# Relativistic transformation of thermodynamic parameters and refined Saha
equation

Xin Hao1, Shaofan Liu2 and Liu Zhao2
1College of Physics, Hebei Normal University, Shijiazhuang 050024, China
2School of Physics, Nankai University, Tianjin 300071, China
email<EMAIL_ADDRESS><EMAIL_ADDRESS>
and<EMAIL_ADDRESS>Correspondence author.

###### Abstract

The relativistic transformation rule for temperature is a subject under debate
for more than 110 years. Several incompatible proposals exist in the
literature, but a final resolution is still missing. In this work, we
reconsider the problem of relativistic transformation rules for a number of
thermodynamic parameters, including temperature, chemical potential, pressure,
entropy and enthalpy densities for a relativistic perfect fluid using
relativistic kinetic theory. The analysis is carried out in a fully
relativistic covariant manner, and the explicit transformation rules for the
above quantities are obtained in both Minkowski and Rindler spacetimes. Our
results suggest that the temperature of a moving fluid appears to be colder,
supporting the proposal by de Broglie, Einstein and Planck in contrast to
other proposals. Moreover, in the case of Rindler fluid, our work indicates
that, the total number of particles and the total entropy of a perfect fluid
in a box whose bottom is parallel to the Rindler horizon are proportional to
the area of the bottom, but are independent of the height of the box, provided
the bottom of the box is sufficiently close to the Rindler horizon. The area
dependence of the particle number implies that the particles tend to be
gathered toward the bottom of the box and hence implicitly determines the
distribution of chemical potential of the system, whereas the area dependence
of the entropy indicates that the entropy is still additive and may find some
applications in explaining the area law of black hole entropy. As a by
product, we also obtain a relativistically refined version of the famous Saha
equation which holds in both Minkowski and Rindler spacetimes.

##  1 Introduction

Of all branches of modern physics, classical thermodynamics and relativity are
outstanding in the sense that they describe the universal rules that every
physical system must obey, irrespective of the detailed matter contents of the
system. There are only two requirements for classical thermodynamics to hold:
i) the physical system needs to be macroscopic, i.e. containing a large number
of microscopic degrees of freedom, ii) the system needs to be at thermodynamic
equilibrium with uniform temperature and pressure. In contrast, there seems to
be no requirement for any physical system to obey the principles of
relativity, although the special relativistic effects can only become manifest
when the system undergoes very fast motion in comparison to the speed of
light, and the general relativistic effects can only become manifest when the
system contains a huge amount of mass and/or energy.

It has long been fascinating to consider situations when both the principles
of classical thermodynamics and relativity apply. Such situations involve
macroscopic system which either undergoes relativistic motion or moves in
curved spacetime. The endeavors in combining thermodynamics and relativity
have lasted for over 110 years. However, the outcome is quite controversial.
Even without considering the general relativistic effects, the combination of
classical thermodynamics and special relativity has led to several
contradictory results on the transformation rule for temperature. Basically,
there are four major views on such transformations, each labeled by the names
of the corresponding researchers below (wherein $\gamma>1$ is the Lorentz
factor):

(i) de Broglie [1], Einstein [2] and Planck [3]: moving bodies appear cooler,
$T^{\prime}=\gamma^{-1}T$;

(ii) Eddington [4], Ott [5] and Arzelies [6]: moving bodies appear hotter,
$T^{\prime}=\gamma T$;

(iii) Landsberg [7, 8]: temperature is a relativistic invariant,
$T^{\prime}=T$;

(iv) Cavalleri, Salgarelli [9] and Newburgh [10]: no unique such
transformation because thermodynamics is defined only in rest frame.

Quite notably, Einstein seems to have supported each of the four views in his
life [11], and Lansberg turned to the fourth view in his later career [12,
13]. The debates between all these different views remain open [14] and a huge
number of papers have been published on the same or related subjects. It is
remarkable that the standing point of the fourth view by Cavalleri et al lies
in that a system at thermodynamic equilibrium must be static [9, 10] and hence
excludes the existence of macroscopic flow which is inherently implied by
global relativistic motion of the system, and that a moving observer in a heat
reservoir cannot detect a blackbody spectrum [12, 13], which implies the
nonexistence of a uniform temperature. Such reasonings, however, should not be
taken to be sufficient justifications for the fourth view, because there are
situations beyond thermodynamic equilibrium when one could talk about the
temperature, pressure and entropy _etc._ , at least locally, of a given
macroscopic system, for instance for systems which are not in global but in
local equilibrium, or for systems under detailed balance. For such systems,
the classical equilibrium thermodynamics does not apply, however, a
description using kinetic theory still works well.

Among the existing papers on related subjects (not necessarily considering the
temperature transformations), some considerations from the point of view of
statistical mechanics or kinetic theory have been introduced. Some authors
[15, 16, 17, 18] started right from equilibrium statistical mechanics or Gibbs
distributions and the formulations were often not presented in explicitly
relativistic covariant fashion, hence not best suited for analyzing the
transformation rules for macroscopic parameters. Some other works either dealt
with the debates about the correct relativistic distribution function [19, 20]
or introduced some modifications to the distribution function [21].

In order to solve the puzzles on the relativistic transformation rules for
macroscopic parameters, the necessary statistical mechanics tool needs to be
relativistic covariant and applicable to systems out of thermodynamic
equilibrium. Such a theory exists and is known as the relativistic kinetic
theory. It was established almost right after the first view by de Broglie,
Einstein and Planck was proposed [22]. Therefore, it is tempting to reconsider
the relativistic transformations for macroscopic parameters like temperature
and pressure from the point of view of relativistic kinetic theory. As far as
we know, a fully covariant treatment for this problem using relativistic
kinetic theory has not been reported before in literature, so we decide to
work it out by ourselves.

Before dwelling into the detailed analysis, it is worth pointing out that the
aforementioned debates stem largely from the way that the question is raised.
All previous works on this subject prescribe the question as follows: Assuming
the temperature of a system in (local) thermodynamic equilibrium is $T$ in the
rest frame. What is its temperature $T^{\prime}$ in a frame in which the
system undergoes fast motion?  An alternative prescription of the question
which does not rely on the choice of coordinate frames can be given as
follows. Assuming the system is at (local) thermodynamic equilibrium at
temperature $T$ with respect to the comoving observer. What is its temperature
$T^{\prime}$ with respect to a non-comoving observer? The two prescriptions
differ from each other in the reason why such changes happen. The first
prescription attributes the change of temperature to the change of coordinate
frames, while the second prescription attributes the change to the change of
observers. Even so, both prescriptions quest the change of the temperature of
the system at same macrostate, and the change of temperature in both
prescriptions arises purely from kinematic effects. Therefore, both
prescriptions can be dubbed as the kinematic version of the question. There is
a dynamic version of the question which quests for the temperature of the
system which is initially at rest and then pushed into fast motion. This
version breaks the initial macrostate and will not be discussed here.

In this work, we will take the second kinematic prescription as the starting
point. The reason to take the second rather than the first prescription is due
to the following considerations. First of all, most thermodynamic parameters
have phenomenological interpretations and their values are naturally observer
dependent. The first kinematic prescription does not provide information about
such dependences. Second, we hope to understand the relativistic
transformation rules for thermodynamic parameters in more generic spacetimes
rather than just in Minkowski spacetime. Therefore, the coordinate changes do
not necessarily belong to the set of Lorentz transformations. Last but not
least, we will show that most of the thermodynamic parameters (or densities
thereof) can be defined as scalars with respect to the coordinate
transformations, whereas their transformation rules under change of observers
are still nontrivial. This last reasoning indicates that the first
prescription is actually ill-posed.

As will be shown in the main context, our analysis indicates that the
transformation rule of temperature agrees with the view of de Broglie,
Einstein and Planck, but with the addition of the transformation rules for a
number of other thermodynamic parameters, notably including the chemical
potential $\mu$, the particle number density $n$ and the enthalpy density $w$.
Our analysis indicates that the transformation rules of those parameters are
identical in both Minkowski and Rindler spacetimes, and we expect that the
same rules should also be valid in other backgrounds as well due to the fully
relativistic covariant formalism. In the case of Rindler background, we shall
also show that the total number of particles and the total entropy of a
perfect fluid system in a box are proportional to the area of the bottom of
the box which is parallel to the Rindler horizon, but are independent of the
height of the box, provided the bottom of the box is sufficiently close to the
Rindler horizon. Moreover, since the chemical potential is explicitly
calculated in our considerations, it is straightforward to obtain a
relativistically refined version of the famous Saha equation [23] which
characterizes the local chemical equilibrium in the ultra relativistic regime.

##  2 Elements of relativistic kinetic theory

Our main tool is relativistic kinetic theory based on a covariant
generalization of Boltzmann equation. This theory is a subfield of non-
equilibrium statistical physics, and the application of this theory in our
analysis implies some microscopic considerations are involved. Different from
the Gibbs method for equilibrium system, in kinetic theory, the distribution
function is taken to be the one particle distribution function (1PDF) $f(x,p)$
which is defined to be the local particle number density in the one particle
phase space. In relativistic settings, one often enlarges the one particle
phase space to the tangent bundle of the full spacetime111For the sake of
generality, we do not require the spacetime to be flat, thus the formulation
of relativistic kinetic theory to be described below applies to both special
and general relativistic cases., of which the fibre space is spanned by the
proper momentum vector $p^{\mu}$ for individual particles which obey the mass
shell constraint $p^{\mu}p_{\mu}=-m^{2}c^{2}$, where $m$ is the rest mass of
the particle. The enlarged one particle phase space is endowed with a
relativistic invariant measure [24, 25]
$\bm{\Omega}=\bm{\varpi}\wedge\bm{\epsilon}$, where
$\bm{\varpi}=\frac{\sqrt{g}}{|p_{0}|}\mathrm{d}^{d}p$ is the momentum space
volume element222The appearance of $\mathrm{d}^{d}p$ rather than
$\mathrm{d}^{d+1}p$ is due to the mass shell constraint.,
$\bm{\epsilon}=\sqrt{g}\mathrm{d}^{d+1}x$ is the spacetime volume element,
$g=|\mathrm{det}(g_{\mu\nu})|$, and $d+1$ is the spacetime dimension. For a
dilute gas system, in much of the region in phase space, the 1PDF is locally
conserved

