## 1 Introduction

Relativistic thermodynamics is needed, because in relativity the mass of a body depends on how hot it is and the temperature is not necessarily homogeneous in equilibrium. But unlike classical thermodynamics the relativistic theory cannot be constructed on the intuitive notions of heat and work, because our intuition does not work well with relativistic effects. Therefore we must rely upon logic, or what would seem logical: the cautious and careful extrapolation of the tenets of non-relativistic thermodynamics.

A pioneer of this strategy was Carl Eckart [141516] who – as early as 1940 – established the thermodynamics of irreversible processes, a theory now universally known by the acronym TIP. The third of Eckart’s three papers addresses the relativistic theory of a fluid. Eckart’s theory is an important step away from equilibria toward non-equilibrium processes. It provides the Navier–Stokes equations for the deviatoric stress and a generalization of Fourier’s law of heat conduction. The latter permits a heat flux to be generated by an acceleration, or a temperature gradient to be equilibrated by a gravitational field.

But Eckart’s theories – the relativistic and non-relativistic ones – have one draw-back: They lead to parabolic equations for the temperature and velocity and thus predict infinite pulse speeds. Naturally relativists, who know that no speed can exceed c, are particularly disturbed by this result and they like to call it a paradox.

Cattaneo [7] proposed a solution of the paradox as far as it concerns heat conduction. He reasoned that under rapid changes of temperature the heat flux is somewhat influenced by the history of the temperature gradient and he was thus able to produce a hyperbolic equation for the temperature – actually a telegraph equation. Müller [3537] incorporated this idea into TIP and came up with a fully hyperbolic system for temperature and velocity. He calculated the pulse speeds and found them to be of the order of magnitude of the speed of sound, far removed from c. And indeed, neither Cattaneo’s nor Müller’s arguments have anything to do with relativity, although Müller [35] also formulated his theory relativistically. The theory became known as Extended Thermodynamics, because the canonical list of fields – density, velocity, temperature – is extended in this theory to include stress and heat flux, 14 fields altogether.

The pulse speed problem may not be the most important question in thermodynamics but it is a question that can be answered, and has to be answered, and so there was a series of papers on the problem all using extended thermodynamics of 14 fields. Israel [21] – who reinvented extended thermodynamics in 1976 – and Kranys [24] and Stewart [46], and Boillat [2], and Seccia & Strumia [44] all calculate the pulse speed for classical as well as for relativistic gases, degenerate and non-degenerate, for Bosons and Fermions, and for the ultra-relativistic case. Actually in some of these gases the pulse speed reaches the order of magnitude of c but it never exceeds it.

So far, so good! But now consider this: The 14 fields mentioned above are the first moments in the kinetic theory of gases and the kinetic theory knows many more moments. In fact, in the kinetic theory we may define infinitely many moments of an increasing tensorial rank. And so Müller and his co-workers, particularly Kremer [2526], Weiss [495150] and Struchtrup [47], came to realize that the original extended thermodynamics was not extended far enough. Guided by the kinetic theory of gases they formulated many-moment theories. These theories have proved their validity and relevance for quickly changing processes and processes with steep gradients, in particular for light scattering, sound dispersion, shock wave structure and radiation thermodynamics. And each theory predicts a new pulse speed. Weiss [49], working with the non-relativistic kinetic theory of gases, demonstrated that the pulse speed increases with an increasing number of moments.

Boillat & Ruggeri [4] proved this observation and – very recently – Boillat & Ruggeri [5] also proved that the pulse speed tends to infinity in the non-relativistic kinetic theory as the number of moments becomes infinite. As yet unpublished is the corresponding result by Boillat & Ruggeri [63] in the relativistic case by which the pulse speed tends to c as the number of moments increases. These results put an end to the long-standing paradox of pulse speeds – 50 years after Cattaneo; they are reviewed in Section 3 and 4.

The quest for macroscopic field theories with finite pulse speeds has proved heuristically useful for the discovery of the formal structure of thermodynamics, relativistic and otherwise. This structure implies

• basis equations are of balance type; hence there is the possibility of weak solutions and shocks,
• constitutive equations are local in space-time; hence follow quasilinear first-order field equations,
• entropy inequality with a concave entropy density; this implies symmetric hyperbolic field equations.

The latter property is essential for finite speeds and for the well-posedness of initial value problems which is a feature at least as desirable as finite speeds. The formal structure of the theory is described in Section 2; it was constructed by Ruggeri and his co-workers, particularly Strumia and Boillat, see [43414]. A convenient presentation may be found in the book by Müller & Ruggeri [39] of which a second edition has just appeared [40].

Section 5 presents extended thermodynamics of viscous, heat-conducting gases due to Liu, Müller & Ruggeri [31], a theory of 14 fields. That section demonstrates the restrictive character of the thermodynamic constitutive theory by showing that most constitutive coefficients can be reduced to the thermal equation of state. Also new insight is provided into the form of the transport coefficients: bulk- and shear-viscosity, and thermal conductivity, which are all explicitly related here to the relaxation times of the gas.

This whole review is concerned with a macroscopic theory: Extended thermodynamics. It is true that some of the tenets of extended thermodynamics are strongly motivated by the kinetic theory of gases, for instance the choice of moments as variables. But even so, extended thermodynamics is a field theory in its own right, it is not kinetic theory.

The kinetic theory, complete with Boltzmann equation and Stoßzahlansatz, offers another possibility of discussing finite propagation speeds – or speeds smaller than c in the relativistic case. Such discussions are more directly based on the observation that the atoms cannot be faster than c. Thus Cercignani [8] has directly linked the phase speed of small harmonic waves to the speed of particles and proved that the phase speeds are smaller than c. Cercignani & Majorana in a follow-up paper [9] have exploited the full dispersion relation to calculate phase speeds and attenuation as functions of frequency, albeit for a simplified collision term. Earlier works on the kinetic theory which address the question of propagation speeds include Sirovich & Thurber [45] and Wang Chang & Uhlenbeck [48]. These works, however, are not subjects of this review.