma_{0}:=\Gamma_{0}(\zeta,G_{0})$ and $G_{0}:=(\zeta^{\alpha}\zeta_{\alpha})$, it was argued in \refciteMul4, \refciteMul6 that these three relations becomes consistent by identifying $\zeta_{E}$ and $G_{0E}:=(\zeta^{\alpha}\zeta_{\alpha})_{E}$ with the fugacity $\alpha$ and the absolute temperature $T$ and moreover set: $\Gamma_{0}(\zeta_{E},G_{0E})=-P$ with $P=P(n,\rho)$ the thermal equation of state. In details

[TABLE]

Moreover the identification: $T^{-2}c^{2}=G_{0E}=(\zeta^{\mu}\zeta_{\mu})_{E}$, implies that $(\zeta_{\mu})_{E}=-cu_{\mu}{T^{-1}}$ and thus in summary the equilibrium values of the multipliers are given by:

[TABLE]

Using these equilibrium values for the multipliers, subsequently they perturb them according to the following scheme:

[TABLE]

where $(\chi,\lambda,\lambda_{\alpha},\sigma,\sigma_{\alpha},\sigma_{\alpha\beta})$ are the non equilibrium parts of the multipliers and for notational simplicity we omitted the subscript $E$ describing the leading equilibrium parts. Inserting (197) back to (190, 191) they obtain a linear system for $(\chi,\lambda,\lambda_{\alpha},\sigma,\sigma_{\alpha},\sigma_{\alpha\beta})$ whose solutions are linear functions of $(q^{\mu},\pi,\pi^{\mu\nu})$ and below, we provide a sample of the resulting solutions

[TABLE]

[TABLE]

[TABLE]

where an over dot signifies derivative with respect to $\alpha$ while $D_{3}$ is the determinant of a $(3\times 3)$ matrix involving derivatives of the equilibrium equation of state $P(\alpha,T)$ and the derivatives of $\Gamma_{1}(\alpha,T)$. The other parts of the Lagrange multipliers appearing in (197) are longer expressions that we do not report here but they can be found in eqs. $(2.39)$ page $(115)$ of Ref. \refciteMul4.

Substituting these linearized representations of the Lagrange multipliers in (188) results a long expression for the components $A^{\mu\nu\lambda}$ (too long to be given here) that can be found on page $116$ formula $(2.41)$ in Ref. \refciteMul4. This representation of $A^{\mu\nu\lambda}$ has the same structure as the one shown in (175) and a comparison between these two yields a representation for the $4$ coefficients $(C^{0}_{\ 1},C^{\pi}_{\ 1},C_{3},C_{5})$ in terms of an equilibrium equation of state $P=P(\alpha,T)$ and the functions $\Gamma_{1}(\alpha,T)$ and $\Gamma_{2}(\alpha,T)$. The explicit formulas797979Recall that although in these formula appear terms $\Gamma_{1}(\alpha,T)$ and $\Gamma_{2}(\alpha,T)$, these functions are determined by $P=P(\alpha,T)$ and two functions $A_{1}(\alpha)$, $A_{2}(\alpha)$ of the fugacity coefficient $\alpha$ (see eqs. (2.68), (2.69) on page $122$). for the coefficients $(C^{0}_{\ 1},C^{\pi}_{\ 1},C_{3},C_{5})$ can be found in eq. $2.43$ on page $116$.

The determination of the coefficients $(A^{\pi}_{\ 1},A^{\pi^{2}}_{\ 1},A^{q}_{\ 1},...,A^{0}_{\ 3})$ that appear in the entropy vector $S^{\mu}$ in (176) are calculated in[21, 22], by noting that (178) combined with (182) imply

[TABLE]

Eliminating the Lagrange multipliers using their linearized versions in (197) and the expansions in (172), the above formula can be re-writen (201) in the form

[TABLE]

so that

[TABLE]

where $(f^{\mu},g^{\mu}{}_{\nu},G^{\mu}{}_{\nu\lambda})$ are well defined functions. An integration of these relations and comparison of the resulting form of $S^{\mu}$ with the one shown in (176), fixes the coefficients $(A^{\pi}_{\ 1},A^{\pi^{2}}_{\ 1},A^{q}_{\ 1},A^{t}_{\ 1},A^{0}_{\ 2},A^{\pi}_{\ 2},A^{0}_{\ 3})$ and their explicit form can be found on page $117$ of ref.\refciteMul4.

To complete the analysis, the coefficients $(B^{\pi}_{\ 1},B_{3},B_{4})$ that appear in the dissipation tensor $I^{\mu\nu}$ in (174) need to be considered. However these coefficients appear to be underdetermined except that they are restricted to obey certain inequalities. These inequalities arise by noting that the entropy production $\sigma=-\zeta_{\mu\nu}I^{\mu\nu}$ upon using $I^{\mu\nu}$ in (174) and substituting $\zeta_{\mu\nu}$ for the one in (197), yields

[TABLE]

The non-negativity808080Notice that even though in the right hand side appear coefficients like $\sigma_{\pi}$, $\sigma_{q}$ and $\Gamma^{-1}_{1}$, these coefficients are considered as known and their form can be found in page $115$ in [21]. of the right hand side of this $\sigma$ puts restrictions in the form of inequalities in the coefficients $(B^{\pi}_{\ 1},B_{4},B_{3})$. Otherwise they remain undetermined and as we shall see below, these coefficients are related to bulk, shear viscosity and the coefficient of heat conduction of the fluid. As a side remark, (204) shows that there exist three dissipative mechanism generating entropy in the fluid, namely the heat flux and the two stresses.

Finally, we mention briefly the important issue of the symmetric-hyperbolic and causality property of the dynamical equations within the relativistic (REIT). As shown in more details in Refs. \refciteMul4,Mul6, these requirement puts restrictions on the coefficients $C$ and $A$ that appear in $A^{\mu\nu\lambda}$ and $S^{\mu}$ (see (175,176)). Amongst these is the restrictions of the form of the equilibrium equation of state $P=P(\alpha,T)$ although this is not the only restriction. In effect when these restrictions hold, an open vicinity of around the equilibrium state the dynamical equations a symmetric-hyperbolic and causal system. Although from the physical standpoint causality is a very important for any theory, we shall not discuss its implications upon relativistic (REIT). In the next section, we shall address this issue from a wider angle.

This concludes a brief summary of the Liu-Müller-Ruggeri relativistic (REIT). All of the above described work aimed at specifying the structure of the constitutive functions $A^{\mu\nu\lambda},I^{\mu\nu}$ and $S^{\mu}$ for states near equilibrium so the set of $14$ dynamical equations for the auxiliary variables $(\rho,P+\pi,q^{a},\pi^{\mu\nu},n,u^{\mu})$ can be written down. Solutions of these equations describe states near equilibrium and these equations form a closed system provided the thermal equilibrium equation of state $P=P(\alpha,T)$ a-priori has been specified818181In that regard, it should be pointed out the philosophy underlying the work in Refs.\refciteMul4,Mul6. Since for relativistic systems it seems hard to determine observationally a thermal equation of state, the authors appeal to relativistic kinetic theory of dilute gases to specify a reliable equation of state. The restriction to states of relativistic gases in equilibrium comes from the property that for such media one may specify families of physically reliable equilibrium equations of state by appealing to Jüttner equilibrium distribution. It should mentioned however, that except this technicality their treatment is quite general..

The dynamical equations for the fields $(n,\rho,P,\pi,q^{\mu},\pi^{\mu\nu})$ are

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where in the last three equations the components of $A^{\alpha\beta\gamma}$ are those in (175) with the ”constants” $(C^{0}_{\ 1},C^{\pi}_{\ 1},C_{3},C_{5})$ determined by $P=P(\alpha,T)$ (and two arbitrary functions of $\alpha$). These equations contain in addition the three unspecified functions $(B^{\pi}_{\ 1},B_{4},B_{3})$ that determine the entropy production.

It is worth noting that by evaluating the left hand-sides (207-209) on equilibrium states and denote their right hand-sides as $(\pi^{1}_{\mu\nu},\pi^{1},q^{1}_{\mu})$, one recovers Eckart’s phenomenological equations that we derived in previous sections. Thus relativistic (REIT) in an appropriate limit, contains Eckart’s first order theory.

If on the other hand on the left hand-sides of (207-209) one employs the form of $A^{\mu\nu\lambda}$ that contains linear contribution $(\pi_{\mu\nu},\pi,q_{\mu})$ one recovers phenomenological equations whose form is structurally analogous to the phenomenological equations (161,162,163) derived in section $11$ within the Israel-Stewart transient thermodynamics. However there is a pronounced difference between the two families of phenomenological sets. Whereas the equations (161,162,163) contain eight unspecified coefficients of two variables namely, $(\zeta_{u},\beta_{0},\bar{\alpha}_{0},k,\bar{\beta}_{1},\bar{\alpha}_{1},\zeta,\beta_{2})$ the corresponding equations within the Liu-Müller-Ruggeri theory contain only three unknown functions $(B^{\pi}_{\ 1},B_{4},B_{3})$ of two variables and two unknown functions of a single variable (correspondingly one unknown function of a single variable for the case of a relativistic gas whose equilibrium thermal equation of state arises from a Jüttner distribution). So from this view point, the Liu-Müller-Ruggeri theory restrict considerably the number of free functions appearing in the phenomenological equations.

Explicit result for viscosities and coefficient of heat conductivity for near equilibrium states of a relativistic gas are discussed in sections $(4.1-4.7)$ in Ref. \refciteMul4. They specified equilibrium equations of state for the relativistic gas by employing a Jüttner equilibrium distribution and for various degrees of degeneracy they work out the corresponding (equilibrium) equation of state. We refer the reader to monograph of Ref. \refciteMul4 page $(128)$ for a exhaustive analysis of different degrees of degeneracies exhibited by a relativistic gas in equilibrium.

13 Dissipative relativistic fluids of divergence type

In the previous section, we discussed at same length the Liu-Müller-Ruggeri theory and we have seen that it is a restrictive theory in the sense that the resulting phenomenological equations involve just three arbitrary functions of two variables (modulo one or two functions of one variable depending upon the nature of the medio under study). However, as we have seen, the implementation of the Entropy Principle and the Principle of Relativity on the constitutive relations involves tedious computations so that the nice features of this theory are not very transparent. Pennisi in Ref.\refcitePen and Geroch and Lindblom in Ref. \refciteGer1, motivated by the structure of the fields equations in (167,168), introduced a class of relativistic fluid theories satisfying the following three properties:

a) The variables specifying the fluid are the components of the conserved particle current $J^{\mu}$ and the components of the conserved and symmetric energy momentum tensor $T^{\mu\nu}.$

b) The dynamical equations are the same as in (167,168), but now the tensor fields $A^{\mu\nu\lambda}$ and $I^{\nu\lambda}$ are symmetric and traceless828282 Thus the treatment of Pennisi and Geroch-Lindblom generalizes slightly the Liu-Müller-Ruggeri theory. While in the, latter theory $A^{\mu\nu\lambda}$ is assumed to be totally symmetric in the Pennisi, Geroch-Lindblom theory, this tensor is symmetric only in the indices $\nu,\lambda$. Also for the later theory, the constraint $A^{\mu\nu}{}_{\nu}=m^{2}J^{\mu}$, it is not any longer imposed. in the indices $\nu,\lambda$. Moreover, as for the Liu-Müller-Ruggeri theory, $A^{\mu\nu\lambda}$ and $I^{\nu\lambda}$ are considered to be constitutive functions i.e. algebraic functions of the basic variables $J^{\mu}$ and $T^{\mu\nu}$.

c) There exists an entropy current $S^{\mu}(J,T)$ whose dependance upon $(J,T)$ is such that the entropy principle holds i.e. for any solution $(J,T)$ of the field equations (167,168), the current satisfies838383 For the theories analyzed in [24], the authors imposed two additional restrictions: in addition to $\sigma(J,T)\geq 0,$ they assumed the theory to be “generic” in the sense that the “dissipation tensor” $I^{\mu\nu}$ and of the derivatives of the generating function $\chi$ (to be defined below) satisfy suitable inequalities to be introduced further ahead.:

[TABLE]

where $\sigma(J,T)$ some algebraic function of $(J,T)$.