$\displaystyle\mathscr{L}_{{\mathcal{H}}}f=0,$ (1)

where $\mathscr{L}_{{\mathcal{H}}}$ denotes the Lie derivatives along the
Hamiltonian vector field $\mathcal{H}$. This is known as the relativistic
Liouville equation. However, taking into account the contribution from the
inter-particle scatterings, the Liouville equation should be replaced by the
Boltzmann equation

$\displaystyle\mathscr{L_{\mathcal{H}}}f=\mathcal{C}(x,p),$ (2)

where the scattering integral $\mathcal{C}(x,p)$ is a non-local integral in
terms of the 1PDF $f(x,p)$ and the local transition rate
$W(x|p_{1},p_{2};p_{3},p_{4})$ if two particle scatterings are dominated.
Assuming that the above equation is solved, all macroscopic evolutions of the
non-equilibrium system will be determined by the 1PDF, including the particle
number current $N^{\mu}$, the energy-momentum tensor $T^{\mu\nu}$ and the
entropy current $S^{\mu}$:

$N^{\mu}=c\int\bm{\varpi}p^{\mu}f,\qquad
T^{\mu\nu}=c\int\bm{\varpi}p^{\mu}p^{\nu}f,\qquad
S^{\mu}=-k_{\mathrm{B}}c\int\bm{\varpi}p^{\mu}f\left[\log(h^{d}f)-1\right],$

wherein $h$ is the Planck constant and $k_{\mathrm{B}}$ is the Boltzmann
constant. Since Boltzmann equation is an integro-differential equation,
finding a solution is a highly nontrivial task. Fortunately, we can draw some
interesting conclusions without too much effort on solving the equation. As
long as Boltzmann equation holds, the particle number is conserved
$\nabla_{\mu}N^{\mu}=0$, the hydrodynamic equation is established
$\nabla_{\mu}T^{\mu\nu}=0$, and the entropy never decrease
$\mathcal{G}\equiv\nabla_{\mu}S^{\mu}\geq 0$. $\mathcal{G}$ is known as the
local entropy production rate, and when $\mathcal{G}=0$, the system is in
detailed balance.

Under detailed balance, one can show that $\log(h^{d}f)$ is additive and must
be linearly dependent on additive conserved quantities of the macroscopic
system. Meanwhile, the Boltzmann equation degenerates into the Liouville
equation (1). With all these conditions we can conclude that the general form
of the 1PDF under detailed balance is

$\displaystyle f_{0}=\frac{1}{h^{d}}\exp(-\alpha+\mathcal{B}_{\mu}p^{\mu}),$
(3)

where $\alpha$ and $\mathcal{B}_{\mu}$ are undetermined coefficients which
satisfy

$\displaystyle-p^{\mu}\partial_{\mu}\alpha+p^{\mu}p^{\nu}\nabla_{\mu}\mathcal{B}_{\nu}=0.$
(4)

We assume that the motion of individual particles is completely random, then
the coefficients in front of the linear and quadratic terms in the momentum
must separately be zero. Hence we have

$\displaystyle\partial_{\mu}\alpha=0,\qquad\text{and}\qquad\nabla_{(\mu}\mathcal{B}_{\nu)}=0,$
(5)

Ignoring the trivial solution $\mathcal{B}^{\mu}=0$, it follows from eq. (5)
that $\mathcal{B}^{\mu}$ must be a Killing vector field. Let us remind that
the 1PDF (3) applies to any classical macroscopic system in any spacetime
admitting a Killing vector field $\mathcal{B}^{\mu}$, provided the system is
under detailed balance. The form of the 1PDF (3) reminds us of the famous
Jüttner distribution [22, 26, 28, 29]

$\displaystyle f=\frac{1}{h^{d}}\exp(-\alpha+\beta U_{\mu}p^{\mu})$ (6)

for relativistic fluid in equilibrium, wherein $U_{\mu}$ is the average proper
velocity of the fluid element and $\beta$ is the inverse temperature. However,
at this point, we do not attempt to link $\mathcal{B}_{\mu}$ with $U_{\mu}$,
and do not introduce a priori an inverse temperature. The system described by
the 1PDF (3) is only in detailed balance but not in thermodynamic equilibrium.
As will be shown later, if we choose the instantaneously comoving observer
while describing the motion of the fluid, the distribution (3) indeed reduces
to the standard Jüttner distribution (6). However, if we choose an arbitrary
non-comoving observer, eq.(3) will differ from the Jüttner distribution (6),
which will allow us to uncover the transformation rules for macroscopic
quantities.

The solution to eq. (5) can be non-unique because the spacetime may admits
several independent Killing vector fields. In such cases,
${\mathcal{B}}^{\mu}$ can be either timelike or spacelike. Accordingly, the
quantity ${\mathcal{B}}_{\mu}p^{\mu}$ can be proportional either to the single
particle energy or to certain momentum component(s) of the single particle. It
is not surprising that the 1PDF can depend on the momentum component(s) of the
particle. Even in non-relativistic statistical mechanics, the distribution
function can depend on the particle’s momentum if the system maintains spatial
translation symmetry. However, there may be surprising cases if the spacetime
has no timelike Killing vector but do have several spacelike ones. In such
spacetimes, the detailed balance distribution of the form (3) can still be
achieved. This possibility is of little interest if one considers the fact
that the macroscopic system needs to be spatially confined by a potential
which breaks all spatial translational symmetries. In short, although the
solutions to eq.(5) depends solely on the spacetime diffeomorphism symmetry,
the physical choice of the solution can only be determined by considering the
symmetry of the potential as well as the boundary conditions of the system. In
the present work we assume that the spacetime and the boundary conditions of
the system altogether admit at least one timelike $\mathcal{B}^{\mu}$. We also
assume that $\mathcal{B}^{\mu}$ is normalized as
$\mathcal{B}^{2}=-\beta^{2}c^{2}$, in which the physical meaning of the scalar
parameter $\beta$ is yet to be interpreted. In order to have an intuitive
understanding about this parameter, let us temporarily consider a two-
component mixture consisting of species (I) and (II) between which only
elastic scatterings could occur. For such a system, the detailed balance
condition reduces to the equation [26]

$\displaystyle
f_{\text{I}}(p_{\text{I}})\,f_{\text{II}}(p_{\text{II}})=f^{\prime}_{\text{I}}(p^{\prime}_{\text{I}})\,f^{\prime}_{\text{II}}(p^{\prime}_{\text{II}}),$

where unprimed and primed symbols represent respectively the corresponding
quantities before and after the elastic scattering. Accordingly

$\displaystyle\mathcal{B}_{(\text{I})\mu}p_{({\text{I}})}{}^{\mu}+\mathcal{B}_{(\text{II})\mu}p_{({\text{II}})}{}^{\mu}=\mathcal{B}_{(\text{I})\mu}p^{\prime}_{({\text{I}})}{}^{\mu}+\mathcal{B}_{(\text{II})\mu}p^{\prime}_{({\text{II}})}{}^{\mu}$

is satisfied for any elastic binary collision,
$p_{({\text{I}})}{}^{\mu}+p_{({\text{II}})}{}^{\mu}=p^{\prime}_{({\text{I}})}{}^{\mu}+p^{\prime}_{({\text{II}})}{}^{\mu}$.
Further, with the same boundary conditions, $\mathcal{B}_{(\text{I})}^{\mu}$
and $\mathcal{B}_{(\text{II})}^{\mu}$ must be collinear. Therefore
$\beta_{(\text{I})}$ and $\beta_{(\text{II})}$ must be equal, which means that
locally there is a commonly shared scalar for two comoving systems under
detailed balance. In this sense, $\beta$ may be used while defining
temperature. This argument agrees in spirits with Ref. [19]. Later, we will
show that $\beta$ is indeed connected to the inverse temperature observed by
comoving observers in more generic settings rather than just in the two
component mixture described above.

Using the detailed balance distribution (3), we can rewrite the particle
number current, the energy-momentum tensor and the entropy current under
detailed balance as follows,

$\displaystyle N^{\mu}$
$\displaystyle=\frac{c\,\mathrm{e}^{-\alpha}}{h^{d}}\int\frac{\sqrt{g}\,\mathrm{d}^{d}p}{|p_{0}|}p^{\mu}\mathrm{e}^{\mathcal{B}_{\rho}p^{\rho}},\quad
T^{\mu\nu}=\frac{c\,\mathrm{e}^{-\alpha}}{h^{d}}\int\frac{\sqrt{g}\,\mathrm{d}^{d}p}{|p_{0}|}p^{\mu}p^{\nu}\mathrm{e}^{\mathcal{B}_{\rho}p^{\rho}},$
(7) $\displaystyle S^{\mu}$
$\displaystyle=\frac{k_{\mathrm{B}}c\,\mathrm{e}^{-\alpha}}{h^{d}}\int\frac{\sqrt{g}\,\mathrm{d}^{d}p}{|p_{0}|}\left(1+\alpha-\mathcal{B}_{\nu}p^{\nu}\right)p^{\mu}\mathrm{e}^{\mathcal{B}_{\rho}p^{\rho}}=k_{\mathrm{B}}\left(1+\alpha\right)N^{\mu}-k_{\mathrm{B}}\,\mathcal{B}_{\nu}T^{\mu\nu}.$
(8)

It is evident that provided the conditions (5) are satisfied, the hydrodynamic
equations $\nabla_{\mu}N^{\mu}=0,\nabla_{\mu}T^{\mu\nu}=0$ and the detailed
balance condition $\nabla_{\mu}S^{\mu}=0$ are explicitly satisfied. Therefore,
the macroscopic system is non-dissipative and can be modeled as a perfect
fluid. Notice, however, since $N^{\mu}$, $T^{\mu\nu}$ and $S^{\mu}$ are all
tensorial objects, the components of these objects depend explicitly on the
choices of the spacetime geometry and the coordinate system. According to the
principles of relativity, the characteristic properties of the perfect fluid
need to be described in a way which is independent of the coordinate choice.
This can be done by introducing a number of scalar density parameters. Even
so, an explicit specification for the spacetime metric $g_{\mu\nu}$ is
inevitable. In the next section, we shall calculate the physical quantities
for the perfect fluid in Minkowski spacetime and uncover the relativistic
transformation rules for a number of scalar observables of the perfect fluid.

