# Local thermodynamical equilibrium and relativistic dissipation

J. Félix Salazar<EMAIL_ADDRESS>Instituto de Ciencias
Nucleares, Universidad Nacional Autónoma de México
AP 70543, Ciudad de México 04510, México  Thomas Zannias
<EMAIL_ADDRESS>Instituto de Física y Matemáticas, Universidad
Michoacana de San Nicolás de Hidalgo,
Edificio C-3, Ciudad Universitaria, 58040 Morelia, Michoacán, México.

###### Abstract

We introduce a class of relativistic fluid states satisfying the relativistic
local thermodynamical equilibrium postulate (abbreviated as relativistic (LTE)
postulate). States satisfying this postulate, are states “near equilibrium” (a
term defined precisely in the course of the paper) and permit us to attach a
fictitious “local thermodynamical equilibrium” state that fits event by event
the actual fluid state. They single out an admissible class of rest frames
relative to which thermodynamical variables like the energy density,
thermodynamical pressure, stresses, particle number density (or densities)
measured by observers at rest relative to these frames are becoming frame
independent provided second (or higher) order deviations from the fictitious
state of ”local thermodynamical equilibrium” are ignored. We have verified
this property for a large class of theories of relativistic dissipation that
include the Hiscock-Lindblom class of first order theories, the Eckart and
Landau-Lifshitz theories, the Israel-Stewart transient thermodynamics, the
Liu-Müller-Ruggeri theory, fluids of divergence type and the latest developed
(BDNK) theory. Moreover, the phenomenological equations describing first order
deviations from the fictitious ”local thermodynamical equilibrium” state
satisfy equations that remain form invariant under change of frame within the
class of admissible frames. We proved this property for the Hiscock-Lindblom
class of first order theories the Eckart and Landau-Lifshitz theories, the
Israel-Stewart transient thermodynamics and the Liu-Müller-Ruggeri theory of
relativistic dissipation the (BDNK) theory and we expect that the same
property to hold for the class of relativistic fluids of divergence type.

##  I Introduction

We begin this paper, by recalling the essential features of the first theory
of Classical Irreversible Thermodynamics (CIT), developed by Onsager OS1 ; OS2
and independently by Eckart EC1 ; EC2  in the decade of $1930s$-$1940s$. The
theory deals with states of continuous Newtonian media that obey (or are
compatible with) the Local Thermodynamical Equilibrium (LTE) postulate. This
postulate primary provides a recipe that assigns an entropy to off-equilibrium
states and presupposes that thermodynamical equilibrium holds but only
locally. The latter means that for any short time interval of the Newtonian
time $t$ and for any point within the medium, there exists a subsystem of
volume111This volume $V$ is considered to be macroscopically small but
microscopically large. $V$ so that within this $V$ thermodynamical equilibrium
prevails. Evidence supporting the existence of such states comes from the
diversity that continuous media appear in nature. For instance, whenever
states admit an inter-particle collision time scale and this scale is much
shorter than any time scale that determines the evolution of macroscopic
parameters, then space time inhomogeneities in the macroscopic variables are
ironed out by collisions so that locally the state appears to be uniform
implying that (locally) thermodynamical equilibrium holds.

The hypothesis that within each subsystem $V$ thermodynamical equilibrium
prevails has important consequences. It asserts the existence of local
extensive variables $(X_{1},X_{2},...,X_{n})$ and a macroscopic entropy
$S=S(X_{1},X_{2},...,X_{n})$ having the same functional form as the
fundamental equation of state describing states in global thermodynamical
equilibrium. The additivity property of $S(X_{1},X_{2},...,X_{n})$ implies
that one can define an entropy density $s(x_{1},x_{2},...,x_{n})$ function of
$x_{i}=\frac{X_{i}}{V},i\in(1,2,....n)$ and this $s(x_{1},x_{2},...,x_{n})$
plays a key role within the Onsager-Eckart theory of (CIT). The (LTE)
postulate asserts that this $s(x_{1},x_{2},...,x_{n})$ represents the physical
entropy density of the underlying off-equilibrium state and this property
combined with a set of balance laws, constitutive relations and by imposing
the second law, closes the system of equations for these off equilibrium
states.

To get more insights into the implications of the (LTE) postulate and for
later needs of the paper, we take the underlying medium to be a collection of
(classical) electrically neutral particles, so that within each subsystem of
volume $V$ labelled by $(t,\vec{x})$, an inter-particle collision time scale
$\tau_{C}(t,\vec{x})$ is defined obeying
$\tau_{C}(t,\vec{x})<<\tau_{M}(t,\vec{x})$ where $\tau_{M}(t,\vec{x})$ is any
macroscopic time scale where the system change appreciably. The inequality
$\tau_{C}(t,\vec{x})<<\tau_{M}(t,\vec{x})$ asserts that within each $V$, local
thermodynamical equilibrium prevails and identifies the regime where the
framework of fluid dynamics LLF  is applicable222 Systems obeying
$\tau_{C}(t,\vec{x})>>\tau_{M}(t,\vec{x})$ are described by the one particle
distribution function and in that case one enters into the province of the
kinetic theory of gases (for an introduction see for instance Huang ). We also
take the opportunity to clarify the difference between the term local
thermodynamical equilibrium and the (LTE)-postulate employed within (CIT).
Validity of the first simply requires
$\tau_{C}(t,\vec{x})<<\tau_{M}(t,\vec{x})$ while validity of (LTE)-postulate
at first presupposes $\tau_{C}(t,\vec{x})<<\tau_{M}(t,\vec{x})$ but most
importantly, the postulate assigns the non equilibrium entropy to the
underlying state.. In that regime, the subsystems of volume $V$ are the
familiar fluid elements or fluid cells and they are labeled by a continuous
real variables $\vec{x}=(x^{1},x^{2},x^{3})$ so that $(t,\vec{x})$ labels the
fluid element at time $t$ centered around $\vec{x}$. If $K$ is an inertial
frame chosen so that the $(t,\vec{x})$ cell is momentarily at rest, then
relative to this frame one can introduce (local) extensive variables such as
the internal energy $U(t,\vec{x})$, the total particle number $N(t,\vec{x})$
and an additive entropy $S(t,\vec{x})$ function of $(U(t,\vec{x}),V,N(t,x))$.
Via these variables, the (LTE) postulate assigns a physical entropy to the
underlying state and as a consequence of this property, these variables
satisfy a host of thermodynamical relations that we now briefly mention.
For their derivation, let us for the moment consider a (global) equilibrium
fluid state at rest relative to a global inertial frame, and denote by
$(S,U,V,N)$ the spatially and temporally homogenous total entropy $S$,
internal energy $U$, and particle number $N$ within a macroscopic volume $V$.
These variables, under an infinitesimal reversible transformation, satisfy the
Gibbs equilibrium relation

$dS=\frac{1}{T}dU+\frac{P}{T}dV-\frac{\mu}{T}dN$ (1)

where $(T,P,{\mu})$ are the temperature $T$, pressure $P$ and chemical
potential $\mu$ of the particle species. The homogeneity property of
$S(U,V,N)$ under rescaling implies that

$S(\lambda U,\lambda V,\lambda N)=\lambda S(U,V,N),\quad\lambda>0$ (2)

and by choosing $\lambda=1+\epsilon$ and expanding the identity

$S(U+\epsilon U,V+\epsilon V,N+\epsilon N)=(1+\epsilon)S(U,V,N)$

then one gets the fundamental relation:

$U=TS-PV+\mu N,$ (3)

which implies the Gibbs-Duhem relation:

$SdT-VdP+Nd\mu=0.$ (4)

Dividing (3) by $V$ and introducing the internal energy density $e$, entropy
density $s$ and particle density $n$, then (3) yields

$e=Ts-P+\mu n$ (5)

which via differentiation and in combination with the densitized form of the
Gibbs-Duhem relation (4) i.e. $sdT-dP+nd\mu=0$, implies

$de=Tds+\mu dn.$ (6)

Thus knowledge of the equation of state $e=e(s,n)$ determines via
differentiation the equilibrium temperature and chemical potential $\mu$ of
the underlying equilibrium fluid state.
Formulas (1-6) are the well known relations that fluid states in global
thermodynamical equilibrium must satisfy and we derive them here just to
illustrate the implications that the (LTE) postulate has upon the description
of hydrodynamical states. Whenever a state is compatible with this postulate,,
then $U(t,\vec{x}),N(t,\vec{x}),S(t,\vec{x})$ and the corresponding densities
$(e(t,\vec{x}),n(t,\vec{x}),s(t,\vec{x}))$ satisfy relations (1-6) i.e. the
same relations as if the state was in global thermodynamical equilibrium.
However a major difference should be noted: the variables
$(U(t,\vec{x}),N(t,\vec{x}),s(t,\vec{x}),e(t,\vec{x}),n(t,\vec{x}),s(t,\vec{x}))$
obey (1-6) only locally i.e. at the interior of (any) fluid cell.

The (CIT) of Onsager-Eckart has been proven to be a successful theory. It
leads to the Fourier-Navier-Stokes fluid system which is the standard theory
that describes terrestrial and a large class of astrophysical fluid flows and
the classical book by de Groot and Mazur dGM  deals extensively with the
development and applications of the Onsager-Eckart theory of (CIT) to linear
thermodynamics of irreversible processes (the interest reader is refereed to
that book for further discussion and additional references).
However, despite these successes, years of efforts and experience lead to the
realization that not all goes that well with the dynamics of states compatible
with the (LTE) postulate within the (CIT). The postulate dictates a very rigid
functional dependance of the physical entropy density $s$ upon the local
thermodynamical variables, and this leads to the prediction that disturbances
in temperature and stresses propagate with an unbounded speed, a highly
unexpected and counterintuitive prediction.
Müller (Mul1, ) in a fundamental paper written in $(1967)$, suggested a way
out of this problematic issue. He proposed that the physical entropy density
$s$ of non equilibrium fluid states should receive contributions from
dissipative variables such as heat flux, bulk and shear stresses, a
proposition which is in a blatant contradiction to the spirit of the (LTE)
postulate.
Müller $(1967)$ suggestion lead to the development of extended theories of
irreversible thermodynamics and pave the way to the formulation of the entropy
principle introduced by Coleman and Noll ColN1 ; ColN2 , Müller MulN1 ; MulN2
and others, as well as lead to the realization that systems of symmetric-
hyperbolic equations should be the appropriate set of dynamical equations
describing irreversible thermodynamics (for a historical development,
references and current status of extended theories, consult refs JVL ; Mul4 ;
Mul7 ).
Although the developments of extended theories remove the assertion of the
(LTE) postulate that dictates the dependance of the physical entropy density
$s$ upon the local thermodynamical variables, nevertheless in these extended
theories the validity of the hydrodynamical regime is left intact and thus
also the validity of local thermodynamical equilibrium is preserved. This in
turn implies the existence of (local) hydrodynamical variables
$(X_{1},X_{2},...,X_{n})$ associated with the underlying fluid state. Using
these variables, one may still employ $s(x_{1},x_{2},...,x_{n})$ as an
”equilibrium equation of state” and via the Gibbs formalism, attaches a
”fictitious333 We refer to this state as a ”fictitious” state because the
entropy that is defined by $s(x_{1},x_{2},...,x_{n})$ bears no relation to the
physical entropy of the underlying state. ”local thermodynamical equilibrium”
state to the underlying fluid state. This fictitious ”local thermodynamical
equilibrium state” it is not always of physical relevance but nevertheless it
can be very convenient444Traces of this fictitious ”local thermodynamical
equilibrium state” can be seen in the introductory treatments of extended
thermodynamics (see for example JVL , RZ ), where one often encounters terms
like the equilibrium temperature, pressure, etc. These equilibrium quantities
are generated by employing the equilibrium equation of state
$s(x_{1},x_{2},...,x_{n})$ alluded above.. As we shall see further ahead, upon
passing to the relativistic description, the idea of attaching a fictitious
local ”thermodynamical equilibrium state” to a fluid state, will be proven
essential in the development of this work.
The present paper discuss an adaptation of the (LTE) postulate to the
relativistic regime so that the postulate becomes a tool for the analysis of
relativistic fluids states. The relativistic framework offers the possibility
to introduce a special class of relativistic fluid states refereed as states
compatible with the relativistic (LTE) postulate and their precise definition
will be introduced in the following sections. Here, we offer a few comments
regarding the motivations that lead us to introduce this special class of
states.
For this, it is of relevance to recall the observational breakthroughs that
took place within the last ten years or so, confirming the minute predictions
of Einstein’s general relativity or of relativistic physics. It is sufficient
to recall the monumental detection of gravitational waves by the LIGO
observatory, the binary neutron star inspiral and merger GW170817 detected by
the LIGO/VIRGO gravitational wave observatory network, the photographs
released by the Event Horizon Telescope (EHT) collaboration of the event
horizon” of the supermassive black hole at the center of the $M87$ and at the
center of our galaxy, the ongoing experiments at the Relativistic Heavy Ion
Collision at BNL and at Large Hadron Collider at CERN, and the formation of
the terrestrial “mini bing-bangs” in the form of quark-gluon plasma.
Although presently, the scientific community is in the process of digesting
these observational breakthroughs, nevertheless one message comes across
clearly: relativists and high energy physicists alike need to develop reliable
theories of relativistic continuous media in order to confront the new
observational realities. It has been however realized and in fact long long
ago, that relativistic dissipation is a tough mathematical problem which is
still wide open (refs.Nor2 ,RZ , RR  offers un update on the latest
developments). As we shall discuss further ahead, all so far proposed theories
are highly non linear and they are plagued by conceptual issues such as
causality violation His2 , instability of equilibrium states (for a recent
discussion see GAV2  GAV3 , GAV6 ) lack of a sensible definition of a fluid
four velocity (see for instance vKAM , Isr1 , Isr2  and the next two sections
of this paper) and often the dynamical evolution equations fail to constitute
a symmetric-hyperbolic and causal set of equations (see for example Mul6 ,
Ger1 ).
In this rough territory, progress has accomplished by restricting attention to
equilibrium states and the analysis of their perturbations. The work in His2
studies the stability properties of first order theories, while in refs. His2
,Olson  the stability properties of transient thermodynamics has been
addressed. The stability of equilibrium states and causality properties of
Carter’s theory of heat conducting relativistic fluids (for an introduction to
this theory see Car1  Car2  Car3 ) has been the focus of refs HisO ,GAV13
,GAV7 .

Even though this brief survey, shows a sizable amount of scientific activity
on equilibrium states and their properties, still others aspects of
relativistic dissipation needed to be investigated. The aim of the present
work is to introduce a family of states as an alternative to equilibrium
states. This class has been motivated by the properties of states satisfying
the (LTE) postulate within the (CIT) and they will be referred as states
satisfying the relativistic (LTE) postulate. They are states that on the one
hand, go beyond the class of the equilibrium states and on the other hand,
avoid (partially) the complexities associated with the non linear nature of
the underlying fluid equations. The central idea that underlies this special
class of states is their property that they are states ”near equilibrium” a
term introduced by Israel long time ago and used as the building block in the
development of transient thermodynamics. Although the term states ”near
equilibrium” at a first side suggest that such states could be just
perturbations of equilibrium states, that is not the case and that will become
clear in the course of the paper. The identification of states satisfying the
relativistic (LTE) postulate is almost independent upon the underlying fluid
theory. Their definition requires only that within the the underlying theory,
states to be described by a conserved symmetric energy momentum tensor and a
collection of conserved particle currents (this last requirement is optional).
The key ingredient that allows us to introduce this special class of states is
the possibility to attach a fictitious ”local thermodynamical equilibrium”
state to an underlying fluid state and this attachment proceeds along the same
reasonig as for the Newtonian framework. However, in the relativistic domain
local observers and their measurements in combination to an ”equilibrium
equation of state” are the key elements that lead to this fictitious ”local
thermodynamical equilibrium” state. This fictitious state in combination with
states near thermodynamical equilibrium (the precise coordinate free
definition of such states is discussed in latter sections) yield the class of
fluid states compatible with the relativistic (LTE) postulate and details of
their construction are discussed in the following sections.

##  II  On dissipative relativistic fluid theories

In this section, we prepare the ground for the formulation of the relativistic
version of the (LTE) postulate and for this purpose, at first, we discuss
briefly the salient features of a few theories of dissipative relativistic
fluids developed so far.

Early attempts to construct viable theories initiated with the work of Eckart
(Eck, ) in the $(1940)s$, the work by Landau and Lifshitz LaL  in the
$(1950)s$ and culminated in the formulation of the class of first order
theories developed by Hiscock and Lindblom His2  in the $(1980)$s. At around
$1970$s, Israel Isr1  and Israel and Stewart (Isr2, ) formulated the transient
thermodynamics which is a second order theory a term that will become clear
further ahead. Within these theories, states of a relativistic fluid are
described by a symmetric energy momentum tensor $T^{\mu\nu}$ a particle
current555For simplicity, in this work we treat the case of a simple fluid.
For a fluid mixture, one may introduce $n$ conserved particle currents
$J_{i}^{\mu}$, $i\in(1,2,...n)$ and proceeds in a similar manner.  $J^{\mu}$
obeying the conservation laws

$\nabla_{\mu}T^{\mu\nu}=\nabla_{\mu}J^{\mu}=0,$ (7)

and accompanied by an entropy flux four vector $S^{\alpha}$ obeying

$\nabla_{\alpha}S^{\alpha}=\sigma,$ (8)

where $\sigma\geq 0$ is the entropy production scalar.
With the development of Rational Extended Irreversible Thermodynamics in the
late $1980s$, by Müller, Liu, Ruggeri and collaborators, a new trend in the
field of relativistic dissipative fluids opened up (for and introduction
consult ref.Mul4 ; Mul7 ). New theories have been developed whose states are
described by a collection of tensor fields obeying a manifestly symmetric-
hyperbolic (or causal) systems of dynamical equations. Such theories include
the theory developed by Liu, Müller and Ruggeri (see Mul6 ; Mul4 ), the theory
of relativistic fluids of divergence type developed by Pennisi Pen , and
independently by Geroch and Lindblom Ger1 , Carter’s theory of heat conducting
fluids Car1 ; Car2 ; Car3 , the Unified Extended Irreversible Thermodynamics
(UEIT) developed by Gavasinno and Antonelli GAV7  ,GAV1 .
Finally, we should mention the enormous impact the recent experiments on heavy
ion collisions had upon the development of theories of relativistic
dissipation (for details on this connection consult for instance Hei ). Here
the unexpected realization that models of relativistic viscous hydrodynamics
describe reliably observational dada acted as a stimulus for the development
of new theories. Such theories include the Denicol-Niemi-Molnar-Rischke (DNMR)
theory DNMR , the Baier-Romatschke-Son-Starinets-Stephanov theory BRSSS , or
anisotropic hydrodynamics theories (for an introduction see see AH1 , AH2 )
etc. These theories are second order theories derived either as limits of the
relativistic Boltzmann equation case of (DNMR) theory, or as effective
theories obtained from the gradient expansion within the quantum field theory,
truncated at second order, case of Baier-Romatschke-Son-Starinets-Stephanov
(BRSSS) theory.
The quest to develop theories with physically relevant characteristics, for
instance theories that their equilibrium states are stable, respect causality,
the corresponding dynamical equations admit a well posed initial value
formulation, lead to the development of the (BDNK) formalism (for an
introduction see (Kov1 , Kov2 , Nor1 , Nor2  ) which is a formalism that lead
to the development of first order theories with remarkable properties and in
section $(IV)$ we shall have the opportunity to comment more on this
formalism.