The theories defined by these three conditions are refereed as (relativistic) fluid theories of divergence type and in this section we shall highlight their most important properties.

Already the analysis of the relativistic (REIT) carried out in the previous section, gave us a good insights regarding the structure of fluids described by relativistic (REIT)848484Ought to be mentioned, that except for a minor point regarding the symmetries of $A^{\mu\nu\lambda}$, relativistic (REIT) and fluid theories of of divergence type are considered as been ”closely related”. In this work we say that a fluid belongs to the class of divergence type whenever the energy momentum tensor $T^{\mu\nu}$ is symmetric and of course obey the field equations cited above. As we shall see in this section, these class of divergence fluids can be treated via a scalar generating function which simplifies mater considerably. However ought to be stressed that the powerful treatment of Liu-Müller-Ruggeri theory should not disregarded. There are many physical important settings where a relativistic fluid interacts with external fields or with other fluids and in these cases one cannot treat them via a scalar generating function., at least for fluid states near equilibrium. However, Pennisi[23] in $1989$ and independently Geroch and Lindblom [24] in $(1990)$ offer an alternative treatment to fluids obeying conditions $a)-c)$ cited above and in this section we analyze their approach.

The Pennisi[23] - Geroch and Lindblom [24] approach introduces again the field of Lagrange multipliers858585For comparison purposes, we denote these multipliers by the same symbols as those employed for Liu-Müller-Ruggeri theory. $(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ obeying $\zeta_{\mu\nu}=\zeta_{\nu\mu}$ and $g^{\mu\nu}\zeta_{\mu\nu}=0$. As for the case of Liu-Müller-Ruggeri treatment, they also introduce the vector function $S^{\prime\alpha}$ as in (178) and the transformation shown in (180). They noticed however, that the symmetries of $T^{\mu\nu}$ imply that this transformation may be generated by a some scalar function $\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ related to the vector function $S^{\prime\alpha}$ via:

[TABLE]

and this fundamental relation makes the important difference in the formulation of divergence fluids. By arguments similar to those in [51] and in [22], (see also discussion in Appendix B and [76]), any fluid theory obeying conditions $a)-c)$ above is determined from the knowledge of the dissipation tensor $I^{\mu\nu}(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ and from a smooth scalar function $\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ referred as the generating function. This scalar function $\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ as a consequence of (211) combined with (182) satisfies:

[TABLE]

and generates the entropy current $S^{\mu}$ via

[TABLE]

while the source term $\sigma$ in (210) is defined from the dissipation tensor $I^{\mu\nu}$ and $\zeta_{\mu\nu}$ via:

[TABLE]

The relations in (212) are the fundamental relations in the Penissi-Geroch-Lindblom formalism and show that the components $(J^{\mu},T^{\mu\nu},A^{\mu\nu\lambda})$ can be regarded as functions of the Lagrange multipliers $(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$, and moreover the dynamical equations take the form:

[TABLE]

[TABLE]

[TABLE]

Introducing capital indices $(A,B,C,...)$ so that $(\zeta_{A}$, $I^{A})$ stand for $\zeta_{A}:=(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ respectively for $I^{A}:=(0,0,I^{\mu\nu})$, the above equations can be written in the compact form:

[TABLE]

where the Einstein’s summation convention has been extended over the capital indices as well.

Thus in the Penissi-Geroch-Lindblom formalism, of prime importance is the pair $(\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu}),I^{\mu\nu}(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$. Once this pair has been specified, then (218) is a manifestly symmetric868686In this section, we follow the terminology and definitions employed in Ref.\refciteGer1. Thus the system $M^{AB\mu}\nabla_{\mu}\zeta_{B}=I^{A}$ is said to be symmetric provided: $M^{AB\mu}=M^{BA\mu}$. A symmetric system is hyperbolic in an open set of a fluid states, if the matrix $M^{AB\mu}w_{\mu}$ is negative definite for some (possibly state-dependent) future directed timelike $w_{\mu}$. Finally, if $M^{AB\mu}\nabla_{\mu}\zeta_{B}=I^{A}$ is symmetric, then it said to be causal in an open set of a fluid states, (i.e. hyperbolic with no fluid signals propagating faster than light) if $M^{AB\mu}w_{\mu}$ is negative definite for all future directed timelike $w_{\mu}$. In these definitions, the contracted principal symbol $M^{AB\mu}w_{\mu}$ is evaluated typically on an equilibrium state (these equilibrium states are defined precisely further ahead). Via continuity arguments, it follows that the principal symbol maintains the same sign over an open vicinity of fluid states around the equilibrium state. system of quasi linear equations for the dynamical variables $(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$. However, whether this system is hyperbolic or (and) causal depends upon the nature of the generating function $\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu}).$ Below, we give a few examples of physically relevant generating functions $\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ and briefly discuss878787Within the Pennisi, Geroch-Lindblom formalism, the relativistic (REIT) of Liu-Müller-Ruggeri that we discussed in the last section, can be generated at least for states near equilibrium, by the dissipation function $I^{\mu\nu}$ shown in (174) and by a generating function $\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ that can be obtained by integrating $S^{\prime\alpha}$ with respect to $\zeta^{\mu}$., the properties of the resulting fluid theories.

The first example consists of the class of functions $\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ that generate states of a simple perfect fluid. Perfect fluid states are generated by setting the dissipation tensor to zero i.e. $I^{\mu\nu}=0$ and choosing $\chi$ to be a smooth function only of $(\zeta,\zeta^{\mu})$ i.e.

[TABLE]

For this choice, it follows from (212) that $A^{\mu\nu\lambda}:=0$, while a simple calculation shows that the particle current $J_{PF}^{\mu}$, the symmetric energy momentum tensor $T_{PF}^{\mu\nu}$ and the entropy current $S_{PF}^{\mu}$ are given by:

[TABLE]

[TABLE]

By comparing the right hand sides of these expressions to the standard formulas for the particle current the energy momentum tensor and entropy current of a perfect fluid, shows that (219) generates states of a simple perfect fluid whose particle number density $n$, energy density $\rho$, isotropic pressure $P$, as measured relative to the rest frame of the flow are given by:

[TABLE]

while the fields $(\zeta,\zeta_{\mu})$ are related to the thermal potential $\Theta$, four velocity $u^{\mu}$, local temperature $T$, and entropy per particle $\hat{s}=sn^{-1}$, via:

[TABLE]

Finally, for completeness, we mention that the well known property that the dynamical equations for a simple perfect fluid form a symmetric-hyperbolic888888This important property of perfect fluids is discussed for instance in ref.[122]. respectively symmetric-causal system, provided that suitable restrictions are imposed upon the equation of state can also established within the Geroch-Lindblom formalism. By examining the contraction of the principal symbol $M^{AB\mu}$ using the generating function in (219) it follows that causality holds on any fluid state if and only if the function $\alpha(\zeta,\mu)$, in (219) is restricted so that the resulting $(\rho,P)$ satisfy following inequalities:

[TABLE]

(for details regarding the derivation of these inequalities see Refs. \refciteGer1,Ger3).

The second example discusses the structure of the generating functions that within the class of relativistic fluids of divergence type generate equilibrium states. Geroch and Lindblom in Ref.\refciteGer1, define a fluid state to be an equilibrium state if and only if it is described by a solution $(J^{\mu},T^{\mu\nu})$ of (167,168) having the property that $I^{\mu\nu}(J,T)=0$. We denoted by $(J^{\mu}_{E},T^{\mu\nu}_{E})$ these equilibrium states and here after we use the subscript $E$ on any quantity to indicate that the underlying quantity is evaluated on an equilibrium state. Clearly on such states $\sigma_{E}(J_{E},T_{E})=-\zeta_{\mu\nu}I_{E}^{\mu\nu}=0$. Under the additional assumption that any solution $(J^{\mu},T^{\mu\nu})$ of (167, 168) satisfies $\sigma(J,T)\geq 0$, it was shown in [24] that on any equilibrium state $\zeta_{\mu\nu}=0$ a conclusion that holds also for the Liu-Müller-Ruggeri relativistic (REIT). Properties of equilibrium states can be determined by considering a series expansion of the generating function $\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ around $(\zeta,\zeta_{\mu},\zeta_{\mu\nu}=0)$. As long as considerations are restricted on tensor fields evaluated on equilibrium states, an expansion of $\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ up and including terms linear in $\zeta^{\mu\nu}$ is sufficient. Such expansion can be presented in the form:

[TABLE]

where $\alpha(\zeta,\mu)$ is an arbitrary smooth algebraic function of $(\zeta,\mu)$, while the derivative of $\chi$ with respect to $\zeta_{\mu\nu}$ evaluated on equilibrium states is taken in the form:

[TABLE]

These properties of the $\chi$ combined with (212), imply that the particle current $J_{E}^{\mu}$, the symmetric energy momentum tensor $T_{E}^{\mu\nu}$ and the entropy current $S_{E}^{\mu}$ have identical forms as those shown in (220, 221) while a bit of algebra shows that

[TABLE]

Therefore the particle number density $n_{E}$, energy density $\rho_{E}$, isotropic pressure $P_{E}$, as measured relative to the rest frame of the flow are identical to those in (222) while the fields $(\zeta,\zeta_{\mu})$ are related to the thermal potential $\Theta$, four velocity $u^{\mu}$ local temperature $T$, and entropy per particle $s$ are as in (223). States in equilibrium impose additional restrictions upon the functions $\alpha(\zeta,\mu)$ and $\beta(\zeta,\mu)$ in (226). These restrictions arise from the equations:

[TABLE]

which yield (for more details see Ref. \refciteGer1)

[TABLE]

[TABLE]

[TABLE]

These three equations can be written in the form:

[TABLE]

where the $(3\times 3)$ matrix $A$ can be read from equations (229-231) and its elements are formed from the derivatives of the functions $\alpha(\zeta,\mu),\beta(\zeta,\mu)$ defined in (225, 226). If these function are chosen so that their derivatives satisfy $detA\neq 0$, then one concludes that necessary

[TABLE]

Furthermore, the equation $\nabla_{\mu}A_{E}^{\mu\nu\lambda}=0$ yields [24]

[TABLE]

and under the assumption that $\beta(\zeta,\mu)$ is non vanishing in $U$ it follows that $\zeta_{\mu}$ is a timelike Killing898989It is interesting to mention here that for the particular class dissipative fluids exhibiting conformal invariance one finds that $\zeta_{\mu}$ satisfies the conformal Killing equation equation (see discussion further ahead). vector field which means that equilibrium states within the class of relativistic fluid theories of divergent type are characterized by the presence of a timelike Killing and a uniform i.e a constant $\zeta$.

The so far analysis shows that the space of relativistic fluids of divergence type it is not empty. But do there exist relativistic dissipative fluids of divergence type that are physically relevant in the sense that their equations are of symmetric-hyperbolic type and their evolution respect causality?

The answer to this question is in the affirmative, and below we discuss a family of relativistic fluids that share this property. This family is a suitable extension of the Eckart theory that we discussed in section 10. As we have seen there, and for states near equilibrium, the particle current $J^{\mu}$, stress tensor $T^{\mu\nu}$ of Eckart’s fluids are described in (132), while the constitutive relations are described in (134). Within the context of a divergent type theory, it is shown in [24] that Eckart’s theory is described by:

[TABLE]

where $I^{\mu\nu}$ stands for the dissipation tensor, $(\hat{\zeta},\hat{\zeta_{v}})$ are the coefficients of bulk and shear viscosity defined in (131,134) (here we used hated version for these coefficients to avoid confusion with the Lagrange multiplier $\zeta$), $u^{\mu}$ is the fluid four velocity, $\pi$ the bulk pressure, $\pi^{\mu\nu}$ the shear stress, $q^{\mu}$ the heat flux and $T$ the temperature all of them measured relative to the Eckart frame. The entropy vector $S^{\mu}$ in (239) is the same as in (127) for the choice $n^{\mu}=0$ provided that we eliminate the entropy density $s$ in favor of the entropy density per particle $sn$ (we use the same symbol for $s$).

