# The maximum entropy principle of self-gravitating fluid system and field
equations

Hikaru Yumisaki

Gunma University, 3-39-22 Showa, Maebashi 371-8511, Japan e-mail:
<EMAIL_ADDRESS>
###### Abstract

We investigate the maximum entropy principle for general field theory,
including a metric tensor $g_{\mu\nu}$, a vector field $A_{\mu}$, and a scalar
field $\varphi$ as the fundamental fields, and find (i) imposing an ordinary
constraint relation on $\varepsilon$, the field equations, which is
constructed of the Euler-Lagrange derivative of an arbitrary Lagrangian
density, the stress tensor of a perfect fluid, and the electric current
vector, in ordinary manner, are compatible with the maximum entropy principle,
and (ii) varying also the constraint relation on $\varepsilon$, the maximum
entropy principle requires an extra scalar field $\eta$, which is introduced
as the difference from the ordinary constraint relation on $\varepsilon$. The
field $\eta$ is also interpreted as the difference between two geometries,
i.e., one is the geometry, defined by $g_{\mu\nu}$, in which the
thermodynamical relations are written in the ordinary and simplest form, and
the other is the geometry, defined by

$\tilde{g}_{\mu\nu}=e^{2\eta}g_{\mu\nu},$

in which a force-free fluid flows along a geodesic orbit, instead the
thermodynamical relations, the first law of thermodynamics and the Gibbs-Duhem
relation, are modified.

We also calculate the variation of the entropy $\delta S$ in the Lovelock
theory, in which $\delta S$ is expressed as a linear combination of the
variations of the Kodama energy and the size of the system. Finally, with the
field $\eta$ introduced, we propose a set of field equations in the
thermodynamical and kinematical geometries, which possesses appropriate
scaling properties, and point out that in vacuum spacetime they resemble those
of the dilaton gravity in the string and Einstein frame, respectively.

###### Contents

  1. 1 Introduction
    1. 1.1 Background and motivation
    2. 1.2 Conventions and notations
  2. 2 Thermodynamical relations and the second law
  3. 3 The Euler-Lagrange operator
    1. 3.1 Definition
    2. 3.2 The variation of $g^{\mu\nu}$ and $u^{\mu}$
  4. 4 Volume term
    1. 4.1 From the maximum entropy principle to field equations
    2. 4.2 Consistency conditions
    3. 4.3 Redefinition of the physical quantities
  5. 5 Surface term
    1. 5.1 Spherically symmetric spacetime and the generalized Kodama flux
    2. 5.2 The Lovelock theory
    3. 5.3 The variation of the entropy in the Lovelock theory
  6. 6 Constraints on the Lagrangian
  7. 7 Conclusion and discussion
  8. A The homogeneity of the extensive quantities
  9. B Noether’s second theorem
    1. B.1 Coordinate transformation
    2. B.2 Identities derived from the coordinate transformation invariance
    3. B.3 Identities from the $U(1)$ gauge transformation

##  1 Introduction

###  1.1 Background and motivation

One of the properties of gravity that leads to difficulties in studying
thermodynamics of self-gravitating fluid is that it generates long range self-
interactions. In statistical picture, long range self-interactions affect the
additivity of the macroscopic quantities, and it becomes impossible to
understand the phenomena of the whole system as a simple sum of small parts.
Therefore, the traditional statistical procedure does not guarantee the
validity of the simple combination of thermodynamics and a gravity theory. On
the other hand, in purely thermodynamical picture, since the presence of long
range self-interaction relates the thermodynamical phenomena and dynamics of
fields, requiring the compatibility of the two theories, namely thermodynamics
and field theory, may give us some perspectives for macroscopic long range
self-interacting system. In this work, we focus on features of the final state
of a system in thermodynamical and field theoretical pictures, namely
equilibrium and stationary states, respectively, and investigate their
compatibility.

In thermodynamical picture, an essential feature of the final state of a
system is stated by the concept of “entropy”. The second law of thermodynamics
states that, given an arbitrary isolated system $\mathcal{V}$, there exist a
quantity $S$, which is called entropy, for each states such that the
irreversibility of the system is represented by forbidding the entropy to
decrease. Therefore, if there exists the final state of the system, called the
equilibrium state, then the equilibrium state is the state of maximum entropy
among all the allowed states, which is called “the maximum entropy principle”.
The range of the allowed states mentioned above depends on the give situation.

On the other hand, including also kinematical picture, we believe that the
final state of a system is static or more generally stationary. For example,
in an ordinary field theory with a self-interact fluid system, if a state is
not static nor stationary, namely dynamical, then the entropy of a system can
increase without violating the energy and the other conservation laws by
converting the kinetic energy into heat energy.111We have assumed the
positivity of kinetic energy of the fields including the gravity, the
conservation of total energy, and that the entropy is monotonically increasing
function for heat energy. It is well known that the first two assumptions are
delicate problems in gravitational field theories [1]. Therefore, it is
obvious that the entropy of a static state is a maximum among those of all the
states that satisfy the given field equations, so-called “on-shell” states,
with the total energy and the other conserved charges fixed. However, it is a
nontrivial problem whether the entropy of a static state is maximum among
those of all the states that contains also the states that do not satisfy the
field equations, namely “off-shell” states.

Is it necessary that the entropy of a static solution of the given field
equations is maximum among those of configurations including off-shell ones?
Suppose that an infinitesimal variation increase the entropy of a static
solution, then the second law of thermodynamics indicates that the “static”
solution varies so as to increase its own entropy222In general, a force acting
in a system resulting from the tendency to increase the total entropy is
called an “entropic force”., that is to say, the requirement of field theory
conflicts with that of thermodynamics. One can interpret the situation as an
instability of the static solution to thermal fluctuations.

In this work, we adopt as the maximum entropy principle for field theory that
the entropy of an arbitrary static solution of the field equations should be
maximum among those of all the configurations that satisfy given appropriate
boundary conditions. The aim of this study is to reveal the condition for
field theories to be compatible with the maximum entropy principle. Indeed,
one finds below that in such a field theory not only static solutions but also
stationary ones satisfy the maximum entropy principle.

Related calculations are found in literature. In [2], it is stated that
requiring to vanish the variation of the entropy among spherically symmetric
solutions of the Einstein equation is consistent with the equation of
hydrostatic equilibrium. In [3], for a spherically symmetric and static
radiation fluid system in general relativity, it is shown that the initial
value constraint equations (equivalent to the time-time component of the
Einstein equation) and requiring for the total entropy to be maximum with the
total mass fixed derive the Tolman-Oppenheimer-Volkoff (TOV) equation of
hydrostatic equilibrium. Generalizations have been done, i.e., for arbitrary
equation of state of fluids [4], non-spherically symmetric system [5], and
attempts to apply other field theories, namely, the Lovelock theory [6], the
$f(R)$ theory [7], Einstein-Maxwell theory [8], and generally covariant purely
gravitational theories [9]. However, since these calculations are based on
variations restricted by some constraint conditions such as spherical or time
translational symmetries, Tolman’s law, or the time-time component of the
gravitational field equation, calculations for general field theories and
based on general variations have not yet done.

In this paper, in order to keep the generality, we do not use any assumptions
about the field equations, except for that the thermodynamical quantities,
including the entropy, are functions of a symmetric second-order tensor field
$g^{\mu\nu}$, a vector field $A_{\mu}$, some scalar fields, and their
arbitrarily high order derivatives.

This paper is organized as follows. In section 2, using thermodynamical
relations, we calculate the first order variation of the entropy. In section
3, the Euler-Lagrange operator is generalized to deal with functions that
contain arbitrarily high order derivatives of fields. In section 4, we
consider the volume term of the variation of the entropy, and derive the
conditions for the stationary states to be equilibrium. Redefining the metric
tensor and extensive quantities in order for them to have physically
reasonable features, for example force-free fluids flow along geodesic orbits,
we introduce the kinematical geometry, which is described by the redefined
metric tensor $\tilde{g}_{\mu\nu}$. In this frame, we obtain the field
equations (172) - (175) for arbitrary Lagrangian that are compatible with the
maximum entropy principle. In section 5, we consider the surface term of the
variation of the entropy. Each surface term is interpreted as the product of a
“chemical potential” and the variation of the conjugate quantity. As an
example, we explicitly calculate the variation of the entropy of a spherically
symmetric system in the Lovelock theory and show that the variation of the
entropy is expressed as a linear combination of those of the Kodama energy and
the size of the system. In section 6, a Lagrangian which has thermodynamically
reasonable properties is presented. In section 7, we summarize the
conclusions, and briefly discuss their applications to other topics, such as
cosmology, thermodynamical aspects of black hole, and so on.

###  1.2 Conventions and notations

In this paper, we use symbols in the following conventions. The dimension of
spacetime is denoted by $d$. The signature of Lorentzian metric tensor obeys
the mostly plus convention $(-,+,\cdots,+)$. Greek indexes
$\alpha,\beta,\cdots$ span the full dimension of spacetime, namely
$\\{0,1,\cdots,d-1\\}$. The index $0$ often represents a timelike coordinate,
whereas Latin indexes $a,b,\cdots$ do spacelike ones, so
$a,b,\cdots\in\\{1,\cdots,d\\}$.

The Levi-Civita symbol $\epsilon_{\mu_{1}\cdots\mu_{d}}$ is defined by

$\displaystyle\epsilon_{\mu_{1}\cdots\mu_{d}}=\begin{cases}+1\ \ \ \ \ \ {\rm
if\ }(\mu_{1},\cdots,\mu_{d}){\rm\ is\ an\ even\ permutation\ of\
}(0,1,\cdots,d-1),&\\\ -1\ \ \ \ \ \ {\rm if\ }(\mu_{1},\cdots,\mu_{d}){\rm\
is\ an\ odd\ permutation\ of\ }(0,1,\cdots,d-1),&\\\ \ \ \ 0\ \ \ \ \ \ {\rm
otherwise,}&\end{cases}$ (1)

in any coordinates, which is a $(0,d)$-type covariant tensor density of weight
$-1$.

We use the symbols which have $n$ subscripts for the bases of $(d-n)$-forms
defined by

$\displaystyle({\mathrm{d}}^{d-n}x)_{\mu_{1}\cdots\mu_{n}}:=\frac{1}{(d-n)!}\epsilon_{\mu_{1}\cdots\mu_{n}\nu_{1}\cdots\nu_{d-n}}{\mathrm{d}}x^{\nu_{1}}\wedge\cdots\wedge{\mathrm{d}}x^{\nu_{d-1}},$
(2)

so that a $(n,0)$-type covariant tensor density
$\sqrt{-g}X^{\mu_{1}\cdots\mu_{n}}$ is naturally associated with a
$(d-n)$-form $\mathbb{X}$, i.e.,

$\displaystyle\mathbb{X}=\frac{1}{n!}({\mathrm{d}}^{d-n}x)_{\mu_{1}\cdots\mu_{n}}\sqrt{-g}X^{\mu_{1}\cdots\mu_{n}}.$
(3)

A set of fields which are taken for the fundamental ones are denoted by $f$.
For example, when we think of $g^{\mu\nu}$, $A_{\mu}$, and $\varphi$ as
fundamental fields, then $f=\\{g^{\mu\nu},A_{\mu},\varphi\\}$.

##  2 Thermodynamical relations and the second law

Consider a $(d-1)$-dimensional region $\mathcal{V}$ of a fluid system in
$d$-dimensional spacetime $\mathcal{M}$. The $d$-velocity of the fluid is
denoted by $u^{\mu}$. The entropy $S$ of the region $\mathcal{V}$ is defied as
a real number such that we can represent the irreversibility of $\mathcal{V}$
by asserting that the quantity $S$ never decrease, which is called the second
law of thermodynamics. We assume that the entropy $S$ always exists and is
written in the form

$\displaystyle
S=\int_{\mathcal{V}}({\mathrm{d}}^{d-1}x)_{\mu}\sqrt{-g}u^{\mu}s(\varepsilon,n),$
(4)

for an arbitrary region $\mathcal{V}$. We employ a second-order symmetric
Lorentzian tensor $g_{\mu\nu}$ and its determinant $g$. The type of the fluid
determines the form of the scalar function $s(\varepsilon,n)$ which depends on
two scalar quantities $\varepsilon$ and $n$. The generalization to the case
that there exist more than one $n$ is straightforward. Since we do not specify
the type of fluid, $s(\varepsilon,n)$ is an arbitrary function. The quantity
$\sqrt{-g}u^{\mu}s$ is called the entropy current density.

The $d$-velocity of the fluid $u^{\mu}$ is normalized by

$\displaystyle g_{\mu\nu}u^{\mu}u^{\nu}=-1.$ (5)

In this paper, we often use a coordinate, called a “co-moving coordinate”, in
which the $d$-velocity $u^{\mu}$ is parallel to the time axis, namely
$u^{\mu}\partial_{\mu}=\sqrt{-g^{00}}\partial_{0}$, and the hypersurface
defied by $x^{0}=0$ contains the region $\mathcal{V}$, i.e.,
$\\{x^{1},\cdots,x^{d-1}\\}$ is a coordinate of $\mathcal{V}$.

Supposing that the conservation law of the entropy current density

$\displaystyle\partial_{\mu}\Big{(}\sqrt{-g}u^{\mu}s\Big{)}=0$ (6)

is satisfied, and the boundary of the system $\partial\mathcal{V}$ is fixed,
then the entropy $S$ does not depend on the choice of the region
$\mathcal{V}$. We also assume the conservation law of $\sqrt{-g}u^{\mu}n$,
namely,

$\displaystyle\partial_{\mu}\Big{(}\sqrt{-g}u^{\mu}n\Big{)}=0,$ (7)

so that the quantity

$\displaystyle
N=\int_{\mathcal{V}}\big{(}{\mathrm{d}}^{d-1}x\big{)}_{\mu}\sqrt{-g}u^{\mu}n$
(8)

is a conserved charge. In section 4.2, we check whether Eqs. (6) and (7) hold,
or not.

Define the “intensive” quantities $T$, $\mu$, and $p$ by

$\displaystyle\frac{1}{T}$ $\displaystyle:=$ $\displaystyle\frac{\partial
s}{\partial\varepsilon}$ (9) $\displaystyle\frac{\mu}{T}$ $\displaystyle:=$
$\displaystyle-\frac{\partial s}{\partial n}$ (10) $\displaystyle p$
$\displaystyle:=$ $\displaystyle Ts+\mu n-\varepsilon.$ (11)

These relations are often represented in the form

$\displaystyle\delta\varepsilon=T\delta s+\mu\delta n$ (12) $\displaystyle
Ts+\mu n=\varepsilon+p.$ (13)

The relation (12) is a local representation of the first law of
thermodynamics, and (13) is the Gibbs-Duhem relation. The relation (13) comes
from the extensive properties of the extensive quantities for sufficiently
small region (See appendix A for details). Comparing Eq. (12) and the
variation of Eq. (13), one finds the constraint relation among the variation
of the intensive quantities:

$\displaystyle\delta p=s\delta T+n\delta\mu.$ (14)

Here, we emphasize that although usually $g_{\mu\nu}$, $\varepsilon$, $n$,
$T$, $\mu$, and $p$ are interpreted as the metric tensor, energy density,
particle number density, temperature, chemical potential, and pressure,
respectively, we have not said about the physical meanings of them yet, except
for the volume element $\sqrt{-g}$, which is introduced respecting the
additivity of the entropy. In this work, we call the geometry defined by
$g_{\mu\nu}$ the “thermodynamical geometry”. In section 4.3, we will discuss
whether the interpretation of these quantities are appropriate even if the
kinematical features are taken into account, or not.

Let us calculate the variation of the entropy

$\displaystyle\delta
S=\int_{\mathcal{V}}({\mathrm{d}}^{d-1}x)_{\mu}\delta\Big{(}\sqrt{-g}u^{\mu}s\Big{)}.$
(15)

Using Eq. (12), (13), and the Leibniz rule of $\delta$, the integrand of
$\delta S$ is arranged as follows:

$\displaystyle\delta\Big{(}\sqrt{-g}u^{\mu}s\Big{)}$ $\displaystyle=$
$\displaystyle\sqrt{-g}u^{\mu}\delta s+s\delta\Big{(}\sqrt{-g}u^{\mu}\Big{)}$
(16) $\displaystyle=$
$\displaystyle\sqrt{-g}\frac{u^{\mu}}{T}(\delta\varepsilon-\mu\delta
n)+\frac{\varepsilon+p-\mu n}{T}\delta\Big{(}\sqrt{-g}u^{\mu}\Big{)}$ (17)
$\displaystyle=$
$\displaystyle\frac{1}{T}\bigg{[}\sqrt{-g}u^{\mu}\delta\varepsilon+(\varepsilon+p)\delta\Big{(}\sqrt{-g}u^{\mu}\Big{)}-\mu\delta\Big{(}\sqrt{-g}u^{\mu}n\Big{)}\bigg{]}$
(18) $\displaystyle=$
$\displaystyle\frac{1}{T}\delta\Big{(}\sqrt{-g}u^{\mu}\varepsilon\Big{)}+\frac{p}{T}\delta\Big{(}\sqrt{-g}u^{\mu}\Big{)}-\frac{\mu}{T}\Big{(}\sqrt{-g}u^{\mu}n\Big{)}.$
(19)

Decomposing the variation by the Leibniz rule,

$\displaystyle\delta\Big{(}\sqrt{-g}u^{\mu}s\Big{)}=-\frac{\mu}{T}\delta\Big{(}\sqrt{-g}u^{\mu}n\Big{)}+\frac{1}{T}\bigg{[}u^{\mu}\delta\Big{(}\sqrt{-g}\varepsilon\Big{)}+\sqrt{-g}\varepsilon\delta
u^{\mu}+p\Big{(}\sqrt{-g}\delta
u^{\mu}+u^{\mu}\delta\sqrt{-g}\Big{)}\bigg{]}.$ (20)

The relations among the variations of $u^{\mu}$, $\sqrt{-g}$, and $g^{\mu\nu}$
are required. Varying the normalization condition (5),

$\displaystyle u_{\mu}\delta u^{\mu}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}u^{\mu}u^{\nu}\delta g_{\mu\nu}$ (21)
$\displaystyle=$ $\displaystyle\frac{1}{2}u_{\mu}u_{\nu}\delta g^{\mu\nu},$
(22)

where we use the relation $g^{\mu\rho}g_{\rho\nu}=\delta^{\mu}_{\nu}$.
Therefore, the variations of $u^{\mu}$ can be written in the form

$\displaystyle\delta u^{\mu}=-\Big{(}\frac{1}{2}u_{\rho}u_{\sigma}\delta
g^{\rho\sigma}\Big{)}u^{\mu}+\big{(}u^{\mu}u_{\nu}+\delta^{\mu}_{\nu}\big{)}\delta
u^{\nu}.$ (23)

Notice that $-u^{\mu}u_{\nu}$ is the projection operator on the direction
parallel to $u^{\mu}$, and $u^{\mu}u_{\nu}+\delta^{\mu}_{\nu}$ is that of the
$(d-1)$-dimensional hypersurface orthogonal to $u^{\mu}$. The first term of
the right hand side of Eq. (23) represents the variation through the change of
the normalization condition, whereas the second term does that of the
direction of $d$-velocity $u^{\mu}$. As we are interested in the variation
with the direction of $d$-velocity $u^{\mu}$ fixed, the second term of Eq.
(23) is ignored, namely

$\displaystyle\delta u^{\mu}=-\Big{(}\frac{1}{2}u_{\rho}u_{\sigma}\delta
g^{\rho\sigma}\Big{)}u^{\mu}.$ (24)

The variation of $u_{\mu}$ is

$\displaystyle\delta u_{\mu}$ $\displaystyle=$
$\displaystyle\delta\big{(}g_{\mu\nu}u^{\nu}\big{)}$ (25) $\displaystyle=$
$\displaystyle-
u_{\rho}\Big{(}\frac{1}{2}u_{\mu}u_{\sigma}+g_{\mu\sigma}\Big{)}\delta
g^{\rho\sigma}.$ (26)

The variation of the projection operator $-u_{\mu}u^{\nu}$ is

$\displaystyle\delta\big{(}-u_{\mu}u^{\nu}\big{)}=u^{\nu}u_{\rho}\big{(}u_{\mu}u_{\sigma}+g_{\mu\sigma}\big{)}\delta
g^{\rho\sigma}.$ (27)

Substituting Eq. (24) and the formula [1]

$\displaystyle\delta\sqrt{-g}=-\frac{1}{2}\sqrt{-g}g_{\mu\nu}\delta
g^{\mu\nu},$ (28)

into Eq. (20), one obtains the following variational equation:

$\displaystyle\delta\Big{(}\sqrt{-g}u^{\mu}s\Big{)}$ $\displaystyle=$
$\displaystyle-\lambda\delta\Big{(}\sqrt{-g}u^{\mu}n\Big{)}+\frac{u^{\mu}}{T}\bigg{\\{}\delta\Big{(}\sqrt{-g}\varepsilon\Big{)}-\frac{1}{2}\sqrt{-g}\Big{[}\varepsilon
u_{\rho}u_{\sigma}+p\big{(}u_{\rho}u_{\sigma}+g_{\rho\sigma}\big{)}\Big{]}\delta
g^{\rho\sigma}\bigg{\\}},$

where the quantity $\lambda$ is introduced by

$\displaystyle\lambda:=\frac{\mu}{T}.$ (30)

##  3 The Euler-Lagrange operator

In this work, we deal with the variations of quantities which contain
arbitrarily high order derivatives of fields, so that the higher-order Euler-
Lagrange procedures often appear in the following calculations. In this
section, we define the Euler-Lagrange operator $\,{}^{\rm
EL}\\!\mathcal{D}_{f}$ and present some formulae.