It is not the purpose of this paper to provide a detailed description of the
above mentioned theories neither to discuss their virtues and failures (for
recent reviews see for instance Nor2 , Mul7 , RMa1 , FT1 , FLO , RR , RZ ,
Isr5 ). Our purpose is to adapt the (LTE) postulate from its natural habitat
i.e. the regime of Newtonian continuous media, to the realm of relativistic
fluids and our approach has been motivated by the development of the Israel-
Stewart transient thermodynamics although the framework that we shall develop
is applicable to arbitrary theories of relativistic dissipation666The only
restriction we shall impose on the structure of the underlying theory is that
amongst the variables that describe fluid states, includes a conserved
particle current $J^{\mu}$ and a symmetric energy momentum tensor $T^{\mu\nu}$
which satisfies the weak energy condition..
As we have already mentioned, the description of relativistic dissipation
presents a few conceptual challanges that are absent, for instance, from
states of the Newtonian Fourier-Navier-Stokes system. For the later, the field
equations are balance laws expressing conservation of mass, energy and linear
momentum and involve the velocity field, mass density, the components of a
stress and the components of the heat flux. By invoking the entropy principle,
constitutive relations are proposed so that one gets a closed system of
equations for the components of the velocity field, energy density and
temperature, all of them defined relative to a fixed global inertial frame.
Unfortunately, attempts to interpret dissipative states of a relativistic
fluid in a similar manner run into difficulties. Terms like particle and
energy densities, stresses, temperature etc. are in general observer dependent
quantities and more dramatic is the realization that the fluid’s four velocity
is either non existent or if it is defined looses its prominence777An
exception is provided by states describing a simple perfect fluid. Here a four
velocity field $u^{\mu}$ is assigned dynamically by the theory and perfect
fluid states are described by the future-directed, timelike velocity field
$u^{\mu}$ and two spacetime scalar fields $(n,\rho)$ representing the particle
density and energy density measured by a comoving with the flow observer. The
particle number and energy momentum conservation laws supplemented by an
equation of state fix these field uniquely.. For fluid theories whose states
are described partially888The use of the world ”partially” reflects the fact
that we have not specified the underlying fluid theory. It may happen that for
an arbitrary theory, one may need besides $(T^{\mu\nu},J^{\mu})$ additional
variables to specify a fluid state for instance the entropy flux $S^{\mu}$ or
other additional fields. by a conserved particle current $J^{\mu}$ and a
conserved energy momentum tensor $T^{\mu\nu}$, one has the liberty to define
the fluid’s four velocity as the unique, future directed timelike eigenvector
$u_{E}^{\mu}$ of the energy momentum tensor $T^{\mu\nu}$ i.e. identify the
fluid’s four velocity with the flow of energy999This $u_{E}^{\mu}$ is well
defined provided the energy momentum tensor obeys the weak energy condition.
or one may identify the fluid four velocity with the $4$-velocity
$u_{N}^{\mu}$ defined via $J^{\mu}=nu_{N}^{\mu}$ where $n$ is the particle
density measured by an observer comoving with this $u^{\mu}_{N}$ and thus
identify the fluid four velocity with the particle motion. Should one identify
the fluid’s four velocity with $u_{E}^{\mu}$, $u_{N}^{\mu}$ or may one employ
an altogether different four velocity field? This ambiguity regarding the
choice of the fluid’s four velocity is a common feature of all so far proposed
theories of relativistic dissipation and to this day, there is not a
satisfactory resolutions of this dilemma101010 Viewing relativistic fluids as
arising from a microscopic descriptions either from relativistic kinetic
theory or from the expectation values of quantum observables via suitable
averaging, it is hoped that this transitions may single our a unique fluid
flow field. Unfortunately up to to now there is no such a satisfactory
prediction..
It is however worth noticing that although the notion of a fluid four velocity
may be ambiguous or even ill defined, the general relativistic framework
allows to introduce a smooth, timelike future directed ”velocity field”
$u^{\mu}$ in the spacetime region occupied by the fluid and view this
$u^{\mu}$ as providing a field of “rest frames” employed by the
$u^{\mu}$-observers who probe the state of the fluid. This liberty in the
choice of a potential “velocity field” $u^{\mu}$, combined with a particular
class of relativistic fluid states defined below, are the building blocks for
the formulation of states satisfying the relativistic (LTE) postulate and in
the next section we address this construction.

##  III  On states compatible with the relativistic (LTE)-postulate

In this section, we consider a test relativistic fluid propagating on a smooth
background spacetime $(M,g)$ and within the region occupied by the fluid, we
introduce a smooth, non singular, future pointing velocity field $u^{\mu}$
whose sole purpose is to provide a family of local observes who perform
measurements upon the fluid’s state. At any event $p$ in the fluids interior
and relative to the local rest frame whose time axis at $p$ coincides with
$u^{\mu}(p)$, we consider an infinitesimal element of a spacelike $3$-volume
$V$, orthogonal a $u^{\alpha}(p)$ and view this $V$ as the local fluid cell.
Furthermore, we assume that it is possible to choose this $u^{\mu}$ so that
relative to the rest frame defined by $u^{\alpha}(p)$, a collision time scale
$\tau_{C}(p)$ is defined that satisfies $\tau_{C}(p)<<\tau_{M}(p)$ where
$\tau_{M}(p)$ is another time scale111111 These time scales are identical to
those that we defined earlier on for Newtonian fluids, although there is
however a pronounced difference between the Newtonian and relativistic case.
While in the former case the inequality $\tau_{C}(p)<<\tau_{M}(p)$ is frame
independent, in the relativistic regime and due to time dilatation effects in
general this inequality is frame dependent. States compatible with the
relativistic version of (LTE) postulate that we are about to introduce, is a
class of states that allows to introduce a preferable class of frames,
refereed as admissible frames, having the property that if
$\tau_{C}(p)<<\tau_{M}(p)$ holds in one frame within this admissible class,
then frame changes within this admissible class, leaves this inequality
intact.  determined by the velocity field, for instance by its expansion,
shear, rotation, etc. As for the Newtonian case, this inequality implies that
the $u^{\alpha}(p)$-observer will conclude that local thermodynamical
equilibrium prevails within $V$ and using $T^{\mu\nu},J^{\mu}$ and $S^{\mu}$,
assigns at $p$ a particle number $n(p)$, energy density $\rho(p)$ and an
entropy density $\hat{s}(p)$ via

$n(p)=-J_{\mu}u^{\mu},\quad\quad\rho(p)=T_{\mu\nu}u^{\mu}u^{\nu},\quad\quad\hat{s}(p)=-S_{\mu}u^{\mu}.$
(9)

Using this $u^{\mu}$ one in general decomposes121212In (10-12), and in the
sequel, we write $\rho(u),P(u),h(u),\tau(u)^{\mu}{}_{\nu}$ etc., in order to
remind the reader that these variables are measured by the $u-$observer. This
notation further signifies that the decompositions in (10-12) are frame
dependent. Often we write $n(p)$, $\rho(p)$ etc to denote results of
measurements by the $u$-observer at a particular event $p$.  $T^{\mu\nu}$ and
$J^{\mu}$ according to:

$T^{\mu\nu}=\rho(u)u^{\mu}u^{\nu}+P(u)\Delta(u)^{\mu\nu}+h(u)^{\mu}u^{\nu}+h(u)^{\nu}u^{\mu}+\tau(u)^{\mu\nu},$
(10) $J^{\mu}=n(u)u^{\mu}+n(u)^{\mu},$ (11)

where $h(u)^{\mu},n(u)^{\mu},\tau(u)^{\mu\nu}$ stand for the energy flow
vector, the particle ”drift” and the spatial symmetric pressure tensor
$\tau(u)^{\mu\nu}$ respectively all of them measured by the $u^{\mu}$
observer. In these expansions, the pressure tensor $\tau(u)^{\mu\nu}$ defines
the bulk pressure $\pi(u)$ and shear stresses $\pi(u)^{\mu\nu}$ according to

$\tau(u)^{\mu\nu}=\pi(u)\Delta(u)^{\mu\nu}+\pi(u)^{\mu\nu},\quad\pi(u)^{\mu}_{\mu}=0,$
(12)

the fields $(h^{\mu},n^{\mu})$ and $\tau(u)^{\mu\nu}$ satisfy

$h(u)^{\mu}u_{\mu}=n(u)^{\mu}u_{\mu}=u_{\mu}\tau(u)^{\mu\nu}=0,$ (13)

while $\Delta^{\mu\nu}(u)=g^{\mu\nu}+u^{\mu}u^{\nu}$ stands for the projection
tensor.
Since as we have already mentioned, the inequality $\tau_{C}(p)<<\tau_{M}(p)$,
implies validity of the local thermodynamical equilibrium within $V$, by
appealing to Gibbs formulation131313For an introduction to this formulation
see for instance Cal . of equilibrium thermodynamics, this $u^{\mu}$ observer
will conclude that $n(p)$ y $\rho(p)$ must satisfy an ”equilibrium equation of
state” of the form $s=s(\rho,n)$. In turn this $s=s(\rho,n)$ implies

$ds=\frac{1}{T}d\rho-\Theta dn,$ (14)

thus defining the local temperature $T(u)$ and the thermal potential
$\Theta(u)$ as measured by the $u^{\mu}$-observer, while the fundamental
relation $s(u)=(\rho(u)+P(u))T^{-1}(u)-\Theta(u)n(u)$ with
$s(u):=s(\rho(u),n(u))$ defines the (local) equilibrium pressure $P(u)$.
Therefore, once a velocity field $u^{\mu}$ and an equation of state
$s=s(\rho,n)$ have been specified, then $(T^{\mu\nu},J^{\mu})$ define at $p$ a
”local thermodynamical equilibrium” state specified by

$(n(p),\rho(p),P(p),s(p),T(p),\Theta(p)).$ (15)

As long as the velocity field $u^{\mu}$ and the fluid state allow the
inequality $\tau_{C}(p)<<\tau_{M}(p)$ to hold at any event $p$ within the
fluid’s interior, then the variables in (15) are defined over the entire
spacetime region occupied by the fluid and combined with $u^{\mu}$ give rise
to a (local) particle current141414It is understood that in (16), the fields
$(n,\rho,P,s)$ stand for $(n(u),\rho(u),P(u),s(u(n(u),\rho(u))$.
$J^{\mu}_{0}$, a tensor $T^{\mu\nu}_{0}$ and an ”entropy flux” vector
$S^{\mu}_{0}$, via

$J^{\mu}_{0}=nu^{\mu},\quad
T^{\mu\nu}_{0}=(\rho+P)u^{\mu}u^{\nu}+Pg^{\mu\nu},\quad
S^{\mu}_{0}=s(\rho,n)u^{\mu}.$ (16)

These fields, in the terminology of Israel-Stewart Isr1 ; Isr2 , define a
“local equilibrium reference state” attached to the physical state described
by the variables $(T^{\mu\nu},J^{\mu},S^{\mu})$. This ”local equilibrium
reference state” depends upon the (almost arbitrarily) chosen velocity field
$u^{\mu}$ and thus at a first sight seem to be of limited utility. Any other
$\hat{u}^{\alpha}(p)$-observer and as long as at $p$ local equilibrium
prevails relative to her/his frame, will measure

$(\hat{n}(p),\hat{\rho}(p),\hat{P}(p),\hat{s}(p),\hat{T}(p),\hat{\Theta}(p))$
(17)

which are related to those in (15) via complicated transformations formulas
induced by a local Lorentz transformation that relates the frames defined by
$u^{\alpha}(p)$ and $\hat{u}^{\alpha}(p)$. However, for a particular states,
the fields in (15, 17) transform under local Lorentz transformations in a
simple manner so they become useful tools in describing properties of the
underlying fluid state.
In order to identify these states, we consider again the unique timelike
eigenvector $u^{\mu}_{E}$ of the energy momentum tensor $T^{\mu\nu}$ which
specifies the energy frame, and the timelike and future directed vector
$u^{\mu}_{N}$ that specifies the particle frame. For an arbitrary fluid state
determined (partially) by $(T^{\mu\nu},J^{\mu})$ the fields
$(u^{\mu}_{E},u^{\mu}_{N})$ are in general distinct and there is not any
obvious relation between the two. However, by appealing to an idea introduced
by Israel long time ago151515Israel suggested that for states near
equilibrium, a theory can be developed that describes the first order
deviations from a local ”equilibrium reference state” and this theory is
invariant under particular change of rest frames within the class of
admissible frames. This idea was elaborated further in a M.Sc. thesis written
by Aitken Ait  then student of Israel. and employed in the formulation of the
transient thermodynamics, we introduce states referred as ”states near
equilibrium” (or ”states close to equilibrium”) in the following manner. At an
event $p$ within the fluid’s region, one sets up an orthonormal frame with a
time axis parallel to $u_{E}$ accompanied by 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 there is freedom in the choice of the triad
$e_{i},~{}i\in(1,2,3)$ , without loss of generality, one may assume:

$\begin{split}u_{N}^{\mu}=&[1-\frac{v^{2}}{c^{2}}]^{-\frac{1}{2}}u_{E}^{\mu}+\frac{v}{c}[1-\frac{v^{2}}{c^{2}}]^{-\frac{1}{2}}e_{1}^{\mu}\\\
=&cosh\epsilon~{}u_{E}^{\mu}+sinh\epsilon~{}e_{1}^{\mu},\quad
g(u_{E},e_{1})=0,\end{split}$ (18)

where $\vec{v}=ve_{1}$ is the “relative velocity” of the particle frame
relative to the energy frame. This relation defines a pseudo angle $\epsilon$
between $u_{E}$ and $u_{N}$ according to:

$cosh\epsilon=-g(u_{E},u_{N})=\left[1-\left(\frac{v^{2}}{c^{2}}\right)\right]^{-\frac{1}{2}},$
(19)

and this angle plays an important role. It is suffice to mention that for all
proposed theories of relativistic dissipation, it holds that for states in a
global equilibrium the fields $u_{E}$ and $u_{N}$ coincide and thus
$\epsilon=0$.
Motivated by this property and in the spirit of transient thermodynamics, we
defines for a simple fluid states near161616For 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 $i\in(1,2,.....,n)$. equilibrium171717Further
below, we shall introduce their close relatives named states satisfying with
the relativistic (LTE) postulate. as those states that have the property that
the pseudo-angle $\epsilon$ in (19) satisfies everywhere within the region
occupied by the fluid the condition $\epsilon=\frac{v}{c}<<1$. For such
states, one notices from

$J^{\mu}=n_{N}u_{N}^{\mu}=n_{N}(cosh\epsilon~{}u_{E}^{\mu}+sinh\epsilon~{}e_{1}^{\mu})=n_{E}u_{E}^{\mu}+n^{\mu},$
(20)

and thus the densities $n_{N}$ and $n_{E}$ measured by the $u^{\mu}_{N}$
respectively the $u^{\mu}_{E}$ observers satisfy

$n_{E}=n_{N}cosh\epsilon=n_{N}+O(\epsilon^{n}),\quad n\geq 2$ (21)

implying that $n_{N}$ and $n_{E}$ are considered to be frame independent
provided terms of order $\epsilon^{2}<<1$ and higher are neglected181818 Here
after we follow the notation of ref.Isr1  often we set $\epsilon:=O_{1}$ while
terms like $O_{2},O_{3}.....$ signify terms of first, second, third order…
deviations.. This (approximately) “invariance property” of the particle
density holds also for other fields that appear in (15) and (17) and in order
to investigate in a systematic manner the transformation properties of the
thermodynamical variables under frame change, let $(u^{\mu},\hat{u}^{\mu})$ be
two (future pointing) unit timelike vectors lying within the “cone” of opening
angle $\epsilon<<1$. These vectors define the time axis of the two rest
frames191919If relative to these frames the inequality
$\tau_{R}(p)<<\tau_{C}(p)$ holds for all $p$ within the fluids interior, then
these class of rest frames is refereed as the admissible class of rest frames,
a term adapted from the terminology employed in transient thermodynamics.
Local Lorentz transformations between such frames are approximately described
by (24). and by complementing them by two triads of spacelike unit vectors
$e_{i},i\in(1,2,3)$ and $\hat{e}_{i},i\in(1,2,3)$, then $(u,e_{i})$
respectively $(\hat{u},\hat{e}_{i})$ constitute an orthonormal bases at the
event under consideration. Accordingly, we have

$\hat{u}=\frac{u}{(1-\frac{v^{2}}{c^{2}})^{\frac{1}{2}}}+\frac{v^{i}e_{i}}{c}\frac{1}{(1-\frac{v^{2}}{c^{2}})^{\frac{1}{2}}},\quad
v^{2}=v^{i}v_{i}.$ (22)

where $v^{i}$ are the components of the three velocity of the frame $u$
relative to the $\hat{u}$ one. Following Israel Isr1 , we write this
transformation law in the equivalent form

$\hat{u}^{\mu}=(1+\hat{\epsilon}^{2})^{1/2}u^{\mu}+\hat{\epsilon}^{\mu},\qquad\hat{\epsilon}^{2}=\hat{\epsilon}^{a}\hat{\epsilon}_{a},\qquad\hat{\epsilon}^{a}u_{a}=0.$
(23)

and assume $\hat{\epsilon}^{\mu}\leq\epsilon^{\mu}=O_{1}$ so that (23) is
approximated by

$\hat{u}^{\mu}=u^{\mu}+\hat{\epsilon}^{\mu}+O(\hat{\epsilon})^{2},\quad\hat{\epsilon}^{\mu}\leq
O_{1},$ (24)

showing that $\hat{\epsilon}$ is a measure of the relative three velocity of
$u^{\mu}$ relative to $\hat{u}^{\mu}$ frame.