In Ref.\refciteGer1, it is shown that the theory in (235-240) can be obtained from the following generating function $\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$:

[TABLE]

where $\alpha(\zeta,\mu)$ is an arbitrary smooth function. Using this $\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ and following similar procedure as for the Liu-Müller-Ruggeri theory, the Lagrange multipliers $(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ are related to the observable fields $(n,\rho,P,u^{\mu},T,\pi,\pi^{\mu\nu})$ in (235-240) via:

[TABLE]

[TABLE]

[TABLE]

while the thermodynamical variables like temperature $T$, particle density $n$, thermodynamical pressure $P$ and density $\rho$ are given by:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Geroch and Lindblom in Ref.\refciteGer1 pointed out that the Eckart theory is an example of a relativistic dissipative fluid theory of divergence type, whose dynamical equations around the equilibrium state, fail to constitute a causal set and thus it cannot be considered to be a satisfactory physical theory (this conclusion was expected due to the results in Refs.\refciteHis2,His1). Indeed for the generating function $\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ defined in (241), the quadratic form $M^{AB\mu}w_{\mu}Z_{A}Z_{B}$ with arbitrary $Z_{A}=(Z,Z_{a},Z_{ab})$ fails to be negative since

[TABLE]

vanishes identically909090It should be mentioned that the structure of the contraction $M^{AB\mu}w_{\mu}Z_{A}Z_{B}$ is rather complicated, since $A,B$ run into the interval $(1,2,..,14)$. But for the case of the $\chi$ in (241) the vanishing of (249) makes the analysis simpler. since for the generating function in (241) is a linear function of $\zeta_{\mu\nu}$.

The advantage of the Pennisi-Geroch-Lindblom formalism is the flexibility that offers to construct dissipative fluid theories with tractable physical properties and this flexibility arises from the freedom in choosing the generating function. In that regard, it it was noticed in Ref. \refciteGer1 that by replacing $\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ in (241) by:

[TABLE]

where $\chi_{2}$ has the form919191Notice that this $\chi$ in (250 ) combined with (251), it is not the most general from of the generating function that contains terms quadratic in $\zeta^{ab}$. For a discussion regarding a more general form see for instance Ref.\refciteNag1,Cal1,Cal2,Cal3,Reu4.:

[TABLE]

results in new theory. Geroch and Lindblom in Ref.\refciteGer1, investigate the causality property of this theory by analyzing the quadratic form

[TABLE]

and they find that this form is negative definitive for all fluid states having $\zeta_{ab}=0$ provided the perfect fluid causality conditions shown in (224) hold and $\frac{\partial\gamma}{\partial\mu}$ is sufficient large. They conclude that the theory is causal for all sufficiently small $\zeta_{ab}$ i.e. in some open neighborhood of the equilibrium states, and this conclusion is welcomed. It demonstrates that a class of dissipative relativistic fluids of divergent are characterized by sensible properties like the causal propagation of disturbances, well posed initial value problem, stability of equilibrium states929292This last property is proven in Ref.\refciteGer1. etc.

The relativistic (REIT) introduced by Liu-Müller and Ruggeri and the family of relativistic dissipative fluid theories of divergence type introduced by Pennisi and Geroch-Lindblom acted as a stimulus for further investigations regarding the properties of this class of fluid theories. A large amount of work in the literature is centered on the nature of the generating function $\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ and how the structure of this function affects the properties of the resulting fluid theory. Since equilibrium states are characterized by $\zeta_{\mu\nu}=0$, therefore states near equilibrium can be studied by a generating function $\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ whose Taylor series expansion around states in equilibrium admit non vanishing quadratic contributions in $\zeta_{\mu\nu}$. The inclusions of such quadratic contributions in $\chi$ (compare the form of $\chi$ in (225)) allows to study the principal part $M^{AB\mu}$ in the dynamical equations shown in (218) and evaluated this principal part on equilibrium states and thus address issues of causality for states near equilibrium. For an explicit representation of such families of generating functions see for instance Refs. \refciteNag1,Cal1,Cal2,Cal3,Reu4.

Another aspect of the class of fluid theories of divergent type that has been addressed in the literature concerns the connection between this class of theories and relativistic kinetic theory of dilute gases. As we have remarked before, for any dilute relativistic gas, the first three moments $\hat{J}^{\mu}$, $\hat{T}^{\mu\nu}$ and $\hat{A}^{\mu\nu\lambda}$ of any distribution function $f(x,p)$ solution of the relativistic Boltzmann equation satisfy: $\nabla_{\mu}\hat{J}^{\mu}=\nabla_{\mu}\hat{T}^{\mu\nu}=0$, $\nabla_{\mu}\hat{A}^{\mu\nu\lambda}=\hat{I}^{\nu\lambda}$ and these equations look similar to the standard equations (167, 168) for the Liu-Müller-Ruggeri and Pennisi-Geroch-Lindblom theories. However, this similarity is deceiving and in Refs. \refciteNag1,Reu4 that question has been discuss at length. In general, the second moment of the collision term $\hat{I}^{\nu\lambda}$ contains the distribution function and thus has different structure than the dissipation tensor $I^{\mu\nu}$ which depends only upon $J^{\mu}$ and $T^{\mu\nu}$ and thus at most depends upon the first two moments of the distribution function. In Refs. \refciteNag1,Reu4, the authors constructed a particular family of dissipative fluid theories of divergence type having the property that the fields $J^{\mu},T^{\mu\nu}$ and $A^{\mu\nu\lambda}$ are expressed as moments of a suitable distribution function. The resulting system of hyperbolic partial differential equations is very simple and allows one to identify a subclass of manifestly causal theories.

Lately, a class of dissipative fluids of divergent type that has been analyzed in the literature concerns relativistic dissipative fluids that exhibits conformal invariance (for a discussion of the implications of this symmetry upon the structure of such fluids consult[88] and further references therein). Physically these fluids could be though as representing the low energy limit of a conformal quantum field theories (for a discussion on this connection see [89]). In Ref. \refciteReu3, the generating function $\chi$ for this class of fluids that includes second order deviations from equilibrium states has been constructed. Due to the underlying conformal symmetry, this $\chi$ depend only upon $(\zeta_{\mu},\zeta_{\mu\nu})$. Equilibrium states within these theories, are states characterized by the condition that $\zeta_{\mu\nu}=0$ and in addition the other Lagrange multipler $\zeta_{\mu}$ is a conformal Killing vector field. It was shown in Ref. \refciteReu3, that whenever such equilibrium states are admitted, then there exist an open set of fluid states around equilibrium so that that hyperbolicity holds.

14 Conclusions

We started this article by pointing out that the recent detection of gravitational waves by the LIGO observatory, the observational evidence for neutron star mergers, the first photograph of the event horizon of the supermassive black hole at the center of the $M87$ galaxy by (EHT) and the experimental data on heavy ion collisions coming from BNL and CERN laboratories, brought the field of irreversible thermodynamics of relativistic continuous media into the frontiers of current research. The irreversible behavior exhibited by such media is encountered everywhere: from the subnuclear scale, to the large astrophysical and cosmological scales and this realization goes well with Penrose ideas regarding time asymmetric physics.

In the course of this review, we have mentioned that many predictions of classical (EIT) and (REIT) have had experimental confirmation and moreover these theories find applications in science and technology. For instance in the treatment of heat transport in micro and nano systems, shock structure of waves propagating on hydrodynamical systems, phenomenological hydrodynamics etc, and the interested reader is refereed Refs. \refciteJVL,Mul4 for overviews successes (and failures) of these theories.

However, at the relativistic level the situation is different. Despite persistence efforts still an accepted theory of non equilibrium thermodynamics of relativistic dissipative fluids or more generally continuous media is not forthcoming although big steps have been taken in the right direction. The recent efforts (and progress) in the field is shifted towards to the construction of theories of relativistic fluids where the dynamical equations constitute a symmetric-hyperbolic and preferably causal set of dynamical equations. This shifting in attitude results into theories of irreversible thermodynamics that respect causality at least in a vicinity of states near equilibrium states and thus these theories eliminate the unphysical infinite propagation of disturbances encountered in the theories of Eckart and Landau-Lifshitz.

The Israel-Stewart transient thermodynamics is a flexible theory and with additional efforts targeting the mathematical structure of the theory, likely to become a useful, practical theory. Even at this level of development, the theory has been applied to a number of physical problems. A tractable system where the effects of transient thermodynamics can be accounted for involves the spatially homogeneous and spatially isotropic cosmological models. As a rule, these cosmologies postulate that the cosmic fluid expands adiabatically but there exist processes in the cosmic evolution where this assumption may be questioned. In process like the GUT-phase transitions, reheating after inflation, decoupling of neutrinos or photons from the matter etc., the cosmic fluid may be modeled as a dissipative fluid where various thermodynamical variables exhibit steep temporal variations so that transient thermodynamics is a more suitable theory to describe the underlying physics. Therefore, the dynamics of a spatially homogeneous and isotropic cosmological model coupled to a dissipative fluid has attracted the interest of cosmologist. Since the energy momentum tensor of a dissipative fluid has to respects the symmetries of the background geometry, it follows that such states are characterized only by a non vanishing bulk viscosity and this property makes the coupled Einstein-dissipative fluid system a tractable system to analyze. For applications of transient thermodynamics to problems in Cosmology the reader is refereed to Refs. [90, 91, 92, 93, 94] and also chapter $(18)$ of Ref. \refciteJVL.

Gravitational collapse also offers many scenarios where transient thermodynamics finds fertile ground for applications. One such scenario corresponds to the phase during the complete gravitational collapse of a star where the escaping neutrinos pass from the free streaming to the trapped regime. During this transition, again many thermodynamics variables exhibit rapid spatial and temporal variations so that transient thermodynamics is applicable. We are not aware of any treatment of this problem within the Einstein-dissipative fluid system although some initial attempts in that directions have pursued in Ref. \refciteMar1. Moreover in Refs. \refciteSch1,Sch2 transient thermodynamics applied to radiation fluids (a mixture of ionized matter and photon) with emphasis to the behavior of density perturbations and their implications upon the structure formation.

It is of interest to mention that in the relativistic version of the Shakura-Sunyaev geometrically thin, optically thick accretion disks the accreting material is modeled by a viscous fluid (for an introduction to the physics of these disks see for instance [98]). The thermodynamical description of this viscous fluid employs the Eckart frame, and assumes constitutive relations on the stresses and heat flux of the conventional Eckart theory. It would be of some theoretical interest to examine whether the equations of transient thermodynamics admit disk like solutions modeling the accreting matter and whether such solutions (if exist) have any physical relevance.

Finally we should mention the efforts by high energy physicists to simulate analytically the quark-gluon plasma generated in heavy ion collisions. Since in this setting, the effects of the spacetime curvature are negligible, many researchers oriented their efforts to construct analytical (or semi analytical) solutions of the equations of the transient thermodynamics on a background Minkowski spacetime. So far only a few families of such solutions have been obtained and the reader is refereed to Ref. \refciteMar1 for progress and further references.

As far as the status of the Liu-Müller-Ruggeri theory and the class of relativistic fluid theories in the Pennisi-Geroch-Lindblom formalism are concerned, we may add that although by design these theories can be made causal theories at least in an open set of fluid states around the equilibrium state, unfortunately it seems that we lack criteria that single out a universally accepted theory of relativistic dissipation. The analysis of Liu-Müller-Ruggeri theory in section $12$ and the discussion in the last section, and in particularly the discussion following equation (251), suggests that there may exist more than one type of divergent type theories that causality holds for states around the equilibrium one. For instance by replacing the generating function $\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ in (251) by a different combination another sensible causal theory can be generated939393 In that regard, we have mentioned earlier on that there exist more general forms of $\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ that includes quadratic contributions of equilibrium (see for instance Ref.\refciteNag1).. Which one, if any of them, is to be considered as the theory preferred by nature?