##  3 Transformation rules in Minkowski spacetime

In this section, we shall fix the spacetime to be Minkowskian, and for
simplicity, we shall set the spacetime dimension to be $3+1$, and take the
coordinates to be cartesian. Under such setting, the Killing vector field
$\mathcal{B}^{\mu}$ can be taken as
$\mathcal{B}^{\mu}=\beta(\partial_{t})^{\mu}=\left(\beta c,0,0,0\right)$, the
particle number current $N^{\mu}$ and the energy-momentum tensor $T^{\mu\nu}$
as given in (7) can be explicitly evaluated to be

$\displaystyle N^{\mu}$
$\displaystyle=\left(\frac{4\pi\mathrm{e}^{-\alpha}}{\lambda_{C}^{3}\zeta}c\,K_{2}(\zeta),0,0,0\right),$
(9) $\displaystyle T^{\mu\nu}$
$\displaystyle=\frac{4\pi\mathrm{e}^{-\alpha}}{\lambda_{C}^{3}\zeta}mc^{2}K_{2}(\zeta)\left(\begin{array}[]{cccc}\left[K_{3}(\zeta)/K_{2}(\zeta)-\zeta^{-1}\right]&0&0&0\\\
0&\zeta^{-1}&0&0\\\ 0&0&\zeta^{-1}&0\\\ 0&0&0&\zeta^{-1}\end{array}\right),$
(14)

where $\lambda_{C}=\frac{h}{mc}$ is the Compton wave length of the constituent
particles, $K_{\nu}(\zeta)$ is the modified Bessel function of the second
kind, and $\zeta=\beta mc^{2}$ is a dimensionless parameter. It follows from
eq.(9) that the particle number current has no spatial component, which
reflects the fact that we are working in a comoving coordinate system. The
particle number density can be read off from the temporal component of
$N^{\mu}$,

$\displaystyle\bar{n}=\frac{1}{c}N^{0}=\frac{4\pi\mathrm{e}^{-\alpha}}{\lambda_{C}^{3}\zeta}K_{2}(\zeta).$
(15)

The energy density is the 00 component of the energy momentum tensor which
reads

$\displaystyle\bar{\epsilon}=T^{00}=\frac{4\pi\mathrm{e}^{-\alpha}}{\lambda_{C}^{3}\zeta}mc^{2}\left[K_{3}(\zeta)-\zeta^{-1}K_{2}(\zeta)\right].$
(16)

Moreover, from eq.(14), one can see that the stress tensor (regarded as the
spatial-spatial part of the energy momentum tensor) is diagonal,
$\bar{\mathcal{S}}^{ij}=\bar{P}\delta^{ij}$, which means that the perfect
fluid is characterized by an isotropic pressure

$\displaystyle\bar{P}=\bar{T}^{ii}=\frac{4\pi\mathrm{e}^{-\alpha}}{\lambda_{C}^{3}\zeta^{2}}mc^{2}K_{2}(\zeta)=\zeta^{-1}mc^{2}\bar{n}=\frac{\bar{n}}{\beta}.$
(17)

The particle number density $\bar{n}$, the pressure $\bar{P}$ and the energy
density $\bar{\epsilon}$ given above are taken to be some specific component
(or sum of components) of the relevant tensors. This may raise concerns about
the relativistic invariance (or covariance) of these quantities. However, the
fact is that there is an observer, say $\bar{\mathcal{O}}$, hidden in the
above result. To be specific, at each given spacetime event, the proper
velocity of $\bar{\mathcal{O}}$ is $\bar{Z}^{\mu}=(\partial_{t})^{\mu}$, and
$\bar{n}$ and $\bar{\epsilon}$ are actually the following scalar observables
measured by the observer $\bar{\mathcal{O}}$,

$\displaystyle\bar{n}=-\frac{1}{c^{2}}\bar{Z}_{\mu}N^{\mu},\qquad\bar{\epsilon}=\frac{1}{c^{2}}\bar{Z}_{\mu}\bar{Z}_{\nu}T^{\mu\nu},$
(18)

and the pressure $\bar{P}$ is simply one third of the trace of the pure stress
tensor $\bar{\mathcal{S}}^{\mu\nu}$ (now regarded as a tensor on the full
spacetime) measured by $\bar{\mathcal{O}}$:

$\displaystyle\bar{\mathcal{S}}^{\mu\nu}\equiv
T^{\rho\sigma}\bar{\Delta}_{\rho}{}^{\mu}\bar{\Delta}_{\sigma}{}^{\nu},\qquad\bar{P}=\frac{1}{3}\bar{\Delta}_{\mu\nu}\bar{\mathcal{S}}^{\mu\nu},$
(19)

where

$\bar{\Delta}^{\mu\nu}=\eta^{\mu\nu}+\frac{1}{c^{2}}\bar{Z}^{\mu}\bar{Z}^{\nu}$

is the normal projection tensor associated to the observer $\bar{\mathcal{O}}$
which satisfies

$\displaystyle\bar{\Delta}_{\mu\nu}\bar{Z}^{\mu}=0,\quad\bar{\Delta}_{\mu\nu}v^{\mu}=v^{\nu},\quad\forall
v^{\mu}\mbox{ such that }\bar{Z}_{\mu}v^{\mu}=0.$

$\bar{\Delta}_{\mu\nu}$ is also the induced metric on the spacelike
hypersurface normal to $\bar{Z}^{\mu}$. From the above point of view,
$\bar{n}$, $\bar{\epsilon}$ and $\bar{P}$ are all scalar observables which are
independent of coordinate choices, but dependent on the choice of observer.

In a static spacetime, an observer whose proper velocity is proportional to
$(\partial_{t})^{\mu}$ is known as a static observer. The choice of the static
observer $\bar{\mathcal{O}}$ leads to the following consequences: (i)
$\mathcal{B}^{\mu}$ and $\bar{Z}^{\mu}$ are colinear,
$\mathcal{B}^{\mu}=\beta\bar{Z}^{\mu}$; (ii) $N^{\mu}$ has no spacial
components with respect to $\bar{Z}^{\mu}$, i.e.
$N^{\mu}\bar{\Delta}_{\mu}{}^{\nu}=0$; (iii) the stress measured by $\bar{O}$
is isotropic. The observer $\bar{\mathcal{O}}$ is actually an instantaneous
comoving observer, which means that the proper velocity of the observer is
identical to that of the fluid element. Indeed, according to eqs.(9) and (14),
the particle number current $N^{\mu}$ and energy-momentum tensor $T^{\mu\nu}$
can be expressed in terms of $\bar{Z}^{\mu}$ as,

$\displaystyle N^{\mu}=\bar{n}\bar{Z}^{\mu},\qquad
T^{\mu\nu}=\bar{P}\eta^{\mu\nu}+\frac{1}{c^{2}}\left(\bar{\epsilon}+\bar{P}\right)\bar{Z}^{\mu}\bar{Z}^{\nu},$
(20)

which recovers the familiar result of the particle number current and energy
momentum tensor for a perfect fluid.

The parametrization of the particle number current and energy momentum tensor
using energy density and pressure is not the only choice. There exist other
sets of variables which fulfill the same purposes, some of which may even be
more preferable in some cases. For instance, when relativistic transformations
are under concern, it may be more reasonable to decompose the above tensorial
objects into irreducible parts which do not mix up under local Lorentz boosts.
For $T^{\mu\nu}$, one such decomposition is given as follows,

$\displaystyle
T^{\mu\nu}=\frac{1}{4}\mathcal{T}\eta^{\mu\nu}+\frac{1}{4}\bar{w}\left(\bar{\Delta}^{\mu\nu}+\frac{3}{c^{2}}\bar{Z}^{\mu}\bar{Z}^{\nu}\right),$
$\displaystyle\mathcal{T}\equiv
T^{\sigma}{}_{\sigma}=-\frac{4\pi\mathrm{e}^{-\alpha}}{\lambda_{C}^{3}\zeta}mc^{2}K_{1}(\zeta),\quad\bar{w}\equiv\bar{\epsilon}+P=\frac{4\pi\mathrm{e}^{-\alpha}}{\lambda_{C}^{3}\zeta}mc^{2}K_{3}(\zeta),$

where $\mathcal{T}$ is the trace of $T^{\mu\nu}$ and $\bar{w}$ represents the
enthalpy density. The decomposition of $T^{\mu\nu}$ now consists in its trace
and traceless parts. Generally speaking, the trace part satisfies333Our
convention on the metric signature is $(-,+,+,+)$. $\mathcal{T}\leq 0$, and
the enthalpy density satisfies $\bar{w}\geq 0$. Now we proceed to study the
entropy density. By use of eq.(20), the entropy current (8) becomes

$\displaystyle
S^{\mu}=\bar{s}\bar{Z}^{\mu},\qquad\bar{s}=k_{\mathrm{B}}\left(\beta\,\bar{w}+\bar{n}\alpha\right).$
(21)

Now we are in a position to analyze the particle number density $n$, enthalpy
density $w$ and entropy density $s$ measured by an arbitrary instantaneous
observer $\mathcal{O}$ with proper velocity $Z^{\mu}$. Of course, $Z^{\mu}$
should still be a normalized timelike vector at each instance. The normal
projection tensor associated to the observer $\mathcal{O}$ is naturally
defined as

$\Delta^{\mu\nu}=\eta^{\mu\nu}+\frac{1}{c^{2}}Z^{\mu}Z^{\nu}.$

Let us recall that, at the same spacetime event, any two instantaneous
observers can be connected by a (local) Lorentz boost. To prove this
statement, let us assume that $\bar{Z}^{\mu}$ and $Z^{\mu}$ are not
proportional to each other. Otherwise, the two observers must be identical. At
the same event, we can always write

$\displaystyle\bar{Z}^{\mu}=\gamma\left(Z^{\mu}+z^{\mu}\right),\quad\text{where}\quad
Z_{\mu}z^{\mu}=0.$ (22)

Clearly, since $z^{\mu}$ is normal to $Z^{\mu}$, one has

$\Delta_{\mu\nu}Z^{\mu}=0,\qquad\Delta_{\mu\nu}z^{\mu}=z_{\nu}.$

From the normalization condition for $\bar{Z}^{\mu}$ and $Z^{\mu}$, it is easy
to see that

$\gamma=\frac{1}{\sqrt{1-z^{2}/c^{2}}},$

so $\gamma$ is nothing but the Lorentz factor and $z^{\mu}$ can be regarded as
the relative coordinate time velocity between the two observers
$\bar{\mathcal{O}}$ and $\mathcal{O}$ (notice that in a coordinate system in
which the observer $\mathcal{O}$ remains static, $z^{\mu}$ has no temporal
component and thus can be considered as a 3-vector). Notice that although the
proper velocity of the observer has been changed, the Killing vector field
$\mathcal{B}^{\mu}$ appearing in the distribution function (3) remains
untouched. Therefore, $\mathcal{B}^{\mu}$ is not parallel to the proper
velocity $Z^{\mu}$ of the new observer $\mathcal{O}$.