Let now the local ”thermodynamical equilibrium state” associated with
$(u^{\mu},s(n,\rho))$ constructed according to (15, 16). If $Z(u)$ stands for
any of the thermodynamical variable in (15) (as measured by the $u$ observer)
and we consider the frame change described in (24), of relevance for the
following analysis is the variation $\delta Z:=Z(\hat{u})-Z(u)$ that suffers
the variable $Z$ under such frame change. Variations of these type worked out
by Israel in Isr1  and also reworked in Ftes  and below we present a summary
of such variations:

$\displaystyle\delta\rho$
$\displaystyle\equiv\rho(\hat{u})-\rho(u)=\hat{\epsilon}O_{1}$ (25)
$\displaystyle\delta h^{\alpha}$ $\displaystyle\equiv
h^{\alpha}(\hat{u})-h^{\alpha}(u)=-(\rho+P)\hat{\epsilon}^{\alpha}$ (26)
$\displaystyle\delta n$ $\displaystyle\equiv
n(\hat{u})-n(u)=\hat{\epsilon}O_{1},$ (27) $\displaystyle\delta P$
$\displaystyle\equiv P(\hat{u})-P(u)=\hat{\epsilon}O_{1},$ (28)
$\displaystyle\delta\tau^{\alpha\beta}$
$\displaystyle\equiv\tau^{\alpha\beta}(\hat{u})-\tau^{\alpha\beta}(u)=\hat{\epsilon}O_{1}.$
(29) $\displaystyle\delta n^{\alpha}$
$\displaystyle\equiv{n}^{\alpha}(\hat{u})-{n}^{\alpha}(u)=-n\hat{\epsilon}^{\alpha},$
(30) $\displaystyle\delta s$ $\displaystyle\equiv
s(\hat{u})-s(u)=\hat{\epsilon}O_{1},$ (31) $\displaystyle\delta T$
$\displaystyle\equiv T(\hat{u})-T(u)=\hat{\epsilon}O_{1},$ (32)
$\displaystyle\delta\Theta$
$\displaystyle\equiv\Theta(\hat{u})-\Theta(u)=\hat{\epsilon}O_{1}.$ (33)

These transformation shows that the variables like $(n,\rho,P,etc)$ behave as
frame independent quantities to an accuracy $\hat{\epsilon}O_{1}\leq O_{2}$
while others variables like $h^{\alpha},n^{\mu}$ are independent only to
$\hat{\epsilon}\leq O_{1}$ accuracy202020For the particular case where
$(u^{\mu},\hat{u}^{\mu})$ are identified as $(u^{\mu}_{E},u^{\mu}_{N})$, the
transformation law in (23) take the form:
$u^{\mu}_{N}\to{u}^{\mu}_{E}=u^{\mu}_{N}+{\epsilon}^{\mu}+O(\epsilon)^{2}$
(34) then one replaces in the right hand sides of (25-33) $\hat{\epsilon}$ by
$\epsilon=O_{1}$ and the resulting formulas describes frame change from the
particle to the energy frame. .
For later use, we mention that the following $4$-vector

$q^{\alpha}(u)=h^{\alpha}(u)-\frac{\rho(u)+P(u)}{n(u)}n^{\alpha}(u),\\\ $ (35)

is frame independent i.e. its variation obeys:

$\delta q^{\alpha}\equiv
q^{\alpha}(\hat{u})-q^{\alpha}(u)=\hat{\epsilon}^{\alpha}O_{1}.\\\ $ (36)

and is refereed as the the invariant heat flux vector.

The formulas in (25-33) express the transformation laws of the various
thermodynamical variables under a frame change described in (24) and there
will be used frequently in the following analysis. Under such frame change,
they show that most of the thermodynamical variables remain practically frame
independent as long as second order and higher order deviations from the state
of “local thermodynamical equilibrium” specified by $(u^{\mu},s(\rho,n))$, are
neglected. Notice that the assumption $\epsilon<<1$ in (19) implies that the
inequality $\tau_{C}(p)<<\tau_{M}(p)$, becomes observer independent in the
sense that as long as $(u^{\mu},\hat{u}^{\mu})$ in (24) are chosen to lie
within the “cone” of the opening pseudo-angle $\epsilon=O_{1}<<1$ then time
dilatation and length contraction are effects considered as been inessential.
Moreover, the ”invariance” of the inequality $\tau_{C}(p)<<\tau_{M}(p)$, under
the frame change in (24) in combination to (25-33), permit us to simply refer
to a state of “local thermodynamical equilibrium” without any further
reference to which particular $(u^{\mu},s(\rho,n))$ this fictitious state’ is
associated with.

The so far analysis used only the property that fluid states are described
(partially) by the fields $(T^{\mu\nu},J^{\mu})$ and all of the above
conclusions are independent of any underlying fluid theory. This observation
allows us to group together states characterized by common properties. To do
so, let us begin with a state described by the fields $(T^{\mu\nu},J^{\mu})$
subject to the restriction that the pseudo-angle $\epsilon$ in (19) satisfies
$\epsilon<<1$ everywhere within the fluid region and let a four velocity
$u^{\mu}$ lying within the cone of opening angle $\epsilon<<1$ chosen so that
$\tau_{C}(p)<<\tau_{M}(p)$ holds. Any such $u^{\mu}$ combined with
$(T^{\mu\nu},J^{\mu})$ defines a particular fluid state which is described by
the fields defined in the expansions (10, 11). We refer to the collection of
such states, as states satisfying the relativistic (LTE) postulate (or often
as states near equilibrium).

One notices that for any two states within this class, specified by $u^{\mu}$,
respectively $\hat{u}^{\mu}$ it holds
$\hat{u}^{\mu}-u^{\mu}\leq\epsilon^{\mu}$ and thus under a frame change
described by (24), the fields in (15, 17) transform according to (25-33).
Therefore even though technically one deals with the two distinct fluid
states, as long as one is interested only in the physics of first order
deviations from the state of ”local thermodynamical equilibrium” one really is
dealing with the same state. Moreover in the next sections, we show that for a
large class of fluid theories, states compatible with this relativistic (LTE)
postulate, the phenomenological equations that describe the dynamics of first
order deviations from the ”local thermodynamical equilibrium” state are
equivalent in the sense that from one solution one generates solutions of the
other equation (or equivalently from one state in (LTE) one generates all the
other states within this class).

As a preparatory step to analyze further properties of states satisfying the
relativistic (LTE) postulate, let us consider one of them described by
$(T^{\mu\nu},J^{\mu})$ and let us introduce also a physical entropy current
$S^{\mu}$ associated to this state. If and in accordance of the above
discussion, this state is specified by $(u^{\mu},s(\rho,n))$, we define the
fields,

$\delta S^{\mu}=S^{\mu}-S_{0},\quad\delta
T^{\mu\nu}=T^{\mu\nu}-T^{\mu\nu}_{0},\quad\delta J^{\mu}=J^{\mu}-J^{\mu}_{0}$
(37)

which describe the deviations of the physical state
$(S^{\mu},T^{\mu\nu},J^{\mu})$ away from the state of ”local thermodynamical
equilibrium” state defined (by $(u^{\mu},s(\rho,n))$) or by
$(S^{\mu}_{0},T^{\mu\nu}_{0},J^{\mu}_{0})$ as defined in according to (16).
Since by construction, $(T^{\mu\nu}_{0},J^{\mu}_{0})$ satisfy the fitting
conditions

$(J^{\mu}-J_{(0)}{}^{\mu})u_{\mu}=(T^{\mu\nu}-T_{(0)}{}^{\mu\nu})u_{\mu}u_{\nu}=0,$
(38)

the perturbations $\delta J^{\mu}$ and $\delta T^{\mu\nu}$ are described by

$\delta J^{\mu}=n(u)^{\mu}$ (39) $\begin{split}\delta
T^{\mu\nu}&=h(u)^{\mu}u^{\nu}+h(u)^{\nu}u^{\mu}+\tau(u)^{\mu\nu}\\\
&=\pi(u)\Delta(u)^{\mu\nu}+\pi(u)^{\mu\nu}+h(u)^{\mu}u^{\nu}+h(u)^{\nu}u^{\mu},\end{split}$
(40)

and these forms are actually independent of the dynamics of the underlying
fluid theory. However that is not any longer the case for the entropy
perturbation $\delta S^{\mu}=S^{\mu}-S^{\mu}_{0}$, the structure of $S^{\mu}$
plays an important role in specifying he underlying theory.

Before we leave this section, it is worth stressing a point. For states
satisfying the relativistic (LTE) postulate, specified by
$(u^{\mu},s(\rho,n))$ even though the equilibrium equation of state
$s(\rho,n)$ combined with the velocity field $u^{\mu}$ defines an entropy like
current $S^{\mu}_{0}=s(\rho,n)u^{\mu}$ as defined in (16) this current is
formal and has nothing to do with the physical entropy $S^{\mu}$ of the
underlying state. Of relevance in the analysis of relativistic fluid states is
the dependance of the physical entropy current $S^{\mu}$ upon other fluid
variables and below, we shall have the opportunity to see how this dependance
leads to alternative theories of relativistic dissipation.

##  IV On states satisfying the relativistic (LTE) postulate and first order
theories

In this section, we study states compatible with the relativistic (LTE)
postulate within the context of the Hiscock-Lindblom class of first order
theories212121The term ”first order theories” coined by Hiscock and Lindblom
in His2  His3 ) and describes theories where the entropy current $S^{\mu}$
receives only first order contributions from a suitably defined state of
”local thermodynamical equilibrium”. Initially in this section we shall be
concerned with the Hiscock-Lindblom class, but at the end of the section we
shall introduce the (BDNK) class of first order theories that differs from the
Hiscock-Lindblom class.. We recall first that this class has been introduced
in His2  His3  and states within this class, are described by the fields
$(T^{\mu\nu},J^{\mu},S^{\mu})$ satisfying (7,8). Even though states within the
theory do not make no reference to any fluid four velocity, nevertheless, in
all treatments of this theory, a velocity field $u^{\mu}$ creeps in (see for
instance the analysis of Lindblom and Hiscock in His2  His3 ). Once a choice
of a four velocity222222Just to avoid confusion, we stress that the choice of
this velocity field $u^{\mu}$ is arbitrary, (except that it is restricted so
that relative to the family of the rest frames that it defines, the inequality
$\tau_{C}(p)<<\tau_{M}(p)$ holds) and this $u^{\mu}$ bears no relation to the
four velocity field entering in the specification of states compatible with
the (LTE) postulate. For instance, we have not yet introduced the pseudo angle
$\epsilon$ defined in (19) restricted to obey $\epsilon<<1$. $u^{\mu}$ has
been made, it is implicitly assumed that relative to the family of rest frames
defined by this $u^{\mu}$, the inequality $\tau_{C}(p)<<\tau_{M}(p)$ holds and
thus local thermodynamical equilibrium prevails. In turn this allows one to
postulate the existence of an ”equilibrium equation of state” $s$ which
combined with $(T^{\mu\nu},J^{\mu},S^{\mu})$ defines a ”local thermodynamical
equilibrium” state attached to $(u^{\mu},s(\rho,n))$ in the manner discussed
in the previous section.
A key ingredient that characterizes the Hiscock-Lindblom class is the
structure of the physical entropy current $S^{\mu}$ which is postulated to
have the form (see discussion in His2 )

$S^{\mu}=su^{\mu}+\hat{\beta}h^{\mu}(u)-\hat{\Theta}n^{\mu}(u)=S^{\mu}_{0}+\hat{\beta}h^{\mu}(u)-\hat{\Theta}n^{\mu}(u),$
(41)

where $(\hat{\beta},\hat{\Theta})$ are undetermined functions. The choice
$(u^{\mu}=u^{\mu}_{E},h^{\mu}(u)=0)$ generates the Landau- Lifshitz theory,
while $(u^{\mu}=u^{\mu}_{N},n^{\mu}=0)$ generates the Eckart theory.
With reference to the velocity field $u^{\mu}$ in (41), the expansions of
$(T^{\mu\nu},J^{\mu})$ in (10,11), introduce the fields
$(\rho(u),n(u),P(u),h^{\mu}(u),\tau^{\mu\nu}(u),n^{\mu}(u))$ which are
required to obey:

$\begin{split}\nabla_{\mu}T^{\mu\nu}=\nabla_{\mu}[&\rho(u)u^{\mu}u^{\nu}+P(u)\Delta(u)^{\mu\nu}+h(u)^{\mu}u^{\nu}\\\
&+h(u)^{\nu}u^{\mu}+\tau(u)^{\mu\nu}]=0,\end{split}$ (42)

$\nabla_{\mu}J^{\mu}=\nabla_{\mu}[n(u)u^{\mu}+n(u)^{\mu}]=0.$ (43)

Moreover, by imposing the second law $\nabla_{\mu}S^{\mu}\geq 0$ and recalling
the decomposition $\tau(u)^{\mu\nu}=\pi(u)\Delta(u)^{\mu\nu}+\pi(u)^{\mu\nu}$,
$\pi(u)^{\mu}_{\mu}=0$ in (12), a calculation shows that this law can be
fulfilled whenever the following equations hold (for a derivations see His2 ):

$h^{\mu}(u)=-kT(u)\Delta^{\mu\nu}(u)[\frac{1}{T(u)}\nabla_{\nu}T(u)+u^{\alpha}\nabla_{\alpha}u_{\nu}],$
(44)

$n^{\mu}(u)=-\sigma T^{2}(u)\Delta(u)^{\mu\nu}\nabla_{\nu}\Theta(u),$ (45)

$\pi^{\mu\nu}(u)=-2\eta<\nabla^{\mu}u^{\nu}>,$ (46)

$\quad\pi(u)=-\zeta\nabla_{\mu}u^{\mu},$ (47)

with the coefficients $(\hat{\beta},\hat{\Theta})$ in (41) given by:

$\hat{\beta}(u)=T^{-1}(u):=\beta(u),\hskip
4.26773pt\hat{\Theta}(u)=\frac{\rho(u)+P(u)}{n(u)T(u)}-s(u):=\Theta(u),$ (48)

and the angular bracket in (46), and here after, signifies symmetric, purely
spatial, trace free part of the enclosed tensor. From these equations one gets

$T\nabla_{\mu}S^{\mu}=\frac{\pi^{2}}{\zeta}+\frac{h^{\mu}h_{\mu}}{kT}+\frac{n^{\mu}n_{\mu}}{\sigma
T}+\frac{\pi^{\mu\nu}\pi_{\mu\nu}}{2\eta}\geq 0,$ (49)

implying that the entropy production is manifestly non negative provided the
four coefficients $(\zeta,\eta,k,\sigma)$ are chosen to be positive and these
coefficients are identified as the bulk viscosity $\zeta$, the thermal
conductivity $k$, a particle diffusion constant $\sigma$ and the shear
viscosity $\eta$.

The system (42-48) constitutes a closed system of equations whose solutions
describe arbitrary fluid states within the Hiscock-Lindblom class of first
order theories. From the mathematical view point, this system is a mixed
parabolic-hyperbolic-elliptic system and as we shall discuss further ahead,
its solutions are characterized by a number of undesirable properties.
Nevertheless, viewed as a closed system, and given suitable initial data,
determines the unknowns variables and solutions possessing distinct velocity
fields are considered to be distinct solutions.

However, matters differ when emphasis is restricted to states satisfying the
relativistic (LTE) postulate. Primary for such states, the fields $T^{\mu\nu}$
and $J^{\mu}$ are restricted so that the pseudo-angle $\epsilon$ between
$u_{E}$ and $u_{N}$ (see (19)) satisfies everywhere within the region occupied
by the fluid the condition $\epsilon<<1$ and moreover exists a (highly non
unique) velocity field $u^{\mu}$ that lies within the cone of the opening
angle $\epsilon<<1$ that generates a state of the ”local thermodynamical
equilibrium” state specified by $(u^{\mu},s(\rho,n))$ in the manner discussed
in the previous section. Clearly there exist infinitely many states satisfying
the (LTE) postulate, with each one of them is specified by the same pair
$(T^{\mu\nu},J^{\mu})$ but by a different velocity field.
We show below, that all such states are in effect equivalent to each other as
long as quadratic and higher order deviations from the state of the ”local
thermodynamical equilibrium” specified by $(u^{\mu},s(\rho,n))$ are omitted.
To establish this equivalence, let us begin with two velocity fields
$(u^{\mu},\hat{u}^{\mu})$ that specify two states satisfying the relativistic
(LTE) postulate. Using $u^{\mu}$, let

$(n(u),\rho(u),P(u),h(u)^{\mu},n(u)^{\mu},\tau(u)^{\mu\nu})$ (50)

be the fields obtained by expanding $(T^{\mu\nu},J^{\mu})$ according to
(10,11) and let us assume that these fields satisfy the exact equations
(42-48) for a set on non negative coefficients $(\zeta,\eta,k,\sigma)$. Using
this solution as a reference, we shall generate all other states compatible
with the relativistic (LTE) postulate. For this, let the frame change

$u^{\mu}\to\hat{u}^{\mu}=u^{\mu}+\hat{\epsilon}^{\mu},\quad\hat{\epsilon}\leq\epsilon,$
(51)

then (25-33) combined with the fields in (50), generate

$(\hat{u}^{\mu},\rho(\hat{u}),n(\hat{u}),P(\hat{u}),h^{\mu}(\hat{u}),\tau^{\mu\nu}(\hat{u}),n^{\mu}(\hat{u})).$

The claim is, that this new state satisfy (42-47), as long as deviations from
a the state of ”local thermodynamical equilibrium” specified by
$(u^{\mu},s(\rho,n))$ are omitted.