Besides this uniqueness issue, there exist another problem related to hyperbolic theories of relativistic dissipation. That problem arose as a consequence of results obtained by Geroch and Lindblom in Refs [25, 26, 78]. In these references, a more general class of relativistic fluid theories have been introduced that encompass the class of divergent theories studied in section $(13)$. By enlarging the number of the dynamical fields and the corresponding field equations in a suitable way the resulting system is a hyperbolic system and thus physically acceptable. What is however perplexing are the results obtained in Refs [26, 78], regarding the observability of the additional fields present in these hyperbolic theories. It was shown in these references that the dynamics of this general class of hyperbolic fluid theories of relativistic dissipation are such that the dynamical fields relax on time scales that is characteristic of the particle interaction (typically the mean free time between collisions), to field configurations that are indistinguishable from the fields present in the much simpler Eckart theory949494This property, for some special configurations, has been established in a mathematically rigorous manner in Ref.[120]..

The conclusion in Refs. \refciteGer3,Lind1 has been contested in Refs \refcitePav1,Pav2,Pav3 who argue that the physical content of hyperbolic theories is, in general, much broader than those of the parabolic type. It is argued in [99, 100, 101] that of relevance is the relaxation time which depending upon the nature of the system can be large and thus hyperbolic dissipation can indeed have observable effects. While the arguments[99, 100, 101] are compelling and in fact Geroch in Ref. \refciteGer4 explicitly states that there may exist substances manifesting dissipation where hyperbolic theories can be of importance for the special case of Navier-Stokes fluids 959595 Following the notation of Ref. \refciteGer4, in this discussion the term Navier-Stokes fluid stands for a collection of five fields $(\rho,n,u^{\mu},q^{\mu},\tau^{\mu\nu})$ interpreted in the standard way i.e. as the fluid mass density, particle-number density, velocity, heat flow and stress with $u^{\mu}$ defining the Eckart frame. Moreover, these fields satisfy the standard Eckart equations see equations 132, 134 discussed in section $10$. that is not the case. For Navier-Stokes fluids, real or gedanken, hyperbolic theories may have a chance to be viable provided the equations satisfied by the heat flux and shear tensor break down on some length scale much larger than the length scale on which the meanings of its variables, $(\rho,n,u^{\mu},q^{\mu},\tau^{\mu\nu})$ break down. However, Navier-Stokes fluids do not exhibit this kind of behavior (see discussion in [102]).

In the phase of this argument, combined with the decay property obtained in Refs. \refciteGer3,Lind1, the state of affairs regarding dissipative Navier-Stokes fluids is as follows: For such a fluid, on large enough length scale where the hydrodynamical description is valid, it seems that there is one system of equations the Navier-Stokes system (Eckart system in our terminology) that is appropriate for the description of the physics of the fluid, and a second family of systems the hyperbolic systems, that are appropriate for the mathematics. Geroch in [102] states: This splitting of the physics and the mathematics is a novel situation, and it takes some getting used to. But, with a little care, ”theories” of this type can be applied as effectively as more traditional physical theories. (Here, we ought to have in mind that for numerical simulation, presumably hyperbolic theories may be of relevance.)

In sum, the description of dissipation in a relativistic fluid is a challenging problem. How eventually that issue will be settled is for the moment unknown.

 We finish this paper by stating that this paper targeted theories of extended thermodynamics describing limited class of Newtonian continuous media and relativistic fluids. As such, the covered material is by no means exhaustive neither we provided a complete list of references. Due to space and time limitations, we left out of considerations many types of interesting classes of continuous media and their thermodynamical behavior. In particularly, we have not discussed Carter’s theory of relativistic heat conduction from a variational view point (for an overview of this theory see see Refs.\refciteCar0,\refciteCMon) neither the structure of a GENERIC which is a general framework aiming to analyze general equations for non-equilibrium reversible-irreversible coupling (see for instance [104, 105, 106]). Also we left out of considerations theories of classical or relativistic (REIT) for polyatomic gases or (REIT) for dense gases (for references of the last two families of theories, consult Ref. \refciteRug1).

For an overview of the spectrum of the existing theories of non equilibrium thermodynamics, like nonlocal theories of thermodynamics of continuous media and other approaches, the reader is recommended to consult Ref.(\refciteIsr5) and other contributed articles in Ref. \refciteAB. Also Ref.\refciteREZ discusses extensively thermodynamical aspects of relativistic fluids.

15 Acknowledgments

This work arose after multiple discussions that the authors have had with the members of the relativity group at the IFM Univ. Michoacana and our warm thanks to all of them. Special thanks to F. Astorga, O. Sarbach, E. Tejeda and U. Nucamendi for their interest in this work. The research of T.Z was supported in part by CONACYT Network Project 280908 Agujeros Negros y Ondas Gravitatorias and by CIC Grant from the Univ. Michoacana. J.F.S thanks CONACYT for a predoctoral fellowship.

The penetrating comments and criticisms of a reviewer on an earlier version of this paper are highly appreciated.

Appendix A On States near Equilibrium

Since by design transient thermodynamics deals with states that are close to thermodynamical equilibrium, it is of relevance to define precisely how this class of states is to be identified. In section $(8)$ and within the context of transient thermodynamics, we identified a special class of fluid states that are interpreted as states in (global or local) thermal equilibrium. In this Appendix, we shall define a class of fluid states that are ”close to states in thermal equilibrium”.

As we discussed in section-$7$, an arbitrary state of a simple fluid is described by a set of primary variables consisting of the conserved and symmetric energy momentum tensor $T$, a conserved timelike particle current $J$ and the entropy four vector $S$ obeying $\nabla_{\mu}S^{\mu}\geq 0$. For classical fluids, the energy momentum tensor $T$ defines a unique timelike eigenvector $u_{E}$ that determines the Landau-Lifshitz (or energy) frame while the particle current $J$ via $J^{\mu}=n_{N}u_{N}^{\mu}$, determines the timelike vector $u_{N}$ that specifies the Eckart frame (or particle frame). These two vectors are in general distinct, unless the state describes a global or local equilibrium. At any event within the region occupied by this simple fluid, we may choose a local Lorentz frame with a time axis parallel to $u_{E}$ and augment this $u_{E}$ with a triad $e_{i},~{}i\in(1,2,3)$ of spacelike vectors so that $(u_{E},e_{i}),~{}i\in(1,2,3)$, constitutes an orthonormal tetrad. Since the triad $e_{i},~{}i\in(1,2,3)$ is not uniquely defined, we may, without loss of generality, choose the spatial vectors, so that:

[TABLE]

where $\vec{v}=ve_{1}$ is the “relative velocity” of the Eckart frame relative to the Landau-Lifshitz frame and introduced the parameter $\epsilon$ as the “pseudo-angle” between $u_{E}$ and $u_{N}$. From this relation it follows that the components of the particle current take the form

[TABLE]

which means that the particle densities $n_{N}$ and $n_{E}$ as seen in the two frames are related via $n_{E}=n_{N}cosh\epsilon$ while the particle drift $\hat{j}^{\mu}$ perceived in the Landau-Lifshitz frame is given by $\vec{\hat{j}}=n_{N}sinh\epsilon~{}\vec{e}_{1}$. The pseudo angle $\epsilon$ between $u_{E}$ and $u_{N}$ satisfies:

[TABLE]

and since in equilibrium (local or global) $u_{E}$ and $u_{N}$ coincide and thus $\epsilon=0$, following Israel [13], it is natural to define states of the simple fluid969696For a fluid mixture consisting of $n-$particle currents $(J_{1},J_{2},....,J_{n})$, one may define $n$-four velocities $(u_{1},u_{2},....,u_{n})$ and thus introduce $n-$pseudo angles $(\epsilon_{1},\epsilon_{2},....,\epsilon_{n})$ between $u_{E}$ and the corresponding $(u_{1},u_{2},....,u_{n})$. A state then is close to equilibrium, whenever $(\epsilon_{1},\epsilon_{2},....,\epsilon_{n})$ obey $\epsilon_{i}\leq 1$ for all $i\in(1,2,.....,n)$. as been close to equilibrium as those states having the property that everywhere within the region occupied by the fluid the pseudo-angle $\epsilon$ in (255) satisfies: $\epsilon=\frac{v}{c}<<1$. For such states, (253) and (254) imply

[TABLE]

where following the notation in Ref.\refciteIsr1, we denote the particle drift $\hat{j}^{\mu}$ relative to the energy frame by $\nu^{\mu}(u_{E})$ i.e. we set: $\nu^{\mu}(u_{E}):=\hat{j}^{\mu}$, and the symbol $O_{1}$ in (256) (and here after) signify a term of first order deviations from equilibrium979797We follow closely the notation of ref.\refciteIsr1. Thus here after $O_{1},O_{2},O_{3}.....$ signify terms of first, second, third order… deviations from equilibrium. Since the phenomenological laws of transient thermodynamics are invariant under first order change of the rest frame, the relation $n_{E}=n_{N}coshe=n_{N}+(O_{1})^{2}$, implies that the difference between $n_{E}$ and $n_{N}$ is second order in deviation from equilibrium and thus the distinction between $n_{E}$ and $n_{N}$ (as well as between $\rho_{E}$ and $\rho_{N}$) will be gradually blurred. That means that (256) can be written in the equivalent form $u_{N}^{\mu}-u_{E}^{\mu}\simeq\frac{\nu^{\mu}(u_{E})}{n_{E}}=O_{1}<<1$. like $\pi(u_{E})^{\mu\nu},\pi(u_{E}),\nu(u_{E})^{\mu})$ etc.

Beyond validity of (256), states close to equilibrium are required to satisfy another constrain: the viscous stresses, denoted collectively by $\tau^{\mu\nu}$, should be small in the sense

[TABLE]

where $(\tau(u_{E})^{\mu\nu},\rho(u_{E}))$ are the components of the viscous stress and the energy density as measured relative to the energy frame. For the motivation behind this condition, see for instance discussion in [13] as well as the Appendix C of the present paper.

The condition (256) imply that on the tangent space of each event within the fluid, an invariant “cone” of opening pseudo-angle $\epsilon\simeq\frac{v}{c}=O_{1}<<1$ with $v$ the speed of the Eckart frame relative to the energy frame can be defined and this invariant ”cone” plays an important role in the description of states near equilibrium. Any four velocity989898In view of the comment $(e)$ on page $26$, this four velocity $u^{\mu}$ could be chosen also as potential four velocity of the fluid. $u$ that falls within this cone can be used as a potential admissible rest frame and observers at rest relative to this frame determine thermodynamical variables associated with the non equilibrium state. Even though the measured thermodynamical variables have a dependance upon the four velocity $u$ and despite the plurality of these rest frames, nevertheless a consistent thermodynamical theory can be developed that is manifestly invariant under first-order changes of the rest-frame $u$, i.e. change in the rest frame described by

[TABLE]

Under this class of frame changes many thermodynamical variables that are frame dependent, transform in a well defined manner (see Appendix D, for a derivation of such transformation). As an example, notice that relation (254) implies that the particle densities $n_{E}$ and $n_{N}$ obey $n_{E}-n_{N}=O_{2}$ and thus to the first order deviations from equilibrium the measured particle densities are independent whether the energy frame or the particle frame is employed (provided we neglect $O_{2}$ and higher order contributions). As we shall show in Appendix D, an analogous property holds for many of the observables measured by observers with corresponding four velocities $(u,\hat{u})$ both future pointing and both lying within the cone spanned by $(u_{E},u_{N})$. In this case the corresponding transformations properties of the thermodynamical variables are discussed in the Appendix D.

We briefly mention here that the cone of opening angle $\epsilon$ offers the means999999In the Appendix C, we introduce another manner to identify states that are ”close” to an equilibrium state. to identify a reference equilibrium state (actually a whole class of such states) specified by $(S^{\mu}_{(0)},J^{\mu}_{(0)},T^{\mu\nu}_{(0)})$ which is”close” to an actual off-equilibrium fluid state described by the primary variables $(S^{\mu},J^{\mu},T^{\mu\nu})$. The details of this identification are discussed in section $9$.

Appendix B Implementation of the entropy principle - Liu’s and Ruggeri’s procedures

As we have seen in the main sections of the paper, the entropy principle plays an important role in the description of the thermodynamical properties of continuous media irrespectively whether such media are treated classically or relativistically. We recall that the principle states that the dependance of any constitutive function upon a set of basic variables should be such that every solution of the balance laws should satisfy the entropy inequality (or to put matters differently should obey the second law).