Inserting eq.(22) into the expressions for $N^{\mu},T^{\mu\nu}$ (20) and
$S^{\mu}$ (21), we get

$\displaystyle N^{\mu}=\gamma\bar{n}Z^{\mu}+\gamma\bar{n}z^{\mu},$
$\displaystyle
T^{\mu\nu}=\bar{P}\eta^{\mu\nu}+\frac{\gamma^{2}}{c^{2}}\bar{w}\left[Z^{\mu}Z^{\nu}+\left(Z^{\mu}z^{\nu}+Z^{\nu}z^{\mu}\right)+z^{\mu}z^{\nu}\right],$
(23) $\displaystyle S^{\mu}=\gamma\bar{s}Z^{\mu}+\gamma\bar{s}z^{\mu}.$

By definition, the particle number density measured by the observer
$\mathcal{O}$ is

$\displaystyle n=-\frac{1}{c^{2}}Z_{\mu}N^{\mu}=\gamma\bar{n},$ (24)

which makes perfect sense in terms of the length contraction effect and
considering that on the classical level the total number of particles in the
system is invariant. Similarly, the transformation for entropy density is

$\displaystyle s=-\frac{1}{c^{2}}Z_{\mu}S^{\mu}=\gamma\bar{s},$ (25)

where again the $\gamma$ factor can be attributed to the length contraction
effect. In result, we can conclude that the total entropy should be invariant
under Lorentz boost after a volume integration.

The definition for the pressure $P$ observed by the observer $\mathcal{O}$
contains some subtleties. We need first to separate the energy momentum
4-vector $\mathcal{P}^{\mu}$ and the pure stress tensor $\mathcal{S}^{\mu\nu}$
from the energy momentum tensor (23):

$\displaystyle\mathcal{P}^{\mu}\equiv-\frac{1}{c^{2}}Z_{\rho}T^{\rho\nu}\Delta_{\nu}{}^{\mu}=\frac{\gamma^{2}\bar{w}}{c^{2}}z^{\mu},$
(26) $\displaystyle\mathcal{S}^{\mu\nu}\equiv
T^{\rho\sigma}\Delta_{\rho}{}^{\mu}\Delta_{\sigma}{}^{\nu}=\bar{P}\Delta^{\mu\nu}+\frac{\gamma^{2}\bar{w}}{c^{2}}z^{\mu}z^{\nu}.$
(27)

The existence of a spacelike momentum $\mathcal{P}^{\mu}$ reflects the fact
that the observer $\mathcal{O}$ is not comoving with the fluid. In other
words, $\mathcal{P}^{\mu}$ is purely a kinematic effect. Notice also that, the
second term in the expression for $\mathcal{S}^{\mu\nu}$, i.e.

$\tilde{\mathcal{S}}^{\mu\nu}\equiv\frac{\gamma^{2}\bar{w}}{c^{2}}z^{\mu}z^{\nu}=\mathcal{P}^{\mu}z^{\nu},$

is nothing but the kinematic momentum flow, and hence is also purely
kinematically originated, so it may be best referred to as kinematic stress
tensor. This part of the total stress tensor introduces an anisotropy in the
diagonal part, yielding different pressures in different spatial directions.
However, since the thermodynamic effects has nothing to do with the global
kinematics of the fluid, it is better to define the thermodynamic pressure as
the trace of the difference between the total stress and the kinematic stress
divided by the dimension of the space. In the present case, we have

$\displaystyle
P=\frac{1}{3}\Delta_{\mu\nu}(\mathcal{S}^{\mu\nu}-\tilde{\mathcal{S}}^{\mu\nu})=\bar{P},$
(28)

which indicates that the thermodynamic pressure is a relativistic invariant.

The definition of pressure is essential to determine the correct enthalpy
density, and therefore settles the transformation rules connecting
thermodynamic parameters measured by arbitrary observer $\mathcal{O}$ and
comoving observer $\bar{\mathcal{O}}$. For enthalpy density, equation (23) and
the invariance of pressure (28) imply

$\displaystyle w\equiv\frac{1}{c^{2}}Z_{\mu}Z_{\nu}T^{\mu\nu}+P,\qquad\quad
w=\gamma^{2}\bar{w}.$ (29)

In view of the explicit expressions for particle density, entropy density,
pressure and the enthalpy density, we find that the local equilibrium state
for arbitrary observer can be parameterized by $\alpha$, $\beta$ and $\gamma$.
And It is important to emphasize that the thermodynamic pressure
$P(\alpha,\beta)$ is isotropic and observer independent whose derivative
yields the following relations:

$\displaystyle n=-\gamma\beta\frac{\partial P}{\partial\alpha},\quad
w=-\gamma^{2}\beta\frac{\partial P}{\partial\beta},\quad s=-\gamma
k_{B}\beta\left(\beta\frac{\partial P}{\partial\beta}+\alpha\frac{\partial
P}{\partial\alpha}\right).$

Therefore, the total derivative of $P(\alpha,\beta)$ yields

$\displaystyle-\gamma^{-1}s{\rm d}\left(\frac{1}{k_{B}\beta}\right)+{\rm
d}P+\gamma^{-1}n{\rm d}\left(\frac{\alpha}{\beta}\right)=0,$ (30)

for a comoving observer, i.e. when $\gamma=1$, this equation is reminiscent to
the local version of Gibbs-Duhem relation, provided the parameters $\beta$ and
$\alpha$ are respectively connected with the inverse temperature and the
chemical potential appropriately. In fact, such a reminiscence is true,
because a system under detailed balance can be considered to be in local
thermodynamic equilibrium, and hence we can use the local Gibbs-Duhem relation
to identify the local temperature and chemical potential as

$\displaystyle\bar{T}=\frac{1}{k_{\mathrm{B}}\beta},\qquad\bar{\mu}=-\frac{\alpha}{\beta}.$
(31)

According to the basic principles of relativity, the choice of observers
should not affect physical identities obeyed by physical observables, although
the value of each observable may be affected by different choices. Among the
thermodynamic relations, in the light of the standpoint of Israel’s theory
[27], we now make our fundamental assumption that the local Gibbs-Duhem
relation is invariant for different observers. In the present context,
equation (30) for different observers should correspond to the same physical
identity. In other words, (30) should be the invariant Gibbs-Duhem relation.
The direct consequences of the invariance of Gibbs-Duhem relation are the
transformation rules of temperature and chemical potential:

$\displaystyle
T=\gamma^{-1}\frac{1}{k_{B}\beta}=\gamma^{-1}\bar{T},\qquad\mu=-\gamma^{-1}\frac{\alpha}{\beta}=\gamma^{-1}\bar{\mu}.$
(32)

It should be mentioned that, although the temperature transformation given
above has already been suggested by de Broglie, Einstein and Planck, the one
for chemical potential is, to the best of our knowledge, a new result.
Finally, from eqs. (24), (25), (29), (32) and the conventional definitions for
Gibbs free energy and Helmholtz free energy, we find that the transformation
rules for the densities of all thermodynamic potentials can be expressed in an
heuristic unified form, $\varphi=\bar{\varphi}+z_{\mu}\mathcal{P}^{\mu}$.

Let us conclude this section by adding two extra remarks.

Remark 1. In relativistic physics, the spacetime dimension is often treated as
an adjustable parameter. Whenever one draws some conclusion in relativistic
physics, it is necessary to check the conclusion holds whether in generic
spacetime dimensions or in some specific dimension. On the other hand, the
behaviors of thermodynamic quantities are very sensitive to the dimension of
the underlying space. Therefore, it makes sense to check whether the
transformation rules uncovered in the present section is specific to 3+1
dimensional Minkowski spacetime or they hold in arbitrary spacetime
dimensions. In order to answer this question, it is necessary to extend the
formulation to arbitrary spacetime dimension $(d+1)$. In this regard, it is
important to note that the fluid configuration is completely determined by
$\left(\alpha,\mathcal{B}^{\mu},n,w,\mathcal{T}\right)$, wherein, for perfect
fluid, $\alpha$ is constant, $\mathcal{B}^{\mu}$ is a Killing vector field,
and, in Minkowskian backgrounds (see Appendix),

$\displaystyle
n=\gamma\left[\frac{2\,\mathrm{e}^{-\alpha}}{\lambda_{C}^{d}}\left(\frac{2\pi}{\zeta}\right)^{\frac{d-1}{2}}\right]K_{\frac{d+1}{2}}(\zeta)=\gamma\bar{n},$
$\displaystyle
w=\gamma^{2}mc^{2}\left[\frac{2\,\mathrm{e}^{-\alpha}}{\lambda_{C}^{d}}\left(\frac{2\pi}{\zeta}\right)^{\frac{d-1}{2}}\right]K_{\frac{d+3}{2}}(\zeta)=\gamma^{2}\bar{w},$
$\displaystyle\mathcal{T}=-mc^{2}\left[\frac{2\,\mathrm{e}^{-\alpha}}{\lambda_{C}^{d}}\left(\frac{2\pi}{\zeta}\right)^{\frac{d-1}{2}}\right]K_{\frac{d-1}{2}}(\zeta).$

The exact results for the particle number density, the enthalpy density and
the trace of the energy momentum tensor for perfect relativistic fluid in
arbitrary spacetime dimensions are, to our knowledge, not reported before in
the literature. Using these results, it will not be difficult to check that
all the transformation rules obtained in the present section are independent
of the spacetime dimension.