###  3.1 Definition

Consider the variation of a quantity $A$ that depends on arbitrarily high
order derivatives of fundamental fields $f$. Using the Leibniz’s rule of the
partial derivative $\partial_{\mu}$ and the commutative relation
$\delta\partial_{\mu}=\partial_{\mu}\delta$, the first-order variation of a
function $A$ is expanded in the following form:

$\displaystyle\delta A$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\frac{\partial
A}{\partial(\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}f)}\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}f$
(33) $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\Bigg{[}\partial_{\mu_{1}}\bigg{(}\frac{\partial
A}{\partial(\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}f)}\delta\partial_{\mu_{2}}\cdots\partial_{\mu_{j}}f\bigg{)}-\partial_{\mu_{1}}\frac{\partial
A}{\partial(\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}f)}\delta\partial_{\mu_{2}}\cdots\partial_{\mu_{j}}f\Bigg{]}$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\Bigg{[}\partial_{\mu_{1}}\bigg{(}\frac{\partial
A}{\partial(\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}f)}\delta\partial_{\mu_{2}}\cdots\partial_{\mu_{j}}f\bigg{)}-\partial_{\mu_{2}}\bigg{(}\partial_{\mu_{1}}\frac{\partial
A}{\partial(\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}f)}\delta\partial_{\mu_{3}}\cdots\partial_{\mu_{j}}f\bigg{)}$
$\displaystyle+\partial_{\mu_{2}}\partial_{\mu_{1}}\frac{\partial
A}{\partial(\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}f)}\delta\partial_{\mu_{3}}\cdots\partial_{\mu_{j}}f\Bigg{]}$
$\displaystyle=$ $\displaystyle\cdots$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\Bigg{[}\sum_{i=0}^{j-1}\partial_{\mu_{i+1}}\bigg{(}(-1)^{i}\partial_{\mu_{i}}\cdots\partial_{\mu_{1}}\frac{\partial
A}{\partial(\partial_{\mu_{1}}\cdots\partial_{\mu_{i+1}}\cdots\partial_{\mu_{j}}f)}\delta\partial_{\mu_{i+2}}\cdots\partial_{\mu_{j}}f\bigg{)}$
$\displaystyle+(-1)^{j}\partial_{\mu_{j}}\cdots\partial_{\mu_{1}}\frac{\partial
A}{\partial(\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}f)}\delta f\Bigg{]}.$

Notice that in variational calculation, the quantities
$\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}f$ are regarded as independent
quantities each other, after that those quantities are identified. Therefore,
there are ambiguities in the partial derivatives in terms of
$\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}f$. In this work, those ambiguities
are removed so that all the (anti-)symmetries in the arguments are inherited
by the partial derivatives, for example

$\displaystyle\frac{\partial
A}{\partial(\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}f)}=\frac{\partial
A}{\partial(\partial_{\mu_{P(1)}}\cdots\partial_{\mu_{P(j)}}f)},$ (34)

where $P(i)$ is a permutation of $i$.

Introducing a subscript $k$ by

$\displaystyle j=i+k+1,$ (35)

the variation of $A$ is written in the following form

$\displaystyle\delta A$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\sum_{i=0}^{\infty}\partial_{\mu_{i+1}}\bigg{(}(-1)^{i}\partial_{\mu_{i}}\cdots\partial_{\mu_{1}}\frac{\partial
A}{\partial(\partial_{\mu_{1}}\cdots\partial_{\mu_{i+1}}\cdots\partial_{\mu_{i+k+1}}f)}\delta\partial_{\mu_{i+2}}\cdots\partial_{\mu_{i+k+1}}f\bigg{)}$
(36)
$\displaystyle+\sum_{j=0}^{\infty}(-1)^{j}\partial_{\mu_{j}}\cdots\partial_{\mu_{1}}\frac{\partial
A}{\partial(\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}f)}\delta f.$

Defining the Euler-Lagrange operator $\,{}^{\rm EL}\\!\mathcal{D}_{f}$ by

$\,{}^{\rm
EL}\\!\mathcal{D}_{f}A:=\sum_{j=0}^{\infty}(-1)^{j}\partial_{\mu_{j}}\cdots\partial_{\mu_{1}}\frac{\partial
A}{\partial(\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}f)},$ (37)

the variation of $A$ can be written as

$\displaystyle\delta A=\,^{\rm EL}\\!\mathcal{D}_{f}A\cdot\delta
f+\partial_{\mu}\bigg{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}A\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)}.$
(38)

We call $\,{}^{\rm EL}\\!\mathcal{D}_{f}A$ the Euler-Lagrange “derivative”
333The Euler-Lagrange derivative $\,{}^{\rm EL}\\!\mathcal{D}_{f}$ does not
satisfy the Leibniz’s rule. of $A$ with respect to $f$.

The commutative relation $\delta$ and $\partial_{\mu}$ derives some formulae.
From Eq. (33) and (38), the variation of $\partial_{\mu}A$ and the divergence
of $\delta A^{\mu}$ are written in the following forms respectively:

$\displaystyle\delta\partial_{\mu}A^{\mu}=\,^{\rm
EL}\\!\mathcal{D}_{f}\partial_{\mu}A^{\mu}\cdot\delta
f+\partial_{\rho}\bigg{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\partial_{\mu}A^{\mu}\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)},$
(39)

and

$\displaystyle\delta\partial_{\mu}A^{\mu}=\partial_{\rho}\delta
A^{\rho}=\partial_{\rho}\bigg{(}\sum_{k=0}^{\infty}\frac{\partial
A^{\rho}}{\partial(\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f)}\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)}.$
(40)

Comparing the two representations, one obtains following two formulae:

$\,{}^{\rm EL}\\!\mathcal{D}_{f}\partial_{\mu}A^{\mu}$ $\displaystyle=$
$\displaystyle 0,$ (41) $\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\partial_{\mu}A^{\mu}$
$\displaystyle=$ $\displaystyle\frac{\partial
A^{\rho}}{\partial(\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f)}.$ (42)

When the variable $f$ is a function of $h$ that does not depend on the
derivatives of $h$, namely $f=f(h)$,

$\displaystyle\delta A$ $\displaystyle=$ $\,{}^{\rm
EL}\\!\mathcal{D}_{f}A\cdot\delta
f+\partial_{\mu}\bigg{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}A\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)}$
(43) $\displaystyle=$ $\,{}^{\rm EL}\\!\mathcal{D}_{f}A\cdot f_{,h}\delta
h+\partial_{\mu}\bigg{[}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}A\cdot\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\big{(}f_{,h}\delta
h\big{)}\bigg{]}$ (44) $\displaystyle=$ $\,{}^{\rm EL}\\!\mathcal{D}_{f}A\cdot
f_{,h}\delta
h+\partial_{\mu}\bigg{[}\sum_{k=0}^{\infty}\sum_{j=0}^{k}\bigg{(}\begin{array}[]{c}k\\\
j\end{array}\bigg{)}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}A\cdot\partial_{\mu_{j+1}}\cdots\partial_{\mu_{k}}f_{,h}\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}h\bigg{]}.$
(47)

Comparing Eq. (LABEL:varyAh) with

$\displaystyle\delta A=\,^{\rm EL}\\!\mathcal{D}_{h}A\cdot\delta
h+\partial_{\mu}\bigg{(}\sum_{j=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}h}A\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}h\bigg{)},$
(49)

one obtains the transformation formulae for change of variables:

$\,{}^{\rm EL}\\!\mathcal{D}_{h}A$ $\displaystyle=$ $\displaystyle
f_{,h}\,^{\rm EL}\\!\mathcal{D}_{f}A$ (50) $\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu_{1}}\cdots\partial_{\mu_{i}}h}A$
$\displaystyle=$
$\displaystyle\sum_{k=i}^{\infty}\bigg{(}\begin{array}[]{c}k\\\
i\end{array}\bigg{)}\big{(}\partial_{\mu_{i+1}}\cdots\partial_{\mu_{k}}f_{,h}\big{)}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}A\ \ \ \ \ \
(i\geq 1).$ (53)

More generally, when the variable $f$ is a function of $h$, $\partial_{\mu}h$,
and their higher derivatives, namely $f=f(h,\partial_{\mu}h,\cdots)$, the
variations of $f$, $h$, and their derivatives are related by

$\displaystyle\delta f=\frac{\partial f}{\partial h}\delta h+\frac{\partial
f}{\partial(\partial_{\mu}h)}\delta\partial_{\mu}h+\cdots.$ (54)

The variation of a quantity $A$ is rearranged as

$\displaystyle\delta A$ $\displaystyle=$ $\displaystyle A_{f}\delta
f+\partial_{\mu}\bigg{(}\sum_{k=0}^{\infty}A_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\cdot\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)}$
(56) $\displaystyle=$ $\displaystyle\bigg{[}A_{f}\frac{\partial f}{\partial
h}-\partial_{\mu}\bigg{(}A_{f}\frac{\partial
f}{\partial(\partial_{\mu}h)}\bigg{)}+\cdots\bigg{]}\delta h$
$\displaystyle+\partial_{\mu}\bigg{[}\sum_{k=0}^{\infty}A_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\cdot\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f+A_{f}\frac{\partial
f}{\partial(\partial_{\mu}h)}\delta h+\cdots\bigg{]}$ $\displaystyle=:$
$\displaystyle B_{h}\delta
h+\partial_{\mu}\bigg{(}\sum_{k=0}^{\infty}B_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}h}\cdot\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}h\bigg{)},$
(57)

where $A_{f}$, $A_{\partial_{\mu}f}$, $\cdots$ are arbitrary coefficients. In
general, two quantities

$\displaystyle\partial_{\mu}\bigg{(}\sum_{k=0}^{\infty}A_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\cdot\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)}$
(58)

and

$\displaystyle\partial_{\mu}\bigg{(}\sum_{k=0}^{\infty}B_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}h}\cdot\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}h\bigg{)}$
(59)

are different. However, when the coefficient $A_{f}$ vanishes, then $B_{h}$
also vanishes and the above two coincide:

$\displaystyle\partial_{\mu}\bigg{(}\sum_{k=0}^{\infty}A_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\cdot\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)}=\partial_{\mu}\bigg{(}\sum_{k=0}^{\infty}B_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}h}\cdot\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}h\bigg{)}.$
(60)

###  3.2 The variation of $g^{\mu\nu}$ and $u^{\mu}$

Since the normalization condition of $u^{\mu}$ depends on $g^{\mu\nu}$, the
variation of quantities that depend on $u^{\mu}$ and $g^{\mu\nu}$ should be
calculated carefully.

Using the relation (38), the variation of $A=A[g^{\mu\nu},u^{\mu}]$ is
arranged as

$\displaystyle\delta A$ $\displaystyle=$ $\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\mu\nu}}A\cdot\delta g^{\mu\nu}+\,^{\rm
EL}\\!\mathcal{D}_{u^{\mu}}A\cdot\delta
u^{\mu}+\partial_{\rho}\bigg{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}A\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)}$
$\displaystyle=$ $\displaystyle\Big{(}\,^{\rm
EL}\\!\mathcal{D}_{g^{\mu\nu}}A-\frac{1}{2}u_{\mu}u_{\nu}u^{\rho}\,{}^{\rm
EL}\\!\mathcal{D}_{u^{\rho}}A\Big{)}\cdot\delta
g^{\mu\nu}+\partial_{\rho}\bigg{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}A\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)}.$

Defining the new operator $\,{}^{\rm EL}\\!\bar{\mathcal{D}}_{g^{\mu\nu}}$ by

$\,{}^{\rm EL}\\!\bar{\mathcal{D}}_{g^{\mu\nu}}:=\,^{\rm
EL}\\!\mathcal{D}_{g^{\mu\nu}}-\frac{1}{2}u_{\mu}u_{\nu}u^{\rho}\,{}^{\rm
EL}\\!\mathcal{D}_{u^{\rho}},$ (63)

the variation of $A$ is written in the form

$\displaystyle\delta A$ $\displaystyle=$ $\,{}^{\rm
EL}\\!\bar{\mathcal{D}}_{g^{\mu\nu}}A\cdot\delta
g^{\mu\nu}+\partial_{\rho}\bigg{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}A\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)}.$
(64)

##  4 Volume term

Let us consider the variation of the entropy current density in terms of
fundamental fields $f=\\{g^{\rho\sigma},A_{\alpha},\varphi\\}$ with the
direction of the $d$-velocity $u^{\mu}$ fixed. To avoid unnecessary
complexity, anti-symmetric second-order tensor $B_{\mu\nu}$ is removed, and
including them is straightforward.

As seen in the last of section 2, the first-order variation of the entropy
current density is

$\displaystyle\delta\Big{(}\sqrt{-g}u^{\mu}s\Big{)}$ $\displaystyle=$
$\displaystyle-\lambda\delta\Big{(}\sqrt{-g}u^{\mu}n\Big{)}+\frac{u^{\mu}}{T}\bigg{\\{}\delta\Big{(}\sqrt{-g}\varepsilon\Big{)}-\frac{1}{2}\sqrt{-g}\Big{[}\varepsilon
u_{\rho}u_{\sigma}+p\big{(}u_{\rho}u_{\sigma}+g_{\rho\sigma}\big{)}\Big{]}\delta
g^{\rho\sigma}\bigg{\\}}.$

If the second term of the right hand side of Eq. (LABEL:varyS2) becomes a
$(d-1)$-dimensional total derivative of a quantity, namely

$\displaystyle\delta\Big{(}\sqrt{-g}\varepsilon\Big{)}-\frac{1}{2}\sqrt{-g}\Big{[}\varepsilon
u_{\rho}u_{\sigma}+p\big{(}u_{\rho}u_{\sigma}+g_{\rho\sigma}\big{)}\Big{]}\delta
g^{\rho\sigma}\overset{\mathrm{stationary}}{=}\partial_{a}\Big{(}\cdots^{a}\Big{)}$
(66)

and the coefficients $\lambda$ and $u^{\mu}/T$ are constants in the stationary
states

$\displaystyle\lambda\overset{\mathrm{stationary}}{=}{\rm const.}$ (67)
$\displaystyle\frac{u^{\mu}}{T}\overset{\mathrm{stationary}}{=}{\rm const.},$
(68)

then the maximum entropy principle is satisfied at least in the first-order
variation.

However, this set of the conditions (66) - (68) is not the only one for the
maximum entropy principle because the coefficients $u^{\mu}/T$ and $\lambda$
are not necessarily constants. For example, dividing $\varepsilon$ into two
parts

$\displaystyle\varepsilon=e^{-\eta}\xi,$ (69)

the variation of the entropy density (LABEL:varyS2) can be arranged as

$\displaystyle\delta\Big{(}\sqrt{-g}u^{\mu}s\Big{)}$ $\displaystyle=$
$\displaystyle-\lambda\delta\Big{(}\sqrt{-g}u^{\mu}n\Big{)}+\frac{u^{\mu}}{T}\bigg{\\{}\delta\Big{(}\sqrt{-g}e^{-\eta}\xi\Big{)}-\frac{1}{2}\sqrt{-g}e^{-\eta}e^{\eta}\Big{[}\varepsilon
u_{\rho}u_{\sigma}+p\big{(}u_{\rho}u_{\sigma}+g_{\rho\sigma}\big{)}\Big{]}\delta
g^{\rho\sigma}\bigg{\\}}$ $\displaystyle=$
$\displaystyle-\lambda\delta\Big{(}\sqrt{-g}u^{\mu}n\Big{)}+\frac{u^{\mu}}{T}e^{-\eta}\bigg{\\{}\delta\Big{(}\sqrt{-g}\xi\Big{)}-\sqrt{-g}\xi\delta\eta-\frac{1}{2}\sqrt{-g}e^{\eta}\Big{[}\varepsilon
u_{\rho}u_{\sigma}+p\big{(}u_{\rho}u_{\sigma}+g_{\rho\sigma}\big{)}\Big{]}\delta
g^{\rho\sigma}\bigg{\\}}.$

By this expression, the conditions

$\displaystyle\delta\Big{(}\sqrt{-g}\xi\Big{)}-\sqrt{-g}\xi\delta\eta-\frac{1}{2}\sqrt{-g}e^{\eta}\Big{[}\varepsilon
u_{\rho}u_{\sigma}+p\big{(}u_{\rho}u_{\sigma}+g_{\rho\sigma}\big{)}\Big{]}\delta
g^{\rho\sigma}\overset{\mathrm{stationary}}{=}\partial_{a}\Big{(}\cdots^{a}\Big{)}$
(72)

and

$\displaystyle\lambda\overset{\mathrm{stationary}}{=}{\rm const.}$ (73)
$\displaystyle\frac{u^{\mu}}{T}e^{-\eta}\overset{\mathrm{stationary}}{=}{\rm
const.}$ (74)

also make the maximum entropy principle be satisfied.

We regard the scalar field $\eta$ as a fundamental field, and every cases in
which the field $\eta$ is not fundamental can be considered as a case with an
extra constraint condition on $\varepsilon$.

###  4.1 From the maximum entropy principle to field equations

Let us deform $\sqrt{-g}\varepsilon$ as

$\displaystyle\sqrt{-g}\varepsilon=-e^{-\eta}u^{\rho}u^{\sigma}\,{}^{\rm
EL}\\!\bar{\mathcal{D}}_{g^{\rho\sigma}}\sqrt{-g}\mathcal{L},$ (75)

where $\mathcal{L}$ is an arbitrary function of the fundamental fields and
$u^{\mu}$. If $\eta$ is not restricted to a fundamental field, this
decomposition is possible for arbitrary $\varepsilon$, but not unique.
Although in general the quantity $\eta$ is not a fundamental field, i.e., may
be a function of the other fundamental fields, firstly we take $\eta$ for a
independent fundamental field. The case in which $\eta$ is an general function
can be obtained by imposing an extra constraint relation upon $\eta$.

Using Eq. (LABEL:varyS2) and (75),

$\displaystyle\delta\Big{(}\sqrt{-g}u^{\mu}s\Big{)}$ $\displaystyle=$
$\displaystyle-\lambda\delta\Big{(}\sqrt{-g}u^{\mu}n\Big{)}$
$\displaystyle-\frac{u^{\mu}}{2T}e^{-\eta}\bigg{\\{}2\delta\Big{(}u^{\rho}u^{\sigma}\,{}^{\rm
EL}\\!\bar{\mathcal{D}}_{g^{\rho\sigma}}\sqrt{-g}\mathcal{L}\Big{)}+2\sqrt{-g}e^{\eta}\varepsilon\delta\eta+\sqrt{-g}e^{\eta}\Big{[}\varepsilon
u_{\rho}u_{\sigma}+p\big{(}u_{\rho}u_{\sigma}+g_{\rho\sigma}\big{)}\Big{]}\delta
g^{\rho\sigma}\bigg{\\}}.$

The Noether’s identity (LABEL:id3bar) for $\mathcal{F}=\mathcal{L}$ and
$f=\\{g^{\mu\nu},A_{\mu},u^{\mu},\eta,\varphi\\}$, namely

$\displaystyle 2u^{\rho}u^{\nu}\,{}^{\rm
EL}\\!\bar{\mathcal{D}}_{g^{\rho\nu}}\sqrt{-g}\mathcal{L}-u^{\nu}A_{\nu}u_{\mu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{L}-\sqrt{-g}\mathcal{L}$
$\displaystyle\equiv$ $\displaystyle-
u_{\mu}u^{\nu}\partial_{\rho}\Omega^{\rho\mu}_{\ \ \
\nu}[\sqrt{-g}\mathcal{L}]+\sum_{k=0}^{\infty}u_{\mu}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot
u^{\nu}\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f.$

is useful for our calculation. The variation of Eq. (LABEL:id3barLag) is
arranges in the form,

$\displaystyle 2\delta\Big{(}u^{\rho}u^{\sigma}\,{}^{\rm
EL}\\!\bar{\mathcal{D}}_{g^{\rho\sigma}}\sqrt{-g}\mathcal{L}\Big{)}$
$\displaystyle\equiv$
$\displaystyle\delta\Big{(}\sqrt{-g}\mathcal{L}\Big{)}+A_{\alpha}\delta\Big{(}u^{\alpha}u_{\beta}\,^{\rm
EL}\\!\mathcal{D}_{A_{\beta}}\sqrt{-g}\mathcal{L}\Big{)}+u^{\alpha}u_{\beta}\,^{\rm
EL}\\!\mathcal{D}_{A_{\beta}}\sqrt{-g}\mathcal{L}\cdot\delta A_{\alpha}$
$\displaystyle-\delta\Big{(}u_{\mu}u^{\nu}\partial_{\rho}\Omega^{\rho\mu}_{\ \
\
\nu}[\sqrt{-g}\mathcal{L}]\Big{)}+\sum_{k=0}^{\infty}\bigg{[}\delta\big{(}u_{\mu}u^{\nu}\big{)}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f$
$\displaystyle+u_{\mu}u^{\nu}\partial_{\nu}\Big{(}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\Big{)}$
$\displaystyle+u_{\mu}\delta\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot
u^{\nu}\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f-u_{\mu}u^{\nu}\partial_{\nu}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{]}.$

Substituting Eq. (LABEL:varyDL) into Eq. (LABEL:varyS3), we obtain the
following expression of the variation of the entropy current:

$\displaystyle\delta\Big{(}\sqrt{-g}u^{\mu}s\Big{)}$ $\displaystyle=$
$\displaystyle-\lambda\delta\Big{(}\sqrt{-g}u^{\mu}n\Big{)}-\frac{u^{\mu}}{2T}e^{-\eta}A_{\alpha}\delta\Big{(}u^{\alpha}u_{\beta}\,^{\rm
EL}\\!\mathcal{D}_{A_{\beta}}\sqrt{-g}\mathcal{L}\Big{)}$
$\displaystyle-\frac{u^{\mu}}{2T}e^{-\eta}\Bigg{\\{}\delta\Big{(}\sqrt{-g}\mathcal{L}\Big{)}+2\sqrt{-g}e^{\eta}\varepsilon\delta\eta+\sqrt{-g}e^{\eta}\Big{[}\varepsilon
u_{\rho}u_{\sigma}+p\big{(}u_{\rho}u_{\sigma}+g_{\rho\sigma}\big{)}\Big{]}\delta
g^{\rho\sigma}$ $\displaystyle+u^{\alpha}u_{\beta}\,^{\rm
EL}\\!\mathcal{D}_{A_{\beta}}\sqrt{-g}\mathcal{L}\cdot\delta
A_{\alpha}-\delta\Big{(}u_{\alpha}u^{\beta}\partial_{\rho}\Omega^{\rho\alpha}_{\
\ \ \beta}[\sqrt{-g}\mathcal{L}]\Big{)}$
$\displaystyle+u_{\rho}u^{\sigma}\partial_{\sigma}\bigg{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)}$
$\displaystyle+\sum_{k=0}^{\infty}\bigg{[}\delta\big{(}u_{\rho}u^{\sigma}\big{)}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot\partial_{\sigma}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f$
$\displaystyle+u_{\rho}\delta\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot
u^{\sigma}\partial_{\sigma}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f-u_{\rho}u^{\sigma}\partial_{\sigma}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{]}\Bigg{\\}}$

Expanding $\delta(\sqrt{-g}\mathcal{L})$ as

$\displaystyle\delta\Big{(}\sqrt{-g}\mathcal{L}\Big{)}$ (80) $\displaystyle=$
$\,{}^{\rm
EL}\\!\bar{\mathcal{D}}_{g^{\rho\sigma}}\sqrt{-g}\mathcal{L}\cdot\delta
g^{\rho\sigma}+\,^{\rm
EL}\\!\mathcal{D}_{A_{\alpha}}\sqrt{-g}\mathcal{L}\cdot\delta
A_{\alpha}+\,^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}\cdot\delta\eta+\,^{\rm
EL}\\!\mathcal{D}_{\varphi}\sqrt{-g}\mathcal{L}\cdot\delta\varphi$
$\displaystyle+\partial_{\rho}\bigg{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)},$

then Eq. (LABEL:varyS4) is

$\displaystyle\delta\Big{(}\sqrt{-g}u^{\mu}s\Big{)}$ $\displaystyle=$
$\displaystyle-\lambda\delta\Big{(}\sqrt{-g}u^{\mu}n\Big{)}-\frac{u^{\mu}}{2T}e^{-\eta}A_{\alpha}\delta\Big{(}u^{\alpha}u_{\beta}\,^{\rm
EL}\\!\mathcal{D}_{A_{\beta}}\sqrt{-g}\mathcal{L}\Big{)}$
$\displaystyle-\frac{u^{\mu}}{2T}e^{-\eta}\Bigg{\\{}\Big{(}\,^{\rm
EL}\\!\bar{\mathcal{D}}_{g^{\rho\sigma}}\sqrt{-g}\mathcal{L}+\sqrt{-g}e^{\eta}\big{[}\varepsilon
u_{\rho}u_{\sigma}+p(u_{\rho}u_{\sigma}+g_{\rho\sigma})\big{]}\Big{)}\delta
g^{\rho\sigma}$ $\displaystyle+\Big{(}\,^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}+2\sqrt{-g}e^{\eta}\varepsilon\Big{)}\delta\eta+\big{(}\delta^{\alpha}_{\beta}+u^{\alpha}u_{\beta}\big{)}\,^{\rm
EL}\\!\mathcal{D}_{A_{\beta}}\sqrt{-g}\mathcal{L}\cdot\delta
A_{\alpha}+\,^{\rm
EL}\\!\mathcal{D}_{\varphi}\sqrt{-g}\mathcal{L}\cdot\delta\varphi$
$\displaystyle-\delta\Big{(}u_{\alpha}u^{\beta}\partial_{\rho}\Omega^{\rho\alpha}_{\
\ \
\beta}[\sqrt{-g}\mathcal{L}]\Big{)}+\big{(}\delta^{\sigma}_{\rho}+u^{\sigma}u_{\rho}\big{)}\partial_{\sigma}\bigg{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)}$
$\displaystyle+\sum_{k=0}^{\infty}\bigg{[}\delta\big{(}u_{\rho}u^{\sigma}\big{)}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot\partial_{\sigma}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f$
$\displaystyle+u_{\rho}\delta\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot
u^{\sigma}\partial_{\sigma}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f-u_{\rho}u^{\sigma}\partial_{\sigma}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{]}\Bigg{\\}}.$

This is the most general expression of the first-order variation of the
entropy current.

Here, we introduce the “co-moving coordinate” in which the $x^{0}$-axis is
parallel to the $d$-velocity $u^{\mu}$ so that $u^{a}=0$, $u^{0}u_{0}=-1$, and
$\partial_{0}X$ for an arbitrary quantity $X$ in a stationary state.
Therefore, in this coordinate, the last two lines of Eq. (LABEL:varySgeneral)
vanish in a stationary state:

$\displaystyle\delta\Big{(}\sqrt{-g}u^{\mu}s\Big{)}$ $\displaystyle=$
$\displaystyle-\lambda\delta\Big{(}\sqrt{-g}u^{\mu}n\Big{)}+\frac{u^{\mu}}{2T}e^{-\eta}A_{0}\delta\Big{(}\,^{\rm
EL}\\!\mathcal{D}_{A_{0}}\sqrt{-g}\mathcal{L}\Big{)}$
$\displaystyle-\frac{u^{\mu}}{2T}e^{-\eta}\Bigg{\\{}\Big{(}\,^{\rm
EL}\\!\bar{\mathcal{D}}_{g^{\rho\sigma}}\sqrt{-g}\mathcal{L}+\sqrt{-g}e^{\eta}\big{[}\varepsilon
u_{\rho}u_{\sigma}+p(u_{\rho}u_{\sigma}+g_{\rho\sigma})\big{]}\Big{)}\delta
g^{\rho\sigma}$ $\displaystyle+\Big{(}\,^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}+2\sqrt{-g}e^{\eta}\varepsilon\Big{)}\delta\eta+\,^{\rm
EL}\\!\mathcal{D}_{A_{a}}\sqrt{-g}\mathcal{L}\cdot\delta A_{a}+\,^{\rm
EL}\\!\mathcal{D}_{\varphi}\sqrt{-g}\mathcal{L}\cdot\delta\varphi$
$\displaystyle+\delta\Big{(}\partial_{\rho}\Omega^{\rho 0}_{\ \ \
0}[\sqrt{-g}\mathcal{L}]\Big{)}+\partial_{a}\bigg{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{a}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)}$
$\displaystyle-\sum_{k=0}^{\infty}\bigg{[}\delta\,^{\rm
EL}\\!\mathcal{D}_{\partial_{0}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot\partial_{0}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f-\partial_{0}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{0}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{]}\Bigg{\\}}$
$\displaystyle\overset{\mathrm{stationary}}{=}$
$\displaystyle-\lambda\delta\Big{(}\sqrt{-g}u^{\mu}n\Big{)}+\frac{u^{\mu}}{2T}e^{-\eta}A_{0}\delta\Big{(}\,^{\rm
EL}\\!\mathcal{D}_{A_{0}}\sqrt{-g}\mathcal{L}\Big{)}$
$\displaystyle-\frac{u^{\mu}}{2T}e^{-\eta}\Bigg{\\{}\Big{(}\,^{\rm
EL}\\!\bar{\mathcal{D}}_{g^{\rho\sigma}}\sqrt{-g}\mathcal{L}+\sqrt{-g}e^{\eta}\big{[}\varepsilon
u_{\rho}u_{\sigma}+p(u_{\rho}u_{\sigma}+g_{\rho\sigma})\big{]}\Big{)}\delta
g^{\rho\sigma}$ $\displaystyle+\Big{(}\,^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}+2\sqrt{-g}e^{\eta}\varepsilon\Big{)}\delta\eta+\,^{\rm
EL}\\!\mathcal{D}_{A_{a}}\sqrt{-g}\mathcal{L}\cdot\delta A_{a}+\,^{\rm
EL}\\!\mathcal{D}_{\varphi}\sqrt{-g}\mathcal{L}\cdot\delta\varphi$
$\displaystyle+\delta\Big{(}\partial_{\rho}\Omega^{\rho 0}_{\ \ \
0}[\sqrt{-g}\mathcal{L}]\Big{)}+\partial_{a}\bigg{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{a}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)}.$

If the field equations

$\,{}^{\rm EL}\\!\bar{\mathcal{D}}_{g^{\rho\sigma}}\sqrt{-g}\mathcal{L}$
$\displaystyle\overset{\mathrm{stationary}}{=}$
$\displaystyle-\sqrt{-g}e^{\eta}\big{[}\varepsilon
u_{\rho}u_{\sigma}+p(u_{\rho}u_{\sigma}+g_{\rho\sigma})\big{]}$ (84)
$\,{}^{\rm EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}$
$\displaystyle\overset{\mathrm{stationary}}{=}$
$\displaystyle-2\sqrt{-g}e^{\eta}\varepsilon$ (85) $\,{}^{\rm
EL}\\!\mathcal{D}_{A_{\rho}}\sqrt{-g}\mathcal{L}$
$\displaystyle\overset{\mathrm{stationary}}{=}$
$\displaystyle-\sqrt{-g}u^{\rho}qn$ (86) $\,{}^{\rm
EL}\\!\mathcal{D}_{\varphi}\sqrt{-g}\mathcal{L}$
$\displaystyle\overset{\mathrm{stationary}}{=}$ $\displaystyle 0,$ (87)

and the “consistency conditions”

$\displaystyle\lambda+\frac{u^{0}}{2T}e^{-\eta}qA_{0}\overset{\mathrm{stationary}}{=}{\rm
const.}$ (88)
$\displaystyle\frac{u^{0}}{T}e^{-\eta}\overset{\mathrm{stationary}}{=}{\rm
const.}$ (89)

hold for any stationary states, any variation of the fundamental fields with
the conserved charges fixed and appropriate boundary conditions held does not
change the total entropy, i.e.,

$\displaystyle\delta S$ $\displaystyle\overset{\mathrm{stationary}}{=}$
$\displaystyle-\bigg{(}\lambda+\frac{u^{0}}{2T}e^{-\eta}qA_{0}\bigg{)}\delta
N+\frac{u^{0}}{2T}e^{-\eta}\delta M_{\rm Noether}$
$\displaystyle-\frac{u^{0}}{2T}e^{-\eta}\oint_{\partial\mathcal{V}}({\mathrm{d}}^{d-2}x)_{0a}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{a}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{L}\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f,$

where
$({\mathrm{d}}^{d-2}x)_{\mu\nu}:=\frac{1}{(d-2)!}{\mathrm{d}}x^{\mu_{1}}\wedge\cdots\wedge{\mathrm{d}}x^{\mu_{d-2}}\epsilon_{\mu\nu\mu_{1}\cdots\mu_{d-2}}$.
From only the maximum entropy principle, one cannot know what equations hold
in non-stationary states. In section 4.2, one finds that if Eqs. (84) - (87)
hold in not only stationary but also non-stationary states, the entropy
current density is conserved, i.e., entropy production is forbidden.
Therefore, such a fluid is called a perfect fluid.

It should be noticed that the field equations (84) - (87) are compatible with
Eq. (75), i.e., contracting with $-u^{\rho}u^{\sigma}$, Eq. (84) is

$\displaystyle-u^{\rho}u^{\sigma}\,{}^{\rm
EL}\\!\bar{\mathcal{D}}_{g^{\rho\sigma}}\sqrt{-g}\mathcal{L}\overset{\mathrm{stationary}}{=}\sqrt{-g}e^{\eta}\varepsilon,$
(91)

that coincides with Eq. (75) in a stationary state. This is a nontrivial
feature of $\varepsilon$ that is described by Eq. (75) with a fundamental
scalar field $\eta$.

The current density $\,{}^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{L}=-\sqrt{-g}u^{\mu}qn$ should
also conserve, i.e.,

$\displaystyle\partial_{\mu}\Big{(}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{L}\Big{)}=0.$ (92)

If one imposes the $U(1)$ gauge symmetry on the Lagrangian, the identity (383)
shows that the conservation law (92) requires the field $\eta$ to be
electrically neutral (See appendix B.3).

###  4.2 Consistency conditions

In this subsection, we see that the field equations (84) - (87) are consistent
with the relations which we have assumed above:

$\displaystyle\nabla_{\mu}\big{(}u^{\mu}s\big{)}=0,$ (93)
$\displaystyle\nabla_{\mu}\big{(}u^{\mu}n\big{)}=0,$ (94)
$\displaystyle\frac{u^{\mu}}{T}e^{-\eta}\overset{\mathrm{stationary}}{=}{\rm
const.\ in\ \mathcal{V}},$ (95)
$\displaystyle\lambda\delta^{\mu}_{\nu}+\frac{q}{2T}e^{-\eta}u^{\mu}A_{\nu}\overset{\mathrm{stationary}}{=}{\rm
const.\ in\ \mathcal{V}},$ (96)

where $\nabla_{\mu}$ is the covariant derivative compatible with $g_{\mu\nu}$.
In this subsection, we use $\nabla_{\mu}$ rather than $\partial_{\mu}$ for
simpler calculation. We call Eqs. (93) - (96) consistency conditions.

From Noether’s theorem (LABEL:id1arrange) with replacing $\mathcal{F}$ into
$\mathcal{L}$, the following identity holds:

$\displaystyle
2\sqrt{-g}\nabla_{\nu}\bar{E}^{\nu}_{\mu}[\sqrt{-g}\mathcal{L}]$
$\displaystyle\equiv$ $\displaystyle A_{\mu}\partial_{\nu}\Big{(}\,^{\rm
EL}\\!\mathcal{D}_{A_{\nu}}\sqrt{-g}\mathcal{L}\Big{)}-F_{\mu\nu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\nu}}\sqrt{-g}\mathcal{L}$
$\displaystyle-\partial_{\nu}\Big{[}u^{\nu}\big{(}u_{\mu}u^{\rho}+\delta^{\rho}_{\mu}\big{)}\,^{\rm
EL}\\!\mathcal{D}_{u^{\rho}}\sqrt{-g}\mathcal{L}\Big{]}-\bigg{[}\partial_{\mu}u^{\rho}-\frac{1}{2}\big{(}\partial_{\mu}g_{\alpha\beta}\big{)}u^{\alpha}u^{\beta}u^{\rho}\bigg{]}\,^{\rm
EL}\\!\mathcal{D}_{u^{\rho}}\sqrt{-g}\mathcal{L}$
$\displaystyle-(\partial_{\mu}\varphi)\,^{\rm
EL}\\!\mathcal{D}_{\varphi}\sqrt{-g}\mathcal{L}-(\partial_{\mu}\eta)\,^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L},$

where

$\displaystyle E_{\mu\nu}[\sqrt{-g}\mathcal{L}]$ $\displaystyle:=$
$\displaystyle\frac{1}{\sqrt{-g}}\,^{\rm
EL}\\!\mathcal{D}_{g^{\mu\nu}}\sqrt{-g}\mathcal{L}$ (98)
$\displaystyle\bar{E}_{\mu\nu}[\sqrt{-g}\mathcal{L}]$ $\displaystyle:=$
$\displaystyle\frac{1}{\sqrt{-g}}\,^{\rm
EL}\\!\bar{\mathcal{D}}_{g^{\mu\nu}}\sqrt{-g}\mathcal{L}$ (99)
$\displaystyle=$ $\displaystyle
E_{\mu\nu}[\sqrt{-g}\mathcal{L}]-\frac{1}{2\sqrt{-g}}u_{\mu}u_{\nu}u^{\rho}\,{}^{\rm
EL}\\!\mathcal{D}_{u^{\rho}}\sqrt{-g}\mathcal{L}$ (100) $\displaystyle
F_{\mu\nu}$ $\displaystyle:=$
$\displaystyle\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.$ (101)

Notice that the third line of Eq. (4.2) vanishes in a stationary state. It is
easier to check by direct calculation in a co-moving coordinate, in which
$u^{a}=0$, $u^{0}=(-g_{00})^{-1/2}$, and $\partial_{0}X=0$ for an arbitrary
quantity $X$. Since the third line of Eq. (4.2) is a covariant vector density,
denoted by $-\sqrt{-g}V_{\mu}$, the following equation holds in any
coordinates:

$\displaystyle\sqrt{-g}V_{\mu}$ $\displaystyle:=$
$\displaystyle\partial_{\nu}\Big{[}u^{\nu}\big{(}u_{\mu}u^{\rho}+\delta^{\rho}_{\mu}\big{)}\,^{\rm
EL}\\!\mathcal{D}_{u^{\rho}}\sqrt{-g}\mathcal{L}\Big{]}+\bigg{[}\partial_{\mu}u^{\rho}-\frac{1}{2}\big{(}\partial_{\mu}g_{\alpha\beta}\big{)}u^{\alpha}u^{\beta}u^{\rho}\bigg{]}\,^{\rm
EL}\\!\mathcal{D}_{u^{\rho}}\sqrt{-g}\mathcal{L}$ (102)
$\displaystyle\overset{\mathrm{stationary}}{=}$ $\displaystyle 0.$ (103)

In more detail, one can check as well that the parallel part to $u_{\mu}$
vanishes in an arbitrary state:

$\displaystyle\sqrt{-g}u^{\mu}V_{\mu}=0.$ (104)

Substituting Eqs. (84), (87), and (4.2) into Eq. (4.2),

$\displaystyle\sqrt{-g}\nabla_{\nu}\Big{\\{}e^{\eta}\Big{[}\varepsilon
u^{\nu}u_{\mu}+p(u^{\nu}u_{\mu}+\delta^{\nu}_{\mu})\Big{]}\Big{\\}}=\frac{1}{2}F_{\mu\nu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\nu}}\sqrt{-g}\mathcal{L}+\frac{1}{2}(\partial_{\mu}\eta)\,^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}+\frac{1}{2}\sqrt{-g}V_{\mu}.$

Transposing all the terms in the left hand side which contain $\eta$ to the
right hand side, and separating them into parallel and orthogonal parts to
$u_{\mu}$, Eq. (4.2) is arranged as follows:

$\displaystyle\sqrt{-g}\nabla_{\nu}\Big{[}\varepsilon
u^{\nu}u_{\mu}+p\big{(}u^{\nu}u_{\mu}+\delta^{\nu}_{\mu}\big{)}\Big{]}$
$\displaystyle=$
$\displaystyle-\sqrt{-g}(\partial_{\nu}\eta)\Big{[}\varepsilon
u^{\nu}u_{\mu}+p\big{(}u^{\nu}u_{\mu}+\delta^{\nu}_{\mu}\big{)}\Big{]}+\frac{1}{2}(\partial_{\mu}\eta)e^{-\eta}\,{}^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}+\frac{1}{2}\sqrt{-g}e^{-\eta}V_{\mu}$
$\displaystyle+\frac{1}{2}e^{-\eta}F_{\mu\nu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\nu}}\sqrt{-g}\mathcal{L}$ $\displaystyle=$
$\displaystyle-
u_{\mu}u^{\nu}\Big{(}\sqrt{-g}\varepsilon+\frac{1}{2}e^{-\eta}\,{}^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}\Big{)}\partial_{\nu}\eta$
$\displaystyle-\big{(}\delta^{\nu}_{\mu}+u^{\nu}u_{\mu}\big{)}\bigg{[}\Big{(}\sqrt{-g}p-\frac{1}{2}e^{-\eta}\,{}^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}\Big{)}\partial_{\nu}\eta-\frac{1}{2}\sqrt{-g}e^{-\eta}V_{\nu}\bigg{]}$
$\displaystyle+\frac{1}{2}e^{-\eta}F_{\mu\nu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\nu}}\sqrt{-g}\mathcal{L}$ $\displaystyle=$
$\displaystyle-
u_{\mu}u^{\nu}\Big{(}\sqrt{-g}\varepsilon+\frac{1}{2}e^{-\eta}\,{}^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}\Big{)}\partial_{\nu}\eta$
$\displaystyle-\big{(}\delta^{\nu}_{\mu}+u^{\nu}u_{\mu}\big{)}\bigg{[}\Big{(}\sqrt{-g}p-\frac{1}{2}e^{-\eta}\,{}^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}\Big{)}\partial_{\nu}\eta-\frac{1}{2}\sqrt{-g}e^{-\eta}V_{\nu}-\frac{1}{2}e^{-\eta}F_{\nu\rho}\,^{\rm
EL}\\!\mathcal{D}_{A_{\rho}}\sqrt{-g}\mathcal{L}\bigg{]},$

where we use the properties Eq. (104) and $\,{}^{\rm
EL}\\!\mathcal{D}_{A_{\rho}}\sqrt{-g}\mathcal{L}\propto u^{\rho}$. Similarly,
the first line of Eq. (4.2) is also separated into two parts:

$\displaystyle\nabla_{\nu}\Big{[}\varepsilon
u^{\nu}u_{\mu}+p\big{(}u^{\nu}u_{\mu}+\delta^{\nu}_{\mu}\big{)}\Big{]}$ (111)
$\displaystyle=$
$\displaystyle\Big{[}u^{\nu}\partial_{\nu}(\varepsilon+p)+(\varepsilon+p)\nabla_{\nu}u^{\nu}\Big{]}u_{\mu}+(\varepsilon+p)u^{\nu}\nabla_{\nu}u_{\mu}+\partial_{\mu}p$
$\displaystyle=$
$\displaystyle\Big{[}u^{\nu}\partial_{\nu}\varepsilon+(\varepsilon+p)\nabla_{\nu}u^{\nu}\Big{]}u_{\mu}$
$\displaystyle+(\varepsilon+p)u^{\nu}\nabla_{\nu}u_{\mu}+\big{(}\delta^{\nu}_{\mu}+u^{\nu}u_{\mu}\big{)}\partial_{\nu}p$
$\displaystyle=$
$\displaystyle\Big{[}u^{\nu}\partial_{\nu}\varepsilon+(\varepsilon+p)\nabla_{\nu}u^{\nu}\Big{]}u_{\mu}$
$\displaystyle+\big{(}\delta^{\nu}_{\mu}+u^{\nu}u_{\mu}\big{)}\Big{[}(\varepsilon+p)u^{\rho}\nabla_{\rho}u_{\nu}+\partial_{\nu}p\Big{]}.$

The following calculation shows that the first term of the right hand side
(111) is related to the conservation law of the entropy and particle number:

$\displaystyle T\nabla_{\nu}\big{(}u^{\nu}s\big{)}$ $\displaystyle=$
$\displaystyle Tu^{\nu}\partial_{\nu}s+Ts\nabla_{\nu}u^{\nu}$ (112)
$\displaystyle=$ $\displaystyle Tu^{\nu}\partial_{\nu}\frac{\varepsilon+p-\mu
n}{T}+(\varepsilon+p-\mu n)\nabla_{\nu}u^{\nu}$ $\displaystyle=$
$\displaystyle u^{\nu}\partial_{\nu}(\varepsilon+p-\mu n)-(\varepsilon+p-\mu
n)\frac{u^{\nu}\partial_{\nu}T}{T}+(\varepsilon+p-\mu n)\nabla_{\nu}u^{\nu}$
$\displaystyle=$ $\displaystyle
u^{\nu}\partial_{\nu}\varepsilon+(\varepsilon+p)\nabla_{\nu}u^{\nu}+u^{\nu}\big{(}\partial_{\nu}p-s\partial_{\nu}T-n\partial_{\nu}\mu\big{)}-\mu\nabla_{\nu}\big{(}u^{\nu}n\big{)}$
$\displaystyle=$ $\displaystyle
u^{\nu}\partial_{\nu}\varepsilon+(\varepsilon+p)\nabla_{\nu}u^{\nu}-\mu\nabla_{\nu}\big{(}u^{\nu}n\big{)},$

where we use the relation
$\partial_{\nu}p-s\partial_{\nu}T-n\partial_{\nu}\mu=0$, which is equivalent
to Eq. (14). Thus, one obtains the relation

$\displaystyle
u^{\nu}\partial_{\nu}\varepsilon+(\varepsilon+p)\nabla_{\nu}u^{\nu}=T\nabla_{\nu}\big{(}u^{\nu}s\big{)}+\mu\nabla_{\nu}\big{(}u^{\nu}n\big{)}.$
(113)

Substituting Eq. (113) into (111),

$\displaystyle\nabla_{\nu}\Big{[}(\varepsilon+p)u^{\nu}u_{\mu}+p\delta^{\nu}_{\mu}\Big{]}=\Big{[}T\nabla_{\nu}\big{(}u^{\nu}s\big{)}+\mu\nabla_{\nu}\big{(}u^{\nu}n\big{)}\Big{]}u_{\mu}+\big{(}\delta^{\nu}_{\mu}+u^{\nu}u_{\mu}\big{)}\Big{[}(\varepsilon+p)u^{\rho}\nabla_{\rho}u_{\nu}+\partial_{\nu}p\Big{]}.$

Combining Eq. (4.2) and Eq. (4.2), we obtain the equation

$\displaystyle\sqrt{-g}\bigg{[}T\nabla_{\nu}\big{(}u^{\nu}s\big{)}+\mu\nabla_{\nu}\big{(}u^{\nu}n\big{)}+\Big{(}\varepsilon+\frac{1}{2\sqrt{-g}}e^{-\eta}\,{}^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}\Big{)}u^{\nu}\partial_{\nu}\eta\bigg{]}u_{\mu}$
$\displaystyle+\sqrt{-g}\big{(}\delta^{\nu}_{\mu}+u^{\nu}u_{\mu}\big{)}\bigg{[}(\varepsilon+p)u^{\rho}\nabla_{\rho}u_{\nu}+\partial_{\nu}p+\Big{(}p-\frac{1}{2\sqrt{-g}}e^{-\eta}\,{}^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}\Big{)}\partial_{\nu}\eta$
$\displaystyle-\frac{1}{2}e^{-\eta}V_{\nu}-\frac{1}{2\sqrt{-g}}e^{-\eta}F_{\nu\rho}\,^{\rm
EL}\\!\mathcal{D}_{A_{\rho}}\sqrt{-g}\mathcal{L}\bigg{]}=0.$ (115)

Since the first term of the left hand side of Eq. (115) is parallel to
$u_{\mu}$ and the second is orthogonal, both the terms should be $0$,
respectively:

$\displaystyle
T\nabla_{\nu}\big{(}u^{\nu}s\big{)}+\mu\nabla_{\nu}\big{(}u^{\nu}n\big{)}+\Big{(}\varepsilon+\frac{1}{2\sqrt{-g}}e^{-\eta}\,{}^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}\Big{)}u^{\nu}\partial_{\nu}\eta=0,$
(116)
$\displaystyle\big{(}\delta^{\nu}_{\mu}+u^{\nu}u_{\mu}\big{)}\bigg{[}(\varepsilon+p)u^{\rho}\nabla_{\rho}u_{\nu}+\partial_{\nu}p$
$\displaystyle\ \ \ +\Big{(}p-\frac{1}{2\sqrt{-g}}e^{-\eta}\,{}^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}\Big{)}\partial_{\nu}\eta-\frac{1}{2}e^{-\eta}V_{\nu}-\frac{1}{2\sqrt{-g}}e^{-\eta}F_{\nu\rho}\,^{\rm
EL}\\!\mathcal{D}_{A_{\rho}}\sqrt{-g}\mathcal{L}\bigg{]}=0.$

From Eq. (116), the conservation laws of $u^{\mu}s$ and $u^{\mu}n$ require the
relation

$\displaystyle\Big{(}\varepsilon+\frac{1}{2\sqrt{-g}}e^{-\eta}\,{}^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}\Big{)}u^{\nu}\partial_{\nu}\eta=0,$
(118)

that is,

$\,{}^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}=-2\sqrt{-g}e^{\eta}\varepsilon$
(119)

or

$\displaystyle u^{\nu}\partial_{\nu}\eta=0.$ (120)

Next, let us consider Eq. (4.2). Using the relation among the variations $p$,
$T$, and $\mu$,

$\displaystyle\partial_{\nu}p$ $\displaystyle=$ $\displaystyle
s\partial_{\nu}T+n\partial_{\nu}\mu$ (121) $\displaystyle=$ $\displaystyle
s\partial_{\nu}T+n(\lambda\partial_{\nu}T+T\partial_{\nu}\lambda)$ (122)
$\displaystyle=$
$\displaystyle(\varepsilon+p)\frac{\partial_{\nu}T}{T}+nT\partial_{\nu}\lambda.$
(123)

Writing the definition of the covariant derivative explicitly,

$\displaystyle u^{\nu}\nabla_{\nu}u_{\mu}$ $\displaystyle=$ $\displaystyle
u^{\nu}\partial_{\nu}u_{\mu}-u^{\nu}\Gamma^{\rho}_{\nu\mu}u_{\rho}$ (124)
$\displaystyle=$ $\displaystyle
u^{\nu}\partial_{\nu}u_{\mu}-\frac{1}{2}u^{\nu}u_{\rho}g^{\rho\sigma}\big{(}\partial_{\mu}g_{\sigma\nu}+\partial_{\nu}g_{\sigma\mu}-\partial_{\sigma}g_{\nu\mu}\big{)}$
(125) $\displaystyle=$ $\displaystyle
u^{\nu}\partial_{\nu}u_{\mu}-\frac{1}{2}u^{\nu}u^{\sigma}\partial_{\mu}g_{\sigma\nu},$
(126)

where $\Gamma^{\rho}_{\nu\mu}$ is the Christoffel symbol. In a co-moving
coordinate, the $d$-velocity of a stationary fluid satisfies
$u^{\rho}\partial_{\rho}u_{\nu}\overset{\mathrm{stationary}}{=}0$, so that

$\displaystyle u^{\rho}\nabla_{\rho}u_{\nu}$
$\displaystyle\overset{\mathrm{stationary}}{=}$
$\displaystyle-\frac{1}{2}u^{\rho}u^{\sigma}\partial_{\nu}g_{\rho\sigma}$
(127) $\displaystyle=$
$\displaystyle-\frac{1}{2}u^{0}u^{0}\partial_{\nu}g_{00}$ (128)
$\displaystyle=$
$\displaystyle\frac{1}{2}\big{(}u^{0}\big{)}^{2}\partial_{\nu}\big{(}u^{0}\big{)}^{-2}$
(129) $\displaystyle=$ $\displaystyle-\partial_{\nu}\ln u^{0},$ (130)

Vanishing the left hand side of Eq. (4.2) for stationary fluids in a co-moving
coordinate is equivalent to

$\displaystyle\partial_{a}\ln
u^{0}-\frac{\partial_{a}T}{T}-\frac{1}{\varepsilon+p}\Big{(}p-\frac{1}{2\sqrt{-g}}e^{-\eta}\,{}^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}\Big{)}\partial_{a}\eta$
$\displaystyle-\frac{nT}{\varepsilon+p}\partial_{a}\lambda+\frac{e^{-\eta}}{2(\varepsilon+p)}F_{a\nu}\frac{1}{\sqrt{-g}}\,^{\rm
EL}\\!\mathcal{D}_{A_{\nu}}\sqrt{-g}\mathcal{L}\overset{\mathrm{stationary}}{=}0,$
(131)

where we use Eq. (103). Due to the relations $\,{}^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{L}=-2\sqrt{-g}e^{\eta}\varepsilon$,
$\,{}^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{L}=-\sqrt{-g}u^{\mu}qn$, and
$\partial_{0}A_{a}=0$, the following relation is obtained:

$\displaystyle\bigg{[}\bigg{(}\frac{u^{0}}{T}e^{-\eta}\bigg{)}^{-1}+\frac{qnTA_{0}}{2(\varepsilon+p)}\bigg{]}\partial_{a}\bigg{(}\frac{u^{0}}{T}e^{-\eta}\bigg{)}-\frac{nT}{\varepsilon+p}\partial_{a}\bigg{(}\lambda+\frac{u^{0}}{2T}e^{-\eta}qA_{0}\bigg{)}\overset{\mathrm{stationary}}{=}0.$

This equation and Eq. (116) show that the field equations (84) - (87) are
compatible with the consistency conditions

$\displaystyle\nabla_{\mu}\big{(}u^{\mu}s\big{)}=0,$ (133)
$\displaystyle\nabla_{\mu}\big{(}u^{\mu}n\big{)}=0,$ (134)
$\displaystyle\partial_{a}\bigg{(}\frac{u^{0}}{T}e^{-\eta}\bigg{)}\overset{\mathrm{stationary}}{=}0,$
(135)
$\displaystyle\partial_{a}\bigg{(}\lambda+\frac{u^{0}}{2T}e^{-\eta}qA_{0}\bigg{)}\overset{\mathrm{stationary}}{=}0.$
(136)

One finds the constraint condition $\eta=0$ works well as below. Since
$\delta\eta=0$ in Eq. (4.1), one find the following field equations
straightforwardly:

$\,{}^{\rm EL}\\!\bar{\mathcal{D}}_{g^{\rho\sigma}}\sqrt{-g}\mathcal{L}$
$\displaystyle=$ $\displaystyle-\sqrt{-g}\big{[}\varepsilon
u_{\rho}u_{\sigma}+p(u_{\rho}u_{\sigma}+g_{\rho\sigma})\big{]}$ (137)
$\,{}^{\rm EL}\\!\mathcal{D}_{A_{\rho}}\sqrt{-g}\mathcal{L}$ $\displaystyle=$
$\displaystyle-\sqrt{-g}u^{\rho}qn$ (138) $\,{}^{\rm
EL}\\!\mathcal{D}_{\varphi}\sqrt{-g}\mathcal{L}$ $\displaystyle=$
$\displaystyle 0.$ (139)

###  4.3 Redefinition of the physical quantities

Using the field equations (85) and (86), Eq. (4.2) is arranged as

$\displaystyle\big{(}u_{\mu}u^{\nu}+\delta^{\nu}_{\mu}\big{)}\bigg{[}u^{\rho}\nabla_{\rho}u_{\nu}+\frac{\partial_{\nu}p}{\varepsilon+p}+\partial_{\nu}\eta+\frac{qn}{2(\varepsilon+p)}e^{-\eta}F_{\nu\rho}u^{\rho}\bigg{]}=0.$
(140)

Consider a situation in which $\partial_{a}p=q=0$ and $\partial_{a}\eta\neq
0$, then

$\displaystyle u^{\rho}\nabla_{\rho}u_{a}+\partial_{a}\eta=0,$ (141)

in a co-moving coordinate. This equation seems strange in that the
electrically neutral fluid does not flow along geodesic orbits even when there
are no gradient of the pressure, namely the fluid is force-free.

In this subsection, we redefine the metric tensor and other physical
quantities so that a force-free fluid flows along geodesic orbits.

One finds that the new metric tensor $\tilde{g}_{\mu\nu}$ defined by

$\displaystyle\tilde{g}_{\mu\nu}$ $\displaystyle:=$ $\displaystyle
e^{2\eta}g_{\mu\nu}$ (142)

and $\tilde{g}^{\mu\nu}$, $\tilde{u}^{\mu}$, and $\tilde{u}_{\mu}$

$\displaystyle\tilde{g}^{\mu\nu}$ $\displaystyle:=$ $\displaystyle
e^{-2\eta}g^{\mu\nu}$ (143) $\displaystyle\tilde{u}^{\mu}$ $\displaystyle:=$
$\displaystyle e^{-\eta}u^{\mu}$ (144) $\displaystyle\tilde{u}_{\mu}$
$\displaystyle:=$ $\displaystyle e^{\eta}u_{\mu}$ (145) $\displaystyle=$
$\displaystyle\tilde{g}_{\mu\nu}\tilde{u}^{\nu}$ (146)

are appropriate for our purpose. The normalization condition of the new
$d$-velocity is

$\displaystyle\tilde{g}_{\mu\nu}\tilde{u}^{\mu}\tilde{u}^{\nu}=-1.$ (147)

Indeed, $\tilde{u}^{\rho}\tilde{\nabla}_{\rho}\tilde{u}_{\nu}$ is deformed to
yield

$\displaystyle\tilde{u}^{\rho}\tilde{\nabla}_{\rho}\tilde{u}_{\nu}$
$\displaystyle=$ $\displaystyle
u^{\rho}\tilde{\nabla}_{\rho}u_{\nu}+\tilde{u}^{\rho}u_{\nu}\partial_{\rho}e^{\eta}$
(148) $\displaystyle=$ $\displaystyle
u^{\rho}\partial_{\rho}u_{\nu}-\frac{1}{2}u^{\rho}\tilde{\Gamma}^{\sigma}_{\rho\nu}u_{\sigma}+u_{\nu}u^{\rho}\partial_{\rho}\eta$
(149) $\displaystyle=$ $\displaystyle
u^{\rho}\partial_{\rho}u_{\nu}-\frac{1}{2}u^{\rho}u_{\sigma}\tilde{g}^{\sigma\omega}\big{(}\partial_{\nu}\tilde{g}_{\omega\rho}+\partial_{\rho}\tilde{g}_{\omega\nu}-\partial_{\omega}\tilde{g}_{\rho\nu}\big{)}+u_{\nu}u^{\rho}\partial_{\rho}\eta$
$\displaystyle=$ $\displaystyle
u^{\rho}\partial_{\rho}u_{\nu}-\frac{1}{2}u^{\rho}u_{\sigma}g^{\sigma\omega}\partial_{\nu}g_{\omega\rho}-u^{\rho}u_{\sigma}\tilde{g}^{\sigma\omega}g_{\omega\rho}e^{2\eta}\partial_{\nu}\eta+u_{\nu}u^{\rho}\partial_{\rho}\eta$
$\displaystyle=$ $\displaystyle
u^{\rho}\nabla_{\rho}u_{\nu}+\partial_{\nu}\eta+u_{\nu}u^{\rho}\partial_{\rho}\eta$
(151) $\displaystyle=$ $\displaystyle
u^{\rho}\nabla_{\rho}u_{\nu}+\big{(}\delta^{\rho}_{\nu}+u_{\nu}u^{\rho}\big{)}\partial_{\rho}\eta,$
(152)

so that Eq. (140) is

$\displaystyle\big{(}\delta^{\nu}_{\mu}+\tilde{u}^{\nu}\tilde{u}_{\mu}\big{)}\bigg{[}\tilde{u}^{\rho}\tilde{\nabla}_{\rho}\tilde{u}_{\nu}+\frac{\partial_{\nu}p}{\varepsilon+p}+\frac{qn}{2(\varepsilon+p)}F_{\nu\rho}\tilde{u}^{\rho}\bigg{]}=0.$
(153)

In the situation $\partial_{\nu}p=q=0$,

$\displaystyle\tilde{u}^{\rho}\tilde{\nabla}_{\rho}\tilde{u}_{\nu}=0.$ (154)

Therefore, with constant pressure, the neutral fluid flows geodesic orbits
defined by the new metric tensor $\tilde{g}_{\mu\nu}$. Notice that the
relation (154) also holds for the metric tensor
$\tilde{g}_{\mu\nu}=e^{2(\eta+a)}g_{\mu\nu}$, where $a$ is an arbitrary
constant.

The determinant of $\tilde{g}_{\mu\nu}$ is

$\displaystyle\sqrt{-g}=e^{-d\eta}\sqrt{-\tilde{g}}.$ (155)

The extensive quantities $s$, $n$ and $\varepsilon$ are redefined by

$\displaystyle\tilde{v}$ $\displaystyle:=$ $\displaystyle e^{-(d-1)\eta}$
(156) $\displaystyle\tilde{s}$ $\displaystyle:=$ $\displaystyle s\tilde{v}$
(157) $\displaystyle\tilde{n}$ $\displaystyle:=$ $\displaystyle n\tilde{v}$
(158) $\displaystyle\tilde{\varepsilon}$ $\displaystyle:=$
$\displaystyle\varepsilon\tilde{v},$ (159)

so that their current densities are invariant under the redefinition:

$\displaystyle\sqrt{-\tilde{g}}\tilde{u}^{\mu}\tilde{s}$ $\displaystyle=$
$\displaystyle\sqrt{-g}u^{\mu}s$ (160)
$\displaystyle\sqrt{-\tilde{g}}\tilde{u}^{\mu}\tilde{n}$ $\displaystyle=$
$\displaystyle\sqrt{-g}u^{\mu}n.$ (161)

The new Lagrangian $\mathcal{\tilde{L}}$ is defined in order to make the
Lagrangian density $\sqrt{-g}\mathcal{L}$ be invariant:

$\displaystyle\mathcal{\tilde{L}}$ $\displaystyle:=$ $\displaystyle
e^{-d\eta}\mathcal{L}.$ (162)

The relation between $\varepsilon$ and fundamental fields, Eq. (75),
transforms into

$\displaystyle\sqrt{-\tilde{g}}\tilde{\varepsilon}=-\tilde{u}^{\mu}\tilde{u}^{\nu}\,{}^{\rm
EL}\\!\mathcal{D}_{\tilde{g}^{\mu\nu}}\sqrt{-\tilde{g}}\mathcal{\tilde{L}}.$
(163)

From Eq. (157) - (159), the first law of thermodynamics and Gibbs-Duhem
relation are modified:

$\displaystyle\delta\tilde{\varepsilon}=T\delta\tilde{s}-p\delta\tilde{v}+\mu\delta\tilde{n}$
(164) $\displaystyle T\tilde{s}+\mu\tilde{n}=\tilde{\varepsilon}+p\tilde{v}.$
(165)

The equation (153) is rewritten as

$\displaystyle\big{(}\delta^{\nu}_{\mu}+\tilde{u}^{\nu}\tilde{u}_{\mu}\big{)}\bigg{[}\tilde{u}^{\rho}\tilde{\nabla}_{\rho}\tilde{u}_{\nu}+\frac{\tilde{v}\partial_{\nu}p}{\tilde{\varepsilon}+p\tilde{v}}+\frac{q\tilde{n}}{2(\tilde{\varepsilon}+p\tilde{v})}F_{\mu\rho}\tilde{u}^{\rho}\bigg{]}=0.$
(166)

We have two kinds of geometry defined by two metric tensors $g_{\mu\nu}$ and
$\tilde{g}_{\mu\nu}$, respectively, those are related by the conformal
transformation (142). While the former, that we call the thermodynamical
geometry, makes the description of the thermodynamics be simple, the latter,
we call the kinematical geometry, does that of kinematics of the fluid be
simple. The quantity $\tilde{v}$, equivalently $\eta$, represents the
conformal factor between the two spatial volume elements $\sqrt{h}$ and
$\sqrt{\tilde{h}}$ defined by $g_{\mu\nu}$ and $\tilde{g}_{\mu\nu}$,
respectively, i.e.,

$\displaystyle\sqrt{h}=\tilde{v}\sqrt{\tilde{h}}.$ (167)

We can interpret $\tilde{v}$ as the thermodynamical volume element measured by
kinematical one.