We also pointed out the need for procedures that lead into the implementation of this principle. Furthermore, in the sections $(12),(13)$ the fields of the Lagrange multipliers appeared as means to implement this principle and in that regard the origin and significance of these multipliers were rather mysterious.

This Appendix is dedicated on the problem of implementing the entropy principle and in that regard, we discuss two related algorithms for its implementation. The first one invented by Liu and is known as Liu’s procedure while the second one introduced by Boillat, Ruggeri and coworkers. Although in both of these methods the field of the Lagrange multipliers appers, the latter approach provides a more transparent interpretation of these multipliers.

We begin by outlining first Liu’s [51] procedure and in this procedure the following Lemma is of key importance:

Lemma B.1**.**

Let $A=A_{\Delta\gamma}$, $\Delta\in(1,2,.....p)$, $\gamma\in(1,2,....n)$ be a real $(p\times n)$ matrix and $\vec{X}=(X^{1},X^{2},....X^{n})\in R^{n}$, $B=(B^{1},B^{2},....B^{p})\in R^{p}$ $\vec{\alpha}=(\alpha^{1},\alpha^{2},....\alpha^{n})\in R^{n}$. Assume that $\vec{\alpha}\neq 0$ and $\beta\in R$ are given and moreover the following set:

[TABLE]

*is non empty. Then the following statements are equivalent:

*(1) For all $\vec{X}\in~{}S$ it holds: $\alpha_{\gamma}~{}X^{\gamma}+\beta\geq 0$

(2) There exist $\vec{\Lambda}=(\Lambda^{1},\Lambda^{2}......\Lambda^{p})\neq 0\in R^{p}$, refereed as Lagrange multipliers, such that:

[TABLE]

(3) There exist $\vec{\Lambda}=(\Lambda^{1},\Lambda^{2}......\Lambda^{p})\neq 0\in R^{p}$ such that

[TABLE]

As it stands, it is not clear how this Lemma is related to the entropy principle. Below, we shall outline briefly this connection and often refer the reader to the original article by Liu in Ref.\refciteLiu for a detailed discussion, proofs and applications of this Lemma.

A key element that connects this Lemma to the entropy principle is the notion of admissible sets. Following Liu’s approach, these sets are defined first for a system of second order quasilinear partial differential equations of the form:

[TABLE]

accompanied by an inequality

[TABLE]

The system is defined in an open set $D$ of $R^{4}$, and $U_{\hat{a}}$ stand for $N$- unknown thermodynamical fields while $(U_{\hat{a},\mu},U_{\hat{a},\mu\nu})$ denote their first and second partial derivatives100100100It is understood that the $M$-function $G_{A}$ and $I$ in (260, 261) are smooth bounded functions of their arguments. Notice also that in the terminology of thermodynamists any solution $V_{\hat{a}}$ of (260) is refereed as a thermodynamical process. For the system (260, 261), we may interpret (260) as a set of balance laws while (261) as an ”entropy like inequality”. In the present context, the entropy principle dictates that the function $I$ in (261) should be chosen so that for any thermodynamical process (261) must hold. with respect to the local coordinates $x^{\mu}$ covering $D$.

Let now an arbitrary point $p$ in $D$ and let a set of real constants:

[TABLE]

chosen so that there exists an open vicinity $O$ of $p$ in $D$ and a process $V_{\hat{a}}(x)$ (i.e solution of (260)) such that

[TABLE]

In that event, the set (262), provided non empty, constitutes an admissible set for the system (260) at $p\in D$. The punch line in Liu’s procedure for the implementation of the entropy principle, is the observation that if one demands validity of the entropy principle, i.e. one demands that for any process $V_{\hat{a}}(x)$ of (260) over $D$, the inequality (261) must hold, then (261) must hold for any admissible set101101101This presupposes that the system (260) admits admissible sets. Liu in Ref.\refciteLiu employed the Cauchy-Kowalewski theorem (and thus employed analyticity) to demonstrate that (260) admits admissible sets. However, as he also stated, this conclusion can be reached under much weaker conditions such as the Cauchy problem for (260) is well posed. of (260) and at any $p\in D$. The existence of admissible sets at $p\in D$ implies that for suitably defined matrix $A$ and vector $B$ related to (260, 261), the set $S$ in the above Lemma is non empty which in turn allows to take advantage of conditions $(2),(3)$ of that Lemma and thus demonstrate the existence of the Lagrange multipliers $\vec{\Lambda}=(\Lambda^{1},\Lambda^{2}......\Lambda^{p})\neq 0$ defined initially over $p\in D$. These multipliers are subsequently extended as fields over the open vicinity $O$ of $p$ in $D$. By appealing to the above Lemma, condition $(1)$ is replaced by conditions $(2)$ and $(3)$ and this replacement sets restrictions upon dependance of the Lagrange multipliers upon the $U_{\hat{a}}$ fields (and possibly their derivatives).

Liu in Ref.\refciteLiu, applied this procedure to a simple heat conducting, viscous fluid. For this system the basic102102102Recall that by the term basic variable, we mean a set of variables that have the distinct property that any other constitutive function should be depending upon these variables. variables are considered to be the density $\rho(t,\vec{x})$, the components $u^{i}(t,\vec{x}),~{}i\in(1,2,3)$ of the velocity field and the empirical temperature $T(t,\vec{x})$. The balance laws of mass, momentum and internal energy, were obtained in sections $(2-4)$, and contain the components of the Cauchy stress tensor $\sigma_{ij}$, the heat flux vector $q^{i}$ and internal energy $e$ which are considered to be constitutive functions i.e. smooth functions of $(\rho,u^{i},T,\dot{T},T,_{i})$. Here $\dot{T}$ stands for

[TABLE]

and for typographical convenience we have suppressed the dependance of the variables $(\rho,u^{i},T,\dot{T},T,_{i})$ upon the spacetime coordinates. For this fluid, the entropy inequality in (33) is written in the equivalent form

[TABLE]

where both $s$ and $\vec{J}$ are constitutive functions i.e.

[TABLE]

Once the dependancies of $(\sigma_{ij},q^{i},e)$ and $(s,J^{i})$ upon the basic variables have been specified, they are substituted in the balance laws and the entropy inequality. Liu observes that the resulting system although rather lengthy, it can be written in the following compact form:

[TABLE]

where the nine components $y_{\lambda}$ of $\vec{y}$ are

[TABLE]

the $5$-components of $B_{\Delta}$ and the constant $\beta$ are given by

[TABLE]

while the $(5\times 26)$ dimensional matrix $A_{\Delta\gamma}$ and the $26$ components $X^{\gamma}$ of the vector $\vec{X}$ in (265) are very complicated long expressions whose explicit form can be found in Ref.\refciteLiu.

Based on these expressions, Liu demonstrates the existence of admissible sets and thus at a fixed point $p\in D$ the set $S$ in Liu’s Lemma is non empty and thus he infers the existence of the Lagrange multipliers. Based on the conditions $(2),(3)$ of this Lemma he deduces restrictions upon the structure of the Lagrange multipliers $(\Lambda^{\rho},\Lambda^{u^{i}},\Lambda^{e})$. We shall not discuss any further this example (the reader who is interested for more details is refereed to the original article by Liu in Ref.\refciteLiu). Instead, below, we discuss an alternative procedure for implementing the entropy principle.

In this alternative procedure, the Lagrange multipliers re-appear but from the perspective of the mathematical theory of hyperbolic systems. This alternative procedure invented after the recognition by Boillat in Ref.\refciteBoi and Ruggeri-Strumia in Ref.\refciteRugStr that whenever the Lagrange multipliers in Liu’s procedure are chosen as thermodynamical variables (and under certain additional requirement upon the entropy vector like convexity), then the field equations (i.e. the balance laws) can be turned into a symmetric, hyperbolic system. Various aspects of the Boillat, Ruggeri-Strumia procedure have developed in Refs. \refciteBoi, \refciteRugStr, \refciteRugF1, \refciteRugF2, \refciteRugF3. For completeness purposes, below, we briefly outline this powerful method and follow closely the treatment in Ref.\refciteRugStr.

The Boillat-Ruggeri-Strumia procedure pre-assumes a medium that is governed by $n-$balance laws of the form:

[TABLE]

where $\vec{F}^{\alpha}$ for $\alpha\in(0,1,2,3)$ stand for four column of $n$-components:

[TABLE]

and

[TABLE]

and in above $T$ signifies transpose.

The $n-$functions $\vec{u}(t,\vec{x}):=(u_{1}(t,\vec{x}),u_{2}(t,\vec{x}),..........,u_{n}(t,\vec{x}))^{T}$, are assumed to be the basic variables describing the medium under consideration and thus (266) stands for a system of $n$- equations for the $n$-unknowns functions $\vec{u}(t,\vec{x}).$

The system (266) is accompanied by a supplementary equation, interpreted as an entropy law, of the form

[TABLE]

where $S^{\alpha}$ are the components of a $4$-dim. vector field and $\Sigma$ is a scalar, both depending smoothly upon $\vec{u}(t,\vec{x})$. The inclusion of this additional equation implies that (266) and (267) becomes an overdetermined system of $(n+1)$ equations for $n$ unknowns.

Within the field of the thermodynamics of continuous media, overdetermined systems of ”conservation laws” are encountered frequently. The dynamical equations for the Fourier-Navier-Stokes simple fluid can be cast in the form (266, 267) (see for instance Refs. \refciteRug1, \refciteRug4) while other physical systems described by systems of the form (266, 267) are discussed in ref. (\refciteMul4). Moreover as we have already seen in sections $(12-13)$, the dynamical equations of the Liu-Müller-Ruggeri and the family of relativistic dissipative fluids of divergence type are particular cases of the systems (266, 267). For these theories, the unknowns $(u_{1}(t,\vec{x}),u_{2}(t,\vec{x}),..........,u_{14}(t,\vec{x}))$ are identified with the $14$ unknown components of $J^{\mu}$ and $T^{\mu\nu}$ and these $14$ variables satisfy $15$ equations (see eqs (167, 168, 170) in the main text). Thus it is worthwhile to briefly describe the features of systems described by (266, 267).

In general one does not expect that overdetermined system of equations to admit solutions. However under suitable restrictions, Friedrichs in Refs. \refciteFried0, \refciteFried1 (see also Ref.\refciteFried2) has shown that such systems can be turned into symmetric hyperbolic system of equations. That will be the case provided a ”main dependency relation103103103In the following, we employ the terminology and notation of (167, 168).” holds (see equation (268)) and in addition a certain quadratic form is positive definite (see relation (275)).

If one rewrites the system (266, 267) in the form:

[TABLE]

and denote by a dot $\cdot$ the Euclidean inner product in $R^{n+1}$, then Friedrich ”main dependency relation” holds provided that there exists104104104In a practical problem one may appeal to Liu’s lemma to establish the existence of the multipliers fields $(\Lambda^{1}(\vec{u}),\Lambda^{2}(\vec{u}),......,\Lambda^{n}(\vec{u}))$ and then proceed to check of Friedrich’s conditions. For an example see Ref. (\refciteFT1).

an $(n+1)$ dimensional vector field $\vec{Z}(\vec{u})$ that satisfies

[TABLE]

and this equation should hold for all $\vec{u}$ and $\partial_{\alpha}u^{\beta},\alpha\in(0,1,2,3),\beta\in(1,2,...,n)$. Choosing the field $\vec{Z}$ in the form: $\vec{Z}=(-\vec{\Lambda}(\vec{u}),1)$ where: $\vec{\Lambda}(\vec{u})=(\Lambda^{1}(\vec{u}),\Lambda^{2}(\vec{u}),......,\Lambda^{n}(\vec{u}))$ with the $\Lambda^{i},i\in(1,2,...n)$ depending smoothly upon $\vec{u}(t,\vec{x})=(u_{1}(t,\vec{x}),u_{2}(t,\vec{x}),..........,u_{n}(t,\vec{x}))$, it follows that (268) is equivalent to

[TABLE]

and this relation shows that $\vec{\Lambda}(\vec{u})$ plays the same role of the Lagrange multipliers in the Liu’s approach.