Remark 2. The temperature, chemical potential, particle number density,
entropy and enthalpy densities and the pressure of the perfect fluid are all
defined as observer-dependent scalars (or scalar densities). Their
transformation rules arise purely from the different choices of observers and
have nothing to do with the coordinate choices. It is not surprising that at
the same spacetime event, any two instantaneous observers can differ at most
from each other by a local Lorentz boost (which is not a coordinate
transformation of the spacetime). Such differences are independent of the
choice of spacetime geometry. Therefore, it is highly expected that the same
transformation rules should hold in other spacetimes, and we shall verify this
expectation in Rindler spacetime in the next section.

##  4 Perfect Rindler fluid and the area law of entropy

The 1PDF under detailed balance given in eq.(3) is valid not only in Minkowski
spacetime, but also in any spacetime admitting a Killing vector field
$\mathcal{B}_{\mu}$. In order to show the influence of spacetime geometry on
the description of perfect fluid, let us move on to another familiar
spacetime, i.e. the Rindler spacetime with the line element

$\displaystyle\mathrm{d}s^{2}=g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}=-\left(1+\frac{\kappa
x}{c^{2}}\right)^{2}c^{2}\mathrm{d}t^{2}+\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2}.$
(33)

The coordinates used in writing the line element (33) is henceforth referred
to as Rindler coordinates. As is well known, the Rindler spacetime contains an
accelerating horizon which is located at $x_{h}=-c^{2}\kappa^{-1}$ in the
Rindler coordinate system. In this spacetime, the static observer
$\bar{\mathcal{O}}$ has proper velocity $\bar{Z}^{\mu}=\left(1+\frac{\kappa
x}{c^{2}}\right)^{-1}(\partial_{t})^{\mu}$, which is timelike but non-Killing.
The timelike Killing vector field $\mathcal{B}^{\mu}$ appearing in the 1PDF
(3) normalized as $\mathcal{B}^{2}=-\beta^{2}c^{2}$ must be proportional to
$\bar{Z}^{\mu}$, i.e.
$\mathcal{B}^{\mu}=\beta\bar{Z}^{\mu}=\beta\left(1+\frac{\kappa
x}{c^{2}}\right)^{-1}(\partial_{t})^{\mu}$, which, together with the fact that
$\bar{Z}^{\mu}$ is non-Killing, implies that $\beta$ must not be constant,
rather, it has an explicit $x$-dependence,

$\displaystyle\beta(x)=-\beta_{0}(x-x_{h})/x_{h},$ (34)

where $\beta_{0}$ is a constant which equals to the value of $\beta$ at $x=0$.
Demanding that $\beta_{0}$ has a finite value yields that $\beta$ becomes very
small as the system gets very close to the Rindler horizon. On the other hand,
if $\beta(x)$ approaches a finite nonvanishing value as $x\to x_{h}$, then
$\beta_{0}$ could be infinitely large.

With the above choices of spacetime metric and the Killing vector field
$\mathcal{B}^{\mu}$, the particle number current $N^{\mu}$ and the energy
momentum tensor $T^{\mu\nu}$ can be explicitly calculated using eq.(7),
yielding

$\displaystyle N^{\mu}$ $\displaystyle=\left[c\left(1+\frac{\kappa
x}{c^{2}}\right)^{-1}\frac{4\pi\mathrm{e}^{-\alpha}}{\lambda_{C}^{3}\zeta}K_{2}(\zeta),0,0,0\right],$
(35) $\displaystyle T^{\mu\nu}$
$\displaystyle=\frac{4\pi\mathrm{e}^{-\alpha}}{\lambda_{C}^{3}\zeta}mc^{2}K_{2}(\zeta)\left(\begin{array}[]{cccc}\frac{\zeta
K_{3}(\zeta)-K_{2}(\zeta)}{\left(1+\frac{\kappa x}{c^{2}}\right)^{2}\zeta
K_{2}(\zeta)}&0&0&0\\\ 0&\zeta^{-1}&0&0\\\ 0&0&\zeta^{-1}&0\\\
0&0&0&\zeta^{-1}\end{array}\right),$ (40)

where $\zeta$ is still defined as $\zeta=\beta mc^{2}$ and hence is non-
constant.

Unlike the case of Minkowski spacetime, now we should not expect to read off
the particle number density $\bar{n}$, the energy density $\bar{\epsilon}$ and
the pressure $\bar{P}$ directly from the appropriate components of $N^{\mu}$
and $T^{\mu\nu}$ respectively as we did in eqs.(15), (17) and (16). Rather, we
should think of these objects as scalar densities defined in eq.(18) and (19).
Without much effort it can be shown that $\bar{n}$, $\bar{\epsilon}$ and
$\bar{P}$ for the Rindler fluid are given respectively by the following
expressions,

$\displaystyle\bar{n}=-\frac{1}{c^{2}}\bar{Z}_{\mu}N^{\mu}=\frac{4\pi\mathrm{e}^{-\alpha}}{\lambda_{C}^{3}\zeta}K_{2}(\zeta),$
(41)
$\displaystyle\bar{\epsilon}=\frac{1}{c^{2}}\bar{Z}_{\mu}\bar{Z}_{\nu}T^{\mu\nu}=\frac{4\pi\mathrm{e}^{-\alpha}}{\lambda_{C}^{3}\zeta}mc^{2}\left[K_{3}(\zeta)-\zeta^{-1}K_{2}(\zeta)\right],$
(42)
$\displaystyle\bar{P}=\frac{1}{3}\bar{\Delta}_{\mu\nu}\bar{\mathcal{S}}^{\mu\nu}=\frac{4\pi\mathrm{e}^{-\alpha}}{\lambda_{C}^{3}\zeta^{2}}mc^{2}K_{2}(\zeta)=\frac{\bar{n}}{\beta},$
(43)

and the decomposition like (20) for $N^{\mu}$ and $T^{\mu\nu}$ still holds,
while the only difference comes from the different choice of spacetime metric:

$\displaystyle N^{\mu}=\bar{n}\bar{Z}^{\mu},\qquad
T^{\mu\nu}=\bar{P}g^{\mu\nu}+\frac{1}{c^{2}}\left(\bar{\epsilon}+\bar{P}\right)\bar{Z}^{\mu}\bar{Z}^{\nu}.$
(44)

These results indicate that the static observer $\bar{\mathcal{O}}$ is
instantaneously comoving with the fluid. Moreover, the entropy density
$\bar{s}$ can be evaluated explicitly,

$\displaystyle\bar{s}$
$\displaystyle=-\frac{1}{c^{2}}\bar{Z}_{\mu}S^{\mu}=-\frac{k_{\mathrm{B}}}{c^{2}}\bar{Z}_{\mu}\left[\left(1+\alpha\right)N^{\mu}-\mathcal{B}_{\nu}T^{\mu\nu}\right]$
$\displaystyle=k_{\mathrm{B}}\frac{4\pi\mathrm{e}^{-\alpha}}{\lambda_{C}^{3}}\left[K_{3}(\zeta)+\alpha\frac{K_{2}(\zeta)}{\zeta}\right]$
$\displaystyle=k_{\mathrm{B}}\left(\beta\,\bar{w}+\bar{n}\alpha\right).$ (45)

All the results given in eqs.(41)-(45) are in perfect agreement with their
Minkowski analogues, as shown in eqs.(15)-(17), (20) and (21). If we proceed
to an arbitrary Rindler observer, the same transformation rules for
$T,\mu,P,s$ and $w$ will be recovered, with the same interpretations for the
parameters $\alpha,\beta$. The $x$ dependence of $\beta$ is then attributed to
the Tolman-Ehrenfest effect in Rindler spacetime. We will not bother with
details on the transformation rules in Rindler spacetime but only mention
that, in this case, the relation (22) between the proper velocities of
different observers results in a position dependent boost factor $\gamma$,
because now $z^{2}=g_{\mu\nu}z^{\mu}z^{\nu}$ is position dependent. Such
boosts must be considered to be performed in the tangent space of the
spacetime at each event, and are known as local Lorentz boosts.

One thing to be noted, however, is that all the transformation rules mentioned
above reflect only the behaviors of the corresponding quantities measured at
the same spacetime event by different observers. If one wishes to compare the
same quantities at different spacetime events, or add up some of the local
densities to get the corresponding total quantities, things would become
drastically different between Minkowski and Rindler cases. For the Minkowski
case, nothing special needs to be taken care of, because the inverse
temperature $\beta$ is constant and all the local densities are uniform. For
the Rindler case, however, the first thing to be noted is the non-constancy of
the inverse temperature $\beta$. At different spatial locations with $x$
coordinates $x_{1}$ and $x_{2}$, one has

$\displaystyle\frac{\zeta_{1}}{\zeta_{2}}=\frac{\beta_{1}}{\beta_{2}}=\frac{x_{1}-x_{h}}{x_{2}-x_{h}}.$

This is the well known Tolman-Ehrenfest effect [30] and it implies that the
local densities (41), (42), (45) as well as the pressure (43) in Rindler
spacetime are all non-uniform. Due to the constancy of the parameter $\alpha$,
the chemical potential $\bar{\mu}$ in Rindler spacetime is also non-uniform.

Now let us consider a box of gas, of which the height and the bottom area are
respectively $L$ and $A$, and orient the box such that the bottom is parallel
to the horizon. Let the coordinate distance of the bottom of the box to the
horizon be $\delta$. The relativistic coldness at the bottom of the box is
denoted by $\zeta_{\mathrm{bot}}$, then at any height within the box we have
$\zeta=(x-x_{h})\zeta_{\mathrm{bot}}/\delta$, and at the top of the box we
have $\zeta_{\mathrm{top}}=(L+\delta)\zeta_{\mathrm{bot}}/\delta$.