Let us rewrite the variation of the entropy current density. The variation of
$g^{\mu\nu}$ is separated as

$\displaystyle\delta g^{\mu\nu}$ $\displaystyle=$
$\displaystyle\delta\big{(}e^{2\eta}\tilde{g}^{\mu\nu}\big{)}$ (168)
$\displaystyle=$ $\displaystyle
e^{2\eta}\delta\tilde{g}^{\mu\nu}+2e^{2\eta}\tilde{g}^{\mu\nu}\delta\eta.$
(169)

Therefore, substituting Eq. (169) into Eq. (4.1),

$\displaystyle\delta\Big{(}\sqrt{-\tilde{g}}\tilde{u}^{0}\tilde{s}\Big{)}+\tilde{\lambda}_{0}\delta\Big{(}\sqrt{-\tilde{g}}\tilde{u}^{0}\tilde{n}\Big{)}$
(170) $\displaystyle\overset{\mathrm{stationary}}{=}$
$\displaystyle-\frac{u^{0}}{2T}e^{-\eta}\bigg{\\{}\delta\Big{(}\sqrt{-\tilde{g}}\mathcal{\tilde{L}}\Big{)}-\,^{\rm
EL}\\!\mathcal{D}_{A_{0}}\sqrt{-\tilde{g}}\mathcal{\tilde{L}}\cdot\delta
A_{0}+2(d-1)\sqrt{-\tilde{g}}p\tilde{v}\delta\eta$
$\displaystyle+\sqrt{-\tilde{g}}\Big{[}\tilde{\varepsilon}\tilde{u}_{\rho}\tilde{u}_{\sigma}+p\tilde{v}\big{(}\tilde{u}_{\rho}\tilde{u}_{\sigma}+\tilde{g}_{\rho\sigma}\big{)}\Big{]}\delta\tilde{g}^{\rho\sigma}+\delta\Big{(}\partial_{\rho}\tilde{\Omega}^{\rho
0}_{\ \ \ 0}[\sqrt{-\tilde{g}}\mathcal{\tilde{L}}]\Big{)}$
$\displaystyle+\partial_{a}\bigg{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{a}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\tilde{f}}\sqrt{-\tilde{g}}\mathcal{\tilde{L}}\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\tilde{f}\bigg{)}\bigg{\\}},$

where

$\displaystyle\tilde{\lambda}_{0}=\frac{\mu}{T}+\frac{q}{2T}\tilde{u}^{0}A_{0},$
(171)

$\tilde{f}=\\{\tilde{g}^{\rho\sigma},A_{\mu},\eta,\varphi\\}$, and
$\tilde{\Omega}^{\rho 0}_{\ \ \ 0}$ is defined by Eq. (361) with
$g^{\rho\sigma}$ and $u^{\mu}$ replaced by $\tilde{g}^{\rho\sigma}$ and
$\tilde{u}^{\mu}$, respectively. The following field equations are derived by
the maximum entropy principle:

$\,{}^{\rm
EL}\\!\bar{\mathcal{D}}_{\tilde{g}^{\mu\nu}}\sqrt{-\tilde{g}}\mathcal{\tilde{L}}$
$\displaystyle=$
$\displaystyle-\sqrt{-\tilde{g}}\Big{[}\tilde{\varepsilon}\tilde{u}_{\mu}\tilde{u}_{\nu}+p\tilde{v}\big{(}\tilde{u}_{\mu}\tilde{u}_{\nu}+\tilde{g}_{\mu\nu}\big{)}\Big{]}$
(172) $\,{}^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-\tilde{g}}\mathcal{\tilde{L}}$
$\displaystyle=$ $\displaystyle-\sqrt{-\tilde{g}}q\tilde{n}\tilde{u}^{\mu}$
(173) $\,{}^{\rm EL}\\!\mathcal{D}_{\eta}\sqrt{-\tilde{g}}\mathcal{\tilde{L}}$
$\displaystyle=$ $\displaystyle-2(d-1)\sqrt{-\tilde{g}}p\tilde{v}$ (174)
$\,{}^{\rm EL}\\!\mathcal{D}_{\varphi}\sqrt{-\tilde{g}}\mathcal{\tilde{L}}$
$\displaystyle=$ $\displaystyle 0.$ (175)

The consistency conditions are

$\displaystyle\frac{\tilde{u}^{0}}{T}\overset{\mathrm{stationary}}{=}{\rm
const.}$ (176)
$\displaystyle\tilde{\lambda}_{0}\overset{\mathrm{stationary}}{=}{\rm
const.}.$ (177)

The condition (176) coincides with Tolman’s law in the kinematical geometry.

##  5 Surface term

In section 4, we see that the field equations (172) - (175) maximize the total
entropy $S$ in appropriate boundary conditions. When a configuration satisfies
the field equations, the variation of the entropy is written in the following
form:

$\displaystyle\delta S$ $\displaystyle\overset{\mathrm{stationary}}{=}$
$\displaystyle-\bigg{(}\lambda+\frac{\tilde{u}^{0}}{2T}A_{0}\bigg{)}\delta
N+\frac{\tilde{u}^{0}}{2T}\delta\tilde{M}_{\rm Noether}$
$\displaystyle-\frac{\tilde{u}^{0}}{2T}\oint_{\partial\mathcal{V}}({\mathrm{d}}^{d-2}x)_{0a}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{a}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\tilde{f}}\sqrt{-\tilde{g}}\mathcal{\tilde{L}}\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\tilde{f},$

where the quantities

$\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{a}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\tilde{f}}\sqrt{-\tilde{g}}\mathcal{\tilde{L}}$
(179)

can be regarded as the chemical potentials444Notice that the variations
$\delta\tilde{f}\big{|}_{\partial\mathcal{V}}$ and
$\delta\partial_{i_{1}}\cdots\partial_{i_{k}}\tilde{f}\big{|}_{\partial\mathcal{V}}$
are not independent.  conjugate to
$\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\tilde{f}$.

In this section, as an example, we calculate the variation of the entropy in
the Lovelock theory including the surface terms, and see that the variation
can be represented by those of the total Kodama energy and area of the
boundary. For simplicity, in this section, the tilde “$\tilde{\ }$” is
omitted.

###  5.1 Spherically symmetric spacetime and the generalized Kodama flux

In the Lovelock theory, it is convenient to describe the spherically symmetric
system by the warped product coordinate (181) due to the relation between the
Lovelock tensors and Kodama flux. In this subsection, we introduce the warped
product coordinate and briefly summarize the properties of the (generalized)
Kodama flux [10, 11].

In any spherically symmetric spacetime, there are coordinates such that the
line element ${\mathrm{d}}l$ is written in the form

$\displaystyle{\mathrm{d}}l^{2}$ $\displaystyle=$ $\displaystyle
g_{\mu\nu}{\mathrm{d}}x^{\mu}{\mathrm{d}}x^{\nu}$ (180) $\displaystyle=$
${}^{\mathrm{B}}\\!\\!g_{\eta\xi}{\mathrm{d}}x^{\eta}{\mathrm{d}}x^{\xi}+\chi(r)^{2}\
{}^{\mathrm{F}}\\!\\!g_{ij}{\mathrm{d}}x^{i}{\mathrm{d}}x^{j},$ (181)

where $\eta$ and $\xi$ run over $\\{0,\,1\\}$, while $i$ and $j$ do
$\\{2,\,\dots,\,d-1\\}$, and $r$ is the radial component of the coordinate. In
the restricted coordinate, one can see the total spacetime
$(\mathcal{M}^{d},\,g_{\mu\nu})$ as a “warped product” manifold with the
$2$-dimensional “base space”
$(\,^{\mathrm{B}}\\!\mathcal{M}^{2},\,\,^{\mathrm{B}}\\!g_{\eta\xi})$ and the
$(d-2)$-dimensional “fibers”
$(^{\mathrm{F}}\\!\mathcal{M}^{d-2},\,\,^{\mathrm{F}}\\!g_{ij})$. The scalar
function on the base space $\chi(r)$ is called the “warp factor”.

The square root of the determinant of the metric $\sqrt{-g}$ is decomposed as

$\sqrt{-g}=\chi^{d-2}\sqrt{\,{}^{\mathrm{F}}\\!g}\sqrt{-^{\mathrm{B}}\\!g}.$

Define the following quantity:

$\displaystyle\epsilon^{\mu\nu}$ $\displaystyle:=$
$\displaystyle\left(\begin{array}[]{cc}^{\mathrm{B}}\\!\epsilon^{\eta\xi}&0\\\
0&0\end{array}\right)$ (184) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}0&1&0&\ldots\\\ -1&0&0&\ldots\\\
0&0&0&\ldots\\\ \vdots&\vdots&\vdots&\ddots\end{array}\right),$ (189)

where ${}^{\mathrm{B}}\\!\epsilon^{\eta\xi}$ denotes the Levi-civita symbol on
the base space, that transforms as a tensor density of weight $1$ on the base
space, and $\epsilon^{\mu\nu}$ transforms as a tensor density of weight $1$ on
the total spacetime under the restricted coordinate transformation. The
quantity

${}^{\mathrm{B}}\\!e^{\eta\xi}:=\frac{{}^{\mathrm{B}}\\!\epsilon^{\eta\xi}}{\sqrt{-^{\mathrm{B}}\\!g}}$
(190)

is the Levi-Civita tensor on the base space.

Note that under the restricted coordinate transformation, there are more
fields that transform as tensor fields than ordinary tensor fields. In this
section, the action of the covariant derivative $\nabla_{\mu}$ on the
generalized tensor fields are defied by the same way as the ordinary ones.

Consider the following covariant derivative:

$\displaystyle\nabla_{\mu}\bigg{(}\frac{1}{\chi^{d-2}}\frac{\epsilon^{\mu\nu}}{\sqrt{-\,^{\mathrm{B}}\\!g}}\bigg{)}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\mu}\bigg{(}\frac{\sqrt{-g}}{\chi^{d-2}}\frac{\epsilon^{\mu\nu}}{\sqrt{-\,^{\mathrm{B}}\\!g}}\bigg{)}$
(191) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\mu}\Big{(}\sqrt{\,{}^{\mathrm{F}}\\!g}\epsilon^{\mu\nu}\Big{)}$
(192) $\displaystyle=$
$\displaystyle\frac{\sqrt{\,{}^{\mathrm{F}}\\!g}}{\sqrt{-g}}\partial_{\mu}\epsilon^{\mu\nu}$
(193) $\displaystyle=$ $\displaystyle 0.$ (194)

Therefore, defining
$e^{\mu\nu}:=\epsilon^{\mu\nu}/\sqrt{-\,^{\mathrm{B}}\\!g}$, the quantity

$\displaystyle\frac{e^{\mu\nu}}{\chi^{d-2}}$ (195)

is a covariantly divergence-free second-order anti-symmetric (generalized)
tensor field. Using these features and an arbitrary (generalized) scalar field
$\Psi$, one can construct a current $J_{[\Psi]}^{\mu}$, called the generalized
Kodama flux:

$\displaystyle J_{[\Psi]}^{\mu}$ $\displaystyle:=$
$\displaystyle\frac{e^{\mu\nu}}{\chi^{d-2}}\nabla_{\nu}\Psi$ (196)
$\displaystyle=$
$\displaystyle\sqrt{\,{}^{\mathrm{F}}\\!g}\frac{\epsilon^{\mu\nu}}{\sqrt{-g}}\partial_{\nu}\Psi.$
(197)

Due to the covariantly divergent-freeness of $e^{\mu\nu}/\chi^{d-2}$, the
divergence of $J_{[\Psi]}^{\mu}$ vanishes, i.e.,

$\displaystyle\nabla_{\mu}J_{[\Psi]}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{e^{\mu\nu}}{\chi^{d-2}}\nabla_{\mu}\nabla_{\nu}\Psi$ (198)
$\displaystyle=$ $\displaystyle 0.$ (199)

For $\Psi=\frac{1}{d-1}\chi^{d-1}$,

$\displaystyle k^{\mu}$ $\displaystyle:=$ $\displaystyle
J^{\mu}_{[\frac{1}{d-1}\chi^{d-1}]}$ (200) $\displaystyle=$
$\displaystyle\sqrt{\,{}^{\mathrm{F}}\\!g}\frac{\epsilon^{\mu\nu}}{\sqrt{-g}}\chi^{d-2}\partial_{\nu}\chi$
(201) $\displaystyle=$
$\displaystyle\frac{\epsilon^{\mu\nu}}{\sqrt{-\,^{\mathrm{B}}\\!g}}\partial_{\nu}\chi$
(202) $\displaystyle=$ $\displaystyle e^{\mu\nu}\partial_{\mu}\chi$ (203)

is called the “Kodama vector”.

###  5.2 The Lovelock theory

In this subsection, we consider the Lovelock theory of gravity in the warped
product coordinate (181). The Lagrangian of the Lovelock theory [12] is

$\displaystyle\mathcal{L}_{\rm Love.}$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa^{2}}L_{\rm Love.}$ (204) $\displaystyle L_{\rm
Love.}$ $\displaystyle:=$ $\displaystyle\sum_{n=0}^{[d/2]}a_{(n)}L_{(n)}$
(205) $\displaystyle L_{(n)}$ $\displaystyle:=$
$\displaystyle\frac{1}{2^{n}}\delta^{\mu_{1}\cdots\mu_{n}\nu_{1}\cdots\nu_{n}}_{\rho_{1}\cdots\rho_{n}\sigma_{1}\cdots\sigma_{n}}R_{\mu_{1}\nu_{1}}^{\
\ \ \ \ \ \rho_{1}\sigma_{1}}\cdots R_{\mu_{n}\nu_{n}}^{\ \ \ \ \ \
\rho_{n}\sigma_{n}},$ (206)

where

$\displaystyle\delta^{\mu_{1}\cdots\mu_{n}}_{\rho_{1}\cdots\rho_{n}}:=n!\delta^{\mu_{1}}_{[\nu_{1}}\cdots\delta^{\mu_{n}}_{\nu_{n}]},$
(207) $\displaystyle R^{\mu}_{\ \
\nu\rho\sigma}:=\partial_{\rho}\Gamma^{\mu}_{\nu\sigma}-\partial_{\sigma}\Gamma^{\mu}_{\nu\rho}+\Gamma^{\mu}_{\alpha\rho}\Gamma^{\alpha}_{\nu\sigma}-\Gamma^{\mu}_{\alpha\sigma}\Gamma^{\alpha}_{\nu\rho},$
(208)

and $a_{(n)}$, $n=0,\,1,\,\cdots,\,[d/2]$ are the parameters of the theory.
The parameter $a_{(0)}$ is related to the cosmological constant $\Lambda$ by
$a_{(0)}=-2\Lambda$. The Euler-Lagrange derivative of
$\sqrt{-g}\mathcal{L}_{\rm Love.}$ is

$\displaystyle G^{\mu}_{\ \ \nu}$ $\displaystyle:=$ $\displaystyle
g^{\mu\rho}\,{}^{\rm EL}\\!\mathcal{D}_{g^{\rho\nu}}\sqrt{-g}\mathcal{L}_{\rm
Love.}$ (209) $\displaystyle=$ $\displaystyle\sum_{n=0}^{[n/2]}a_{(n)}G_{(n)\
\nu}^{\ \ \ \mu},$ (210)

where

$\displaystyle G_{(n)\ \nu}^{\ \ \ \mu}$ $\displaystyle=$
$\displaystyle-\frac{1}{2^{n+1}}\delta^{\mu\mu_{1}\cdots\mu_{n}\rho_{1}\cdots\rho_{n}}_{\nu\nu_{1}\cdots\nu_{n}\sigma_{1}\cdots\sigma_{n}}R_{\mu_{1}\rho_{1}}^{\
\ \ \ \ \ \nu_{1}\sigma_{1}}\cdots R_{\mu_{n}\rho_{n}}^{\ \ \ \ \ \
\nu_{n}\sigma_{n}}.$ (211)

The tensor $G_{(n)\ \nu}^{\ \ \ \mu}$ is called the $n$-th order Lovelock
tensor.

In a spherically symmetric spacetime with a warped product structure of the
metric (181), the $(\eta,\,\xi)$ components of the $n$-th order Lovelock
tensor is written in the form [13]

$\displaystyle G_{(n)\ \xi}^{\ \ \ \eta}$ $\displaystyle=$
$\displaystyle-\frac{n(d-2)!}{(d-1-2n)!}\frac{{}^{\mathrm{B}}\\!\nabla^{\eta}\\!\,{}^{\mathrm{B}}\\!\nabla_{\xi}\chi-(^{\mathrm{B}}\\!\nabla^{\zeta}\\!\,{}^{\mathrm{B}}\\!\nabla_{\zeta}\chi)\delta^{\eta}_{\xi}}{\chi}\bigg{(}\frac{k_{\mathrm{F}}-|^{\mathrm{B}}\\!\nabla\chi|^{2}}{\chi^{2}}\bigg{)}^{n-1}$
(212)
$\displaystyle-\frac{(d-2)!}{2(d-2-2n)!}\delta^{\eta}_{\xi}\bigg{(}\frac{k_{\mathrm{F}}-|^{\mathrm{B}}\\!\nabla\chi|^{2}}{\chi^{2}}\bigg{)}^{n},$

where $\eta,\,\xi=0,\,1$ and
$|\,^{\mathrm{B}}\\!\nabla\chi|^{2}:=\,^{\mathrm{B}}\\!g^{\eta\xi}\,{}^{\mathrm{B}}\\!\nabla_{\eta}\chi\,^{\mathrm{B}}\\!\nabla_{\xi}\chi$.
Consider the quantity

$\displaystyle G_{(n)\ \xi}^{\ \ \ \eta}k^{\xi}$ $\displaystyle=$
$\displaystyle\sqrt{\,{}^{\mathrm{F}}\\!g}\frac{\epsilon^{\eta\xi}}{\sqrt{-g}}\frac{(d-2)!}{2(d-1-2n)!}\Big{[}n\chi^{d-1-2n}\partial_{\xi}\big{(}|\,^{\mathrm{B}}\\!\nabla\chi|^{2}\big{)}\big{(}k_{\mathrm{F}}-|\,^{\mathrm{B}}\\!\nabla\chi|^{2}\big{)}^{n-1}$
(213)
$\displaystyle-(d-1-2n)\chi^{d-2-2n}\partial_{\xi}\chi\big{(}k_{\mathrm{F}}-|\,^{\mathrm{B}}\\!\nabla\chi|^{2}\big{)}^{n}\Big{]}$
$\displaystyle=$
$\displaystyle\sqrt{\,{}^{\mathrm{F}}\\!g}\frac{\epsilon^{\eta\xi}}{\sqrt{-g}}\partial_{\xi}\bigg{[}-\frac{(d-2)!}{(d-1-2n)!}\frac{\chi^{d-1-2n}}{2}\big{(}k_{\mathrm{F}}-|\,^{\mathrm{B}}\\!\nabla\chi|^{2}\big{)}^{n}\bigg{]}$
(214) $\displaystyle=$
$\displaystyle\sqrt{\,{}^{\mathrm{F}}\\!g}\frac{\epsilon^{\eta\xi}}{\sqrt{-g}}\partial_{\xi}\bigg{(}-\frac{\kappa^{2}}{A_{\mathrm{F}}}m_{(n)}\bigg{)},$
(215)

where

$\displaystyle
m_{(n)}:=\frac{A_{\mathrm{F}}(d-2)!}{2\kappa^{2}(d-1-2n)!}\chi^{d-1-2n}\big{(}k_{\mathrm{F}}-|\,^{\mathrm{B}}\\!\nabla\chi|^{2}\big{)}^{n}.$
(216)

Define the following quantity $m$, called the generalized Misner-Sharp mass
[13], by

$\displaystyle m$ $\displaystyle:=$
$\displaystyle\sum_{n=0}^{[d/2]}a_{(n)}m_{(n)}$ (217) $\displaystyle=$
$\displaystyle\sum_{n=0}^{[d/2]}\frac{a_{(n)}A_{\mathrm{F}}(d-2)!}{2\kappa^{2}(d-1-2n)!}\chi^{d-1-2n}\big{(}k_{\mathrm{F}}-|\,^{\mathrm{B}}\\!\nabla\chi|^{2}\big{)}^{n}$
(218) $\displaystyle\overset{\mathrm{static}}{=}$
$\displaystyle\sum_{n=0}^{[d/2]}b_{(n)}\chi^{d-1-2n}\bigg{(}k_{\mathrm{F}}-\frac{(\partial_{1}\chi)^{2}}{g_{11}}\bigg{)}^{n},$
(219)

and

$\displaystyle
b_{(n)}:=\frac{a_{(n)}A_{\mathrm{F}}(d-2)!}{2\kappa^{2}(d-1-2n)!}.$ (220)

The equation (219) relates $m$, $\chi$, $\partial_{1}\chi$, and $m$. Note that
since in Eq. (219), $g_{11}$ and $\partial_{1}\chi$ appear only in the form
$(\partial_{1}\chi)^{2}/g_{11}$, the quantity $(\partial_{1}\chi)^{2}/g_{11}$
depends only on $m$ and $\chi$:

$\displaystyle\frac{(\partial_{1}\chi)^{2}}{g_{11}}=W(m,\chi),$ (221)

where $W(m,\chi)$ is a function of $m$ and $\chi$. From Eq. (221), the partial
derivative of $g_{11}$ with respect to $\partial_{1}\chi$ is obtained:

$\displaystyle\frac{\partial
g_{11}}{\partial\partial_{1}\chi}=\frac{2g_{11}}{\partial_{1}\chi}$ (222)
$\displaystyle\frac{\partial
g^{11}}{\partial\partial_{1}\chi}=-\frac{2g^{11}}{\partial_{1}\chi}.$ (223)

When the coefficients $a_{(n)}$ are constants, the contraction of the Lovelock
tensor and the Kodama vector is arranged in the form

$\displaystyle G^{\eta}_{\ \ \xi}k^{\xi}$ $\displaystyle=$
$\displaystyle\sqrt{\,{}^{\mathrm{F}}\\!g}\frac{\epsilon^{\eta\xi}}{\sqrt{-g}}\partial_{\xi}\bigg{(}-\frac{\kappa^{2}}{A_{\mathrm{F}}}m\bigg{)}.$
(224)

Comparing with the definition of the generalized Kodama flux (197),

$\displaystyle G^{\ \ \ \mu}_{(n)\ \nu}k^{\nu}$ $\displaystyle=$
$\displaystyle J_{[-\kappa^{2}m_{(n)}/A_{\mathrm{F}}]}^{\mu}$ (225)
$\displaystyle=:$ $\displaystyle J_{(n)}^{\mu}$ (226) $\displaystyle
G^{\mu}_{\ \ \nu}k^{\nu}$ $\displaystyle=$ $\displaystyle
J_{[-\kappa^{2}m/A_{\mathrm{F}}]}^{\mu}$ (227) $\displaystyle=:$
$\displaystyle J^{\mu}.$ (228)

The conserved current $J^{\mu}$ is called the Kodama current for the Lovelock
theory [10].