Introducing the $n$-dimensional gradient operator $\hat{\nabla}$ via

[TABLE]

and requiring that condition (269) to hold for all $\vec{u}$ and $\partial_{\alpha}u^{\beta}$ it follows

[TABLE]

Making use of the $n$-dimensional exterior differential operator $d$ in Euclidean $R^{n}$, these conditions can be written in the equivalent form

[TABLE]

These two equations are consequence of Friedrich’s ”main dependency relation” and would be worth to have an understanding of their deeper significance. We shall not address their deeper significance in this work, but we shall exploit some of their consequences.

Suppose that in (266) it is possible to choose: $\vec{u}=\vec{F}^{0}$. Then for this choice, (270) implies that

[TABLE]

and this is a set of important relations.

Suppose for this particular choice i.e. $\vec{u}=\vec{F}^{0}$, the entropy density105105105This property is well defined for the case of Newtonian media. However for relativistic media issues of covariance are hidden in this property. $S^{0}(\vec{u})$ is a concave function of $\vec{u}$ i.e. the matrix $\frac{\partial^{2}S^{0}}{{\partial{\vec{u}}}{\partial{\vec{u}}}}$ is negative definite, clearly from the above relations it follows that the matrix $(\frac{\partial{\vec{\Lambda}}}{\partial{\vec{u}}})$ is also negative definite. By appealing to standard theorems on Jacobian matrices, one concludes that the transformation $\vec{u}\to\Lambda(\vec{u})$ is globally invertible. In turn this property allows to view

[TABLE]

Combining these relations with (270) one concludes that exist four scalars $S^{\prime\alpha},\alpha\in(0,1,2,3)$ such that

[TABLE]

Thus the system (266) is compatible with the complementary law (267), provided the components $\vec{F}^{\alpha}$ are the gradient of $S^{\prime\alpha}$ with the later fields viewed as functions of the field $\vec{\Lambda}$.

Returning to the system (266) and using the representation of ${\vec{F}}^{\alpha}$ shown in (273), one gets

[TABLE]

where the four $(n\times n)$ matrices:

[TABLE]

are the Hessian matrices of $S^{\prime\alpha}(\vec{\Lambda})$ (with respect to the components of $\vec{\Lambda})$ and thus by construction are all symmetric106106106 Notice that this property does not hold on general whenever the components of the field $\vec{u}$ are employed as basic variables..

Beyond the manifestly symmetric property of (274), the system has another remarkable property. As long as the entropy density $S^{0}$ is a concave function of $\vec{u}(t,\vec{x})$, the system (274) is a symmetric-hyperbolic system. The hyperbolic nature can be established either by appealing to Friedrichs second condition or checking directly whether the definition of a symmetric-hyperbolic system of equations hold true for the system (274).

We recall that Friedrichs second condition requires that there exist a co-vector $\xi=(\xi_{0},\xi_{1},\xi_{2},\xi_{3})$ so that the quadratic form

[TABLE]

is positive definite for all smooth variations $\delta u(t,\vec{x})$ of a background $\vec{u}(\vec{x},t)$. Validity of this condition combined with (270) yields the desired conclusion i.e. the symmetric-hyperbolic nature of (274).

Notice however, that we can reach the same conclusion by appealing to the second alternative i.e. checking directly whether Friedrichs definition of a symmetric-hyperbolic system holds true.

Writing the system (274) in the form $A^{\alpha}{}_{ij}\partial_{\alpha}\Lambda^{i}=f_{j}$, then this is a symmetric-hyperbolic system in the sense of Friedrich, provided the four matrices $A^{\alpha}{}_{ij}$ are symmetric i.e. $A^{\alpha}{}_{ij}=A^{\alpha}{}_{ji}$ for all $\alpha\in(0,1,2,3)$ and in addition a ”positivity condition” should hold. This later condition requires that there should exist co-vectors $\xi=(\xi_{0},\xi_{1},\xi_{2},\xi_{3})$ so that the matrix $\xi_{\mu}A^{\mu}_{ij}$ should be positive definite. For the system (274), we have already seen that the four matrices $A^{\alpha}{}_{ij}$ are all symmetric. To establish positivity, it is sufficient to choose the co-vector $\xi=(\xi_{0},\xi_{1},\xi_{2},\xi_{3})=(1,0,0,0,0)$. For this choice, positivity holds provided the matrix

[TABLE]

is definite. This conclusion however follows by recalling that for the choice $\vec{u}=\vec{F}^{0}$ the relation $S^{\prime\alpha}=\vec{\Lambda}\cdot\vec{F}^{\alpha}-S^{\alpha}$ implies that $S^{\prime 0}=\vec{\Lambda}\cdot\vec{u}-S^{0}$ and thus $S^{\prime 0}$ is the Legendre transform of $S^{0}$. This implies that $S^{\prime 0}$ is a concave function of $\vec{\Lambda}$ since $S^{0}$ has been assumed to be a concave function of $\vec{u}$. This conclusion proves that the symmetric system (274) is hyperbolic.

In the terminology adopted by Ruggeri and collaborators, the four fields $S^{\prime\mu}(\vec{\Lambda})$ are refereed as the generator field, while the Lagrange multipliers $\vec{\Lambda}(t,\vec{x})$ constitute the main field. From the so far analysis, it follows that if one would be able to identify the generator field $S^{\prime\mu}(\vec{\Lambda})$, then the right constitutive relations i.e. the functions $F^{\alpha}$ compatible to the entropy inequality (267), subject to $\Sigma(\vec{u})\geq 0$, are those obtained via differentiating the generator with respect to the components of the main field. Here we see a more concrete implementation of the entropy principle. The dependance of the constitutive functions $F^{\alpha}$ upon the basic variables, taken here as the Lagrange multipliers, should be special in order to be compatible with the entropy inequality.

As far as applications of the above results are concerned, it is suffice to stress that of relevance here is the construction of the generator field i.e. $S^{\prime a},\alpha\in(0,1,2,3)$. For the case of the Liu, Müller and Ruggeri theory, developed in section $(12)$, the fields $S^{\prime a},\alpha\in(0,1,2,3)$ are defined in (178). Approximate expression for $S^{\prime\alpha}$ as a function of the Lagrange multipliers (see eq.(185)), has been constructed by appealing on the principle of relativity. For the case of the dissipative relativistic fluids of divergence type developed in section $(13)$, the approach was different. There the components of $S^{\prime a}$ are the gradients of the smooth generating scalar function $\chi(\zeta,\zeta_{\mu},\zeta_{\mu\nu})$ i.e. $S^{\prime\alpha}=\frac{\partial\chi}{\partial\zeta_{\alpha}}$ and this formula makes clear the economy in the Pennisi-Geroch-Lindblom formalism.

It should be mentioned here that in both of the above cited approaches the analysis is local. The transformation $\Lambda^{i}\to u^{i}$ employed in the Liu, Müller and Ruggeri theory and also for the fluids of divergence type, is only locally invertible and this local invertibility is a consequence of Friedrichs ”main dependency relation” and the positive definite character of the quadratic form shown in (275). Validity of these two conditions lead to a local invertibility of the transformation $\Lambda^{i}\to u^{i}$ (see discussion in Ref. (\refciteFT1)).

Appendix C On the thermodynamics of relativistic continuous media

In this Appendix, we discuss a few basics aspects of the thermodynamics of relativistic continuous media. Because such media exhibit a great diversity and their thermodynamical description requires a great deal of the theory of constitutive relations and the entropy principle, therefore in this Appendix, we only offer a few remarks concerning the structure of the first and the second law of thermodynamics applied to states of such media and comment on the identification of their equilibrium states. For the case of simple fluids, we provide some details regarding the thermodynamics of such media.

As we have mentioned in the introduction, the laws of thermodynamics, as applied to classical (Newtonian) continuous media have been established long ago, but their formulation assumed a Newtonian setting. Upon the arrival of the special theory of relativity (and later of the general theory) these laws had to be reformulated so they become compatible with the requirements of the Poincare (or general) covariance. That problem attracted the attention of many leading physics of that epoch including Einstein, Planck, Pauli amongst others and these early attempts were meet with partial success, disagreements and confusions. It is sufficient to recall the “Planck-Ott controversy” regarding the definition and transformation properties of the relativistic temperature, and the “Abraham-Minkowski controversy” concerning the definitions of stresses and momenta in polarized media (a historical account of these early efforts is described in Pauli’s book[117]).

Modern approaches to thermodynamics of relativistic media employ the fields of local Lorentz frames or the proper reference frames associated to a world line of an observer, and in general, these classes of frames are specified by defining a four velocity field in the spacetime region occupied by the fluid. Although one can develop the thermodynamical properties of a medium relative to an arbitrary chossen velocity field, in practice one presuppose that the medium under consideration defines a preferred four velocity or families of preferred four velocities107107107For the case of a fluid, a preferred four velocity $u$ may be identified as the four velocity that defines the energy frame or the four velocity that defines the particle frame. More generally preferred velocity fields may be restricted to lie within the interior of the ”cone” of opening pseudo-angle $\epsilon$ defined in the Appendix A.. In addition to the specification of a preferred velocity field, the derivation of the thermodynamical laws presented in this Appendix relies on the validity of the local thermodynamical equilibrium hypothesis ((LTE) in short). These two ingredients together lead to the deduction of the first law and the formulation of second law. Based on these laws we comment also on the nature of relativistic equilibrium for relativistic fluids.

We begin by assuming that the medium is propagating on a smooth four dimensions spacetime $(M,g)$ so that at any event $p$ within the medium is defined a future directed four velocity $u$ normalized according to $g(u,u)=-1$. For any such $p$, one may introduce a local Lorentz frame whose time axis is determined by the four velocity $u$. Notice however that at the same $p$, one may introduce the proper reference frame108108108For a definition and properties of proper frames see for instance discussion in ref. \refciteMTW. attached to an observer comoving with $u$ . Even though the resulting local coordinate systems have the same time axis they are not identical. The local Lorentz frame is a free falling frame and is best suited for implementing Einstein’s principle of equivalence while the proper frame is in general an accelerating frame. The later, is a more convenient since it can be extended along the trajectory of the fluid element and this proper frame often is referred as the rest frame.

The medium is assumed to be described by a set of tensor fields, denoted collectively by $\zeta_{A}$, with the index $A\in(1,2,...,N)$ enumerating the various field. These fields would satisfy a set of dynamical equations of tensorial nature and furthermore rules are supplied so that a conserved energy momentum tensor $T^{\mu\nu}(\zeta_{A})$ and other conserved currents may be constructed out of the dynamical equations.

Moreover for any $p\in(M,g)$, it is assumed that the allowed values of $\zeta_{A}$ define a smooth manifold $Y$ describing the possible states of the medium over $p$ and these fibers join smoothly so that they define a smooth fiber bundle, with base manifold the spacetime $(M,g)$. Geroch in ref. \refciteGer3, (see also ref.\refciteGer2) used this framework to analyze a large class of relativistic fluid theories whose dynamic equations constitute a symmetric-hyperbolic (causal) system and this framework contains as a special case the class of relativistic fluids of divergence type discussed in section $(13)$. We shall not enter on this mathematically beautiful formalism but we only mention that by appropriate restrictions upon the structure of the field equations, for a simple fluid Geroch identified a class of equilibrium states spanning a five dimensional submanifold $\hat{E}$ within each fiber $Y$. Moreover, he showed that the causality restriction upon the structure of the field equations induces on each fiber $Y$ a positive-definite metric $G_{AB}$ and this metric can be employed to give a precise meaning to the notion that a fluid state finds itself near equilibrium109109109 As we have already seen in Appendix A, transient thermodynamics identifies states close to thermal equilibrium by making use of the invariant cone of opening pseudo-angle $\epsilon$ formed by $u_{E}$ and $u_{N}$. It would be interesting to investigate whrther these two methods of defining states close to equilibrium are equivalent and whether the mathematical setting in Geroch approach in defining ”closeness” offers additional advantages..