The total number of particles and the total entropy in the box can be
calculated by a direct spatial volume integration of (41) and (45), yielding

$\displaystyle\bar{N}$
$\displaystyle=A\int_{x_{h}+\delta}^{x_{h}+\delta+L}\bar{n}(x)\mathrm{d}x=A\frac{4\pi\mathrm{e}^{-\alpha}}{\lambda_{C}^{3}}\frac{\delta}{\zeta_{\mathrm{bot}}}\left[\frac{K_{1}(\zeta_{\mathrm{bot}})}{\zeta_{\mathrm{bot}}}-\frac{K_{1}(\zeta_{\mathrm{top}})}{\zeta_{\mathrm{top}}}\right],$
$\displaystyle\bar{S}$
$\displaystyle=A\int_{x_{h}+\delta}^{x_{h}+\delta+L}\bar{s}(x)\mathrm{d}x=A\frac{4\pi
k_{\mathrm{B}}\mathrm{e}^{-\alpha}}{\lambda_{C}^{3}}\frac{\delta}{\zeta_{\mathrm{bot}}}\left[K_{2}(\zeta_{\mathrm{bot}})-K_{2}(\zeta_{\mathrm{top}})\right]+(\alpha+2)k_{\mathrm{B}}\bar{N}.$

When the bottom of the box is very close to the horizon, e.g. $\delta\ll L$,
the coldness at the bottom and top of the box behave as
$\zeta_{\mathrm{bot}}\ll\zeta_{\mathrm{top}}$. Recalling that both
$K_{1}(\zeta)$ and $K_{2}(\zeta)$ are monotonically decreasing functions, we
have $K_{1}(\zeta_{\mathrm{bot}})\gg K_{1}(\zeta_{\mathrm{top}})$,
$K_{2}(\zeta_{\mathrm{bot}})\gg K_{2}(\zeta_{\mathrm{top}})$. Then from
eq.(41) one finds that the particles tend to be gathered at the bottom of the
box, and the total number of particles and entropy can be approximated as

$\displaystyle\bar{N}$ $\displaystyle\approx
A\frac{4\pi\mathrm{e}^{-\alpha}}{\lambda_{C}^{3}}\frac{\delta}{\zeta_{\mathrm{bot}}^{2}}K_{1}(\zeta_{\mathrm{bot}}),$
$\displaystyle\bar{S}$ $\displaystyle\approx A\frac{4\pi
k_{\mathrm{B}}\mathrm{e}^{-\alpha}}{\lambda_{C}^{3}}\frac{\delta}{\zeta_{\mathrm{bot}}^{2}}\left[(\alpha+2)K_{1}(\zeta_{\mathrm{bot}})+\zeta_{\mathrm{bot}}K_{2}(\zeta_{\mathrm{bot}})\right],$

which are both independent of the hight of the box and are proportional to the
area of the bottom of the box. These results are similar to those in [15, 16,
17] where the particle number and entropy are obtained through completely
different approaches. For closed classical systems, the number of particles
needs to be constant, and the expression for $\bar{N}$ actually determines the
chemical potential distribution implicitly. Comparing the values of $\bar{N}$
and $\bar{S}$ in the above case, we get

$\displaystyle\bar{S}\approx\bar{N}k_{\mathrm{B}}\left[(\alpha+2)+\frac{\zeta_{\mathrm{bot}}K_{2}(\zeta_{\mathrm{bot}})}{K_{1}(\zeta_{\mathrm{bot}})}\right],$
(46)

wherein the factor besides $\bar{N}k_{\mathrm{B}}$ depends purely on the
coldness at the bottom of the box but not on any other physical parameters
because of the constancy of $\alpha$. The relationship (46) indicates that the
area dependence of the entropy does not break its additivity.

##  5 Refined Saha equation

As we have shown in the last two sections, the local particle number density
$\bar{n}$ takes the same form in both Minkowski and Rindler spacetimes. By
means of the explicit expressions for $\bar{n}$, see eqs. (15) or (41), we
find that the fugacity can be written as

$\displaystyle\mathfrak{f}=\mathrm{e}^{-\alpha-\zeta}=\bar{n}\lambda_{T}^{3},\qquad\lambda_{T}=\frac{\lambda_{C}}{\sqrt[3]{4\pi\mathrm{e}^{\zeta}\zeta^{-1}K_{2}(\zeta)}}.$
(47)

To understand the physical meaning of $\lambda_{T}$, we now focus on two
extremal cases, $\zeta\gg 1$ and $\zeta\ll 1$. The former choice corresponds
to the non-relativistic limit which applies to baryons, and the latter choice
corresponds to the ultra-relativistic limit, which is attained if the
temperature is ultra high or the rest mass is very small. It is easy to check
that, in the non-relativistic limit and ultra-relativistic limit,
$\lambda_{T}$ coincides respectively with the thermal de Broglie wavelength
for massive and massless particles

$\displaystyle\lambda_{T}\approx\lambda_{\text{NR}}=\frac{h}{\sqrt{2\pi
mk_{\mathrm{B}}\bar{T}}},~{}~{}~{}\left(\zeta\gg 1\right),$ (48)
$\displaystyle\lambda_{T}\approx\lambda_{\text{UR}}=\frac{ch}{2\pi^{1/3}k_{\mathrm{B}}\bar{T}},~{}~{}~{}~{}~{}\left(\zeta\ll
1\right).$ (49)

In this sense, $\lambda_{T}$ may be referred to as the relativistic thermal de
Broglie wavelength. It is worth mentioning that, at generic coldness $\zeta$,
the relativistic thermal wavelength $\lambda_{T}$ can be significantly
different from either $\lambda_{\text{NR}}$ or $\lambda_{\text{UR}}$, as can
be seen in Fig.1.

Figure 1: Comparison between $\lambda_{T}$ and
$\lambda_{\text{NR}},\lambda_{\text{UR}}$

By use of the relativistic thermal de Broglie wavelength $\lambda_{T}$ , we
can recast the expressions for $\bar{n}$ in the form

$\displaystyle\bar{n}=\frac{1}{(\lambda_{T})^{3}}\mathrm{e}^{-\alpha-\zeta}=\frac{1}{(\lambda_{T})^{3}}\mathrm{e}^{(\bar{\mu}-mc^{2})/k_{\mathrm{B}}\bar{T}}.$

If the particle under consideration has some intrinsic quantum number, then
the above expression needs to be multiplied by the corresponding intrinsic
quantum degeneracy $\mathfrak{g}$, i.e.

$\displaystyle\bar{n}=\frac{\mathfrak{g}}{(\lambda_{T})^{3}}\mathrm{e}^{-\alpha-\zeta}=\frac{\mathfrak{g}}{(\lambda_{T})^{3}}\mathrm{e}^{(\bar{\mu}-mc^{2})/k_{\mathrm{B}}\bar{T}}.$
(50)

This expression allows us to get a refined version of the famous Saha equation
[23] which is related to the local “chemical equilibrium” for systems
containing several different species of particles among which chemical
reactions (or ionizations/recombinations) may take place. Let us assume that
the system contains 3 different particle species $A,B,C$ among which $C=AB$ is
the composite particle may come as the result of the chemical reaction

$A+B\rightleftharpoons C.$

Assume also that the interaction between different particles is negligible
unless they collide and/or make the above chemical reaction. Under the above
assumptions, the chemical potentials for different particle species satisfy
the condition $\bar{\mu}_{A}+\bar{\mu}_{B}=\bar{\mu}_{C}$ if chemical reaction
reaches a local equilibrium. Therefore, it follows from eq.(50) that

$\displaystyle\frac{\bar{n}_{A}\bar{n}_{B}}{\bar{n}_{C}}$
$\displaystyle=\frac{\mathfrak{g}_{A}\mathfrak{g}_{B}}{\mathfrak{g}_{C}}\left(\frac{\lambda_{T}^{(C)}}{\lambda_{T}^{(A)}\lambda_{T}^{(B)}}\right)^{3}\mathrm{e}^{-\Delta
E/k_{\mathrm{B}}\bar{T}},$ (51)

where $\Delta E=(m_{A}+m_{B}-m_{C})c^{2}$ is the binding energy of the
composite particle $C$. If $\lambda_{T}$ takes the form of eq.(48), then
eq.(51) is precisely the famous Saha equation. However, as we have shown in
Section 3, $\lambda_{T}$ actually takes the form (47), which is different from
(48) unless $\zeta\gg 1$. Therefore, at smaller coldness (or higher
temperature), eq.(51) gives a relativistically refined version of the standard
Saha equation. For gaseous systems with a single species of atoms which have
several ionized states, eq.(51) reduces into

$\frac{\bar{n}_{i+1}\bar{n}_{e}}{\bar{n}_{i}}=2\frac{\mathfrak{g}_{i+1}}{\mathfrak{g}_{i}}\frac{1}{\left(\lambda_{T}^{(e)}\right)^{3}}\mathrm{e}^{-\Delta
E_{i+1}/k_{\mathrm{B}}\bar{T}},$

where $\bar{n}_{i}$ is the number density of the atom in the $i^{th}$ ionized
state, $\Delta E_{i+1}$ is the energy required to remove the $(i+1)^{th}$
electron, and $\lambda_{T}^{(e)}$ is the relativistic thermal de Broglie
wavelength for the electron.