###  5.3 The variation of the entropy in the Lovelock theory

Here, we calculate the variation of the entropy of a spherically symmetric and
static neutral fluid system using the special form of the Lovelock tensor.

In a co-moving coordinate, the energy density $\sqrt{-g}\varepsilon$ is
arranged as

$\displaystyle\sqrt{-g}\varepsilon$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa^{2}}\sqrt{-g}G^{0}_{\ \ 0}$ (229)
$\displaystyle\overset{\mathrm{static}}{=}$
$\displaystyle\frac{\sqrt{-g}}{A_{\mathrm{F}}\chi^{d-2}\partial_{1}\chi}\partial_{1}m$
(230) $\displaystyle=$
$\displaystyle\frac{\sqrt{\,{}^{\mathrm{F}}\\!g}\sqrt{-\,^{\mathrm{B}}\\!g}\partial_{1}m}{A_{\mathrm{F}}\partial_{1}\chi}$
(231) $\displaystyle=$
$\displaystyle\frac{\sqrt{\,{}^{\mathrm{F}}\\!g}\sqrt{-g_{00}}\sqrt{g_{11}}\partial_{1}m}{A_{\mathrm{F}}\partial_{1}\chi}$
(232) $\displaystyle=$
$\displaystyle\frac{\sqrt{\,{}^{\mathrm{F}}\\!g}\sqrt{-g_{00}}\partial_{1}m}{A_{\mathrm{F}}\sqrt{W(m,\chi)}}.$
(233)

Let us employ $f=\\{g_{00},\,m,\,\chi\\}$ as the fundamental fields. We see in
Eq. (233) that $\sqrt{-g}\varepsilon$ does not depend on neither
$\partial_{1}g_{00}$, $\partial_{1}\partial_{1}m$, $\partial_{1}\chi$ nor
their higher derivatives. Therefore, the variation of the entropy is

$\displaystyle\delta S$ $\displaystyle\overset{\mathrm{static}}{=}$
$\displaystyle\int_{\mathcal{V}}{\mathrm{d}}^{d-1}x\frac{u^{0}}{T}\bigg{\\{}\,^{\rm
EL}\\!\mathcal{D}_{g_{00}}\sqrt{-g}\varepsilon\cdot\delta g_{00}+\,^{\rm
EL}\\!\mathcal{D}_{m}\sqrt{-g}\varepsilon\cdot\delta m+\,^{\rm
EL}\\!\mathcal{D}_{\chi}\sqrt{-g}\varepsilon\cdot\delta\chi$ (234)
$\displaystyle\ \ \ \ \ \ -\frac{1}{2}\sqrt{-g}\Big{[}\varepsilon
u^{\mu}u^{\nu}+p\big{(}u^{\mu}u^{\nu}+g^{\mu\nu}\big{)}\Big{]}\delta
g_{\mu\nu}\bigg{\\}}$
$\displaystyle+\oint_{\partial\mathcal{V}}{\mathrm{d}}^{d-2}x\frac{u^{0}}{T}\bigg{(}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{1}m}\sqrt{-g}\varepsilon\cdot\delta m\bigg{)}.$

From the expression (233), the Euler-Lagrange derivatives of
$\sqrt{-g}G^{0}_{\ \ 0}$ are easily obtained:

$\,{}^{\rm EL}\\!\mathcal{D}_{\partial_{1}m}\sqrt{-g}\varepsilon$
$\displaystyle=$ $\displaystyle\frac{1}{\kappa^{2}}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{1}m}\sqrt{-g}G^{0}_{\ \ 0}$ (235)
$\displaystyle=$
$\displaystyle\frac{\sqrt{\,{}^{\mathrm{F}}\\!g}\sqrt{-\,^{\mathrm{B}}\\!g}}{A_{\mathrm{F}}\partial_{1}\chi}$
(236) $\displaystyle=$
$\displaystyle\frac{\sqrt{\,{}^{\mathrm{F}}\\!g}\sqrt{-g_{00}}\sqrt{g_{11}}}{A_{\mathrm{F}}\partial_{1}\chi}.$
(237)

The second line of Eq. (234) is arranged as

$\displaystyle\frac{1}{2}\sqrt{-g}\Big{[}\varepsilon
u^{\rho}u^{\sigma}+p(u^{\rho}u^{\sigma}+g^{\rho\sigma})\Big{]}\delta
g_{\rho\sigma}$ (238) $\displaystyle\overset{\mathrm{static}}{=}$
$\displaystyle\frac{1}{2}\sqrt{-g}\Big{[}-\varepsilon g^{00}\delta
g_{00}+pg^{11}\delta g_{11}+pg^{ij}\delta g_{ij}\Big{]}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{-g}\Big{[}-\varepsilon\,^{\mathrm{B}}\\!g^{00}\delta\,^{\mathrm{B}}\\!g_{00}+p\,^{\mathrm{B}}\\!g^{11}\delta\,^{\mathrm{B}}\\!g_{11}+p\chi^{-2}\,{}^{\mathrm{F}}\\!g^{ij}\delta\big{(}\chi^{2}\,{}^{\mathrm{F}}\\!g_{ij}\big{)}\Big{]}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{-g}\Big{[}-\varepsilon\,^{\mathrm{B}}\\!g^{00}\delta\,^{\mathrm{B}}\\!g_{00}+p\,^{\mathrm{B}}\\!g^{11}\delta\,^{\mathrm{B}}\\!g_{11}+(d-2)p\chi^{-2}\delta\big{(}\chi^{2}\big{)}\Big{]}$
(245) $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{-g}\bigg{[}-\varepsilon\,^{\mathrm{B}}\\!g^{00}\delta\,^{\mathrm{B}}\\!g_{00}+p\,^{\mathrm{B}}\\!g^{11}\bigg{(}\frac{\partial\,^{\mathrm{B}}\\!g_{11}}{\partial
m}\delta
m+\frac{\partial\,^{\mathrm{B}}\\!g_{11}}{\partial\chi}\delta\chi+\frac{\partial\,^{\mathrm{B}}\\!g_{11}}{\partial\partial_{1}\chi}\delta\partial_{1}\chi\bigg{)}$
$\displaystyle+2(d-2)p\chi^{-1}\delta\chi\bigg{]}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{-g}\bigg{[}-\varepsilon\,^{\mathrm{B}}\\!g^{00}\delta\,^{\mathrm{B}}\\!g_{00}+p\,^{\mathrm{B}}\\!g^{11}\bigg{(}\frac{\partial\,^{\mathrm{B}}\\!g_{11}}{\partial
m}\delta
m+\frac{\partial\,^{\mathrm{B}}\\!g_{11}}{\partial\chi}\delta\chi\bigg{)}+2(d-2)p\chi^{-1}\delta\chi\bigg{]}$
$\displaystyle-\partial_{1}\bigg{(}\frac{1}{2}\sqrt{-g}p\,^{\mathrm{B}}\\!g^{11}\frac{\partial\,^{\mathrm{B}}\\!g_{11}}{\partial\partial_{1}\chi}\bigg{)}\delta\chi+\partial_{1}\bigg{(}\frac{1}{2}\sqrt{-g}p\,^{\mathrm{B}}\\!g^{11}\frac{\partial\,^{\mathrm{B}}\\!g_{11}}{\partial\partial_{1}\chi}\delta\chi\bigg{)}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{-g}\bigg{[}-\varepsilon\,^{\mathrm{B}}\\!g^{00}\delta\,^{\mathrm{B}}\\!g_{00}+p\,^{\mathrm{B}}\\!g^{11}\bigg{(}\frac{\partial\,^{\mathrm{B}}\\!g_{11}}{\partial
m}\delta
m+\frac{\partial\,^{\mathrm{B}}\\!g_{11}}{\partial\chi}\delta\chi\bigg{)}+2(d-2)p\chi^{-1}\delta\chi\bigg{]}$
$\displaystyle-\partial_{1}\bigg{(}\frac{\sqrt{-g}p}{\partial_{1}\chi}\bigg{)}\delta\chi+\partial_{1}\bigg{(}\frac{\sqrt{-g}p}{\partial_{1}\chi}\delta\chi\bigg{)}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{-g}\bigg{[}-\varepsilon\,^{\mathrm{B}}\\!g^{00}\delta\,^{\mathrm{B}}\\!g_{00}+p\,^{\mathrm{B}}\\!g^{11}\bigg{(}\frac{\partial\,^{\mathrm{B}}\\!g_{11}}{\partial
m}\delta
m+\frac{\partial\,^{\mathrm{B}}\\!g_{11}}{\partial\chi}\delta\chi\bigg{)}+2(d-2)p\chi^{-1}\delta\chi\bigg{]}$
$\displaystyle-\partial_{1}\bigg{(}\frac{\sqrt{-g}p}{\partial_{1}\chi}\bigg{)}\delta\chi+\partial_{1}\bigg{(}-t_{0}\frac{n_{1}}{\partial_{1}\chi}\sqrt{\gamma}p\delta\chi\bigg{)}$

The variation of the total entropy $\delta S$ of the system that satisfies the
field equations is

$\displaystyle\delta S+\lambda_{0}\delta N$ $\displaystyle=$
$\displaystyle\oint_{\partial\mathcal{V}}{\mathrm{d}}^{d-2}x\frac{u^{0}}{T}\bigg{(}-\frac{1}{\kappa^{2}}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{1}m}\sqrt{-g}G^{0}_{\ \ 0}\cdot\delta
m-t_{0}\frac{n_{1}}{\partial_{1}\chi}\sqrt{\gamma}p\delta\chi\bigg{)}$ (246)
$\displaystyle\overset{\mathrm{static}}{=}$
$\displaystyle\oint_{\partial\mathcal{V}}{\mathrm{d}}^{d-2}x\frac{u^{0}}{T}(-t_{0})\frac{n_{1}}{\partial_{1}\chi}\bigg{(}\frac{\sqrt{\,{}^{\mathrm{F}}\\!g}}{A_{\mathrm{F}}}\delta
m+p\sqrt{\gamma}\delta\chi\bigg{)}$
$\displaystyle\overset{\mathrm{static}}{=}$
$\displaystyle\bigg{(}\frac{n_{1}}{T\partial_{1}\chi}\delta
m+\frac{n_{1}}{T\partial_{1}\chi}A_{\mathrm{F}}\chi^{d-2}p\delta\chi\bigg{)}\bigg{|}_{\partial\mathcal{V}}$
(247) $\displaystyle=:$ $\displaystyle\frac{1}{T_{\rm global}}\delta
m|_{\partial\mathcal{V}}+\frac{A_{\mathrm{F}}\chi^{d-2}}{T_{\rm global}}P_{\rm
global}\delta\chi|_{\partial\mathcal{V}}.$ (248)

The globally defied temperature $T_{\rm global}$ and pressure $P_{\rm global}$
are

$\displaystyle T_{\rm global}$ $\displaystyle=$
$\displaystyle\frac{\partial_{1}\chi}{n_{1}}T\bigg{|}_{\partial\mathcal{V}}$
(249) $\displaystyle P_{\rm global}$ $\displaystyle=$ $\displaystyle
p|_{\partial\mathcal{V}}.$ (250)

For example, in general relativity without the cosmological constatn in
$4$-dimensional spacetime, $a_{(0)}=0,\ a_{(1)}=1,a_{(2)}=\cdots=0$, $d=4$,
$A_{\mathrm{F}}=4\pi$, and $\kappa^{2}=8\pi$,

$\displaystyle g_{11}$ $\displaystyle\overset{\mathrm{static}}{=}$
$\displaystyle(\partial_{1}\chi)^{2}\bigg{(}1-\frac{2m}{\chi}\bigg{)}^{-1}.$
(251)

The global temperature and pressure in general relativity are

$\displaystyle T_{\rm global}^{\rm GR}$ $\displaystyle=$
$\displaystyle\Bigg{[}\bigg{(}1-\frac{2m}{\chi}\bigg{)}^{1/2}T\Bigg{]}\Bigg{|}_{\partial\mathcal{V}}$
(252) $\displaystyle P_{\rm global}^{\rm GR}$ $\displaystyle=$ $\displaystyle
p|_{\partial\mathcal{V}}.$ (253)

##  6 Constraints on the Lagrangian

In this section, we determine the form of the Lagrangian supposing following
two assumptions:

  1. 1.

The variation of the entropy $\delta S$ is represented by a linear combination
of the variations of a conserved energy $\delta\tilde{M}$, the particle number
$\delta\tilde{N}$, and the size of the system $\delta\tilde{\chi}$ in the
kinematical geometry.

  2. 2.

The uniform rescaling of the kinematical volume element with the
thermodynamical geometry fixed, namely

$\displaystyle\eta\rightarrow\eta+a,\ \ \ \ \ \ a={\rm const.}$ (254)
$\displaystyle g_{\mu\nu}\rightarrow g_{\mu\nu}$ (255)

does not affect the dynamics of the fields in vacuum resion.

In the previous section, we saw that the Lovelock theory satisfies the
assumption 1. However, the assumption 2 is not satisfied, due to the
dependence of the uniform rescaling of Lovelock’s Lagrangian $L_{(n)}$ on
their order $n$, i.e.,

$\displaystyle L_{(n)}\rightarrow e^{-2na}L_{(n)}.$ (256)

Therefore, the gravitational part of the Lagrangian should be one of $n$-th
order Lovelock’s Lagrangians. In this work, we adopt the $1$st order, namely
the Einstein-Hilbert Lagrangian $R$.

Let us consider the Lagrangian including $A_{\mu}$, $\eta$, and $\varphi$. For
satisfying the assumption 2, the Lagrangian should be written in the form

$\displaystyle\mathcal{L}=\frac{1}{\kappa^{2}}e^{k\eta}\mathcal{K}[g^{\mu\nu},A_{\mu},\partial_{\mu}\eta,\varphi],$
(257)

so that the rescaling of $\varepsilon$ and $n$

$\displaystyle\sqrt{-g}\varepsilon$ $\displaystyle=$
$\displaystyle-\frac{1}{\kappa^{2}}u^{\rho}u^{\sigma}e^{-\eta}\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\rho\sigma}}[\sqrt{-g}e^{k\eta}\mathcal{K}]$ (258)
$\displaystyle\rightarrow$
$\displaystyle-\frac{e^{(k-1)a}}{\kappa^{2}}u^{\rho}u^{\sigma}e^{-\eta}\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\rho\sigma}}[\sqrt{-g}e^{k\eta}\mathcal{K}]$ (259)
$\displaystyle\sqrt{-g}n$ $\displaystyle=$
$\displaystyle\frac{1}{q\kappa^{2}}u_{\rho}\,^{\rm
EL}\\!\mathcal{D}_{A_{\rho}}[\sqrt{-g}e^{k\eta}\mathcal{K}]$ (260)
$\displaystyle\rightarrow$
$\displaystyle\frac{e^{ka}}{q\kappa^{2}}u_{\rho}\,^{\rm
EL}\\!\mathcal{D}_{A_{\rho}}[\sqrt{-g}e^{k\eta}\mathcal{K}]$ (261)

can be canceled by re-defining the gravitational constant $\kappa^{2}$ and
electrical charge density $q$, such as
$\kappa^{2}=\kappa^{2}_{0}e^{(k-1)\eta_{0}}$, $q=q_{0}e^{\eta_{0}}$, and
$\eta_{0}$ is the value of $\eta$ at an arbitrary point.

Imposing the $U(1)$ gauge invariance (369) and (370), Noether’s identity (383)
shows that the conservation law of $N$ demands the field $\eta$ to be
electrically neutral. The following Lagrangian density satisfies all the
demands:

$\displaystyle\mathcal{L}=\frac{1}{\kappa^{2}}e^{(d-2)\eta}\bigg{(}R-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\omega\partial_{\mu}\eta\partial^{\mu}\eta-
D_{\mu}\varphi D^{\mu}\varphi\bigg{)},$ (262)

where $D_{\mu}\varphi:=(\partial_{\mu}-ie_{\varphi}A_{\mu})\varphi$ is the
gauge covariant derivative.

Under the conformal transformation (142), the scalar curvature $R$ transforms
into

$\displaystyle
R=e^{2\eta}\Big{[}\tilde{R}+2(d-1)\tilde{\nabla}^{2}\eta-(d-2)(d-1)\partial_{\mu}\eta\tilde{\partial}^{\mu}\eta\Big{]},$
(263)

and the Lagrangian density in thermodynamical frame (262) into kinematical
frame as

$\displaystyle\sqrt{-g}\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa^{2}}\sqrt{-\tilde{g}}\bigg{[}\tilde{R}+2(d-1)\tilde{\nabla}^{2}\eta-(d-2)(d-1)\partial_{\mu}\eta\tilde{\partial}^{\mu}\eta$
(265)
$\displaystyle-\frac{1}{4}e^{2\eta}F_{\mu\nu}\tilde{F}^{\mu\nu}+\omega\tilde{g}^{\mu\nu}\partial_{\mu}\eta\partial_{\nu}\eta-\tilde{g}^{\mu\nu}D_{\mu}\varphi
D_{\nu}\varphi\bigg{]}$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa^{2}}\sqrt{-\tilde{g}}\bigg{\\{}\tilde{R}-\frac{1}{4}e^{2\eta}F_{\mu\nu}\tilde{F}^{\mu\nu}+\big{[}\omega-(d-2)(d-1)\big{]}\partial_{\mu}\eta\tilde{\partial}^{\mu}\eta-
D_{\mu}\varphi\tilde{D}^{\mu}\varphi\bigg{\\}}$
$\displaystyle+\partial_{\mu}\Big{[}2(d-1)\sqrt{-\tilde{g}}\tilde{\partial}^{\mu}\eta\Big{]}.$

Omitting the total derivative, the Lagrangian density in kinematical frame
$\sqrt{-\tilde{g}}\mathcal{\tilde{L}}$ satisfying the assumption 1 and 2 is
written by

$\displaystyle\sqrt{-\tilde{g}}\mathcal{\tilde{L}}=\frac{1}{\kappa^{2}}\sqrt{-\tilde{g}}\bigg{\\{}\tilde{R}-\frac{1}{4}e^{2\eta}F_{\mu\nu}\tilde{F}^{\mu\nu}+\big{[}\omega-(d-2)(d-1)\big{]}\partial_{\mu}\eta\tilde{\partial}^{\mu}\eta-
D_{\mu}\varphi\tilde{D}^{\mu}\varphi\bigg{\\}}.$

##  7 Conclusion and discussion

We calculate the first order variation of the entropy of a self-interacting
system using only the first law of thermodynamics (12), the Gibbs-Duhem
relation (13), and Noether’s theorem for the invariance of the coordinate and
$U(1)$ gauge transformations. We find that as long as there exists a
Lagrangian density $\mathcal{L}$ and the ordinary constraint relation

$\displaystyle\sqrt{-g}\varepsilon=-u^{\mu}u^{\nu}\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\mu\nu}}\sqrt{-g}\mathcal{L}$ (267)

is imposed, every stationary state satisfying the following ordinary field
equations for a charged perfect fluid

$\displaystyle\delta\varepsilon=T\delta s+\mu\delta n$ (268) $\displaystyle
Ts+\mu n=\varepsilon+p$ (269) $\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\mu\nu}}\sqrt{-g}\mathcal{L}=\sqrt{-g}\Big{[}\varepsilon
u_{\mu}u_{\nu}+p(u_{\mu}u_{\nu}+g_{\mu\nu})\Big{]}$ (270) $\,{}^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{L}=\sqrt{-g}u^{\mu}qn$ (271)
$\,{}^{\rm EL}\\!\mathcal{D}_{\varphi}\sqrt{-g}\mathcal{L}=0$ (272)

maximizes the total entropy under appropriate boundary conditions, no matter
what the form of the Lagrangian density $\mathcal{L}$ that depends on
$g^{\mu\nu}$, $A_{\mu}$, $\varphi$, and $u^{\mu}$. Therefore, the
compatibility between ordinary field equations (268) - (272) and the maximum
entropy principle requires a physical necessity for the constraint relation
(267) to hold.