For the purpose of this section, of relevance are the variations $\delta\zeta_{A}$ of the fields $\zeta_{A}$ that are defined along the fiber $Y$ over a fixed $p\in(M,g)$. These variations, refereed as fibered variations, will be important in the formulation of the first law. Moreover we shall use the notion of the a thermodynamic temperature and temperature as defined in Ref. \refciteGer3 and these notions will be explained further below.

We now assume that the medium under consideration is a simple fluid and in the region occupied by the fluid is defined a four velovity $u$ that identifies a family of rest frames. Suppose that relative to a rest frame defined at $p$ by the four velocity $u$, we consider a spatial three volume $V$ residing on the spacelike plane orthogonal to $u(p)$. We denote by $(n,\rho)$ the particle number density, and the density of mass-energy as measured by an observer at rest relative to this frame, so that $N=nV$ and $U=\rho V$ are the total number of particles and total mass-energy within this $V$. As we have already mentioned, the other key hypothesis that will be employed for the derivation of the laws of thermodynamics of continuous media is the validity of local thermodynamical equilibrium hypothesis110110110It should be mentioned that the issue of validity of local thermodynamical equilibrium hypothesis (LTE), within the relativistic regime becomes a subtle issue and this subtleties arise from the basic principles of relativity that neither time intervals nor space intervals by themselves have an absolute meaning. As a consequence validity of (LTE) may become observer dependent. Fortunately within transient thermodynamics and as long as one treats fluid states near equilibrium, then there is $\epsilon$-”cone” worth of admissible observers who have the following property: If relative to the one of these observers (LTE) holds, then it holds for all observers whose four velocity lies within this $\epsilon$-”cone”. This property is discussed in more details in Ref.\refciteFT2..

Within the present context, validity of (LTE) affirms that the state variables within $V$ satisfy the same thermodynamical relations as if this system was in a state of a global thermodynamical equilibrium.

Let us now assume that the state of the fluid and the nature of the congruence associated with the four velocity $u$, are such that within the three volume $V$, (LTE) holds. Thus we may introduce the ”entropy density” $s$ which is a function of $(\rho,n)$ i.e. $s=s(\rho,n)$ and this $s=s(\rho,n)$ is refereed as the equilibrium equation of state 111111111For the case where the fluid is a perfect fluid this $s=s(\rho,n)$ is the equilibrium equation of state defined in section $8$ (see also eqs (83,84)). For this case $S^{\mu}=s(\rho,n)u^{\mu}$ is the Tolman physical entropy flux and $s(\rho,n)=-S^{\mu}u_{\mu}$ is the physical entropy density measured by the observer comoving the fluid’s four velocity. Away from perfect fluids, this $s=s(\rho,n)$ is well defined as a consequence of the validity of (LTE) postulate. In this case, $s(u)=s(\rho(u),n(u))$ is a purely formal quantity and although refereed as ”entropy density” there is no physical reasoning to support this interpretation. Still however the mathematical manipulations that follows by employing this $s(u)=s(\rho(u),n(u))$ are well defined.. Using this ”entropy density” $s$ we refer to $S=sV$ is the total ”entropy” within $V$.

Validity of (LTE) allows us to introduce intensive variables112112112It is understood that all these thermodynamical variables are local and we ought to indicate their explicitly dependance upon either $p$ or a set of local coordinates i.e. ought to write either $n(p),\rho(p),...etc$ or equivalently $n(x^{\mu}),\rho(x^{\mu}),...etc$. For simplicity of the presentation we omit such dependancies. temperature $T$, pressure $P$ and chemical potential $\mu$ so that $(U,S,N,T,P,\mu)$ etc. satisfy the familiar laws of thermodynamics for spatially homogeneous equilibrium states. In particularly, the first law takes the form113113113The variations $dU$, $dQ$.. etc in this first law refer to fibered variations over $p$ and these variations defined above.:

[TABLE]

where $dQ$ is that amount of heat entering the volume $V$ and $PdV$, $\mu dN$ have the interpretation as the work done by expanding (contracting) $V$ or adding (subtracting) particles. For a fluid mixture, one may introduce $n$-chemical potentials $\mu_{i}$ describing the $n$ different particle species and denote by $N_{i}$ the total number of particle of type $i$ within $V$. For such fluids, the first law in (276) is obtained by replacing $\mu dN$ by ${\sum\limits_{i}\mu_{i}dN_{i}}$. Since the system within $V$ is considered as been in equilibrium (thermal, mechanical, chemical) the familiar language of equilibrium thermodynamics like reversible, irreversible, quasistatic transformations, extensive, intensive variables etc. apply to this system but we should keep in mind that these concepts have local character.

If an amount of heat $dQ$ is injected reversibly within $V$ then $dQ=TdS$ and for these reversible transformation (276) yields the familiar Gibbs relation

[TABLE]

The scaling properties of the extensive variables $(U,S,N)$ in this formula leads to

[TABLE]

which in turn imply that the intensive variables $(T,P,\mu)$ satisfy the Gibbs-Duhem relation:

[TABLE]

Dividing (278) by $V$ yields

[TABLE]

which via differentiation and in combination to the the densitized form of the Gibbs-Duhem relation (279) i.e. $sdT-dP+nd\mu=0$, implies

[TABLE]

and thus knowledge of the equation of state $\rho=\rho(s,n)$ for this fluid determines the (local) temperature and (local) chemical potential $\mu$ via differentiation. Relation (281) is the form of the first law for a relativistic fluid as perceived by an observer with four velocity $u$ at $p$.

Often it is convenient to formulate this law by employing “the per particle description”. To do this, we set $U=\rho V=\frac{\rho N}{n}$, $PdV=Pd\left(\frac{N}{n}\right)$ and introduce the ”entropy” per particle $\hat{s}$ via: $s=\hat{s}n$ so that $S=sV=s\frac{N}{n}=\hat{s}N$. In terms of these new variables, (277) reduces to

[TABLE]

and for any transformations that keeps the total particle number $N$ within $V$ fixed, the first law takes the form

[TABLE]

from which one concludes:

[TABLE]

By introducing the mean internal energy $e$ per particle via

[TABLE]

and returning again to (282) keeping $N$ fixed, we obtain

[TABLE]

Moreover substituting (285) in (280), we find that the relativistic chemical potential per particle $\mu$ has the form 114114114The classical chemical potential $\mu_{cl}$ is defined via $\mu_{cl}=e+P\hat{V}-T\hat{s}$ so that $\mu=mc^{2}+\mu_{cl}.$:

[TABLE]

where in the formula for $\mu n$ the entropy per particle $\hat{s}$ and $\hat{V}$ have been eliminated. If in (286), we eliminate $\hat{s}$, $\hat{V}$ and $e$ in favor $(s,n,\rho)$ and use the above relations we return into (281).

If we introduce the (relativistic) thermal potential $\Theta$ and the inverse temperature $b$ via115115115If we restore for a moment $c$ and Boltzmann’ s constant $k$, then the form of the relativistic thermal potential $\Theta$ and the inverse temperature $b$ read: $\Theta=\frac{\mu}{kT},\quad b=\frac{c^{2}}{kT}$. via

[TABLE]

then the formulas $\mu n=\rho+P-Ts$ and (281) take the equivalent forms:

[TABLE]

These two relations imply the important identity116116116Whenever $c\neq 1$ the identity reads: $d(sX)=bd(\rho X)-\Theta d(nX)+b\frac{P}{c^{2}}dX$ (derived first by Israel)

[TABLE]

where $X$ stands for an arbitrary smooth function. Its validity can be seen by expanding both sides of (290) yielding:

[TABLE]

showing that (290) is a consequence of (289). We shall employ (290) further below but it is worth noticing that this identity depends only upon the validity of the (local) first law and the scaling property as expressed in (289) and is independent of the nature of the background fluid.

Finally, it should be mentioned here that the form of the first law deduced above applies to a relativistic fluid whose equation of state has the form $\rho=\rho(n,s)$ (or equivalently $s=s(\rho,n)$). For other type of media, and as long as the local thermodynamical equilibrium hypothesis is assumed to hold, one via similar reasoning may formulate a local version of the first law. Notice however, that the local energy density $\rho$ may depend besides the ”entropy” density $s$, upon other fields such as the state of strain (for an elastic medium) electric or magnetic fields and in such event the relation (281) (or more precisely (276)) would have different structure (for elastic media the structure of the first law is discussed in refs. \refciteCar1,\refciteEhl).

Before we enter into the formulation of the second law, we comment briefly on the notion of the (local) temperature $T(u)$ that appears in the local version of the first law. From the above deductions, it follows that as long as an equilibrium equation of state $s=s(\rho,n)$ (or an equivalent form $\rho=\rho(s,n)$) is specified, then the local equilibrium hypothesis defines the (local) temperature $T(u)$ as measured by the $u$-observer, as the partial derivative of the equation of state and thus as a function of $(\rho,n)$ or $(s,n)$. Often however, a given equation of state $s=s(\rho,n)$ and given temperature profile $T=T(\rho,n)$, are specified simultaneously and in such event Maxwell’s relations demand that these $s=s(\rho,n)$ and $T=T(\rho,n)$ are compatible with the first law, provided $T=T(\rho,n)$ satisfies :

[TABLE]

Geroch in ref. \refciteGer3 refers to such $T=T(\rho,n)$ as the thermodynamic temperature and points out that there is a gauge freedom in the choice of this $T=T(\rho,n)$. If $T=T(\rho,n)$ and $\hat{T}=\hat{T}(\rho,n)$ satisfy this equation then for any constants $(c_{1},c_{2})$ then $c_{1}T+c_{2}\hat{T}$ is also a solution of the above equation. Moreover, if $T=T(\rho,n)$ and $\hat{T}=\hat{T}(\rho,n)$ are solutions of thee above equation, then any $\hat{\hat{T}}$ which is a homogeneous function of degree one i.e. $\hat{\hat{T}}(\lambda T,\lambda\hat{T}))=\lambda\hat{\hat{T}}(T,\hat{T}))$ is also an admissible thermodynamical temperature. This freedom in the choice of the thermodynamical temperature. raises the question which one of these admissible $T$ should be the physically correct temperature. He argues that may be none of them, but for suitably restricted fluid theories, a meaningful temperature comes from the description of dissipative effects and in particularly the structure of the heat flow or energy flow relative to the $u$ observer (for further discussion on this point the reader is referred to Geroch’s article).

We now pass to the formulation of the second law and the assignment of entropy for states describing continuous relativistic media. As we have seen above, the (LTE) postulate allows to introduce the (local) ”entropy” density scalar $s$ so that $S=sV$ is the total ”entropy” within $V$. However, as we have already mentioned both $s$ and $S$ are formal quantities and away from the special case where the medium is a perfect fluid, there are no physical arguments to interpret them as the physical entropy density or total entropy within $V$. Nevertheless, it is suggestive to examine the evolution of the total ”entropy” $S=sV$ as it is transported forward in time along the flow lines (more precisely along the integral curves of $u^{\mu}$). It follows at once that

[TABLE]

and this formula suggests that as long as $(\nabla_{\mu}S^{\mu})\geq 0$ then $S=sV$ does not decrease as it is transported forward in time along the flow lines.

The expression $S^{\mu}=su^{\mu}$ does represent the entropy flux for a perfect fluid, provided $s$ is the equilibrium equation of state and $u^{\mu}$ is the four velocity of the fluid and in fact as we shall see below for this case $(\nabla_{\mu}S^{\mu})=\nabla_{\mu}(su^{\mu})=0$. These considerations, suggest to assign to any state describing an arbitrary fluid (or a relativistic continuous media) an entropy flux vector $S^{\mu}$ which is arbitrary except that is subject to obey the constraint: $\nabla_{\mu}S^{\mu}\geq 0.$ This constraint guarantees that for any two non intersecting spacelike hypersurfaces $\Sigma_{t_{1}},\Sigma_{t_{2}}$ cutting across an asymptotically flat spacetime, with $\Sigma_{t_{2}}$ to the future of $\Sigma_{t_{1}}$, the total entropy $S(t)=\int_{\Sigma_{t}}S^{\mu}d\Sigma_{\mu}$ satisfies $S(t_{2})\geq S({t_{1}})$ and this inequality implements the second law117117117Within the standard equilibrium thermodynamics, the second law deals with the transformation of heat into work and this law is stated either in the Kelvin-Planck or Clausius form (see for instance [118]). The Clausius inequality allows to introduce the entropy $S$ as a state function that has the property that for any isolated system $S$ does not decrease in time. Within the relativistic domain, the second law is implemented by introducing the entropy flux $S^{\mu}$ and ought to be keept in mind that this primary variable $S^{\mu}$ is distinct to the the ”entropy” flux $su^{\mu}$ arising via the local equilibrium postulate. for states of arbitrary relativistic media. Moreover, this $S^{\mu}$ has the property that any observer with four velocity $\hat{u}^{\mu}$, measures an entropy density: $s(\hat{u})=-\hat{u}_{\mu}S^{\mu}$.