##  6 Concluding remarks

Relativistic kinetic theory is a powerful tool for analyzing macroscopic
behaviors for systems undergoing relativistic motion. In this work, using
relativistic kinetic theory, we considered the long standing problem on the
relativistic transformation rules for some basic thermodynamic quantities,
including the temperature $T$ and the chemical potential $\mu$. Our the major
results are listed as follows:

$T=\gamma^{-1}\bar{T},\quad
s=\gamma\bar{s},\quad\mu=\gamma^{-1}\bar{\mu},\quad n=\gamma\bar{n},\quad
P=\bar{P},\quad w=\gamma^{2}\bar{w},$

wherein $\gamma$ is the (local) Lorentz factor, and quantities with/without a
bar are respectively measured by the instantaneous comoving observer
$\bar{\mathcal{O}}$ and an arbitrary instantaneous observer $\mathcal{O}$,
which are interrelated by a (local) Lorentz boost. These transformation rules,
supplemented with the well acknowledged contraction rule $\delta
v=\gamma^{-1}\delta\bar{v}$ for spatial volume element $\delta v$, constitute
a complete set of transformation rules for all thermodynamic parameters which
hold in both Minkowski and Rindler spacetimes, and we do not see any reason
why they would not hold in other spacetimes. The transformation rule for $T$
suggests that a moving body appears to be colder, supporting the first view by
de Broglie, Einstein and Planck _et al_. Moreover, our study adds some novel
elements to the existing transformation rules, for instance, the rules for
chemical potential and enthalpy density have not been reported elsewhere.

In the case of perfect Rindler fluid, if we place the fluid in a box whose
bottom is parallel to the Rindler horizon and let the bottom be located
sufficiently close to the horizon, then our calculation shows that the total
entropy and the total number of particles are proportional to the area of the
bottom of the box, and both are independent of the height of the box. By
extending to spacetimes with event horizons, similar analysis might help to
understand the area law of black hole entropy.

Finally, the exact results for the particle number density allow us to get a
relativistically refined version of the famous Saha equation. At extremely
high temperatures, or in the ultra relativistic limits, the refinement could
be significant, as in these cases the relativistic thermal de Broglie
wavelength (47) could be significantly different from its non-relativistic
counterpart (48).

## Acknowledgement

This work is supported by the National Natural Science Foundation of China
under the grant No. 11575088, Hebei NSF under grant No. A2021205037 and the
fund of Hebei Normal University under grant No. L2020B04. XH would like to
thank Shao-Jiang Wang and Bin Wu for useful discussions.

## Appendix

In the literature on relativistic kinetic theory, the expressions of local
equilibrium quantities are mostly presented in a concrete spacetime dimension,
mostly taken to be 4. In case that one may be interested in physical rules in
generic spacetime dimensions, we now present the detailed calculations for all
the macroscopic densities used in the main text in $(d+1)$-dimensional
Minkowski spacetime.

Let us start with the cartesian coordinates and take the Killing vector field
$\mathcal{B}^{\mu}$ to be
$\mathcal{B}^{\mu}=\beta(\partial_{t})^{\mu}=\beta\bar{Z}^{\mu}$. In this
coordinates we can always parametrize $p^{\mu}$ and $Z^{\mu}$ as

$\displaystyle p^{\mu}$
$\displaystyle=mc(\cosh\vartheta,~{}\sinh\vartheta\cos\theta_{d-1},~{}\sinh\vartheta\sin\theta_{d-1}\cos\theta_{d-2},\cdots,$
$\displaystyle\qquad\qquad~{}\sinh\vartheta\sin\theta_{d-1}\cdots\sin\theta_{2}\cos\theta_{1},~{}\sinh\vartheta\sin\theta_{d-1}\cdots\sin\theta_{2}\sin\theta_{1}),$
$\displaystyle Z^{\mu}$
$\displaystyle=c(\cosh\eta,~{}\sinh\eta\cos\varTheta_{d-1},~{}\sinh\eta\sin\varTheta_{d-1}\cos\varTheta_{d-2},\cdots$
$\displaystyle\qquad\qquad~{}\sinh\vartheta\sin\varTheta_{d-1}\cdots\sin\varTheta_{2}\cos\varTheta_{1},~{}\sinh\vartheta\sin\varTheta_{d-1}\cdots\sin\varTheta_{2}\sin\varTheta_{1}),$

where $\theta_{1},\varTheta_{1}\in[0,2\pi)$, and
$\theta_{i},\varTheta_{i}\in[0,\pi)$ for $i\geq 2$. To make the above
parametrization also work for $d=1$, we also introduce
$\theta_{0}=\varTheta_{0}=0$.

It is straightforward to check that $p^{\mu}$ and $Z^{\mu}$ parametrized as
above automatically satisfy the on-shell conditions $p^{2}=-m^{2}c^{2}$,
$Z^{2}=-c^{2}$, and

$|\vec{p}|=\sqrt{\left(p^{1}\right)^{2}+\left(p^{2}\right)^{2}+\cdots+\left(p^{d}\right)^{2}}=mc\sinh\vartheta,\quad\mathrm{d}|\vec{p}|=p^{0}\mathrm{d}\vartheta=|p_{0}|\mathrm{d}\vartheta,$

so the momentum space volume element reads

$\varpi=\frac{\mathrm{d}^{d}p}{|p_{0}|}=\frac{|\vec{p}|^{d-1}\mathrm{d}|\vec{p}|\,\mathrm{d}\Omega_{d-1}}{|p_{0}|}=(mc\sinh\vartheta)^{d-1}\mathrm{d}\vartheta\,\mathrm{d}\Omega_{d-1},$

where for $d=1$ $\mathrm{d}\Omega_{0}=1$, and for $d\geq 2$,
$\mathrm{d}\Omega_{d-1}=\sin^{d-2}\theta_{d-1}\mathrm{d}\theta_{d-1}\mathrm{d}\Omega_{d-2}$.
The coordinate choice freedom allows us to set $\varTheta_{d-1}=0$, which will
greatly simplify the forthcoming calculations. Moreover, comparing
$\bar{Z}^{\mu}$ with $Z^{\mu}$ and considering the relation (22), one
recognizes that $\gamma=\cosh\eta$ is precisely the Lorentz factor related to
the boost between the observers $\bar{\mathcal{O}}$ and $\mathcal{O}$.

Before delving into the analysis, we recall the following three integral
representations for the modified Bessel function of the second kind, which are
frequently used in our calculations,

$\displaystyle K_{\nu}(\zeta)$
$\displaystyle=\int^{\infty}_{0}\mathrm{e}^{-\zeta\cosh\vartheta}\cosh(\nu\vartheta)\mathrm{d}\vartheta,$
(52) $\displaystyle K_{\nu}(\zeta)$
$\displaystyle=\left(\frac{\zeta}{2}\right)^{\nu}\frac{\sqrt{\pi}}{\Gamma\left(\nu+\frac{1}{2}\right)}\int^{\infty}_{0}\sinh^{2\nu}\vartheta\,\mathrm{e}^{-\zeta\cosh\vartheta}\mathrm{d}\vartheta,$
(53) $\displaystyle K_{\nu}(\zeta)$
$\displaystyle=\left(\frac{\zeta}{2}\right)^{\nu}\frac{\Gamma\left(\frac{1}{2}\right)}{\Gamma\left(n+\frac{1}{2}\right)}\int_{1}^{\infty}\mathrm{e}^{-\zeta\vartheta}\left(\vartheta^{2}-1\right)^{\nu-\frac{1}{2}}\mathrm{d}\vartheta.$
(54)

From the integral representation (53), we can get

$\left(\frac{\mathrm{d}missing}{\mathrm{d}\zeta}\right)^{l}\left[\frac{K_{\nu}(\zeta)}{\zeta^{\nu}}\right]=\frac{(-1)^{l}\sqrt{\pi}}{2^{\nu}\,\Gamma\left(\nu+\frac{1}{2}\right)}\int^{\infty}_{0}\mathrm{d}\vartheta\sinh^{2\nu}\vartheta\cosh^{l}\vartheta\,\mathrm{e}^{-\zeta\cosh\vartheta}.$

On the other hand, from the representation (54) we obtain the recurrence
relations

$\left(\frac{\mathrm{d}missing}{\zeta\mathrm{d}\zeta}\right)^{l}\left[\frac{K_{\nu}(\zeta)}{\zeta^{\nu}}\right]=(-1)^{l}\frac{K_{\nu+l}(\zeta)}{\zeta^{\nu+l}}.$

It follows that

$\displaystyle\int^{\infty}_{0}\mathrm{d}\vartheta\sinh^{2\nu}\vartheta\,\mathrm{e}^{-\zeta\cosh\vartheta}=\frac{1}{\sqrt{\pi}}\left(\frac{2}{\zeta}\right)^{\nu}\Gamma\left(\nu+\frac{1}{2}\right)K_{\nu}(\zeta),$
$\displaystyle\int^{\infty}_{0}\mathrm{d}\vartheta\sinh^{2\nu}\vartheta\cosh\vartheta\,\mathrm{e}^{-\zeta\cosh\vartheta}=\frac{1}{\sqrt{\pi}}\left(\frac{2}{\zeta}\right)^{\nu}\Gamma\left(\nu+\frac{1}{2}\right)K_{\nu+1}(\zeta),$
$\displaystyle\int_{0}^{\infty}\mathrm{d}\vartheta\sinh^{2\nu}\vartheta\cosh^{2}\vartheta\,\mathrm{e}^{-\zeta\cosh\vartheta}=\frac{1}{\sqrt{\pi}}\left(\frac{2}{\zeta}\right)^{\nu}\Gamma\left(\nu+\frac{1}{2}\right)\left[K_{\nu+2}(\zeta)-\zeta^{-1}K_{\nu+1}\right].$

These integrations are sufficient for us to analyze the following observable
quantities,

$\displaystyle n$
$\displaystyle=-\frac{1}{c^{2}}Z_{\mu}N^{\mu}=-\frac{\mathrm{e}^{-\alpha}}{ch^{d}}\int\frac{\mathrm{d}^{d}p}{|p_{0}|}Z_{\mu}p^{\mu}\mathrm{e}^{\mathcal{B}_{\mu}p^{\mu}},$
(55) $\displaystyle\epsilon$
$\displaystyle=\frac{1}{c^{2}}Z_{\mu}Z_{\nu}T^{\mu\nu}=\frac{\mathrm{e}^{-\alpha}}{ch^{d}}\int\frac{\mathrm{d}^{d}p}{|p_{0}|}\left(Z_{\mu}p^{\mu}\right)^{2}\mathrm{e}^{\mathcal{B}_{\mu}p^{\mu}},$
(56) $\displaystyle\mathcal{T}$
$\displaystyle=\frac{c\,\mathrm{e}^{-\alpha}}{h^{d}}\int\frac{\sqrt{g}\,\mathrm{d}^{d}p}{|p_{0}|}p^{2}\mathrm{e}^{\mathcal{B}_{\rho}p^{\rho}}=-\frac{m^{2}c^{3}\mathrm{e}^{-\alpha}}{h^{d}}\int\frac{\mathrm{d}^{d}p}{|p_{0}|}e^{\mathcal{B}_{\mu}p^{\mu}},$
(57)

where $\mathcal{B}_{\mu}p^{\mu}=\zeta\cosh\vartheta$ where $\zeta$ is the
relativistic coldness, and the contraction $Z_{\mu}p^{\mu}$ is simplified
after taking $\varTheta_{d-1}=0$,