In order to show this, we first consider the special case where
$\hat{u}^{\mu}$ is chosen to be the four velocity $u^{\mu}_{E}$ of the energy
frame so that (51) takes the form

$u^{\mu}\to
u^{\mu}_{E}=u^{\mu}+\hat{\epsilon}^{\mu},\quad\hat{\epsilon}^{\mu}\leq\epsilon^{\mu}.$
(52)

Using (25-33), let

$u^{\mu}_{E},n(u_{E}),\rho(u_{E}),P(u_{E}),h(u_{E})^{\mu}=0,n(u_{E})^{\mu},\tau(u_{E})^{\mu\nu}$
(53)

defined according to

$n(u_{E})=n(u)+\hat{\epsilon}O_{1},\quad\rho(u_{E})=\rho(u)+\hat{\epsilon}O_{1},\quad
etc,$

and in this generation process, it is worth noticing that the condition
$h^{\mu}(u_{E})=0$ demands that $\hat{\epsilon}^{\mu}$ to satisfy

$h^{\mu}({u})=(\rho+P)\hat{\epsilon}^{\mu},$ (54)

while $n^{\mu}(u_{E})$ has the the value

$n^{\mu}(u_{E})=n^{\mu}(u)-n\hat{\epsilon}^{\mu}=n^{\mu}(u)-\frac{n(u)h^{\mu}(u)}{(\rho+P)}.$
(55)

The state in (53), approximately satisfy (42-48). Indeed, starting from
eqs.(42) and replacing $u^{\mu}$ by $u^{\mu}_{E}$ and $(n(u),\rho(u),)$ etc by
the fields in (53), we find

$\begin{split}0=&\nabla_{\mu}T^{\mu\nu}(u)\\\
=&\nabla_{\mu}[\rho(u)u^{\mu}u^{\nu}+P(u)\Delta(u)^{\mu\nu}+h(u)^{\mu}u^{\nu}+h(u)^{\nu}u^{\mu}+\tau(u)^{\mu\nu}]\\\
=&\nabla_{\mu}[\rho(u_{E})u^{\mu}_{E}u^{\nu}_{E}+P(u_{E})\Delta(u_{E})^{\mu\nu}+\tau(u_{E})^{\mu\nu}+O(\epsilon
O_{1})],\end{split}$ (56)

where we arrived at the last equality using
$u^{\mu}=u^{\mu}_{E}-\hat{\epsilon}^{\mu}+O(\hat{\epsilon})^{n},n\geq 2$ and
$h^{\mu}(u)=(\rho+P)\hat{\epsilon}^{\mu}+O(\hat{\epsilon})^{n},n\geq 2$. Thus
the state in (53) satisfy the conservation equation
$\nabla_{\mu}T^{\mu\nu}=0$, provided terms of order $(\epsilon O_{1})$ and
their gradients are thrown away. Via a similar reasoning, but now starting
from eq. (43), one finds

$\begin{split}0&=\nabla_{\mu}J^{\mu}=\nabla_{\mu}[n(u)u^{\mu}+n(u)^{\mu}]\\\
&=\nabla_{\mu}[n({u_{E}})u^{\mu}_{E}+n^{\mu}(u_{E})+O(\epsilon
O_{1})],\end{split}$ (57)

where we used
$n^{\mu}(u_{E})=n^{\mu}(u)-n\epsilon^{\mu}+O(\hat{\epsilon})^{n},n\geq 2$ and
thus to an $O_{1}$ accuracy the state in (53) indeed satisfy the conservation
law in (43).

We now examine whether the fields $(n(u_{E})^{\mu},\tau(u_{E})^{\mu\nu})$
satisfy to linear order equations (45-47). The easiest way to prove this
assertion is to start from the entropy current $S^{\mu}$ in (41) and eliminate
$u^{\mu},h^{\mu}(u)$ and $n^{\mu}(u)$ in favor of the fields measured relative
to the energy frame. Using
$u^{\mu}=u^{\mu}_{E}-\hat{\epsilon}^{\mu}+O(\hat{\epsilon})^{n},h^{\mu}(u)=(\rho+P)\hat{\epsilon}^{\mu}+O(\hat{\epsilon})^{n},n\geq
2$, it follows that $S^{\mu}$ takes the form

$\begin{split}S^{\mu}&=s(u^{\mu}_{E}-\hat{\epsilon}^{\mu})+\beta(\rho+P)\hat{\epsilon}^{\mu}-\Theta(n^{\mu}(u_{E})+n\hat{\epsilon}^{\mu})\\\
&=su^{\mu}_{E}-\Theta(u)n^{\mu}_{E}-[s-\beta(\rho+P)+\Theta
n]\hat{\epsilon}^{\mu}+O(\epsilon^{2})\\\
&=s(u_{E})u^{\mu}_{E}-\Theta(u_{E})n^{\mu}_{E}+O(\epsilon^{2})\end{split}$
(58)

where $s(u_{E}):=s(\rho(u_{E}),n(u_{E}))$ and we used the fundamental relation
$s=(\rho+P)T^{-1}-\Theta n$. By imposing the second law on this form of
$S^{\mu}$ but now written relative to the energy frame, one arrives at

$n^{\mu}(u_{E})=-\sigma
T^{2}\Delta(u_{E})^{\mu\nu}\nabla_{\nu}\Theta(u_{E})+O(\hat{\epsilon})^{2},$
(59)

$\pi^{\mu\nu}(u_{E})=-2\eta<\nabla^{\mu}u^{\nu}_{E}+O(\hat{\epsilon})^{2}>,$
(60)

$\quad\pi(u_{E})=\zeta\nabla_{\mu}u^{\mu}_{E}+O(\hat{\epsilon})^{2}.$ (61)

Thus the state in (53) satisfy the equations of the Landau-Lifshitz theory
provided that non linear terms in the deviations from the state of ”local
thermodynamical equilibrium” and gradients of $h^{\mu}$, $\pi$, and
$\pi^{\mu\nu}$ have been depreciated232323 Notice that equations (60-61) could
be derived directly from (46,47) by replacing $u^{\mu}$ by
$u^{\mu}=\hat{u}^{\mu}-\epsilon^{\mu}$ and treating
$(\nabla_{\mu}\epsilon^{\mu},\nabla_{\mu}\epsilon_{\nu})$ as $\epsilon O_{1}$
and thus depreciated them in comparison to the gradients of the velocity field
(this estimate will be employed further ahead).

It is worth noticing that if in this analysis, we replace $u^{\mu}$ by
$u^{\mu}_{N}$ and set $n^{\mu}(u_{N}):=0$ in (50) and $\sigma:=0$ in (45), so
that

$u^{\mu}_{N},n(u_{N}),\rho(u_{N}),P(u_{N}),h^{\mu}(u_{N}),\tau^{\mu\nu}(u_{N}),$

satisfy the exact equations (42-48) for the Eckart theory, then the state
$u^{\mu}_{E},n(u_{E}),\rho(u_{E}),P(u_{E}),n(u_{E})^{\mu},\tau(u_{E})^{\mu\nu}$
satisfy approximately the equations for the Landau-Lifshitz theory. This
process can be also reversed i.e. starting from a state satisfying the the
Landau-Lifshitz theory, one can generate a state that to $O_{1}$ accuracy
satisfy the Eckart theory.
Actually a more general statement holds. Within the context of the Hiscock-
Lindblom class of first order theories, states compatible with the
relativistic (LTE) postulate are to linear order equivalent states. To verify
this claim, we return to the original fields $(u^{\mu},\hat{u}^{\mu})$ and
consider the frame change in (52). We assume that $u^{\mu}$, and the fields in
(50) satisfy (10,11) and generate
$n(\hat{u}),\rho(\hat{u}),P(\hat{u}),h(\hat{u})^{\mu},n(\hat{u})^{\mu},\tau(\hat{u})^{\mu\nu}$
according to

$n(\hat{u})=n(u)+\hat{\epsilon}O_{1},\quad\rho(\hat{u})=\rho(u)+\hat{\epsilon}O_{1},\quad
etc.$

By similar arguments that lead us to (56,57), the conservation eqs yield

$\begin{split}0&=\nabla_{\mu}T^{\mu\nu}\\\
&=\nabla_{\mu}[(\rho+P)\hat{u}^{\mu}\hat{u}^{\nu}+Pg^{\mu\nu}+h(\hat{u})^{\mu}\hat{u}^{\nu}+h(\hat{u})^{\nu}\hat{u}^{\mu}\\\
&+\tau(\hat{u})^{\mu\nu}+\hat{\epsilon}O_{1}]-\nabla_{\mu}[h^{\mu}(\hat{u})\epsilon^{\nu}+h^{\nu}(\hat{u})\epsilon^{\mu}]\end{split}$
(62)

$\begin{split}0&=\nabla_{\mu}J^{\mu}=\nabla_{\mu}[n(u)u^{\mu}+n(u)^{\mu}]\\\
&=\nabla_{\mu}[n(u)\hat{u}^{\mu}+n(\hat{u})^{\mu}+\hat{\epsilon}^{\mu}O_{1}],\end{split}$
(63)

and under the assumptions that gradients of $h^{\mu}$ and of $\epsilon^{\mu}$
are neglected in comparison to the velocity gradients, it follows that the
terms linear in $\epsilon^{\mu}$ in the right hand side of (62, 63) can be
neglected and thus the conservation laws (42, 43) to an $O_{1}$ accuracy holds
true for the state
$(\hat{u}^{\mu},n(\hat{u}),\rho(\hat{u}),P(\hat{u}),h(\hat{u})^{\mu},n(\hat{u})^{\mu},\tau(\hat{u})^{\mu\nu})$.
We now indicate that equations (44-47) remain valid and for this we observe
that the entropy flux $S^{\mu}$ in (41) that under the frame change described
in (52) implies:

$\begin{split}S^{\mu}&=su^{\mu}+\beta(u)h^{\mu}(u)-\Theta(u)n^{\mu}(u)\\\
&=s\hat{u}^{\mu}+\beta(\hat{u})h^{\mu}(\hat{u})-\hat{\Theta}(u)n^{\mu}(\hat{u})+O(\epsilon^{2})\end{split}$
(64)

and this $O_{1}$-invariance property of $S^{\mu}$ implies that the fields
$h(\hat{u})^{\mu},n(\hat{u})^{\mu},\tau(\hat{u})^{\mu\nu}$ satisfy the
analogues of (44-47) to an $O_{1}$ accuracy i.e to linear order in the
deviations from the state of ”local thermodynamical equilibrium” specified by
$(u^{\mu},s(\rho,n))$.

Before we continue, we offer a few comments regarding the constitutive
relations that states satisfying the relativistic (LTE) postulate obey. We
recall first the derivation of the exact relations (44-47). For this
derivation, one begins by imposing the second law on the the entropy current
$S^{\mu}$ in (41) and using the exact eqs
$\nabla_{\mu}T^{\mu\nu}=\nabla_{\mu}J^{\mu}=0$ one arrives at the exact
relations (44-47) with the positive coefficients $(\zeta,\eta,k,\sigma)$
putting in by hand. However for states satisfying the (LTE) postulate, the
constitutive relations obeyed by $(h(u),n^{\mu}(u),\pi^{\mu\nu}(u),\pi(u))$
hold approximately, since in their derivation we used the approximate
expressions for the entropy currents $S^{\mu}$ shown in the last lines of (58)
and (64). It is worth however, to probe this approximation and in particularly
to probe the transition from the constitutive relations of the Eckart theory
to the corresponding relations for the Landau-Lifshtz theory. To do so, we
start from the easily verifiable identity (see for instance eq.$(34)$ in His1
)

$nT\nabla_{\alpha}\Theta=\nabla_{\alpha}P-\frac{\rho+P}{T}\nabla_{\alpha}T$
(65)

and let for generality purpose assume that the fluid is described by a
$T^{\mu\nu}$ shown in (10). For this case the following identity holds

$\Delta^{\mu\nu}\left[\frac{\nabla_{\nu}T}{T}+a_{\nu}\right]=-\frac{nT}{\rho+P}\Delta^{\mu\nu}\nabla_{\nu}\Theta+\frac{F^{\mu}}{\rho+P},\quad
a^{\mu}=u^{\nu}\nabla_{\nu}u^{\mu}$ (66)

where $F^{\mu}$ is defined from the conservation equation
$\Delta^{\mu}{}_{\rho}\nabla_{\nu}T^{\nu\rho}=0$ which can be written in the
equivalent form

$\Delta^{\mu\nu}\nabla_{\nu}P+(\rho+P)a^{\mu}=F^{\mu}$ (67)

where the precise form of $F^{\mu}$ is easily derivable, but it is not really
needed since it contributes terms of $O(\epsilon^{2})$ or terms that make
negligible contribution like $\epsilon^{\mu}\nabla_{a}u^{a}$ etc. It is
understood that in (65-67), the terms $T,\Theta,P,\Delta^{\mu\nu}$ stand for
$T(u),\Theta(u),P(u),\Delta^{\mu\nu}(u)$ etc.

Suppose now we start from the Eckart theory so that the following relations
hols

$h^{\mu}(u_{N})=-kT(u_{N})\Delta^{\mu\nu}(u_{N})\left[\frac{\nabla_{\nu}T(u_{N})}{T(u_{N})}+a_{\nu}\right],\quad\pi^{\mu\nu}(u_{N})=-2n\langle\nabla^{\mu}u^{\nu}\rangle,\quad\pi(u_{N})=-\zeta\nabla_{\mu}u_{N}^{\mu}$

for some non vanishing $(k,\zeta,\eta)$. Let us now perform a frame change
described by

$u_{N}^{\mu}\to u^{\mu}_{E}=u_{N}^{\mu}+\epsilon^{\mu}$

and for this frame change, we evaluate the identity (66) at $u_{N}^{\mu}$ and
combine it with

$h^{\mu}(u_{N})=-kT(u_{N})\Delta^{\mu\nu}(u_{N})\left[\frac{\nabla_{\nu}T(u_{N})}{T(u_{N})}+a_{\nu}\right].$
(68)

Using $h^{\mu}(u_{N})=(\rho+P)\epsilon^{\mu}$ ,$\quad
n^{\mu}(u_{E})=-n\epsilon^{\mu}$ writing
$\Delta^{\mu\nu}(u_{N})=\Delta^{\mu\nu}(u_{E})-u_{E}^{\mu}\epsilon^{\nu}-u_{E}^{\nu}\epsilon^{\mu}+O(\epsilon^{2})$,
$T(u_{N})=T(u_{E})+O(\epsilon^{2})$ etc, one finds after some algebra that
(68) transforms into

$n^{\mu}(u_{E})=-\sigma
T^{2}(u_{E})\Delta^{\mu\nu}(u_{E})\nabla_{\nu}\Theta(u_{E})+O(\epsilon)$ (69)

where $O(\epsilon)$ denote terms involving products of $\epsilon$ with other
other terms which in general considered as negligible in comparison to the
leading term $\sigma
T^{2}(u_{E})\Delta^{\mu\nu(u_{E}}\nabla_{\nu}\Theta(u_{E})$ and in arriving at
(69) we introduced the coefficient $\sigma$ via

$\sigma=k\frac{n^{2}}{(\rho+P)^{2}}$

.

Similarly, staring from

$\pi^{\mu\nu}(u_{N})=-2n\langle\nabla^{\mu}u^{\nu}\rangle,\quad\pi(u_{N})=-\zeta\nabla_{\mu}u_{N}^{\mu},$

then under the frame change $u_{N}^{\mu}\to
u^{\mu}_{E}=u_{N}^{\mu}+\epsilon^{\mu}$, it can be easily seen that they
transform into

$\pi^{\mu\nu}(u_{E})=-2n\langle\nabla^{\mu}u_{E}^{\nu}\rangle,\quad\pi(u_{E})=-\zeta\nabla_{\mu}u_{u_{E}}^{\mu},$

where we have dropped terms linear in $\epsilon^{\mu}$ and its derivatives.
One can also reverse this process i.e. one can start from the constitutive
relation in the Landau-Lifshitz theory to generate the constitutive relations
for the Eckart theory. More generally one can consider frame changes described
for instance by (51) and starting from the constitutive relations relative to
the $u^{\mu}$ frame work out the constitutive relations relative to the
$\hat{u}^{\mu}$. In general one finds to lowest order (44-47) hold but they
are modified by corrections terms of order $O(\epsilon^{\mu})$, etc. and these
correction terms are neglected as long as we are in the regime where
$\epsilon<<1$ (see also a discussion at this approximation, in the review
article by Israel in Isr5  page $(179))$.

In summary, therefore within the Hiscock-Lindblom class of first order
theories, states that satisfy the relativistic (LTE) postulate are equivalent
states as long as one neglects terms of $\epsilon^{2}$ and higher order in the
deviations from the local equilibrium field and gradients of $h^{\mu}$, $\pi$,
and $\pi^{\mu\nu}$ are small compared to thermal and velocity gradients and
this conclusion holds also for the Landau-Lifshitz and Eckart theories.