Further, enlarging the region of the constraint relation to

$\displaystyle\sqrt{-g}\varepsilon=-e^{-\eta}u^{\mu}u^{\nu}\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\mu\nu}}\sqrt{-g}\mathcal{L},$ (273)

and varying the additional field $\eta$ independently, we find that the
modified field equations (84) - (87) maximize the entropy of the stationary
states, instead a force-free fluid do not flow along the geodesic orbits for
$g_{\mu\nu}$ so that another geometry $\tilde{g}_{\mu\nu}$, the kinematical
geometry, should be introduced. The kinematical geometry $\tilde{g}_{\mu\nu}$
and the originally introduced thermodynamical geometry $g_{\mu\nu}$ are
related by the following conformal transformation

$\tilde{g}_{\mu\nu}=e^{2\eta}g_{\mu\nu}.$

In the kinematical geometry, the modified field equations (84) - (87)
transform into

$\displaystyle\delta\tilde{\varepsilon}=T\delta\tilde{s}-p\delta\tilde{v}+\mu\delta\tilde{n}$
(274) $\displaystyle T\tilde{s}+\mu\tilde{n}=\tilde{\varepsilon}+p\tilde{v}$
(275) $\,{}^{\rm
EL}\\!\mathcal{D}_{\tilde{g}^{\mu\nu}}\sqrt{-\tilde{g}}\mathcal{\tilde{L}}=\sqrt{-\tilde{g}}\Big{[}\tilde{\varepsilon}\tilde{u}_{\mu}\tilde{u}_{\nu}+p\tilde{v}(\tilde{u}_{\mu}\tilde{u}_{\nu}+\tilde{g}_{\mu\nu})\Big{]}$
(276) $\,{}^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-\tilde{g}}\mathcal{\tilde{L}}=\sqrt{-\tilde{g}}\tilde{u}^{\mu}q\tilde{n}$
(277) $\,{}^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-\tilde{g}}\mathcal{\tilde{L}}=2(d-1)\sqrt{-\tilde{g}}p\tilde{v}$
(278) $\,{}^{\rm
EL}\\!\mathcal{D}_{\varphi}\sqrt{-\tilde{g}}\mathcal{\tilde{L}}=0,$ (279)

where $\tilde{v}=e^{-(d-1)\eta}$ is interpreted as the (thermodynamical)
volume density measured by the kinematical volume element.

Concrete calculations show that the variation of the entropy in the Lovelock
theory of gravity is expressed by a linear combination of the variations of
the generalized Kodama energy and the size of the system. Further, demanding
that uniform rescaling the volume density $\tilde{v}$ with the thermodynamical
geometry $g_{\mu\nu}$ fixed should not affect the dynamics of the fields in
vacuum region, the appropriate form of Lagrangian density is determined and
expressed in the thermodynamical frame as

$\displaystyle\mathcal{L}=\frac{1}{\kappa^{2}}e^{(d-2)\eta}\bigg{(}R-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\omega\partial_{\mu}\eta\partial^{\mu}\eta-
D_{\mu}\varphi D^{\mu}\varphi\bigg{)},$ (280)

and in the kinematical frame as

$\displaystyle\mathcal{\tilde{L}}=\frac{1}{\kappa^{2}}\bigg{\\{}\tilde{R}-\frac{1}{4}e^{2\eta}F_{\mu\nu}\tilde{F}^{\mu\nu}+\big{[}\omega-(d-2)(d-1)\big{]}\partial_{\mu}\eta\tilde{\partial}^{\mu}\eta-
D_{\mu}\varphi\tilde{D}^{\mu}\varphi\bigg{\\}}.$

Including an anti-symmetric tensor $B_{\mu\nu}$ is straightforward. It is
strange and interesting that these Lagrangians (280) and (7) for
$\omega=(d-2)^{2}$ are analogous to the low energy effective theories of
bosonic string [14] for the critical dimension, namely $d=26$, in the string
frame and Einstein frame, respectively, and the field $\eta$ corresponds to
the dilaton field $\Phi$, even though there is no direct relationship between
the two. In string theory, the dilaton action is obtained by requiring the
quantum conformal anomaly of the Polyakov action on world sheet to vanish.
This analogy implies that the maximum entropy principle in a spacetime be
related to the quantum conformal symmetry of the Polyakov action of wold sheet
embedded into the spacetime as the background geometry.

The field equations (274) - (279) are strange but have some desirable eatures.
The existence of a source term proportional to Gibbs’s free energy density
$p\tilde{v}$ in Eq. (278) leads some interesting properties for application to
cosmology and black hole physics. First, in application to cosmology, since
the source term of Eq. (278) can lead acceleration of expansion of universe,
it may explain the data of cosmological observations without introducing dark
energy, the cosmological constant, nor inhomogeneity of universe. Second, in
application to black hole physics, the non zero source term in Eq. (278) leads
an irremovable scalar hair around a black hole, so that the uniqueness
theorem, or so-called no-hair theorem [15, 16, 17] prevents the black hole
from being spherically symmetric and static, i.e., a black hole has no
spherically symmetric and static state. This can be rephrased as follows: in
the theories that respect the maximal entropy principle, a spherically
symmetric and static black hole, for example a Schwarzchild black hole, has a
kind of instability to thermal fluctuations.

## Acknowledgements

The author would like to thank all who supported this research and gave me
fruitful discussions.

##  Appendix A The homogeneity of the extensive quantities

Consider a $(d-1)$-dimensional region $\Omega$ whose length scale is
sufficiently small than that of the field configuration. The thermodynamical
volume of $\Omega$, measured by $g_{\mu\nu}$, is $\Delta V$. Since the entropy
$\Delta S$, energy $\Delta E$, and particle number $\Delta N$ are extensive
quantities, they are represented by the products of their densities and
$\Delta V$, namely, $\Delta S=s\Delta V$, $\Delta E=\varepsilon\Delta V$, and
$\Delta N=n\Delta V$. The entropy $\Delta S$ is a function of $\Delta E$,
$\Delta N$, and $\Delta V$, i.e.,

$\displaystyle\Delta S$ $\displaystyle=$ $\displaystyle\Delta S(\Delta
E,\Delta N,\Delta V)$ (282) $\displaystyle=$ $\displaystyle s(\Delta E/\Delta
V,\Delta N/\Delta V)\,\Delta V.$ (283)

Thus, the function $s(\varepsilon,n)$ is represented by

$\displaystyle s(\varepsilon,n)=\frac{1}{\Delta V}\Delta S(\varepsilon\Delta
V,n\Delta V,\Delta V).$ (284)

From the definitions of $T$ and $\mu$, namely Eq. (9) and (10),

$\displaystyle\frac{\partial\Delta S}{\partial\Delta E}=\frac{\partial
s}{\partial\varepsilon}=\frac{1}{T}$ (285) $\displaystyle\frac{\partial\Delta
S}{\partial\Delta N}=\frac{\partial s}{\partial n}=-\frac{\mu}{T}.$ (286)

The equation (283) shows that the function $\Delta S(\Delta E,\Delta N,\Delta
V)$ is a homogeneous function of degree $1$.

$\displaystyle\forall a\in\mathbb{R},\ \ \ \ \ \ a\Delta S(\Delta E,\Delta
N,\Delta V)=\Delta S(a\Delta E,a\Delta N,a\Delta V).$ (287)

The entropy density $s(\varepsilon,n)$ is written by

$\displaystyle s(\varepsilon,n)=\Delta S(\varepsilon,n,1).$ (288)

Conversely, it is necessary for constructing $s(\varepsilon,n)$ that the
function $\Delta S(\Delta E,\Delta N,\Delta V)$ is homogeneous of degree $1$.
Therefore, the homogeneity of $\Delta S(\Delta E,\Delta N,\Delta V)$ is a
necessary and sufficient condition of the extensiveness of $s$, $\varepsilon$,
and $n$.

Differentiating Eq. (287) with respect to $a$ and substituting $a=1$, one
obtains following relation:

$\displaystyle\Delta S$ $\displaystyle=$ $\displaystyle\frac{\partial\Delta
S}{\partial\Delta E}\Delta E+\frac{\partial\Delta S}{\partial\Delta N}\Delta
N+\frac{\partial\Delta S}{\partial\Delta V}\Delta V$ (289) $\displaystyle=$
$\displaystyle\frac{1}{T}\Delta E-\frac{\mu}{T}\Delta N+\frac{\partial\Delta
S}{\partial\Delta V}\Delta V.$ (290)

Dividing by $\Delta V$ and multiplying $T$,

$\displaystyle Ts+\mu n=\varepsilon+T\frac{\partial\Delta S}{\partial\Delta
V}.$ (291)

Comparing Eq. (291) with the definition of $p$, namely Eq. (11),

$\displaystyle\frac{\partial\Delta S}{\partial\Delta V}=\frac{p}{T}.$ (292)

From Eq. (290) and (292),

$\displaystyle\Delta S=\frac{1}{T}\Delta E-\frac{\mu}{T}\Delta
N+\frac{p}{T}\Delta V.$ (293)

This is the first law of thermodynamics. Therefore, the definition of $p$, Eq.
(11) is equivalent to the first law.

##  Appendix B Noether’s second theorem

In this section, Noether’s theorem [18] for arbitrarily high order derivative
is presented.

###  B.1 Coordinate transformation

Consider the infinitesimal coordinate transformation

$\displaystyle y^{\mu}=x^{\mu}+\tilde{\delta}x^{\mu}.$ (294)

We use the symbol $\tilde{\delta}$ for the difference between the values of
fields at the same wold point, and the symbol $\delta$ for that of the values
at the same two points whose values of the coordinates coincide, i.e.,

$\displaystyle\tilde{\delta}f(x)$ $\displaystyle:=$ $\displaystyle
f^{\prime}(y)-f(x)$ (295) $\displaystyle=$ $\displaystyle\delta
f(x)+\big{(}\partial_{\mu}f(x)\big{)}\tilde{\delta}x^{\mu}$ (296)

and

$\displaystyle\delta f(x):=f^{\prime}(x)-f(x).$ (297)

The operator $\partial_{\mu}$ commutes with $\delta$, whereas does not with
$\tilde{\delta}$:

$\displaystyle\partial_{\mu}\delta f-\delta\partial_{\mu}f$ $\displaystyle=$
$\displaystyle 0$ (298)
$\displaystyle\partial_{\mu}\tilde{\delta}f-\tilde{\delta}\partial_{\mu}f$
$\displaystyle=$
$\displaystyle(\partial_{\nu}f)\partial_{\mu}\tilde{\delta}x^{\nu}.$ (299)

The commutator of $\partial_{\mu}$ and $\tilde{\delta}$ is

$\displaystyle\big{[}\partial_{\mu},\tilde{\delta}\big{]}=(\partial_{\mu}\tilde{\delta}x^{\nu})\partial_{\nu}.$
(300)

The commutator of $\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}$ and
$\tilde{\delta}$ is calculated as

$\displaystyle\big{[}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}},\tilde{\delta}\big{]}f$
$\displaystyle=$
$\displaystyle\sum_{l=1}^{k}\partial_{\mu_{1}}\cdots\partial_{\mu_{l-1}}\big{[}\partial_{\mu_{l}},\tilde{\delta}\big{]}\partial_{\mu_{l+1}}\cdots\partial_{\mu_{k}}f$
(301) $\displaystyle=$
$\displaystyle\sum_{l=1}^{k}\partial_{\mu_{1}}\cdots\partial_{\mu_{l-1}}\Big{[}(\partial_{\mu_{l}}\tilde{\delta}x^{\nu})\partial_{\nu}\partial_{\mu_{l+1}}\cdots\partial_{\mu_{k}}f\Big{]}.$
(302)

The infinitesimal coordinate transformation of scalar $\varphi$, covariant
vector $A_{\mu}$, contravariant vector $u^{\mu}$, $2$-th order symmetric
cotravariant tensor $g^{\mu\nu}$, and $2$-th order anti-symmetric covariant
tensor $B_{\mu\nu}$ at the same world point are

$\displaystyle\tilde{\delta}\varphi$ $\displaystyle=$ $\displaystyle 0$ (303)
$\displaystyle\tilde{\delta}A_{\mu}$ $\displaystyle=$ $\displaystyle-
A_{\rho}(\partial_{\mu}\tilde{\delta}x^{\rho})$ (304)
$\displaystyle\tilde{\delta}u^{\mu}$ $\displaystyle=$ $\displaystyle
u^{\rho}(\partial_{\rho}\tilde{\delta}x^{\mu})$ (305)
$\displaystyle\tilde{\delta}g^{\mu\nu}$ $\displaystyle=$ $\displaystyle
g^{\rho\nu}(\partial_{\rho}\tilde{\delta}x^{\mu})+g^{\mu\rho}(\partial_{\rho}\tilde{\delta}x^{\nu})$
(306) $\displaystyle\tilde{\delta}B_{\mu\nu}$ $\displaystyle=$ $\displaystyle-
B_{\rho\nu}(\partial_{\mu}\tilde{\delta}x^{\rho})-B_{\mu\rho}(\partial_{\nu}\tilde{\delta}x^{\rho}).$
(307)

The transformation of their $k$-th order derivatives are calculated by
differentiating Eqs. (303) - (307) and using the comutator (302):

$\displaystyle\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\varphi$
(308) $\displaystyle=$
$\displaystyle-\sum_{l=1}^{k}\partial_{\mu_{1}}\cdots\partial_{\mu_{l-1}}\Big{[}(\partial_{\mu_{l}}\tilde{\delta}x^{\nu})\partial_{\nu}\partial_{\mu_{l+1}}\cdots\partial_{\mu_{k}}\varphi\Big{]}$
$\displaystyle=$
$\displaystyle-\sum_{l=1}^{k}\big{(}\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{l-1}\partial_{l+1}\cdots\partial_{\mu_{k}}\varphi\big{)}\partial_{\mu_{l}}\tilde{\delta}x^{\nu}+\cdots$
(309) $\displaystyle=$
$\displaystyle-\sum_{l=1}^{k}\big{(}\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{l-1}\partial_{l+1}\cdots\partial_{\mu_{k}}\varphi\big{)}\delta^{\mu}_{\mu_{l}}\partial_{\mu}\tilde{\delta}x^{\nu}+\cdots,$
(310)
$\displaystyle\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}A_{\alpha}$
(315) $\displaystyle=$
$\displaystyle-\sum_{j=0}^{k}\bigg{(}\begin{array}[]{c}k\\\
j\end{array}\bigg{)}(\partial_{\mu_{j+1}}\cdots\partial_{\mu_{k}}A_{\gamma})\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}\partial_{\alpha}\tilde{\delta}x^{\gamma}$
$\displaystyle-\sum_{l=1}^{k}\partial_{\mu_{1}}\cdots\partial_{\mu_{l-1}}\Big{[}(\partial_{\mu_{l}}\tilde{\delta}x^{\nu})\partial_{\nu}\partial_{\mu_{l+1}}\cdots\partial_{\mu_{k}}A_{\alpha}\Big{]}$
$\displaystyle=$
$\displaystyle-\Big{[}\delta^{\mu}_{\alpha}(\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}A_{\nu})+\sum_{l=1}^{k}(\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{l-1}\partial_{l+1}\cdots\partial_{\mu_{k}}A_{\alpha})\delta^{\mu}_{\mu_{l}}\Big{]}\partial_{\mu}\tilde{\delta}x^{\nu}$
$\displaystyle+\cdots,$
$\displaystyle\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}u^{\alpha}$
(320) $\displaystyle=$
$\displaystyle\sum_{j=0}^{k}\bigg{(}\begin{array}[]{c}k\\\
j\end{array}\bigg{)}(\partial_{\mu_{j+1}}\cdots\partial_{\mu_{k}}u^{\gamma})\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}\partial_{\gamma}\tilde{\delta}x^{\alpha}$
$\displaystyle-\sum_{l=1}^{k}\partial_{\mu_{1}}\cdots\partial_{\mu_{l-1}}\Big{[}(\partial_{\mu_{l}}\tilde{\delta}x^{\nu})\partial_{\nu}\partial_{\mu_{l+1}}\cdots\partial_{\mu_{k}}u^{\alpha}\Big{]}$
$\displaystyle=$
$\displaystyle\Big{[}\delta_{\nu}^{\alpha}(\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}u^{\mu})-\sum_{l=1}^{k}(\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{l-1}\partial_{l+1}\cdots\partial_{\mu_{k}}u^{\alpha})\delta^{\mu}_{\mu_{l}}\Big{]}\partial_{\mu}\tilde{\delta}x^{\nu}$
$\displaystyle+\cdots,$
$\displaystyle\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}g^{\alpha\beta}$
(328) $\displaystyle=$
$\displaystyle\sum_{j=0}^{k}\bigg{(}\begin{array}[]{c}k\\\
j\end{array}\bigg{)}(\partial_{\mu_{j+1}}\cdots\partial_{\mu_{k}}g^{\gamma\beta})\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}\partial_{\gamma}\tilde{\delta}x^{\alpha}$
$\displaystyle+\sum_{j=0}^{k}\bigg{(}\begin{array}[]{c}k\\\
j\end{array}\bigg{)}(\partial_{\mu_{j+1}}\cdots\partial_{\mu_{k}}g^{\alpha\gamma})\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}\partial_{\gamma}\tilde{\delta}x^{\beta}$
$\displaystyle-\sum_{l=1}^{k}\partial_{\mu_{1}}\cdots\partial_{\mu_{l-1}}\Big{[}(\partial_{\mu_{l}}\tilde{\delta}x^{\nu})\partial_{\nu}\partial_{\mu_{l+1}}\cdots\partial_{\mu_{k}}g^{\alpha\beta}\Big{]}$
$\displaystyle=$
$\displaystyle\Big{[}\delta^{\alpha}_{\nu}(\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}g^{\mu\beta})+\delta^{\beta}_{\nu}(\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}g^{\alpha\mu})-\sum_{l=1}^{k}(\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{l-1}\partial_{l+1}\cdots\partial_{\mu_{k}}g^{\alpha\beta})\delta^{\mu}_{\mu_{l}}\Big{]}\partial_{\mu}\tilde{\delta}x^{\nu}$
$\displaystyle+\cdots.$
$\displaystyle\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}B_{\alpha\beta}$
(336) $\displaystyle=$
$\displaystyle-\sum_{j=0}^{k}\bigg{(}\begin{array}[]{c}k\\\
j\end{array}\bigg{)}(\partial_{\mu_{j+1}}\cdots\partial_{\mu_{k}}B_{\gamma\beta})\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}\partial_{\alpha}\tilde{\delta}x^{\gamma}$
$\displaystyle-\sum_{j=0}^{k}\bigg{(}\begin{array}[]{c}k\\\
j\end{array}\bigg{)}(\partial_{\mu_{j+1}}\cdots\partial_{\mu_{k}}B_{\alpha\gamma})\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}\partial_{\beta}\tilde{\delta}x^{\gamma}$
$\displaystyle-\sum_{l=1}^{k}\partial_{\mu_{1}}\cdots\partial_{\mu_{l-1}}\Big{[}(\partial_{\mu_{l}}\tilde{\delta}x^{\nu})\partial_{\nu}\partial_{\mu_{l+1}}\cdots\partial_{\mu_{k}}B_{\alpha\beta}\Big{]}$
$\displaystyle=$
$\displaystyle-\Big{[}\delta^{\mu}_{\alpha}(\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}B_{\nu\beta})+\delta^{\mu}_{\beta}(\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}B_{\alpha\nu})+\sum_{l=1}^{k}(\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{l-1}\partial_{l+1}\cdots\partial_{\mu_{k}}B_{\alpha\beta})\delta^{\mu}_{\mu_{l}}\Big{]}\partial_{\mu}\tilde{\delta}x^{\nu}$
$\displaystyle+\cdots,$

where the last line of each equation is the term which is proportional to the
least derivative, namely the first derivative, of $\tilde{\delta}x^{\nu}$.