$Z_{\mu}p^{\mu}=mc^{2}\left(\sinh\eta\sinh\vartheta\cos\theta_{d-1}-\cosh\eta\cosh\vartheta\right)=mc^{2}\mathcal{C}(\eta,\vartheta,\theta_{d-1}).$

When $d=1$ there is only one spacial direction and the above integrations can
be carried out with ease. First, the trace of energy-momentum tensor reads

$\displaystyle\mathcal{T}$
$\displaystyle=-\frac{\mathrm{e}^{-\alpha}}{\lambda_{C}}mc^{2}\int_{-\infty}^{+\infty}\mathrm{d}\vartheta\mathrm{e}^{-\zeta\cosh\vartheta}=-\frac{2\mathrm{e}^{-\alpha}}{\lambda_{C}}mc^{2}K_{0}(\zeta).$
(58)

For the particle number density and energy density, after a few lines of
simple calculations we obtain

$\displaystyle n$
$\displaystyle=-\frac{\mathrm{e}^{-\alpha}}{\lambda_{C}}\int_{-\infty}^{\infty}\mathrm{d}\vartheta\left(\sinh\eta\sinh\vartheta-\cosh\eta\cosh\vartheta\right)\mathrm{e}^{-\zeta\cosh\vartheta}$
$\displaystyle=\frac{2\mathrm{e}^{-\alpha}}{\lambda_{C}}\cosh\eta\int_{0}^{\infty}\mathrm{d}\vartheta\cosh\vartheta\mathrm{e}^{-\zeta\cosh\vartheta}$
$\displaystyle=\gamma\frac{2\mathrm{e}^{-\alpha}}{\lambda_{C}}K_{1}(\zeta),$
(59)

and

$\displaystyle\epsilon$
$\displaystyle=\frac{\mathrm{e}^{-\alpha}}{\lambda_{C}}mc^{2}\int_{-\infty}^{\infty}\mathrm{d}\vartheta\left(\sinh\eta\sinh\vartheta-\cosh\eta\cosh\vartheta\right)^{2}e^{-\zeta\cosh\vartheta}$
$\displaystyle=\frac{2\mathrm{e}^{-\alpha}}{\lambda_{C}}mc^{2}\int_{0}^{\infty}\mathrm{d}\vartheta\left[\gamma^{2}\cosh
2\vartheta-\sinh^{2}\vartheta\right]e^{-\zeta\cosh\vartheta}$
$\displaystyle=\frac{2\mathrm{e}^{-\alpha}}{\lambda_{C}}mc^{2}\left[\gamma^{2}K_{2}(\zeta)-\frac{K_{1}(\zeta)}{\zeta}\right].$
(60)

When $d\geq 2$, the integrations (55)-(57) can be evaluated in a unified way,

$\displaystyle n$
$\displaystyle=-\frac{\mathrm{e}^{-\alpha}}{\lambda_{C}^{d}}\int\mathrm{d}\Omega_{d-1}\int_{0}^{\infty}\mathrm{d}\vartheta\sinh^{d-1}\vartheta\,\mathcal{C}(\eta,\vartheta,\theta_{d-1})\mathrm{e}^{-\zeta\cosh\vartheta}$
$\displaystyle=\frac{\mathrm{e}^{-\alpha}}{\lambda_{C}^{d}}\cosh\eta\int\mathrm{d}\Omega_{d-1}\int_{0}^{\infty}\mathrm{d}\vartheta\sinh^{d-1}\vartheta\cosh\vartheta\,\mathrm{e}^{-\zeta\cosh\vartheta}$
$\displaystyle=\frac{\mathcal{A}_{d-1}\mathrm{e}^{-\alpha}}{\lambda_{C}^{d}}\gamma\int_{0}^{\infty}\mathrm{d}\vartheta\sinh^{d-1}\vartheta\cosh\vartheta\,\mathrm{e}^{-\zeta\cosh\vartheta}$
$\displaystyle=\gamma\frac{2\,\mathrm{e}^{-\alpha}}{\lambda_{C}^{d}}\left(\frac{2\pi}{\zeta}\right)^{\frac{d-1}{2}}K_{\frac{d+1}{2}}(\zeta),$
(61)

where
$\displaystyle\mathcal{A}_{d-1}=\int\mathrm{d}\Omega_{d-1}=\frac{2\pi^{d/2}}{\Gamma\left(\frac{d}{2}\right)}$
is the area of a $(d-1)$-dimensional unit sphere,

$\displaystyle\epsilon$
$\displaystyle=\frac{\mathrm{e}^{-\alpha}}{\lambda_{C}^{d}}mc^{2}\int\mathrm{d}\Omega_{d-1}\int_{0}^{\infty}\mathrm{d}\vartheta\sinh^{d-1}\vartheta\,\mathcal{C}^{2}(\eta,\vartheta,\theta_{d-1})\,\mathrm{e}^{-\zeta\cosh\vartheta}$
$\displaystyle=\frac{\mathrm{e}^{-\alpha}}{\lambda_{C}^{d}}mc^{2}\left[\gamma^{2}\mathcal{A}_{d-1}\int_{0}^{\infty}\mathrm{d}\vartheta\sinh^{d-1}\vartheta\cosh^{2}\vartheta\,\mathrm{e}^{-\zeta\cosh\vartheta}\right.$
$\displaystyle\qquad\qquad\qquad\left.+(\gamma^{2}-1)\mathcal{A}_{d-2}\frac{\sqrt{\pi}}{2}\frac{\Gamma\left(\frac{d-1}{2}\right)}{\Gamma\left(\frac{d}{2}+1\right)}\int_{0}^{\infty}\mathrm{d}\vartheta\sinh^{d+1}\vartheta\,\mathrm{e}^{-\zeta\cosh\vartheta}\right]$
$\displaystyle=\frac{\mathrm{e}^{-\alpha}}{\lambda_{C}^{d}}mc^{2}\mathcal{A}_{d-1}\int_{0}^{\infty}\mathrm{d}\vartheta\left[\gamma^{2}\sinh^{d-1}\vartheta\cosh^{2}\vartheta+\frac{\gamma^{2}-1}{d}\sinh^{d+1}\vartheta\right]\mathrm{e}^{-\zeta\cosh\vartheta}$
$\displaystyle=\frac{\mathrm{e}^{-\alpha}}{\lambda_{C}^{d}}\frac{\mathcal{A}_{d-1}}{\sqrt{\pi}}\left(\frac{2}{\zeta}\right)^{\frac{d-1}{2}}\Gamma\left(\frac{d}{2}\right)mc^{2}\left[\gamma^{2}\left(K_{\frac{d+3}{2}}-\zeta^{-1}K_{\frac{d+1}{2}}\right)+\frac{\gamma^{2}-1}{d}\frac{2}{\zeta}\frac{d}{2}K_{\frac{d+1}{2}}\right]$
$\displaystyle=\frac{2\,\mathrm{e}^{-\alpha}}{\lambda_{C}^{d}}\left(\frac{2\pi}{\zeta}\right)^{\frac{d-1}{2}}mc^{2}\left[\gamma^{2}K_{\frac{d+3}{2}}(\zeta)-\zeta^{-1}K_{\frac{d+1}{2}}(\zeta)\right],$
(62)

and

$\displaystyle\mathcal{T}$
$\displaystyle=-\frac{\mathrm{e}^{-\alpha}}{\lambda_{C}^{d}}\mathcal{A}_{d-1}mc^{2}\int_{0}^{\infty}\mathrm{d}\vartheta\sinh^{d-1}\vartheta\,\mathrm{e}^{-\zeta\cosh\vartheta}$
$\displaystyle=-\frac{\mathrm{e}^{-\alpha}}{\lambda_{C}^{d}}\frac{\mathcal{A}_{d-1}}{\sqrt{\pi}}\left(\frac{2}{\zeta}\right)^{\frac{d-1}{2}}\Gamma\left(\frac{d}{2}\right)mc^{2}K_{\frac{d-1}{2}}(\zeta)$
$\displaystyle=-\frac{2\,\mathrm{e}^{-\alpha}}{\lambda_{C}^{d}}\left(\frac{2\pi}{\zeta}\right)^{\frac{d-1}{2}}mc^{2}K_{\frac{d-1}{2}}(\zeta).$
(63)

Interestingly, the results (61)-(63) calculated for $d\geq 2$ are also valid
for $d=1$, as we have shown in eqs.(58)-(60) .

For the thermodynamic pressure we refer to (28), with the replacement
$\displaystyle\frac{1}{3}\to\frac{1}{d}$. By evaluating the isotropic pressure
in the rest frame where $\gamma=1$, we get

$\displaystyle
P=\bar{P}=\frac{\mathcal{T}+\bar{\epsilon}}{d}=\frac{2\,\mathrm{e}^{-\alpha}}{\lambda_{C}^{d}}\left(\frac{2\pi}{\zeta}\right)^{\frac{d-1}{2}}\zeta^{-1}mc^{2}K_{\frac{d+1}{2}}(\zeta),$
(64)

finally the enthalpy density is obtained to be

$\displaystyle
w=\epsilon+P=\gamma^{2}\left[\frac{2\,\mathrm{e}^{-\alpha}}{\lambda_{C}^{d}}\left(\frac{2\pi}{\zeta}\right)^{\frac{d-1}{2}}\right]mc^{2}K_{\frac{d+3}{2}}(\zeta).$
(65)

This finishes our calculations for the relevant quantities in arbitrary
dimensions.

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