In the so far analysis, we used the Hiscock-Lindblom class of first order
theories as a test bed, to get insights on the properties of states satisfying
the relativistic (LTE) postulate. However, ought to be mentioned that this
class of theories, including the Eckart and Landau-Lifshitz theories, are
pathological. According to the results in His2 ; His3 , they do not respect
causality and they are unstable in the sense that linear perturbations of
their global equilibrium states become unbounded on a very short time scale.
These important conclusions have been derived in His2 ; His3  by analyzing
linearized perturbations in the form of exponentially plane waves off a
globally homogeneous equilibrium state propagating on a Minkowski spacetime.
As long as one of the coefficients $(\zeta,\eta,k,\sigma)$ in (49) is
different than zero, there exist transversal and longitudinal exponentially
growing modes. Based on this property, other physically acceptable solutions
of the perturbations equations exhibit also this type of instability (for more
details, consult His2 ).

The results in His2 ; His3  revealed the following highly counter-intuitive
property: an equilibrium state appear to be stable when it is observed from a
particular frame, but becomes unstable when is observed from a frame related
to the first one by a Lorentz boost. This behavior can be seen but analyzing
the behavior of the perturbing plane waves modes constructed in His2 . For
instance, whenever the equilibrium state is observed from the comoving frame,
then for the Eckart theory (or more generally for any theory within the
Hiscock-Lindblom class) where $k\neq 0$, these modes contain exponentially
growing modes while for the Landau-Lifshitz theory (where $k=0$) these plane
waves contains only decaying modes. However, when the state is viewed from a
Lorentz frame where the state is in motion, then for both theories i.e. Eckart
and also the Landau-Lifshitz theory (or more generally for all of first order
theories) contain exponentially growing modes and thus exhibit instability.
Here one sees that within the Landau-Lifshitz theory the equilibrium state is
stable when is observed from the comoving with the state frame, but is
unstable when is observed from any other boosted frame.

This counter intuitive behavior has triggered a sizable amount of research
activity (see for example RP1 ; Per ; GAV2 ; Nor2 ; GAV6 ) and these efforts
lead to powerful statements linking causality to stability of equilibrium
states. One of the strongest results describing this interplay is Gavassino’s
criterion GAV2  which asserts that if for a theory causality holds, then
stability of equilibrium sates is a Lorentz-invariance property. In more
intuitive terms, the criterion asserts that if for a causal theory one is able
to prove stability of an equilibrium state relative to one reference frame,
then there cannot be any growing Fourier mode in any other boosted frame. A
slightly different statement has proven also in Nor2  which restricts slightly
however the nature of the field equations.
In view of these developments, an interesting interpretation regarding the
origin of the instability in the Hiscock-Lindblom class seen in His2 ; His3 ,
has been put forward in ref. GAV6 . It was shown in that reference that for
these theories the total entropy as a function of the accessible states fail
to have upper bound a situation which is in sharp contrast to what occurs for
instance for the case of the Israel-Stewart transient thermodynamics where the
total entropy exhibits an absolute maximum (for the stability properties of
that theory consult His1 , His3 , Olson ). Finally very recently, a connection
between causality and thermodynamical stability has been discussed in GAV3 .
It was shown in that work for any theory that is thermodynamically stable i.e.
the total entropy is maximized at equilibrium, it is also causal at least
close to equilibrium, a conclusion that indicates clearly that causality is
tied up thermodynamic stability (and not hydrodynamic stability) that is
commonly believed (see illuminating discussion in GAV3 ).

In view of this perplexing state of affairs centered on causality, stability
and the Hiscock-Lindblom class of first order theories, below we discuss
briefly a few properties of equilibrium states perturbed by states satisfying
the relativistic (LTE) postulate. For this, let again the class of states
satisfying the relativistic (LTE) postulate assuming here that the fields
$T^{\mu\nu}$ and $J^{\mu}$ are defined on a Minkowski spacetime restricted so
that the pseudo-angle $\epsilon$ in (19) satisfies $\epsilon<<1$ everywhere
within the fluid region. Let a four velocity $u_{o}^{\mu}$ is chosen within
the cone of opening angle $\epsilon<<1$ and let the fields in the expansion
(10, 11) satisfy

$h^{\mu}=n^{\mu}=\pi=\pi^{\mu\nu}=0,$ (70)

while $(\rho_{o},n_{o},P_{o})$ are chosen to be homogeneous and isotropic and
relative to a global inertial coordinates $(t,x,y,z)$ the velocity field
$u_{o}^{\mu}$ takes the form $u_{o}^{\mu}=\delta^{\mu}{}_{t}$ i.e. the state
is at rest relative to this global rest frame. The so defined state satisfies
the relativistic (LTE) postulate but it also is an equilibrium state242424Here
after thermodynamical variables describing this equilibrium state are denoted
by subscript $(o)$. In particularly, for emphasis we write
$(T_{o}^{\mu\nu},J_{o}^{\mu})$ instead of writing the correct expressions
$(T^{\mu\nu},J^{\mu})$. since the entropy current $S^{\mu}$ as a consequence
of (70) satisfies $\nabla_{\mu}S^{\mu}=0$ while its energy momentum tensor
$T_{o}^{\mu\nu}$ and particle current $J_{o}^{\mu}$ satisfy

$\nabla_{\mu}T_{o}^{\mu\nu}=\nabla_{\mu}[(\rho_{o}+P_{o})u_{o}^{\mu}u_{o}^{\nu}+P_{o}g^{\mu\nu}]=0,\quad\nabla_{\mu}J_{o}^{\mu}=\nabla_{\mu}(n_{o}u_{o}^{\mu})=0.$
(71)

We perturb the equilibrium state in (70, 71) by a state satisfying the (LTE)
postulate denoted by $(u^{\mu},s(\rho,n))$.
Following the notation in His2 , the (Eulerian) perturbations of the
thermodynamical variables are denoted by $\delta\rho,\delta n,\delta
u^{\mu},\delta\tau,$ etc and represent the difference between the value of the
non equilibrium variable and the corresponding equilibrium one evaluated at
the same spacetime point. It is not difficult to show that the equations for
these perturbations are obtained by linearizing (42-48) around a background
equilibrium state are given by (see also His2 )

$\nabla_{\mu}\delta T^{\mu\nu}=0,\quad\nabla_{\mu}\delta J^{\mu}=0,\quad\delta
h^{\mu}=-kT\Delta^{\mu\nu}[\nabla_{\nu}(\frac{\delta
T}{T})+u^{\lambda}\nabla_{\lambda}\delta u_{\nu}+\delta
u^{\lambda}\nabla_{\lambda}u_{\nu}],\quad\delta\nu^{\mu}=-\sigma
T^{2}\Delta^{\mu\nu}\nabla_{\nu}\delta\Theta$ (72)
$\delta\pi^{\mu\nu}=-2\eta<\nabla^{\mu}\delta u^{\nu}+\delta
u^{\mu}u^{\lambda}\nabla_{\lambda}u^{\nu}>,\quad\delta\pi=-\zeta\nabla_{\mu}\delta
u^{\mu},$ (73)

with the perturbations $\delta T^{\mu\nu}$ and $\delta J^{\mu}$ defined by

$\delta T^{\mu\nu}=(\rho+P)(\delta u^{\mu}u^{\nu}+u^{\mu}\delta
u^{\nu})+\delta\rho u^{\mu}u^{\nu}+(\delta
P+\delta\pi)\Delta^{\mu\nu}+u^{\mu}\delta h^{\nu}+u^{\nu}\delta
h^{\mu}+\delta\pi^{\mu\nu},$ (74) $\delta J^{\mu}=\delta nu^{\mu}+n\delta
u^{\mu}+\delta n^{\mu}.$ (75)

and in these equations and here after, variables without the prefix $\delta$,
denote equilibrium values252525For typographical convenience we have written
$(T,u^{\mu},\Delta^{\mu\nu},P,....)$ instead of
$(T_{o},u_{o}^{\mu},\Delta_{o}^{\mu\nu},P_{o},....)$.. Notice also that as a
consequence of the constraints $u^{\mu}u_{\mu}=-1$, $h^{\mu}u_{\mu}=0$ etc,
the perturbations variables are subject to the constraints (see also His2 ):

$u^{\mu}\delta u_{\mu}=u^{\mu}\delta h_{\mu}=u^{\mu}\delta
n_{\mu}=u^{\mu}\delta\pi_{\mu\nu}=0$ (76)

Since in the derivation of equations (72-76), are actually independent of
whether the perturbing state is an arbitrary state or a state compatible with
the (LTE) postulate, the results in His2  holds true for our problem as well.
Thus for small (Eulerian) perturbations of exponential plane waves of the form

$\delta Q=\delta Q_{0}e^{ik_{x}x+\Gamma t}$ (77)

propagating along the $x$-axis with constant ”frequency” $\Gamma$, the results
in His2  show that for a non vanishing $k_{x}$, exist exponentially growing
(and decreasing) transverse modes with real ”frequencies” $\Gamma_{\pm}$ given
by

$2kT\Gamma_{\pm}=(\rho+P)\pm[(\rho+P)^{2}+4\zeta kTk_{x}^{2}]^{\frac{1}{2}}$
(78)

while for the case of the Landau-Lifshitz theory there exist decreasing
transverse modes with

$\Gamma=-\frac{\eta k_{x}^{2}}{\rho+P},$ (79)

and since this $\Gamma$ is purely real and negative, it follows that the
Landau-Lifshitz theory escapes the instability that is manifest in the Eckart
(or any other first order theory subject to $k>0$). However, as was shown in
His2 , His3  this property disappears once the equilibrium state is viewed
from a Lorentz frame where the equilibrium state is in motion.
The results expressed in (78, 79) hold also for the equilibrium states
perturbed by states satisfying the relativistic (LTE) postulate and thus these
equilibrium states exhibit the same instabilities as the equilibrium states
within the full class of first order theories.

Suppose however, we consider another state satisfying the relativistic (LTE)
postulated specified by $(\hat{u}^{\mu},s(\hat{\rho},\hat{n}))$, which is
initially close to the equilibrium state in (71). We consider again the linear
perturbations denoted here after by a prime i.e.
$\delta^{\prime}\rho,\delta^{\prime}n,\delta^{\prime}u^{\mu},\delta^{\prime}\tau,$
etc induced by $(\hat{u}^{\mu},s(\hat{\rho},\hat{n}))$ on the equilibrium
state in (71). Clearly these new perturbations satisfy again (72-76) (obtained
by linearizing the system (42-48)).
As we have seen in this section although, the states $(u^{\mu},s(\rho,n))$ and
$(\hat{u}^{\mu},s(\hat{\rho},\hat{n}))$ are in fact equivalent and formulas
(25-33), shows that equivalence, below we show that the perturbations induced
by $(u^{\mu},s(\rho,n))$ and $(\hat{u}^{\mu},s(\hat{\rho},\hat{n}))$ on the
background equilibrium state in (71) fail to be equivalent.
Indeed starting from the relation

$\hat{u}^{\mu}-u^{\mu}=\hat{\epsilon}^{\mu}+O(\hat{\epsilon})^{2}$

the following relations hold between the primed and unprimed perturbations:

$\delta
u^{\mu}:=u^{\mu}-u_{o}^{\mu}=\hat{u}^{\mu}-u_{o}^{\mu}+u^{\mu}-\hat{u}^{\mu}=\delta^{\prime}u^{\mu}-\hat{\epsilon}^{\mu}+O(\hat{\epsilon})^{2}$
(80) $\delta
h^{\mu}=h^{\mu}(u)=\delta^{\prime}h^{\mu}+(\rho+P)\hat{\epsilon}^{\mu}+O(\hat{\epsilon})^{2},\quad\delta
n^{\mu}=\delta^{\prime}n^{\mu}+n\hat{\epsilon}^{\mu}+O(\hat{\epsilon})^{2}$
(81)

$\delta\rho=\delta^{\prime}\rho+O(\hat{\epsilon})^{2},\quad\delta
n=\delta^{\prime}n+O(\hat{\epsilon})^{2},\quad\quad\delta
P=\delta^{\prime}P+O(\hat{\epsilon})^{2}\quad\delta\pi=\delta^{\prime}\pi+O(\hat{\epsilon})^{2},\quad\delta\pi^{\mu\nu}=\delta^{\prime}\pi^{\mu\nu}+O(\hat{\epsilon})^{2}$
(82)

Moreover it can be easily seen that if the unprimed perturbations satisfy
equations (72-76). This means that the perturbations induced on the background
equilibrium state by $(u^{\mu},s(\rho,n))$ and
$(\hat{u}^{\mu},s(\hat{\rho},\hat{n}))$ states are in fact related by formulas
(80-82) that are analogous to (25-33). Since however, these relations hold for
arbitrary solutions of the perturbations equations, they also hold for the the
of exponential plane waves solutions described in (77) i.e. propagating along
the $x$-axis and of constant ”frequency” $\Gamma$, as in His2 . However, here
we lead into an impasse. If for instance we choose $(u^{\mu},s(\rho,n))$ to be
described by an Eckart state ( or more generally an arbitrary first order
state with $k>0$) and $(\hat{u}^{\mu},s(\hat{\rho},\hat{n}))$ say a Landau-
Lifshitz state (thus $k=0$), then formulas (80-82) describe an
incompatibility. One perturbation is exponentially growing while the other is
exponentially decreasing. This in turn implies that the equivalence between
the two states $(u^{\mu},s(\rho,n))$ and
$(\hat{u}^{\mu},s(\hat{\rho},\hat{n}))$ that satisfy the (LTE) postulate as
expressed by formulas (25-33), breaks down for the first order perturbations
of an equilibrium state. To put it differently, if the states
$(u^{\mu},s(\rho,n))$ and $(\hat{u}^{\mu},s(\hat{\rho},\hat{n}))$ were truly
equivalent one would have expected that the induced perturbations on a
background equilibrium state of the theory to in fact identical solutions.
Interestingly this kind of equivalence holds for the Israel-Stewart theory. If
one replaces the Hiscock-Libdblom class of first order theories by the Israel-
Stewart theory and the $(u^{\mu},s(\rho,n))$,
$(\hat{u}^{\mu},s(\hat{\rho},\hat{n}))$ states by the Israel-Stewart theory
expressed in the Eckart frame respectively Landau-Lifshitz frame, one would
find that the resulting induced perturbations are indeed equivalent and this
equivalence has been demonstrated262626Our thanks to an anonymous referee who
pointed out to us ref.GAV1  and suggested to probe the interconnection between
Hiscock-Libdblom class of first order theories and the Israel-Stewart theory.
in ref. GAV1  (we shall return to this point at the end of the section $(V)$).
Here we only mentioned that this pronounced difference between the two
theories can be traced in thermodynamical reasoning. Gavasinno, Antonelli and
Haskell in GAV6  have shown that the total entropy between the two theories
behaves differently. Sable point behavior for the Hiscock-Libdblom class of
first order theories versus an absolute maximum for the Israel-Stewart theory.
We shall discuss this point further at the end of the next section.
The so far analysis demonstrates that states satisfying the relativistic (LTE)
postulate with the Hiscock-Libdblom class of first order theories exhibit the
same pathologies as arbitrary states within this class and thus their utility
is very limited.

We shall leave this section, by briefly discussing the connection between
states obeying the relativistic (LTE) postulate and the (BDNK) theory. As we
have already mentioned in section $(II)$ the (BDNK) theory is branded as a
first order theory272727It ought to be stressed that while in the Hiscock-
Lindblom terminology, a first order theory is a theory where the entropy flux
$S^{\mu}$ receives contributions only from first order deviations from a
fictitious ”local thermodynamical equilibrium” state, the term ”first order
theories” within the (BDNK) formalism has a very different meaning as it will
become clear further ahead. A suitably defined series is approximated only by
the first term. and it has some remarkable properties. It was shown in (Kov1 ,
Kov2 , Nor1 , Nor2 ) that under suitable choice of frame, the theory respects
causality and admits stable equilibrium states and moreover in Nor2  it was
shown that the Cauchy problem for the theory is locally well posed, and
strongly hyperbolic and this locally well posedness and strong hyperbolicity
remain intact even when the fluid is dynamically coupled to Einstein’s
equations ( see ref Nor2  for details and references).