The infinitesimal transformation of ${\mathrm{d}}^{n}x$ is

$\displaystyle{\mathrm{d}}^{n}y=\big{(}1+\partial_{\mu}\tilde{\delta}x^{\mu}\big{)}{\mathrm{d}}^{n}x.$
(337)

Consider an arbitrary scalar function $\mathcal{F}$. The infinitesimal
transformation of the integral of $\sqrt{-g}\mathcal{F}$ over $\mathcal{M}$ is
arranged as

$\displaystyle\int_{\mathcal{M}}{\mathrm{d}}^{n}y\sqrt{-g^{\prime}}\mathcal{F}^{\prime}(y)-\int_{\mathcal{M}}{\mathrm{d}}^{n}x\sqrt{-g}\mathcal{F}(x)$
(338) $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}{\mathrm{d}}^{n}x\Big{[}\tilde{\delta}\big{(}\sqrt{-g}\mathcal{F}\big{)}+\sqrt{-g}\mathcal{F}\big{(}\partial_{\mu}\tilde{\delta}x^{\mu}\big{)}\Big{]}$
(339) $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}{\mathrm{d}}^{n}x\bigg{[}\frac{\partial(\sqrt{-g}\mathcal{F})}{\partial
f}\tilde{\delta}f+\frac{\partial(\sqrt{-g}\mathcal{F})}{\partial(\partial_{\mu_{1}}f)}\tilde{\delta}\partial_{\mu_{1}}f+\cdots+\sqrt{-g}\mathcal{F}\partial_{\mu}\tilde{\delta}x^{\mu}\bigg{]}$
(340) $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}{\mathrm{d}}^{n}x\bigg{[}\frac{\partial(\sqrt{-g}\mathcal{F})}{\partial
f}\delta
f+\frac{\partial(\sqrt{-g}\mathcal{F})}{\partial(\partial_{\mu_{1}}f)}\delta\partial_{\mu_{1}}f+\cdots+\partial_{\mu}\big{(}\sqrt{-g}\mathcal{F}\tilde{\delta}x^{\mu}\big{)}\bigg{]}$
$\displaystyle=$
$\displaystyle\int_{\mathcal{M}}{\mathrm{d}}^{n}x\bigg{[}\,^{\rm
EL}\\!\mathcal{D}_{f}\sqrt{-g}\mathcal{F}\cdot\delta f$
$\displaystyle+\partial_{\mu}\bigg{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{F}\cdot\delta\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f+\sqrt{-g}\mathcal{F}\tilde{\delta}x^{\mu}\bigg{)}\bigg{]}$
$\displaystyle=$
$\displaystyle\int_{\mathcal{M}}{\mathrm{d}}^{n}x\bigg{[}\,^{\rm
EL}\\!\mathcal{D}_{f}\sqrt{-g}\mathcal{F}\cdot\delta f$
$\displaystyle+\partial_{\mu}\bigg{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{F}\cdot\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f-\sqrt{-g}T^{\mu}_{\
\ \nu}[\sqrt{-g}\mathcal{F}]\tilde{\delta}x^{\nu}\bigg{)}\bigg{]}$
$\displaystyle=$
$\displaystyle\int_{\mathcal{M}}{\mathrm{d}}^{n}x\bigg{[}\,^{\rm
EL}\\!\mathcal{D}_{f}\sqrt{-g}\mathcal{F}\cdot\tilde{\delta}f-(\partial_{\mu}f)\,^{\rm
EL}\\!\mathcal{D}_{f}\sqrt{-g}\mathcal{F}\cdot\tilde{\delta}x^{\mu}$
$\displaystyle+\partial_{\mu}\bigg{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{F}\cdot\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f-\sqrt{-g}T^{\mu}_{\
\ \nu}[\sqrt{-g}\mathcal{F}]\tilde{\delta}x^{\nu}\bigg{)}\bigg{]},$

where $T^{\mu}_{\ \ \nu}[\sqrt{-g}\mathcal{F}]$ is defined by

$\displaystyle\sqrt{-g}T^{\mu}_{\ \ \nu}[\sqrt{-g}\mathcal{F}]$
$\displaystyle:=$ $\displaystyle\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{F}\cdot\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f-\sqrt{-g}\mathcal{F}\delta^{\mu}_{\nu}.$

When $\mathcal{F}$ is the Lagrangian of a system, $T^{\mu}_{\ \
\nu}[\sqrt{-g}\mathcal{L}]$ is called the canonical energy-momentum affine
tensor.

###  B.2 Identities derived from the coordinate transformation invariance

Because the integral of a scalar density $\sqrt{-g}\mathcal{F}$ is invariant
under any coordinate transformations, the integral (B.1) vanishes for
arbitrary $\tilde{\delta}x^{\mu}$. When
$f=\\{g^{\alpha\beta},B_{\alpha\beta},A_{\alpha},u^{\alpha},\varphi\\}$,
substituting555We use the symmetries $g^{\mu\nu}=g^{\nu\mu}$,
$B_{\mu\nu}=-B_{\nu\mu}$, and  $\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\mu\nu}}\sqrt{-g}\mathcal{F}=\,^{\rm
EL}\\!\mathcal{D}_{g^{\nu\mu}}\sqrt{-g}\mathcal{F}$ $\,{}^{\rm
EL}\\!\mathcal{D}_{B_{\mu\nu}}\sqrt{-g}\mathcal{F}=-\,^{\rm
EL}\\!\mathcal{D}_{B_{\nu\mu}}\sqrt{-g}\mathcal{F}.$ Eqs. (303) - (307) into
Eq. (B.1),

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}{\mathrm{d}}^{n}x\bigg{\\{}-\Big{[}\partial_{\nu}\Big{(}2g^{\nu\beta}\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\mu\beta}}\sqrt{-g}\mathcal{F}-2B_{\mu\beta}\,^{\rm
EL}\\!\mathcal{D}_{B_{\nu\beta}}\sqrt{-g}\mathcal{F}-A_{\mu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\nu}}\sqrt{-g}\mathcal{F}$
$\displaystyle+u^{\nu}\,{}^{\rm
EL}\\!\mathcal{D}_{u^{\mu}}\sqrt{-g}\mathcal{F}\Big{)}+(\partial_{\mu}f)\,^{\rm
EL}\\!\mathcal{D}_{f}\sqrt{-g}\mathcal{F}\Big{]}\tilde{\delta}x^{\mu}$
$\displaystyle+\partial_{\mu}\bigg{[}\Big{(}2g^{\mu\beta}\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\nu\beta}}\sqrt{-g}\mathcal{F}-2B_{\nu\beta}\,^{\rm
EL}\\!\mathcal{D}_{B_{\mu\beta}}\sqrt{-g}\mathcal{F}-A_{\nu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}$
$\displaystyle+u^{\mu}\,{}^{\rm
EL}\\!\mathcal{D}_{u^{\nu}}\sqrt{-g}\mathcal{F}-\sqrt{-g}T^{\mu}_{\ \
\nu}[\sqrt{-g}\mathcal{F}]\Big{)}\tilde{\delta}x^{\nu}$
$\displaystyle+\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{F}\cdot\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{]}\bigg{\\}}$
$\displaystyle=$
$\displaystyle-\int_{\mathcal{M}}{\mathrm{d}}^{n}x\Big{[}\partial_{\nu}\Big{(}2g^{\nu\beta}\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\mu\beta}}\sqrt{-g}\mathcal{F}-2B_{\mu\beta}\,^{\rm
EL}\\!\mathcal{D}_{B_{\nu\beta}}\sqrt{-g}\mathcal{F}-A_{\mu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\nu}}\sqrt{-g}\mathcal{F}$
$\displaystyle+u^{\nu}\,{}^{\rm
EL}\\!\mathcal{D}_{u^{\mu}}\sqrt{-g}\mathcal{F}\Big{)}+(\partial_{\mu}f)\,^{\rm
EL}\\!\mathcal{D}_{f}\sqrt{-g}\mathcal{F}\Big{]}\tilde{\delta}x^{\mu}$
$\displaystyle+\oint_{\partial\mathcal{M}}({\mathrm{d}}^{n-1}x)_{\mu}\bigg{[}\Big{(}2g^{\mu\beta}\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\nu\beta}}\sqrt{-g}\mathcal{F}-2B_{\nu\beta}\,^{\rm
EL}\\!\mathcal{D}_{B_{\mu\beta}}\sqrt{-g}\mathcal{F}-A_{\nu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}$
$\displaystyle+u^{\mu}\,{}^{\rm
EL}\\!\mathcal{D}_{u^{\nu}}\sqrt{-g}\mathcal{F}-\sqrt{-g}T^{\mu}_{\ \
\nu}[\sqrt{-g}\mathcal{F}]\Big{)}\tilde{\delta}x^{\nu}+\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{F}\cdot\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{]}.$

Consider a coordinate transformation such that
$\tilde{\delta}x^{\mu}=\partial_{\nu_{1}}\tilde{\delta}x^{\mu}=\partial_{\nu_{1}}\partial_{\nu_{2}}\tilde{\delta}x^{\mu}=\cdots=0$
on $\partial\mathcal{M}$, in other words, the third and fourth lines of
(LABEL:deltaIntegral2b) vanish. Then, the square bracket in the first and
second lines of (LABEL:deltaIntegral2b) should be identically $0$, i.e.,

$\displaystyle\partial_{\nu}\Big{(}2g^{\nu\beta}\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\mu\beta}}\sqrt{-g}\mathcal{F}-2B_{\mu\beta}\,^{\rm
EL}\\!\mathcal{D}_{B_{\nu\beta}}\sqrt{-g}\mathcal{F}-A_{\mu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\nu}}\sqrt{-g}\mathcal{F}+u^{\nu}\,{}^{\rm
EL}\\!\mathcal{D}_{u^{\mu}}\sqrt{-g}\mathcal{F}\Big{)}$
$\displaystyle+(\partial_{\mu}f)\,^{\rm
EL}\\!\mathcal{D}_{f}\sqrt{-g}\mathcal{F}\equiv 0.$ (348)

Introducing the symbols $E_{\mu\nu}$, $\bar{E}_{\mu\nu}$, and $F_{\mu\nu}$ by

$\displaystyle E_{\mu\nu}[\sqrt{-g}\mathcal{F}]$ $\displaystyle:=$
$\displaystyle\frac{1}{\sqrt{-g}}\,^{\rm
EL}\\!\mathcal{D}_{g^{\mu\nu}}\sqrt{-g}\mathcal{F}$ (349)
$\displaystyle\bar{E}_{\mu\nu}[\sqrt{-g}\mathcal{F}]$ $\displaystyle:=$
$\displaystyle\frac{1}{\sqrt{-g}}\,^{\rm
EL}\\!\bar{\mathcal{D}}_{g^{\mu\nu}}\sqrt{-g}\mathcal{F}$ (350)
$\displaystyle=$ $\displaystyle
E_{\mu\nu}[\sqrt{-g}\mathcal{F}]-\frac{1}{2\sqrt{-g}}u_{\mu}u_{\nu}u^{\rho}\,{}^{\rm
EL}\\!\mathcal{D}_{u^{\rho}}\sqrt{-g}\mathcal{F}$ (351) $\displaystyle
F_{\mu\nu}$ $\displaystyle:=$
$\displaystyle\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ (352)
$\displaystyle H_{\mu\nu\rho}$ $\displaystyle:=$
$\displaystyle\partial_{\rho}B_{\mu\nu}+\partial_{\nu}B_{\rho\mu}+\partial_{\mu}B_{\nu\rho},$
(353)

and Using the relation for the covariant derivative of a symmetric tensor
$S_{\mu\nu}$

$\displaystyle\sqrt{-g}\nabla_{\nu}S^{\nu}_{\mu}=\partial_{\nu}\Big{(}\sqrt{-g}S^{\nu}_{\mu}\Big{)}+\frac{\sqrt{-g}}{2}\big{(}\partial_{\mu}g^{\rho\sigma}\big{)}S_{\rho\sigma},$
(354)

it is found that the identity (348) is equivalent to

$\displaystyle
2\sqrt{-g}\nabla_{\nu}\bar{E}^{\nu}_{\mu}[\sqrt{-g}\mathcal{F}]$
$\displaystyle\equiv$ $\displaystyle 2B_{\mu\rho}\partial_{\nu}\Big{(}\,^{\rm
EL}\\!\mathcal{D}_{\nu\rho}\sqrt{-g}\mathcal{F}\Big{)}-H_{\mu\nu\rho}\,^{\rm
EL}\\!\mathcal{D}_{\nu\rho}\sqrt{-g}\mathcal{F}+A_{\mu}\partial_{\nu}\Big{(}\,^{\rm
EL}\\!\mathcal{D}_{A_{\nu}}\sqrt{-g}\mathcal{F}\Big{)}-F_{\mu\nu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\nu}}\sqrt{-g}\mathcal{F}$
$\displaystyle-\partial_{\nu}\Big{[}u^{\nu}\big{(}u_{\mu}u^{\rho}+\delta^{\rho}_{\mu}\big{)}\,^{\rm
EL}\\!\mathcal{D}_{u^{\rho}}\sqrt{-g}\mathcal{F}\Big{]}-\bigg{[}\partial_{\mu}u^{\rho}-\frac{1}{2}\big{(}\partial_{\mu}g_{\alpha\beta}\big{)}u^{\alpha}u^{\beta}u^{\rho}\bigg{]}\,^{\rm
EL}\\!\mathcal{D}_{u^{\rho}}\sqrt{-g}\mathcal{F}$
$\displaystyle-(\partial_{\mu}\varphi)\,^{\rm
EL}\\!\mathcal{D}_{\varphi}\sqrt{-g}\mathcal{F}-(\partial_{\mu}\eta)\,^{\rm
EL}\\!\mathcal{D}_{\eta}\sqrt{-g}\mathcal{F},$

Substituting the identity (348) into Eq. (B.2),

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}{\mathrm{d}}^{n}x\,\partial_{\mu}\bigg{[}\Big{(}2g^{\mu\beta}\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\nu\beta}}\sqrt{-g}\mathcal{F}-2B_{\nu\beta}\,^{\rm
EL}\\!\mathcal{D}_{B_{\mu\beta}}\sqrt{-g}\mathcal{F}-A_{\nu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}$ (357)
$\displaystyle+u^{\mu}\,{}^{\rm
EL}\\!\mathcal{D}_{u^{\nu}}\sqrt{-g}\mathcal{F}-\sqrt{-g}T^{\mu}_{\ \
\nu}[\sqrt{-g}\mathcal{F}]\Big{)}\tilde{\delta}x^{\nu}+\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{F}\cdot\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{]}$
$\displaystyle=$
$\displaystyle\int_{\mathcal{M}}{\mathrm{d}}^{n}x\,\bigg{[}\partial_{\mu}\Big{(}2g^{\mu\beta}\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\nu\beta}}\sqrt{-g}\mathcal{F}-2B_{\nu\beta}\,^{\rm
EL}\\!\mathcal{D}_{B_{\mu\beta}}\sqrt{-g}\mathcal{F}-A_{\nu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}$
$\displaystyle+u^{\mu}\,{}^{\rm
EL}\\!\mathcal{D}_{u^{\nu}}\sqrt{-g}\mathcal{F}-\sqrt{-g}T^{\mu}_{\ \
\nu}[\sqrt{-g}\mathcal{F}]\Big{)}\tilde{\delta}x^{\nu}$
$\displaystyle+\Big{(}2g^{\mu\beta}\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\nu\beta}}\sqrt{-g}\mathcal{F}-2B_{\nu\beta}\,^{\rm
EL}\\!\mathcal{D}_{\mu\beta}\sqrt{-g}\mathcal{F}-A_{\nu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}$
$\displaystyle+u^{\mu}\,{}^{\rm
EL}\\!\mathcal{D}_{u^{\nu}}\sqrt{-g}\mathcal{F}-\sqrt{-g}T^{\mu}_{\ \
\nu}[\sqrt{-g}\mathcal{F}]\Big{)}\partial_{\mu}\tilde{\delta}x^{\nu}$
$\displaystyle+\partial_{\mu}\Big{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{F}\cdot\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\Big{)}\bigg{]}.$

Consider coordinate transformation such that
$\partial_{\nu_{1}}\tilde{\delta}x^{\mu}=\partial_{\nu_{1}}\partial_{\nu_{2}}\tilde{\delta}x^{\mu}=\cdots=0$
on $\mathcal{M}$. Then, since
$\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f,\ \ \ k=0,\ 1,\ 2,\
\cdots$ do not contain $\tilde{\delta}x^{\nu}$, the second and third line of
(357) vanish. Therefore, the coefficient of $\tilde{\delta}x^{\nu}$ is
identically $0$, i.e.,

$\displaystyle\partial_{\nu}\Big{(}2g^{\nu\beta}\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\mu\beta}}\sqrt{-g}\mathcal{F}-2B_{\mu\beta}\,^{\rm
EL}\\!\mathcal{D}_{\nu\beta}\sqrt{-g}\mathcal{F}-A_{\mu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\nu}}\sqrt{-g}\mathcal{F}$
$\displaystyle+u^{\mu}\,{}^{\rm
EL}\\!\mathcal{D}_{u^{\nu}}\sqrt{-g}\mathcal{F}-\sqrt{-g}T^{\mu}_{\nu}[\sqrt{-g}\mathcal{F}]\Big{)}\equiv
0.$ (358)

Substituting identity (358) into Eq. (357),

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}{\mathrm{d}}^{n}x\,\bigg{[}\Big{(}2g^{\mu\beta}\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\nu\beta}}\sqrt{-g}\mathcal{F}-2B_{\nu\beta}\,^{\rm
EL}\\!\mathcal{D}_{B_{\mu\beta}}\sqrt{-g}\mathcal{F}-A_{\nu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}$ (360)
$\displaystyle+u^{\mu}\,{}^{\rm
EL}\\!\mathcal{D}_{u^{\nu}}\sqrt{-g}\mathcal{F}-\sqrt{-g}T^{\mu}_{\ \
\nu}[\sqrt{-g}\mathcal{F}]\Big{)}\partial_{\mu}\tilde{\delta}x^{\nu}$
$\displaystyle+\partial_{\mu}\Big{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{F}\cdot\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\Big{)}\bigg{]}$
$\displaystyle=$
$\displaystyle\int_{\mathcal{M}}{\mathrm{d}}^{n}x\,\bigg{[}\Big{(}2g^{\mu\beta}\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\nu\beta}}\sqrt{-g}\mathcal{F}-2B_{\nu\beta}\,^{\rm
EL}\\!\mathcal{D}_{B_{\mu\beta}}\sqrt{-g}\mathcal{F}-A_{\nu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}$
$\displaystyle+u^{\mu}\,{}^{\rm
EL}\\!\mathcal{D}_{u^{\nu}}\sqrt{-g}\mathcal{F}-\sqrt{-g}T^{\mu}_{\ \
\nu}[\sqrt{-g}\mathcal{F}]\Big{)}\partial_{\mu}\tilde{\delta}x^{\nu}$
$\displaystyle+\partial_{\rho}\Big{(}\Omega^{\rho\mu}_{\ \ \
\nu}[\sqrt{-g}\mathcal{F}]\partial_{\mu}\tilde{\delta}x^{\nu}+\Xi^{\rho\mu_{1}\mu_{2}}_{\
\ \ \ \ \ \
\nu}[\sqrt{-g}\mathcal{F}]\partial_{\mu_{1}}\partial_{\mu_{2}}\tilde{\delta}x^{\nu}+\cdots\Big{)}\bigg{]}.$

From Eqs. (308) - (336), the coefficients, $\Omega^{\rho\mu}_{\ \ \ \nu}$,
$\Xi^{\rho\mu_{1}\mu_{2}}_{\ \ \ \ \ \ \ \nu}$, $\cdots$, can be calculated.
For example, $\Omega^{\rho\mu}_{\ \ \ \nu}$ is written explicitly:

$\displaystyle\Omega^{\rho\mu}_{\ \ \ \nu}[\sqrt{-g}\mathcal{F}]$ (361)
$\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\Bigg{\\{}\Big{[}\delta^{\alpha}_{\nu}(\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}g^{\mu\beta})+\delta^{\beta}_{\nu}(\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}g^{\alpha\mu})$
$\displaystyle\ \ \ \ \ \ \ \ \ \
-\sum_{l=1}^{k}(\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{l-1}\partial_{l+1}\cdots\partial_{\mu_{k}}g^{\alpha\beta})\delta^{\mu}_{\mu_{l}}\Big{]}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}g^{\alpha\beta}}\sqrt{-g}\mathcal{F}$
$\displaystyle+\Big{[}-\delta_{\alpha}^{\mu}(\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}B_{\nu\beta})-\delta_{\beta}^{\mu}(\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}B_{\alpha\nu})$
$\displaystyle\ \ \ \ \ \ \ \ \ \
-\sum_{l=1}^{k}(\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{l-1}\partial_{l+1}\cdots\partial_{\mu_{k}}B_{\alpha\beta})\delta^{\mu}_{\mu_{l}}\Big{]}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}B_{\alpha\beta}}\sqrt{-g}\mathcal{F}$
$\displaystyle+\Big{[}-\delta^{\mu}_{\alpha}(\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}A_{\nu})-\sum_{l=1}^{k}(\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{l-1}\partial_{l+1}\cdots\partial_{\mu_{k}}A_{\alpha})\delta^{\mu}_{\mu_{l}}\Big{]}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}A_{\alpha}}\sqrt{-g}\mathcal{F}$
$\displaystyle+\Big{[}\delta^{\alpha}_{\nu}(\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}u^{\mu})-\sum_{l=1}^{k}(\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{l-1}\partial_{l+1}\cdots\partial_{\mu_{k}}u^{\alpha})\delta^{\mu}_{\mu_{l}}\Big{]}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}u^{\alpha}}\sqrt{-g}\mathcal{F}$
$\displaystyle-\sum_{l=1}^{k}\big{(}\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{l-1}\partial_{l+1}\cdots\partial_{\mu_{k}}\varphi\big{)}\delta^{\mu}_{\mu_{l}}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\rho}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\varphi}\sqrt{-g}\mathcal{F}\Bigg{\\}}.$

The invariance of Eq. (360) under coordinate transformations such that