The entropy flux vector $S^{\mu}$ introduced above does not have any apparent relation to the energy momentum tensor $T^{\mu\nu}$ (or on the possible particle currents $J_{A}^{\mu}$) characterizing the medium. In fact, as has been discussed in the main sections of the paper, the relation of this $S^{\mu}$ to variables specifying fluid states is the central issue of irreversible thermodynamics. Either via specific postulates of Müller - Israel type or by appealing to entropy principle which asserts that $S^{\mu}$ is a constitutive function, the entropy flux vector $S^{\mu}$, is required to be specified and the condition $\nabla_{\mu}S^{\mu}\geq 0$ becomes a very restrictive.

Just to get some insights into the implications and significance of this $S^{\mu}$, let us consider the particular case of a simple relativistic fluid characterized by the property that $S^{\mu}$ has the Tolman form i.e. $S^{\mu}=su^{\mu}$ with $s$ the ”entropy” density as measured by the $u^{\mu}$-observer. Application of the Gibbs relation (289), implies

[TABLE]

where an over dot signifies differentiation along the flow lines (integral curves of $u^{\mu}$). For a simple perfect fluid, by taking $u^{\mu}$ to be the uniquely defined four velocity of the fluid combined with the balance laws $\nabla_{\mu}T^{\mu\nu}=\nabla_{\mu}J^{\mu}=0$ imply that the right hand side of (293) is identically vanishing. Thus the evolution of perfect fluids states do not generate entropy and the total entropy $S(t)=\int_{\Sigma_{t}}S^{\mu}d\Sigma_{\mu}$ across a spacelike hypersurface $\Sigma_{t}$ remains constant i.e. is independent of $\Sigma_{t}$. In a state of global thermodynamical equilibrium it is expected that the total entropy $S(t)$ to be maximum and thus any first order variation $\delta S(t)$ induced by for small perturbations $\delta T^{\mu\nu}$ and $\delta J^{\mu}$ of the background perfect fluid configurations $T^{\mu\nu}$ and $J^{\mu}$ should be vanishing to first order. By appealing to covariant Gibbs relation it follows that will be the case provided the thermal potential $\Theta$ is uniform and the $b^{\mu}$ is a timelike Killing vector field conditions that we have already seen in section $8$.

Away from perfect fluids, one may postulate an entropy flux $S^{\mu}$ of the form

[TABLE]

where $(\beta,\theta)$ are unspecified coefficients, while $h^{\mu},n^{\mu}$ stand for the energy flow and particle “drift” relative to the $u$-frame (see formulas (110) and (111) in the main text). This choice, defines the class of first order theories that includes the Eckart and Landau-Lifshitz theories as a special case (this class of theories has been introduced in ref. \refciteHis2). For this class the phenomenological equations and the nature of equilibrium states i.e. states obeying $\nabla_{\mu}S^{\mu}=0,$ have been worked out in ref. \refciteHis2.

It is interesting to mention that transient thermodynamics can be developed by enlarging the entropy flux $S^{\mu}$ shown (294), by the inclusion of second order contributions arising from the energy flux and stresses. For this one should first define an admissible four velocity $u^{\mu}$ i.e. a velocity field $u^{\mu}$ lying within the invariant ”cone” of opening pseudo-angle $\epsilon<<1$ that we have defined in Appendix A and write down an extended entropy flux vector $S^{\mu}$ that contains second order deviations from equilibrium. By demanding that the so extended $S^{\mu}$ satisfies $\nabla_{\mu}S^{\mu}\geq 0$, coupled to the structure of the balance laws $\nabla_{\mu}T^{\mu\nu}=\nabla_{\mu}J^{\mu}=0$ one would obtain the phenomenological equations of the theory. However, the issues of the covariance of the theory under change of the four velocity $u$ needs to examined and that is what we have done in the main part of the paper.

Equilibrium states within transient are states obeying $\nabla_{\mu}S^{\mu}=0$. From the structure of the phenomenological equations it follows that this condition demands the heat flux and stresses to be vanishing and this leads to the conclusion that they are perfect fluid states characterized by a uniform thermal potentials and a four velocity parallel to a timelike Killing field i.e. states obeying the conditions outlined in section $8$ (for a derivation of these results consult Ref.\refciteHis1).

For the rest of this Appendix, we present a covariant formulation of the thermodynamics for a perfect fluid i.e. a formulation where the fundamental thermodynamical relations between two fluid sates makes no references to any rest frame nor to quantities defined relative to such frame.

In order to do so, let $u$ be the unique hydrodynamical four velocity defined by the perfect fluid and let $\Delta^{\mu}{}_{\nu}=\delta^{\mu}{}_{\nu}+u^{\mu}u_{\nu}$ be the associated projection tensor. Clearly, relatively to the rest frame $u^{\mu}=\delta^{\mu}_{\ 0}$ and thus $\Delta^{\mu}{}_{\nu}=diag(0,1,1,1)$. Moreover, relative to the rest frame the components of the energy momentum tensor $T_{\mu\nu}$ take the form

[TABLE]

while the particle current $J^{a}$ and entropy current $S^{a}$ read

[TABLE]

Returning to the identity (290) and taking $X=u^{\mu}$ it follows

[TABLE]

where we used $\rho u^{\mu}=-u_{\lambda}T^{\lambda\mu}$ and introduced $b_{\lambda}=bu_{\lambda}=\frac{u_{\lambda}}{T}$. Moreover the relation $s=b\left(\rho+\frac{P}{c^{2}}\right)-\Theta n$ implies that the entropy vector $S^{\mu}=su^{\mu}$ satisfies

[TABLE]

This relation in combination to (297) incorporates the thermodynamics of a simple perfect fluid. They are the covariant versions of the two relations in (289), but in sharp contrast formulas (297, 298) make no reference to any rest frame. Relation (297) is the covariant Gibbs relation that we have encountered in section$-9$ of the paper.

Appendix D Transformations properties of thermodynamical variables

In this Appendix, we outline the proofs of the two lemmas $(9.1,9.2)$ employed in section (9). For this, let $(u,\hat{u})$ be two (future pointing) unit timelike vectors lying within the ”cone” of opening pseudo-angle $\epsilon$ formed by the four velocity of the energy frame $u_{E}$ and the corresponding velocity $u_{N}$ of the particle frame. The vectors $(u,\hat{u})$ define the time axis of the two admissible rest frames and we complement these vectors by two triads of spacelike unit vectors $e_{i},i\in(1,2,3)$ and $\hat{e}_{i},i\in(1,2,3)$ so that $(u,e_{i})$ respectively $(\hat{u},\hat{e}_{i})$ constitute an orthonormal basis at the event under consideration. For this setting, we have

[TABLE]

where $v^{i}$ are the components of the three velocity of the frame $u$ relative to the $\hat{u}$ one. Following Israel in Ref. \refciteIsr1, we re-express formula (299) into the equivalent form

[TABLE]

Since both $u$ and $\hat{u}$ lie within the ”cone” of opening pseudo-angle $\epsilon\simeq\frac{v}{c}=O_{1}$ and for states near equilibrium $u_{E}-u_{N}=\epsilon=O_{1}$, it follows that the components $\delta^{\mu}$ of $\delta$ in ( 300) satisfy $\delta^{\mu}=\hat{\epsilon}^{\mu}:=\hat{\epsilon}\leq\epsilon=O_{1}$ so that to lowest order in $\hat{\epsilon}$, we have: $\hat{u}^{\mu}-u^{\mu}=\hat{\epsilon}\leq O_{1}$. This new smallness parameter $\hat{\epsilon}$ introduced here is a measure of the relative three velocity between $(\hat{u}^{\mu},u^{\mu})$ and will be important further below.

For a frame change implemented by the transformation $u^{\mu}\to{\hat{u}}^{\mu},$ we want to determine the variation induced by this frame change on the various thermodynamical variables as measured relative in these two frames.

To derive these variations, it is convenient to introduce the four velocity $u^{\mu}_{E}$ of the energy frame and invoke the expansions analogues to (300):

[TABLE]

[TABLE]

where in above $\hat{\delta}^{\nu}$ and $\gamma^{\mu}$ are taken to be of order $\epsilon=O_{1}$ but relations (300, 301, 302) imply118118118In order to see the origin of $\hat{\delta}^{\mu}-\gamma^{\mu}=\delta^{\mu}=\hat{\epsilon}\leq O_{1}$ one notices that (301, 302) imply to leading order: $\hat{u}^{\mu}-u_{E}^{\mu}=\hat{\delta}^{\mu}$ and $u^{\mu}-u_{E}^{\mu}=\gamma^{\mu}$ while on the other hand (300) implies $\hat{u}^{\mu}-u^{\mu}=\delta^{\mu}.$ These two estimates lead to the relation $\hat{\delta}^{\mu}-\gamma^{\mu}=\delta^{\mu}$. that $\hat{\delta}^{\mu}-\gamma^{\mu}=\delta^{\mu}=\hat{\epsilon}\leq O_{1}$.

With these preliminaries, we now return to demonstrate formulas (116, 117, 118) in Lemma $(9.1)$ and begin with the proof of (116).

Let $\rho(\hat{u})$, $\rho(u)$ be the energy densities measured by $\hat{u}$ respectively $u$, then the following relations hold:

[TABLE]

Recalling that that states close to equilibrium satisfy $\tau(u_{E})^{\mu\nu}=\rho(u_{E})O_{1}$ (see eq. (257)) it follows that to leading order

[TABLE]

On the other hand, if in the first term

[TABLE]

we eliminate $\hat{\delta}^{\mu}$ in favor of $\delta^{\mu}$ using $\hat{\delta}^{\mu}-\gamma^{\mu}=\delta^{\mu}=\hat{\epsilon}$, and $\hat{\delta}^{\mu}+\gamma^{\mu}=O_{1}$ we obtain to leading order:

[TABLE]

and thus we have the estimate:

[TABLE]

In order to prove (117), we note that the variation $\delta h^{\mu}$ of the energy flow $h^{\mu}$ takes the form:

[TABLE]

However, one notices that the term $P(u_{E})g^{ab}$ gives zero contribution, while the term $\tau^{ab}(u_{E})$ supplies an $O_{1}$ contribution to the right hand side of above. However the lowest order contribution arises from the term:

[TABLE]

Indeed a litle calculus shows that

[TABLE]

and by eliminating $\hat{u}^{\mu}$ using $\hat{u}^{\mu}=u^{\mu}_{E}+\hat{\delta}^{\mu}+O(\delta^{\mu})^{2}$ we finds

[TABLE]

This estimate in turn implies that to leading order

[TABLE]

Via similar procedure one can easily establish formula (118) i.e.

[TABLE]

and for this proof one starts from

[TABLE]

and applies the same steps as the previous two cases.

The proof of the Lemma 2 follows similar steps and is omitted.

We finish this Appendix by mentioning that the thermodynamical variables $(P(u),n(u),s(u),T(u),\Theta(u))$ measured by the $u$-observer, under a change of the frame described in the above proof, i.e. $u^{\mu}\to\hat{u}^{\mu}=u^{\mu}+\hat{\epsilon}^{\mu}$ with $\hat{\epsilon}^{\mu}\leq O_{1}$, all of them change by $\hat{\epsilon}O_{1}$ while the particle drift $n^{\mu}(u)$, like the energy flux $h^{\mu}(u)$, change by $\hat{\epsilon}$. The proof of these variations can be constructed as above or can be found in Ref.\refciteF1.