For a simple, electrically neutral, fluid the dynamical equations for this
theory are $\nabla_{\mu}T^{\mu\nu}=\nabla_{\mu}J^{\mu}=0$ and fluid states are
specified by assigning a ”hydrodynamical frame” or simply a frame282828Here we
are warning the reader of a potential confusion arising from the use of the
term ”frame”. While in this work the term ”frame” signifies an orthonormal
tetrad (or often a collections of orthonormal tetrads defined along an
integral curve (or on all integral curves)) of a smooth velocity field
$u^{\mu}$ and the term ”change of frame” is a transition to a new tetrad
induced by the action of a point wise Lorentz transformation (or a family of
such point wise transformations), within the (BDNK) formalism, a
”hydrodynamical frame” or simply ”frame” signifies a specification of a
triplet $(T,\mu,u^{\mu})$ and a change of the ”hydrodynamical frame” (or a
field redefinition), is a passage to a new ”hydrodynamical frame” i.e. to a
new set of hydrodynamical variables $(\hat{T},\hat{\mu},\hat{u}^{\mu})$. This
change of frame advocated in the (BDNK) formalism is in general different than
the change of frame induced by the action of a local Lorentz transformation in
our terminology, although the two are related as we shall see further ahead
(in reference (Nor2 ) a complete set of references are compiled where the
reader can find the diverse meaning assigned to the term frames and change of
frames). which signifies a specification of a fluid four velocity $u^{\mu}$ a
temperature $T$ and a chemical potential $\mu$ (both of them measured by the
$u^{\mu}$-observer). The theory targets states that are near equilibrium,
although a precise definition of what exactly constitutes a state near thermal
equilibrium has not been addressed in the so far development of the
theory292929 An attempt to define precisely the term equilibrium and states
near equilibrium is pursued in refs Nor1 , Nor2  via the employment of
relativistic kinetic theory. Although a Chapman-Enskog expansion, or a Grad
like or any other expansion yield fluid states that capture the spirit of the
relativistic (LTE) postulate, still we feel that an independent definition of
states compatible with a relativistic (LTE) postulate build entirely within
the field of Relativistic hydrodynamics viewed as a discipline in its own
right, is worth having (and this is precisely the main purpose of the present
work).. Within this theory, the fields $(T^{\mu\nu},J^{\mu})$ depend upon
$(u^{\mu},T,{\mu})$ and the theory utilizes (or better adapts) the gradient
expansion technique from quantum field theory to the hydrodynamical regime
(for an introduction to this technique within the hydrodynamical regime,
consult (Gex1 , Gex2 ) and references therein). The fields
$(T^{\mu\nu},J^{\mu})$ are expanded according to

$T^{\mu\nu}=\mathcal{E}u^{\mu}u^{\nu}+\mathcal{P}\Delta^{\mu\nu}(u)+\left(\mathcal{Q}^{\mu}u^{\nu}+\mathcal{Q}^{\nu}u^{\mu}\right)+\mathcal{T}^{\mu\nu}\\\
$ (83)

$J^{\mu}=\mathcal{N}u^{\mu}+\mathcal{J}^{\mu}$ (84)

which are the same expansions as in (10, 11) except that new symbols303030The
usage of
$(\mathcal{E},\mathcal{P},\mathcal{Q}^{\mu},\mathcal{T}^{\mu\nu},\mathcal{N},\mathcal{J}^{\mu})$
is the standard notation for the practioners of the (BDNK) formalism.
$(\mathcal{E},\mathcal{P},\mathcal{Q}^{\mu},\mathcal{T}^{\mu\nu},\mathcal{N},\mathcal{J}^{\mu})$
are used instead of $(\rho,P,h^{\mu},\tau^{\mu\nu},n,n^{\mu})$ entering in
(10, 11). However, within the (BDNK) formalism, it is postulated that the
fields
$(\mathcal{E},\mathcal{P},\mathcal{Q}^{\mu},\mathcal{T}^{\mu\nu},\mathcal{N},\mathcal{J}^{\mu})$
admit a (convergent) derivative series expansion formed from the derivatives
of $(T,u^{\mu},\mu)$. The series employs the scalars $(T,\mu)$, their
derivatives along the flow of $u^{\mu}$ i.e. $u^{\mu}\nabla_{\mu}T:=\dot{T}$,
$\nabla_{\mu}u^{\mu}$ and $u^{\alpha}\nabla_{\alpha}\mu:=\dot{\mu}$, utilizes
also the transverse vectors $\Delta^{\mu\nu}\nabla_{\nu}T$,
$a^{\mu}:=u^{\nu}\nabla_{\nu}u^{\mu}$, $\Delta^{\mu\nu}\nabla_{\nu}\mu$, and
the transverse second rank, traceless symmetric tensor, $\sigma_{\mu\nu}$ and
in a standard notation, the series expansions have the form (for details
consult (Kov1 , Kov2 , Nor1 , Nor2 ):

$\mathcal{E}=\epsilon+\epsilon_{1}\frac{\dot{T}}{T}+\epsilon_{2}\nabla_{\mu}u^{\mu}+\epsilon_{3}u^{\mu}\nabla_{\mu}(\frac{\mu}{T})+O(\partial^{2})$
(85)

$\mathcal{P}=p+\pi_{1}\frac{\dot{T}}{T}+\pi_{2}\nabla_{\mu}u^{\mu}+\pi_{3}u^{\mu}\nabla_{\mu}(\frac{\mu}{T})+O(\partial^{2})$
(86)

$\mathcal{Q}^{\mu}=\theta_{1}\dot{u}^{\mu}+\frac{\theta_{2}}{T}\Delta^{\mu\nu}\nabla_{\nu}T+\theta_{3}\Delta^{\mu\nu}\nabla_{\nu}(\frac{\mu}{T})+O(\partial^{2})$
(87)

$\mathcal{T}^{\mu\nu}=-\eta\sigma^{\mu\nu}+O(\partial^{2})$ (88)

$\mathcal{N}=n+\nu_{1}\frac{\dot{T}}{T}+\nu_{2}\nabla_{\mu}u^{\mu}+\nu_{3}u^{\mu}\nabla_{\mu}(\frac{\mu}{T})+O(\partial^{2})$
(89)

$\mathcal{J}^{\mu}=\gamma_{1}{\dot{u}}^{\mu}+\frac{\gamma_{2}}{T}\Delta^{\mu\nu}\nabla_{\nu}\frac{\mu}{T}+\gamma_{3}\Delta^{\mu\nu}\nabla_{\nu}\frac{\mu}{T}+O(\partial^{2})$
(90)

where $O(\partial^{2})$ (or more generally $O(\partial^{k})$) signifies terms
of second order (or $k$ order) derivatives in the variables $(T,u^{\mu},\mu)$,
while $(\epsilon_{i},\pi_{i},\theta_{i},\nu_{i})$, $(\gamma_{i},i\in(1,2,3))$
and $\eta$ are considered to be transport coefficients depending upon
$(T,\mu)$. Moreover, it is postulated that the basic fields
$(T(x),u^{\mu}(x),\mu(x))$ are non unique and can be transformed to new fields
$(T^{\prime}(x),u^{\prime\mu}(x),\mu^{\prime}(x))$ according to:

$(T(x),u^{\mu}(x),\mu(x))\to(T^{\prime}(x),u^{\prime\mu}(x),\mu^{\prime}(x))=(T(x)+\delta
T(x),u^{\mu}(x)+\delta u^{\mu}(x),\mu(x)+\delta\mu(x)))$ (91)

where the variations $(\delta T(x),\delta u^{\mu}(x),\delta\mu(x))$ are
defined via:

$\delta
T=\alpha_{1}\frac{\dot{T}}{T}+\alpha_{2}\nabla_{\mu}u^{\mu}+\alpha_{3}u^{\mu}\nabla_{\mu}(\frac{\mu}{T})+O(\partial^{2})$
(92)

$\delta
u^{\mu}=b_{1}{\dot{u}^{\mu}}+b_{2}\Delta^{\mu\nu}\nabla_{\nu}T+b_{3}\Delta^{\mu\nu}\nabla_{\mu}(\frac{\mu}{T})+O(\partial^{2})$
(93)

$\delta\mu=c_{1}\frac{\dot{T}}{T}+c_{2}\nabla_{\mu}u^{\mu}+c_{3}u^{\mu}\nabla_{\mu}(\frac{\mu}{T})+O(\partial^{2})$
(94)

where $(a_{i},b_{i},c_{i},i\in(1,2,3))$ are arbitrary functions of the
$(T,\mu)$ (for a discussion leading to the above transformation see Kov1 ,
Kov2 ). The transformation in (91-94) describes a change of the hydrodynamical
frame and plays an important role within the theory.

We shall not discuss any further properties of this theory, but for the rest
of this section, we shall link states obeying the relativistic (LTE) postulate
to the (BDNK) theory and for this let us begin by choosing a state compatible
with the relativistic (LTE) postulate within however the (BDNK) theory.
Accordingly we choose $T^{\mu\nu}$ and $J^{\mu}$ such that the angle
$\epsilon$ formed by $u^{\mu}_{E}$ and $u^{\mu}_{N}$ (see (19)) satisfies
$\epsilon<<1$ everywhere within the the fluid region and choose a velocity
field $u^{\mu}$ within the cone of the angle $\epsilon<<1$. We denote by
$(u^{\mu},s(\rho,n))$ be the state of ”local thermodynamical equilibrium” and
let $T(u)$ the local temperature and $\mu(u)$ the local chemical potential,
defined from derivatives of $s(\rho(u),n(u))$ (see (14) and remember that
$\Theta(u)=\frac{\mu(u)}{T(u)}$). Thus any state satisfying the relativistic
(LTE) postulate leads to a hydrodynamical frame (in the (BDNK)-sense)
specified by $(T(u),u^{\mu},\mu(u))$.

Under an admissible frame change described by

$u^{\mu}\to\hat{u}^{\mu}=u^{\mu}+\hat{\epsilon}^{\mu}+O(\hat{\epsilon})^{2},\quad\hat{\epsilon}^{\mu}\leq
O_{1}.$ (95)

formulas (25-33), imply

$T^{\prime}:=T(\hat{u})=T(u)+\hat{\epsilon}O_{1},\quad\mu^{\prime}:=\mu^{\prime}(\hat{u})=\mu(u)+\hat{\epsilon}O_{1},\quad
u^{\prime\mu}:=\hat{u}^{\mu}=u^{\mu}+\hat{\epsilon}^{\mu}+(\hat{\epsilon})^{2}$
(96)

where $(T(\hat{u}),\mu(\hat{u}))$ are the local temperatures and local
chemical potentials measured by the $\hat{u}^{\mu}$-observer. Thus any frame
change as in (95), induces a particular field redefinition described in (96)
and this field redefinition can be seen as a particular case of the field
redefinition in (91-94). Indeed it can be generated by choosing
$\alpha_{i}=c_{i}=0,i\in(1,2,3)$, taking $b_{1}=b_{2}=b_{3}=b<<1$ in (92-94),
and identify $\hat{\epsilon}^{\mu}$ in (95) via

$\hat{\epsilon}^{\mu}:=b[{\dot{u}^{\mu}}+\Delta^{\mu\nu}\nabla_{\nu}T+\Delta^{\mu\nu}\nabla_{\mu}(\frac{\mu}{T})+O(\partial^{2})].$
(97)

Under the field redefinition in (96), the series expansion in (85 -90) remain
form invariant, since terms like $\nabla_{\mu}\hat{\epsilon}_{\nu}$ and
$\nabla_{\mu}\hat{\epsilon}^{\mu}$ are considered as second order in the
deviation from the state state of ”local thermodynamical equilibrium” and the
contributions of terms like $u^{\mu}\hat{\epsilon}^{\nu}\nabla_{\nu}T$,
$\hat{\epsilon}^{\mu}\hat{\epsilon}^{\nu}\nabla_{\nu}T$ etc are neglected in
comparison to terms $u^{\mu}u^{\nu}\nabla_{\nu}T$,
$u^{\mu}u^{\nu}\nabla_{\nu}T$ etc. The invariance of the series in (85-90)
implies that the dynamical equations satisfied by $(T,\mu,u^{\mu})$ remain
form invariant under the transformation of frame induced by (95) as long as
second order deviations from a ”local thermodynamical equilibrium state are
neglected. Thus within the (BNTK) formalism states compatible to the
relativistic (LTE) postulate are equivalent provided second order deviations
from from a ”local thermodynamical equilibrium” state are thrown away.
However ought to be stressed that the real power of the the (BDNK) formalism
lies in the freedom of frame change described in (91-94). This liberty, allows
to select particular hydrodynamical frames where the second order equations
$\nabla_{\mu}T^{\mu\nu}=\nabla_{\mu}J^{\mu}=0$ for the basic variables
$(u^{\mu},T,{\mu})$ (or an equivalent set) exhibit causality and stability of
equilibrium states (for details consult (Kov1 , Kov2 , Nor1 , Nor2 ). Thus
within the (BDNK) framework, states satisfying the relativistic (LTE)
postulate are becoming relevant.

##  V Transient thermodynamics and (LTE) states

In this section, we discuss states satisfying the relativistic (LTE) postulate
within the Israel-Stewart Isr1 ; Isr2  transient thermodynamics. We ought to
bear in mind however, that transient thermodynamics, by design deals with
states that are near equilibrium (or in our terminology with states satisfying
the relativistic (LTE) postulate313131 As we have already mentioned, the
definition of states compatible with the relativistic (LTE) postulate proposed
in this work, has been motivated by the properties of states satisfying the
equations of transient thermodynamics. Also we mention here that a state near
equilibrium it is not necessary a state satisfying the relativistic (LTE)
postulate. The four velocity that identifies the former does not necessarily
leads to a state where ”local thermodynamical equilibrium” prevails.). It is
worth remembering this restriction since as pointed in His4  for states with
large deviations from equilibrium the theory it is not always well behaved. In
that regard, is worth remembering some recent results derived in ref. BEM
where sufficient conditions, in the form of algebraic inequalities derived
that ensure that the theory is causal away from equilibrium and a theorem
established on the local existence and uniqueness of solutions of the theory
and these results raise the confidence on the theory.
With this comments in mind, and since the structure of the theory is discussed
in the original articles, by Israel Isr1  and Israel and Stewart Isr2  below,
we shall only provide a brief overview of the principles of transient
thermodynamics and shall offer a few comments regarding the equivalence of the
phenomenological equations as expressed relative to distinct frames.

We begin by considering a fluid state compatible with the relativistic (LTE)
postulate described by $(J^{\mu},T^{\mu\nu},S^{\mu})$ obeying (7, 8) with the
entropy flux $S^{\mu}$ to be defined shortly. We choose a four velocity
$u^{\mu}$ defined within the fluid region and lying within the familiar cone
of opening angle $\epsilon<<1$ and let $s(\rho,n)$ be the corresponding
equilibrium equation of state. To this $(u^{\mu},s(\rho,n)),$ we attach the
”local thermodynamical equilibrium” state
$(S^{\mu}_{0},T^{\mu\nu}_{0},J^{\mu}_{0})$ in the manner discussed in section
$(III)$ (see discussion leading to (16)). The velocity $u^{\mu}$ and the
expansions in (10,11), define the fields

$(n(u),\rho(u),P(u),h(u)^{\mu},n(u)^{\mu},\tau(u)^{\mu\nu})$ (98)

while the thermal potential $\Theta(u)$ and the local inverse temperature
$\beta(u)=T(u)^{-1}$ measured by the $u$-observer are defined by the
fundamental Gibbs relation (14). The velocity $u^{\mu}$ and the fields in (98)
satisfy the eqs of transient thermodynamics provided they obey:

$\nabla_{\mu}T^{\mu\nu}=\nabla_{\mu}[\rho(u)u^{\mu}u^{\nu}+P(u)\Delta(u)^{\mu\nu}+h(u)^{\mu}u^{\nu}+h(u)^{\nu}u^{\mu}+\tau(u)^{\mu\nu}]=0,$
(99)

$\nabla_{\mu}J^{\mu}=\nabla_{\mu}[n(u)u^{\mu}+n(u)^{\mu}]=0,$ (100)

and additional equations arising by imposing the second law. The entropy flux
$S^{\mu}$ is postulated to have the form (see Isr1 ; Isr2  a discussion
leading to this choice):

$S^{\mu}=P(u)\beta^{\mu}(u)-\Theta(u)J^{\mu}-\beta_{\nu}(u)T^{\mu\nu}-Q^{\mu}(\delta
J^{\alpha},\delta T^{\alpha\beta},X^{\alpha\beta\gamma..}_{(i)})$ (101)

where $\beta_{\lambda}=\beta(u)u_{\lambda}=\frac{u_{\lambda}}{T(u)}$ and
$Q^{\mu}$ is a vector field that depends quadratically upon the deviations
$(\delta J^{\alpha},\delta T^{\alpha\beta})$ from the state323232 The
variables $X_{(i)}^{\mu\nu\lambda...}$,$i\in(1,2,....)$ appearing in
$Q^{\mu}$, are additional variables needed to completely specify the non
equilibrium state. In this work, and in fact within transient thermodynamics,
are assumed to be zero and thus will be disregarded here after. of a ”local
thermodynamical equilibrium” specified by $(u^{\mu},s(\rho,n))$.
For the following analysis, it is convenient to transform the right hand side
of (101) in a form so that its relation to the entropy flux $S^{\mu}$ for
first order theories, becomes apparent. For this, starting from
$s(u)=\beta(u)\left(\rho(u)+{P(u)}\right)-\Theta(u)n(u),$ and the fundamental
Gibbs relation $ds=\beta d\rho-\Theta dn$, one gets their covariant versions

$S^{\mu}_{0}=P\beta^{\mu}-\Theta
J^{\mu}_{0}-\beta_{\lambda}T^{\lambda\mu}_{0},\quad dS^{\mu}_{0}=-\Theta
dJ^{\mu}_{0}-\beta_{\lambda}dT^{\lambda\mu}_{0},$ (102)

where for their derivations we used: $S^{\mu}_{0}=su^{\mu}$, $\rho
u^{\mu}=-u_{\lambda}T^{\lambda\mu}_{0}$ and
$(J^{\mu}_{0},T^{\mu\nu}_{0},S^{\mu}_{0})$ are the fields introduced in (16).
Eliminating the contribution of $P(u)\beta^{\mu}$ from the right hand side of
(101) one gets

$\begin{split}S^{\mu}&=S^{\mu}_{0}-\Theta(J^{\mu}-J^{\mu}_{0})-\beta_{\lambda}(T^{\lambda\mu}-T^{\lambda\mu}_{0})-Q^{\mu}\\\
&=S^{\mu}_{0}-\Theta\delta J^{\mu}-\beta_{\lambda}\delta
T^{\lambda\mu}-Q^{\mu}(\delta J^{\alpha},\delta T^{\alpha\beta})\\\
&=su^{\mu}-\Theta(u)n^{\mu}(u)+\beta(u)h^{\mu}(u)-Q^{\mu}(\delta
J^{\alpha},\delta T^{\alpha\beta}),\end{split}$ (103)

which shows that transient thermodynamics is an extension of the Hiscock-
Lindblom class first order theories by incorporating in $S^{\mu}$ second order
deviations from the state of ”local thermodynamical equilibrium” manifesting
themselves in the non vanishing of the $Q^{\mu}(\delta J^{\alpha},\delta
T^{\alpha\beta})$ term.

Motivated from the relativistic kinetic theory of diluted gases, Israel and
Stewart Isr1 ; Isr2  proposed that $S^{\mu}$ (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 thermodynamical
equilibrium”. Within the hydrodynamical approximation333333The name derives
itself from the assumption that $S^{\mu}$ is specified only by the
hydrodynamical variables $(J^{\mu},T^{\mu\nu})$., they proposed that for an
electrically neutral, simple fluid the term $Q^{\mu}(u)$ should have the form:

$Q^{\mu}(u)=\frac{1}{2}u^{\mu}[\beta_{0}\pi^{2}+\beta_{1}q^{\nu}q_{\nu}+\beta_{2}\pi^{\lambda\nu}\pi_{\lambda\nu}]-\alpha_{0}\pi
q^{\mu}-\alpha_{1}\pi^{\mu\nu}q_{\nu}+R^{\mu}(u),$ (104)

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

$R^{\mu}(u)=\frac{1}{T(\rho+P)}\left[\frac{1}{2}u^{\mu}h^{\nu}h_{\nu}+\tau^{\mu\nu}h_{\nu}\right],$
(105)

and in above $q^{\mu}$ stands for the invariant heat flux defined in (35). The
coefficients $\alpha_{j}$, $j\in(0,1)$, $\beta_{i}$, $i\in(0,1,2)$, are
undetermined depending upon $(\rho,n)$ and $(T,\rho,P,\pi,etc)$ are the local
thermodynamical variables measured by the $u^{\mu}$-observer (for simplicity
of the representation we suppressed their implicit dependance upon $u^{\mu}$).
As we mentioned, the dynamical equations for transient thermodynamics, are
those in (99,100) augmented by the phenomenological equations that follow by
imposing the second law on the entropy flux $S^{\mu}$ in (101) and
$Q^{\mu}(u)$ as in (104, 105). Their forms and the detail calculations leading
to their derivations can be found in Isr1 ; Isr2 . Here we shall offer a few
comments regarding their behavior under a frame change. For this, we consider
another velocity field $\hat{u}^{\mu}$ lying within the cone of the opening
pseudo-angle $\epsilon<<1$ and let the frame change

$u^{\mu}\to\hat{u}^{\mu}=u^{\mu}+\hat{\epsilon}^{\mu},\quad\hat{\epsilon}^{\mu}\leq\epsilon^{\mu}.$
(106)

For this frame change, we introduce the new fields
$n(\hat{u}),\rho(\hat{u}),P(\hat{u}),h(\hat{u})^{\mu},n(\hat{u})^{\mu},\tau(\hat{u})^{\mu\nu}$
defined according to

$n(\hat{u})=n(u)+\hat{\epsilon}O_{1},\quad\rho(\hat{u})=\rho(u)+\hat{\epsilon}O_{1},\quad
etc,$

and at first we show that these new fields satisfy to an $O_{1}$ accuracy the
conservation laws (99, 100). Indeed a repetition of the calculation that lead
us to (62,63) for the case of first order theories, yields to the same
conclusion. To an $O_{1}$ accuracy in the deviations from the state of ”local
thermodynamical equilibrium” the conservation laws (99, 100) remain valid.
However proving that the same property holds for the rest of the equations
that follow by imposing the second law on $S^{\mu}$ in (103) becomes more
subtle. One may for instance, start from the equations written say relative to
the $u^{\mu}$ frame and subsequently via (25-33), rewrite them relative to the
$\hat{u}^{\mu}$ frame keeping only terms describing first order deviations.
Unfortunately this procedure is long and time consuming, but there is an
alternative method that avoids this route. For this, we return to the entropy
flux $S^{\mu}$ in (103) and examine its variation under the frame change in
(106). Denoting by $\delta$ the resulting variation, and noting that $\delta
S^{\mu}=\delta J^{\mu}=\delta T^{\mu\nu}=0$, we find that $\delta S^{\mu}$ in
(103) satisfies:

$\begin{split}0=\delta S^{\mu}=u^{\mu}\delta s+s\delta
u^{\mu}-(J^{\mu}-J_{0}^{\mu})\delta\Theta-\Theta\delta(J^{\mu}-J_{0}^{\mu})\\\
-\delta\beta_{\lambda}(T^{\mu\lambda}-T_{0}^{\mu\lambda})-\beta_{\lambda}\delta(T^{\mu\lambda}-T_{0}^{\mu\lambda})-\delta
Q^{\mu}.\end{split}$ (107)

Due to the identity $s(u)=\beta(u)\left(\rho(u)+{P(u)}\right)-\Theta(u)n(u),$
one finds that $s\delta
u^{\mu}-\Theta\delta(J^{\mu}-J_{0}^{\mu})-\beta_{\lambda}\delta(T^{\mu\lambda}-T_{0}^{\mu\lambda})$
is vanishing to linear order in $\epsilon^{\mu}$ and thus (107) implies

$\delta
Q^{\mu}=-(J^{\mu}-J_{0}^{\mu})\delta\Theta-(T^{\mu\lambda}-T_{0}^{\mu\lambda})\delta\beta_{\lambda}+\hat{\epsilon}O_{1}+\hat{\epsilon}O_{2}.$
(108)

Taking into account that

$\begin{split}\delta\Theta&\equiv\Theta(\hat{u})-\Theta(u)=\epsilon O_{1},\\\
\delta\beta^{\mu}&=\frac{\epsilon^{\mu}}{T(u)}+u^{\mu}\delta\left(\frac{1}{T(u)}\right)=\frac{\epsilon^{\mu}}{T(u)}+\epsilon
O_{1},\\\\[10.0pt] \end{split}$ (109)

results

$\delta
Q^{\mu}=-(T^{\mu\lambda}-T_{0}^{\mu\lambda})\delta\beta_{\lambda}+\epsilon
O_{1}=\frac{1}{T(u)}\left[u^{\mu}h^{\lambda}+\tau^{\mu\lambda}\right]\epsilon_{\lambda}+\epsilon
O_{2}.\\\ $ (110)

On the other hand, working out the variation of the $Q^{\mu}(u)$ term arising
(104, 105) one finds that the majority of the terms yield an $\epsilon O_{2}$
contribution except the variation of the $R^{\mu}(u)$ terms. Working out
$\delta R^{\mu}(u)$ using formulas (25 \- 33), one finds an expression
identical to that in (110) which implies that $Q^{\mu}(u)$ has been chosen
consistently in (101) and (105) and this consistency in the choice of the
$Q^{\mu}(u)$ plays an important role in proving the frame invariance property
of the phenomenological equations. Under the frame change in (106), a
calculation shows that $S^{\mu}$ in (103) satisfies the identity:

$\begin{split}S^{\mu}&=su^{\mu}+\beta(u)h^{\mu}(u)-\Theta(u)n^{\mu}(u)-Q^{\mu}(u)\\\
&=\hat{s}\hat{u}^{\mu}+\beta(\hat{u})h^{\mu}(\hat{u})-\Theta(\hat{u})n^{\mu}(\hat{u})-(T^{\mu\lambda}-T_{0}^{\mu\lambda})\delta\beta_{\lambda}\\\
&-Q^{\mu}(\hat{u})-\delta
R^{\mu}+\hat{\epsilon}O_{2}+\hat{\epsilon}O_{1}.\end{split}$ (111)

However the term $-(T^{\mu\lambda}-T_{0}^{\mu\lambda})\delta
b_{\lambda}-\delta R^{\mu}$ cancel out leaving

$\begin{split}S^{\mu}&=su^{\mu}+\beta(u)h^{\mu}(u)-\Theta(u)n^{\mu}(u)-Q^{\mu}(u)\\\
&=\hat{s}\hat{u}^{\mu}+\beta(\hat{u})h^{\mu}(\hat{u})-\Theta(\hat{u})n^{\mu}(\hat{u})-Q^{\mu}(\hat{u})+\hat{\epsilon}O_{2}+\hat{\epsilon}O_{1},\end{split}$
(112)

which establishes the $O_{1}$ invariance of $S^{\mu}$ under frame change
described by (106). This property of $S^{\mu}$ implies that the resulting
phenomenological equations are equivalent under the frame changes described in
(106).

To get insights into the implications of this invariance property of $S^{\mu}$
shown in (112), let us evaluate the $Q^{\mu}(u)$ in the energy frame. Since
relative to this frame

$h^{\mu}(u_{E})=0,\quad q^{\mu}(u_{E})=-\dfrac{\rho+P}{n}n^{\mu}(u_{E})$

using (104, 105), one finds

$Q^{\mu}(u_{E})=\frac{1}{2}u^{\mu}_{E}\left[\beta_{0}\pi^{2}+\beta_{1}n^{\nu}_{E}{n_{E}}_{\nu}+\beta_{2}\pi^{\lambda\mu}\pi_{\lambda\mu}\right]-\alpha_{0}\pi
n^{\mu}_{E}-\alpha_{1}\pi^{\mu\nu}n_{E\nu}$ (113)

where we arrived in this expression by normalizing the $\beta_{1}$,
$\alpha_{0}$ and $\alpha_{1}$ coefficient by absorbing terms like
$(\rho+p)n^{-1}$ into their definition. If on the other hand, we evaluate the
same term relative to the particle frame where

$q^{\mu}=h^{\mu}_{N},\quad n^{\mu}_{N}=0$

and one introduce new coefficients $\bar{\alpha}_{j}$, $j\in(0,1)$,
$\bar{\beta}_{i}$, $i\in(0,1,2)$, one finds

$Q^{\mu}(u_{N})=\frac{u^{\mu}_{N}}{2}\left[\bar{\beta}_{0}\pi^{2}+\left(\bar{\beta}_{1}+\frac{1}{T(\rho+P)}\right)h^{\mu}h^{\nu}+\bar{\beta}_{2}\pi^{\lambda\nu}\pi_{\lambda\nu}\right]-$
(114) $-\left(\bar{\alpha}_{0}-\frac{1}{T(\rho+P)}\right)\pi
h^{\mu}_{N}-\left(\bar{\alpha}_{1}-\frac{1}{T(\rho+P)}\right)\pi^{\mu\nu}h_{\nu}$

A comparison between (113, 114) show identical structures provided one defines

$\beta_{0}=\bar{\beta}_{0},\quad\beta_{1}=\bar{\beta}_{1}+\frac{1}{T(\rho+P)},\quad\beta_{2}=\bar{\beta}_{2},\quad\alpha_{0}=\bar{\alpha}_{0}-\frac{1}{T(\rho+P)},\quad\alpha_{1}=\bar{\alpha}_{1}-\frac{1}{T(\rho+P)}$
(115)

and these relations have been also derived in Israel Isr1  and Israel and
Stewart Isr2  using different means that the ones employed in the present
work. If formally one introduce $q^{\mu}=n^{\mu}_{E}$ and
$q^{\mu}=h^{\mu}_{N}$ as fields variables in (113) respectively (114) and use
(112) to evaluate the phenomenological equations in view of (115), one obtains
identical forms provided terms of $\epsilon O_{1},O_{2}$ have been neglected,
showing the $O_{1}$ equivalence of the phenomenological equations relative to
the energy or particle frame. For arbitrary frame changes generated by
$(u^{\mu},\hat{u}^{\mu})$ in (106), the equivalence of the phenomenological
equations can also established but the situation is more complex. For this
case, the number of thermodynamical variables increase and either one can
treat $(q^{\mu}(u),h^{\mu}(u))$ as independent variables or treat $h^{\mu}(u)$
and the particle drift $n^{\mu}(u)$ as independent variables. In either case
(112) shows their equivalence.

We shall leave this section by offering a few comments regarding the causality
and stability properties of the Israel-Stewart transient thermodynamics.
The classical works by Hiscock and Lindblom in His1 ,His3  and Olson in Olson
, established that linear perturbations about a spatially homogeneous
equilibrium state343434The identification of equilibrium states for the
Israel-Stewrt theory have been worked out in refs. His1 , Olson  (see also
Isr1 , Isr2 ). are bounded and propagate causally as long as suitable
restrictions upon the parameters and the equations of state are imposed. This
conclusion is in sharp contrast to the causality and stability behavior of the
Hiscock-Lindblom class of first order theories as we have already discussed in
sec. (IV). Gavasinno, Antonelli and Haskell in GAV6  convincingly argue that
this difference should be traced in thermodynamics and it is due to the
different structure that the entropy current $S^{\mu}$ has in the two theories
(compare $S^{\mu}$ in (41) to that of Israel-Stewart theory shown in (103). It
was shown in GAV6  that the total entropy based on (41) has no upper bound,
while the inclusion of the quadratic contributions in (103) for the Israel-
Stewart theory implies that the total entropy has an absolute maximum
presumably representing the equilibrium state. Finally in a recent work,
Gavassino, Antonelli and Haskell in GAV1  showed for the Israel-Stewart
transient thermodynamics, perturbations around global equilibrium states on
Minkowski spacetime relative to the Eckart (particle) frame and perturbations
relative to Landau-Lifshitz (energy) satisfy equivalent equations i.e. they
are equivalent theories implying that the stability-causality conditions found
by Olson in Olson  (in the Landau frame) are just a rewrite of those found by
Hiscock and Lindblom in His1 ,His3  (in the Eckart frame). This is to be
contrasted to what occurs for the case of perturbations induced on an
equilibrium state induced by an Eckart theory or a Landau-Lifshitz theory
within the context of the Hiscock-Lindblom class of first order
theories353535Our thanks to an anonymous referee who suggested to probe this
interconnection between the two classes of theories. that we discussed in
section $(IV)$. It would be of interest to examine whether this difference is
traced in hydrodynamics or in thermodynamics and this issue is for the moment
open. Finally, we mention here the recent results obtained in BEM  regarding
causality, local existence, and uniqueness of solutions of the theory within
the full nonlinear regime. Overall the theory stands on a very firm ground.

##  VI On states satisfying the (LTE) postulate and the Liu-Müller-Ruggeri
theory

We finish this paper by turning attention to fluid states compatible with the
relativistic (LTE) postulate and their relation to Liu-Müller-Ruggeri theory
(for an introduction to this theory see Mul6 ; Mul4 ) and the theory of
relativistic fluids of divergence type developed by Pennisi Pen , and
independently by Geroch and Lindblom in Ger1 . Of these two closely related
theories, we shall consider only the relation between states compatible with
the relativistic (LTE) postulate and the Liu-Müller-Ruggeri theory since their
relation to the theory of fluids of divergence type requires to introduce the
generating function and the field of Lagrange multipliers and this needs a
more elaborate treatment that will be presented elsewhere FT2 .

We recall that within the Liu-Müller-Ruggeri theory Mul6 ; Mul4 , arbitrary
fluid states are described by the $10$ components of the symmetric energy
momentum tensor $T^{\mu\nu}$ and the $4$ components of the particle current
$J^{\mu}$ satisfying363636The structure of eqs.(116,117), 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. For an introduction to this theory see refs. Ehl ; An1
; O3 ; O4 .:

$\nabla_{\mu}T^{\mu\nu}=\nabla_{\mu}J^{\mu}=0,$ (116)
$\nabla_{\mu}A^{\mu\nu\lambda}=I^{\nu\lambda},$ (117)

where $A^{\mu\nu\lambda}$ is a completely symmetric tensor field subject to
$A^{\mu\nu}{}_{\nu}=m^{2}c^{2}J^{\mu}$ and thus necessarily
$\nabla_{\mu}A^{\mu\nu}{}_{\nu}=0$, while $I^{\mu\nu}$ is symmetric and
traceless $I^{\mu}{}_{\mu}=0$. The fields $A^{\mu\nu\lambda}$ and $I^{\mu\nu}$
are considered to be constitutive relations i.e.
$A^{\mu\nu\lambda}=A^{\mu\nu\lambda}(J^{\mu},T^{\mu\nu})$,
$I^{\mu\nu}=I^{\mu\nu}(J^{\mu},T^{\mu\nu})$ and this is a special feature of
the theory. In addition to the balance laws (116, 117), the theory employs an
entropy flux vector $S^{\mu}$ which is also a constitutive function i.e.
$S^{\mu}=S^{\mu}(J^{\mu},T^{\mu\nu})$ and for any solution
$(J^{\mu},T^{\mu\nu})$ of (116, 117) obeys:

$\nabla_{\mu}S^{\mu}(J^{\mu},T^{\mu\nu})\geq 0.$ (118)

A major issue in this theory is the specification 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
$S^{\mu}=S^{\mu}(J^{\mu},T^{\mu\nu})$ as well as the implementation of the
entropy inequality in (118). Liu, Müller and Ruggeri in ref.Mul6  (see also
ref.Mul4 ) were able to construct representations of
$A^{\mu\nu\lambda}(J^{\mu},T^{\mu\nu})$,
$I^{\mu\nu}=I^{\mu\nu}(J^{\mu},T^{\mu\nu})$ and
$S^{\mu}=S^{\mu}(J^{\mu},T^{\mu\nu})$ by invoking three principles: the
Entropy Principle, the Principle of Relativity, and the requirement of the
Hyperbolic nature of the dynamical equations. Subsequently, by employing the
Eckart frame373737The analysis in refs.Mul6 , Mul4  has been performed
relative to the Eckart frame. In principle however, one could have chosen an
arbitrary velocity field $u^{\mu}$ and decompose $J^{\mu}$ and $T^{\mu\nu}$
relative to this new $u^{\mu}$ and carry out the analysis relative to this new
frame. We are not aware of any related work on this issue. they worked out
predictions of their theory and compared its predictions of those of transient
theory (for details of this comparison see refs.Mul6 ; Mul4 ).

The purpose in this section is to discuss states compatible with the
relativistic (LTE) postulate within this theory.
For this, let a state satisfying the (LTE) postulate i.e. it is described by
$(J^{\mu},T^{\mu\nu})$ obeying (116-117) and subject to the restriction that
the pseudo-angle $\epsilon$ between $(u^{\mu}_{E},u^{\mu}_{N})$ satisfies
everywhere within the fluid region the condition $\epsilon<<1$. Let $u^{\mu}$
be a velocity field chosen in the region occupied by the fluid and lying
within the cone formed by this $\epsilon$, so that local thermodynamical
equilibrium prevails. If $s(n,\rho)$ denotes the corresponding equilibrium
equation of state, then $(u^{\mu},s(\rho,n))$ introduce the ”local
thermodynamical equilibrium state” specified by
$(S^{\mu}_{0},T^{\mu\nu}_{0},J^{\mu}_{0})$ (see discussion leading to (16)).
This $u^{\mu}$ and the decompositions in (10,11) define the fields

$(n(u),\rho(u),P(u),h(u)^{\mu},n(u)^{\mu},\tau(u)^{\mu\nu})$ (119)

and these fields describe a state within the Liu-Müller-Ruggeri theory,
provided they satisfy (116, 117). However, in order to impose these equations,
we need a representation of $A^{\mu\nu\lambda}(J^{\mu},T^{\mu\nu})$,
$I^{\mu\nu}=I^{\mu\nu}(J^{\mu},T^{\mu\nu})$ and their specification is a
delicate problem. Since in this work, we are interested only in the behavior
of small deviations away from the local ”thermodynamical equilibrium state”
specified by $(u^{\mu},s(\rho,n))$ (or equivalently by
$(S^{\mu}_{0},T^{\mu\nu}_{0},J^{\mu}_{0})$), we set

$T^{\mu\nu}=T^{\mu\nu}_{0}+\delta T^{\mu\nu},\quad J^{\mu}=J^{\mu}_{0}+\delta
J^{\mu}.$ (120)

and take advantage of the constitutive nature of
$A^{\mu\nu\lambda}(J^{\mu},T^{\mu\nu})$,
$I^{\mu\nu}=I^{\mu\nu}(J^{\mu},T^{\mu\nu}).$ This later property allows us to
write:

$\begin{split}A^{\mu\nu\lambda}(T^{\mu\nu},J^{\mu})&=A^{\mu\nu\lambda}(T^{\mu\nu}_{0}+\delta
T^{\mu\nu},J^{\mu}_{0}+\delta J^{\mu})\\\
&=A^{\mu\nu\lambda}(T^{\mu\nu}_{0},J^{\mu}_{0})+\frac{\delta
A^{\mu\nu\lambda}}{\delta T{}^{\alpha\beta}}\delta T^{\alpha\beta}\\\
&+\frac{\delta A^{\mu\nu\lambda}}{\delta J{}^{\alpha}}\delta
J^{\alpha}+O(\delta J^{\mu})^{2}+O(\delta T^{\mu\nu})^{2},\end{split}$ (121)
$\begin{split}I^{\nu\lambda}(T^{\mu\nu},J^{\mu})&=I^{\nu\lambda}(T^{\mu\nu}_{0}+\delta
T^{\mu\nu},J^{\mu}_{0}+\delta J^{\mu})\\\
&=I^{\nu\lambda}(T^{\mu\nu}_{0},J^{\mu}_{0})+\frac{\delta
I^{\nu\lambda}}{\delta T{}^{\alpha\beta}}\delta T^{\alpha\beta}\\\
&+\frac{\delta I^{\nu\lambda}}{\delta J{}^{\alpha}}\delta J^{\alpha}+O(\delta
J^{\mu})^{2}+O(\delta T^{\mu\nu})^{2},\end{split}$ (122)

where in these expansions we kept terms describing only first order deviations
from the state of the local equilibrium field.
The symmetries of $A^{\mu\nu\lambda}$ and the condition
$A^{\mu\nu}{}_{\nu}=m^{2}c^{2}J^{\mu}$ allows us to propose a representation
of $A^{\mu\nu\lambda}(T^{\mu\nu}_{0},J^{\mu}_{0}):=A^{\mu\nu\lambda}_{0}(u)$
involving only terms that appear in the representation of the local
equilibrium field $(S^{\mu}_{0},T^{\mu\nu}_{0},J^{\mu}_{0})$. These symmetries
and the trace condition dictate that $A^{\mu\nu\lambda}_{0}(u)$ should be of
the form383838For this section indices enclosed in a parenthesis indicate
symmetrization, thus
$u^{(\mu}u^{\nu}u^{\lambda)}=\frac{1}{6}(u^{\mu}u^{\nu}u^{\lambda}+u^{\nu}u^{\lambda}u^{\mu}+u^{\lambda}u^{\mu}u^{\nu})$.

$A^{\mu\nu\lambda}_{0}(u)=2{\gamma_{1}}u^{(\mu}u^{\nu}u^{\lambda)}+(nm^{2}+\gamma_{1})g^{(\nu\lambda}u^{\mu)},$
(123)

where $\gamma_{1}$ is a function of $(n,\rho)$ to be determined and $m$ is a
mass scale393939The right hand-side of (123) as well as of (124), are indeed
functions of $T^{\mu\nu}$ and $J^{\mu}$. One can see that by expressing the
fields in (119) in terms of $T^{\mu\nu}$ and $J^{\mu}$.. Similarly the
symmetry requirements upon $I^{\mu\nu}$ permit us to propose that the leading
contributions of the $I^{\nu\lambda}(T^{\mu\nu},J^{\mu})$ to be vanishing i.e.
$I^{\nu\lambda}(T^{\mu\nu}_{0},J^{\mu}_{0}):=0$ and thus to set the expansion
of $I^{\nu\lambda}$ in (124) to have the form

$\begin{split}I^{\nu\lambda}(T^{\mu\nu},J^{\mu})&=\frac{\delta
I^{\nu\lambda}}{\delta T{}^{\alpha\beta}}\delta T^{\alpha\beta}+\frac{\delta
I^{\nu\lambda}}{\delta J{}^{\alpha}}\delta J^{\alpha}\\\
&=\delta_{1}\pi(g^{\mu\nu}+4u^{\mu}u^{\nu})+\delta_{2}\pi^{\mu\nu}\\\
&+\delta_{3}(q^{\mu}u^{\nu}+q^{\nu}u^{\mu}),\end{split}$ (124)

where $q^{\mu}$ is the invariant heat flux defined in (35) and the
coefficients $\delta_{i},i\in(1,2,3)$ are unknown functions of $(n,\rho)$.
With these choices, the fields in (119) satisfy

$\begin{split}\nabla_{\mu}T^{\mu\nu}&=\nabla_{\mu}[\rho(u)u^{\mu}u^{\nu}+P(u)\Delta(u)^{\mu\nu}+h(u)^{\mu}u^{\nu}\\\
&+h(u)^{\nu}u^{\mu}+\tau(u)^{\mu\nu}]=0,\end{split}$ (125)
$\nabla_{\mu}J^{\mu}=\nabla_{\mu}[n(u)u^{\mu}+n(u)^{\mu}]=0$ (126)
$\begin{split}\nabla_{\mu}A^{\mu\nu\lambda}_{0}(u)&=\delta_{1}\pi(g^{\mu\nu}+4u^{\mu}u^{\nu})+\delta_{2}\pi^{\mu\nu}\\\
&+\delta_{3}(q^{\mu}u^{\nu}+q^{\nu}u^{\mu}).\end{split}$ (127)

We leave aside for the moment the implementation of the entropy law in (118)
since as we have already mentioned, its implementation requires to introduce
the generating function, the field of Lagrange multipliers and apply of Liu’s
procedure Liu  for the implementation of of the second law. Since these issue
need altogether a different machinery they will be discussed in a separate
account in connection to fluids of divergence type FT2 . For the rest of this
section we shall treat (125-127) as the equations that fluid states compatible
to the relativistic (LTE) postulate are required to satisfy.

Let us suppose now that the system (125-127) admits a solution
$(u^{\mu},n(u),\rho(u),P(u),h(u)^{\mu},n(u)^{\mu},\tau(u)^{\mu\nu})$ and let
us consider a frame change described by

$u^{\mu}\to\hat{u}^{\mu}=u^{\mu}+\hat{\epsilon}^{\mu},\quad\hat{\epsilon}\leq\epsilon,$
(128)

where $\hat{u}^{\mu}$ has been chosen to lie within the cone of the opening
pseudo-angle $\epsilon<<1$ and relative to this new frame, we define fields
$(\rho(\hat{u}),n(\hat{u}),P(\hat{u}),h^{\mu}(\hat{u}),\tau^{\mu\nu}(\hat{u}),n^{\mu}(\hat{u}))$
by applying the same procedure as for the case of first order theory and the
case of transient theory. It is easily seen that the transformed fields
$(\hat{u}^{\mu},\rho(\hat{u}),n(\hat{u}),P(\hat{u}),h^{\mu}(\hat{u}),\tau^{\mu\nu}(\hat{u}),n^{\mu}(\hat{u}))$
satisfy to an $O_{1}$ accuracy the conservation laws in (116). However matters
complicate in proving that the transformed fields satisfy to an $O_{1}$
accuracy equation (125-127). To facilitate matters, we note that (127) imply

$u_{\nu}u_{\lambda}\nabla_{\mu}A^{\mu\nu\lambda}_{0}(u)=3\delta_{1}\pi(u),$
(129)
$\Delta_{\rho\nu}(u)u_{\lambda}\nabla_{\mu}A^{\mu\nu\lambda}_{0}(u)=\delta_{3}q_{\rho}(u),$
(130)
$\left(\Delta_{\alpha\nu}(u)\Delta_{\beta\lambda}(u)-\frac{1}{3}\Delta_{\alpha\beta}(u)\Delta_{\nu\lambda}(u)\right)\nabla_{\mu}A^{\mu\nu\lambda}_{0}(u)=\delta_{2}\pi_{\alpha\beta}(u),$
(131)

and the idea is to show that these equations remain form $O_{1}$ invariant
under a change of the rest frame described by (127). Based on the
representation of $A^{\mu\nu\lambda}_{0}(u)$ in (123), then a calculus shows
that that under the frame change in (128) the following formula holds:

$\nabla_{\mu}A^{\mu\nu\lambda}_{0}(u)=\nabla_{\mu}A^{\mu\nu\lambda}_{0}(\hat{u})-\nabla_{\mu}[2\gamma_{1}\hat{\epsilon}^{(\mu}\hat{u}^{\nu}\hat{u}^{\lambda)}+(nm^{2}+\gamma_{1})g^{(\nu\lambda}\hat{\epsilon}^{\mu)}+\hat{\epsilon}O_{1}].$
(132)

Using this formula, it is straightforward to verify that (129, 130, 131)
remains form invariant to $O_{1}$ deviation from the state of local
equilibrium taking into account that $\nabla_{\mu}u^{\mu}=O(1)$,
$\nabla_{\mu}\epsilon^{\mu}=O(1)$ and always $u^{\mu}\epsilon_{\mu}=0$.
In summary within the class of the Liu-Müller-Ruggeri theory, states
satisfying the relativistic (LTE) postulate satisfy eqs that remain invariant
under a frame change described in (51) and thus exhibit the same behavior as
states near equilibrium for the first order theories and the transient
thermodynamics. We shall discuss further properties of states satisfying the
relativistic (LTE) postulate in light of some recent advances on the theory of
divergence type fluids reported lately in refs. GAV1  (see also GAV7 ) .

##  VII Discussion

In this article, we have introduced a special class of fluid states satisfying
(or been compatible to) the relativistic (LTE) postulate. Our motivation to
introduce this class has been partially induced by the role of the (LTE)
postulate within the Onsager-Eckart theory and largely by Israel’s ideas on
the description of relativistic fluid states that are considered to be near
equilibrium within the transient thermodynamics.
We started the article by pointing out the implications of the (LTE) postulate
within the Onsager-Eckart theory and stressed that states compatible with the
(LTE) postulate are special. Firstly they permit to introduce local
thermodynamical variables and secondly their physical entropy is determined by
these local variables404040Because of these properties, often in the
literature and even within the context of relativistic fluids, one encounter
the term ”theories based on the local equilibrium hypothesis” . With this term
one understands theories whose independent variables are those that determine
equilibrium states. Clearly this definition is not wide enough, for instance
transient thermodynamics, incorporates fluxes as independent variables and
these variables do not enter in the description of equilibrium states.. The
extension of the classical (LTE) postulate to the relativistic regime proposed
in this paper, define fluid states compatible with the relativistic (LTE)
postulate as states subject to two restrictions.
The first one requires that the state should allow the attachment of a ”local
thermodynamical equilibrium” state and this attachment requires that the four
velocity $u^{\mu}$ ought to be chosen so that this $u$-observer, relative to
her/his local rest frame, detects a collision time scale which is much shorter
than any other macroscopically defined time scale. Under this condition, an
”equilibrium equation of state” of the form $s=s(\rho,n)$ may be postulated to
exist and thus ”thermal” aspects such as local temperature and local chemical
or thermal potential appear414141Notice, that even though any observer with an
arbitrary four velocity $u^{\mu}$, using the state variables
$(J^{\mu},T^{\mu\nu})$ can always define along his/her world line particle
density $n$ energy density $\rho$ energy flux $h^{\mu}$ y stresses
$\tau^{\mu\nu}$, still ”thermal” aspects do not appear unless an ”equilibrium
equation of state” $s=s(\rho,n)$ is postulated to exist.. The second
restriction requires the state to be a state near equilibrium in Israel’s
sense, meaning that everywhere within the region occupied by the fluid, the
”angle $\epsilon$” formed by the four velocity that defines the energy frame
and the four velocity that defines the particle frame, satisfies
$\epsilon<<1$. Whenever any $u^{\mu}$ lies within the cone of opening angle
$\epsilon$, then the appearance of the class of admissible frames and the
treatment of deviations of the physical state, from the state of ”local
thermodynamical equilibrium” follows the root that we presented in sections
$(III,VI)$.
Parenthetically we add, that in order that a fluid state be compatible with
the relativistic (LTE) postulate, it is necessary that it satisfies
simultaneously both of the above mentioned restrictions. One can imagine fluid
states where either the attachment of a ”local thermodynamical equilibrium” it
is not possible or the state it is not a state near equilibrium in Israel’s
sense. In other case, one does not get the benefits arising from the existence
of admissible frames.

Although in this paper, we examined properties of states satisfying the
relativistic (LTE) postulate within the context of the Hiscock-Lindblom class
of first order theories, transient thermodynamics, the (BDNK) theory, and the
Liu-Müller-Ruggeri theory, it would be of interest to include in this list,
the class of relativistic fluids of divergence type in view on some recent
advances in this theory reported by Gavassino, Antonelli and Haskell in refs
GAV1 , GAV7 .
Overall the analysis in this paper shows that that within the Hiscock-Lindblom
class of first order theories, states satisfying the relativistic
(LTE)-postulate they do not seem to offer any advantage. Like arbitrary states
within this theory, they violate causality and equilibrium states are
unstable. However, maters are different within the (BDNK) theory and transient
thermodynamics. As far as the latter theory is concerned, we have argued in
section $V$, that the theory deals exclusively with states obeying the
relativistic (LTE)-postulate and in reaching this conclusion, refs His4 , GAV7
, BEM  are very relevant. As far as the former theory is concerned, in section
$(IV)$, we presented evidences suggesting that states obeying the relativistic
(LTE)-postulate are states referred as states near equilibrium within the
(BDNK) formalism. In fact we believe, that all states within the (BDNK) theory
where the gradient series expansion is meaningful, are in fact states
compatible to the relativistic (LTE)-postulate, although at this point this is
a conjectural claim. In that regard, it would be of interest, starting from
the relativistic Boltzmann equation and via suitable expansion investigate
whether one may generate fluid states where the pseudo-angle $\epsilon$ formed
by $u_{E}^{\mu}$ and $u_{N}^{\mu}$ satisfies everywhere the
condition$\epsilon<<1$. So far the relativistic Boltzmann equation has offered
considerable insights in the derivation of relativistic hydrodynamic equation
(see for instance Isr2 , Nor1 , Den  and references therein) and may be
further efforts one gets insights into the nature of the geometry of possible
rest frames may be helpful. Currently we have that issue under active
investigation.
We finish this paper by mentioning that in the literature there have been
advanced alternative definitions of states obeying the relativistic (LTE)
postulate and these definitions either originate from the quantum statistical
framework (see for instance Bec  and reference therein) It would be of some
interest to compare these definitions and their predictions to the one
proposed in this work.

##  VIII Acknowledgments

Our warm thanks to the members of the relativity group at the IFM Univ.
Michoacana for multiple discussions. Special thanks to O. Sarbach whose
continuing poking lead to the development of this work. The research of T.Z
was supported by a Grant from CIC-UMSNH. During the completion of this work
J.F.S was funded by CONACYT by a post doctoral fellowship through the project
A1 -S-31269.

## References

  * (1) L. Onsager Phys.Rev. 37, 405, (1931)

  * (2) L.Onsager Phys.Rev. 38, 2265, (1931)

  * (3) C.Eckart Phys.Rev. 58, 267, (1940)

  * (4) C.Eckart Phys.Rev. 58, 269, (1940)

  * (5) L. Landau and E. Lifshitz, Fluid Mechanics, (Adison Wesley Pergamon, 1959)

  * (6) K Huang Statistical Physics, Second Edition, J.Wiley and Sons, New York (1987)

  * (7) S. de Groot and P. Mazur, Non-equilibrium Thermodynamics, Dover Books on Physics (Dover Publications, 1984)

  * (8) I. Müller, Zeitschrift fur Physik 198, 329, (1967)

  * (9) B D Coleman and W Noll, Arch. Ration. Mech. Anal, 13, 167, (1963)

  * (10) B D Coleman, M E Gurtin, M.E. J. Chem. Phys, 47, 597, (1967)

  * (11) I Müller I Arch. Rational Mech. Anal. 40, 1, (1971)

  * (12) I Müller I Arch. Rational Mech. Anal. 41, 319, (1971)

  * (13) I Müller and T Ruggeri Rational Extended Thermodynamics, 2th edition, Springer, New York (1998)

  * (14) Jou D, Casas-Vázquez J and Lebon G, Extended Irreversible Thermodynamics, 4th edition, Springer (2010)

  * (15) Rezzolla L and Zanotti O, Relativistic Hydrodynamics (Oxford University Press, New York, 2013)

  * (16) Romatschke P and Romatschke U, Relativistic Fluid Dynamics In and Out of Equilibrium, Cambridge Monographs on Mathematical Physics, CUP, 2019

  * (17) L Gavassino Can we make sense of dissipation without causality?, arXiv: 2111.05254 [gr-qc]

  * (18) L Gavassino M. Antonelli and B. Haskell Thermodynamic stability implies causality , arXiv: 2105.14621 [gr-qc]

  * (19) L. Gavassino, M. Antonelli and B. Haskell  Rev. D 102, 043018 (2020)

  * (20) N.G. van Kampen, Phys. Rev 173, 295 (1968)

  * (21) W Israel Ann. Phys., NY, 100 , 310, (1976)

  * (22) W Israel W and J M Stewart, Ann. Phys., NY ,118 , 341, (1979)