# The $*$-product of domains in several complex variables

Sylwester Zając  Institute of Mathematics, Faculty of Mathematics and Computer
Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
<EMAIL_ADDRESS>
###### Abstract.

In this article we continue the research, carried out in [28], on computing
the $*$-product of domains in $\mathbb{C}^{N}$. Assuming that $0\in
G\subset\mathbb{C}^{N}$ is an arbitrary Runge domain and $0\in
D\subset\mathbb{C}^{N}$ is a bounded, smooth and linearly convex domain (or a
non-decreasing union of such ones), we establish a geometric relation between
$D*G$ and another domain in $\mathbb{C}^{N}$ which is ’extremal’ (in an
appropriate sense) with respect to a special coefficient multiplier dependent
only on the dimension $N$. Next, for $N=2$, we derive a characterization of
the latter domain expressed in terms of planar geometry. These two results,
when combined together, give a formula which allows to calculate $D*G$ for
two-dimensional domains $D$ and $G$ satisfying the outlined assumptions.

###### Key words and phrases:

Hadamard product, analytic continuation, spaces of holomorphic functions

###### 2010 Mathematics Subject Classification:

Primary: 32A05, 32D15

The research was supported by NCN grant SONATA BIS no. 2017/26/E/ST1/00723 of
the National Science Centre, Poland.

##  1\. Introduction

Let $\mathcal{O}_{0}$ be the set of all germs of holomorphic functions at the
origin of $\mathbb{C}^{N}$ and let $\mathcal{O}_{0,D}$, for a domain $0\in
D\subset\mathbb{C}^{N}$, be the subset of $\mathcal{O}_{0}$ consisting of all
germs of elements of $\mathcal{O}(D)$. The latter symbol denotes, as usually,
the Fréchet space of all holomorphic functions on $D$ equipped with the
compact-open topology. The Hadamard product, called also the $*$-product, can
be regarded as a bilinear mapping from $\mathcal{O}_{0}\times\mathcal{O}_{0}$
to $\mathcal{O}_{0}$ given by the formula

$\left(\sum_{\alpha\in\mathbb{N}^{N}}f_{\alpha}z^{\alpha}\right)*\left(\sum_{\alpha\in\mathbb{N}^{N}}g_{\alpha}z^{\alpha}\right):=\sum_{\alpha\in\mathbb{N}^{N}}f_{\alpha}g_{\alpha}z^{\alpha}.$

It was extensively studied in various aspects: as a bilinear form, as a linear
operator with one factor fixed (see the survey [22] and, for instance, the
papers [2], [3], [4], [5], [14], [17], [18], [19], [20], [21], [23], [24],
[25], [28]), and also as a map acting on spaces of real analytic functions
(see [7], [8], [9], [10], [11], [12]). The problem of similar nature as here,
that is of extending the $*$-products to as large domain as possible, was
investigated, for instance, in [1], [13] and [16] for weighted Hadamard
products with certain weights, where results were obtained for starlike
domains and for so-called $p$-convex domains (we refer the reader to [13]).

In [28] it was shown (see Proposition 4.1 there) that if at least one of
domains $0\in D,G\subset\mathbb{C}^{N}$ is Runge domain, then there exists the
largest domain $0\in\Omega\subset\mathbb{C}^{N}$ having the property that the
image of $\mathcal{O}_{0,D}\times\mathcal{O}_{0,G}$ under the $*$-product lies
in $\mathcal{O}_{0,\Omega}$ (here, as well as in [28], we follow [15,
Definition 2.7.1] and consider Runge domains to be pseudoconvex). This largest
$\Omega$, denoted as $D*G$, was the subject of research carried out in [28],
which concluded in finding a description of $D*G$ for domains of a special
class. The methodology employed in [28], as well as the results achieved
there, were of symmetrical nature: both $D$ and $G$ were assumed to belong to
the same family of domains and the fundamental integral formula for $f*g$
relied on geometric properties of $D$ to the same extent as on those of $G$.

The approach presented in this paper is substantially different. Starting from
a non-symmetric integral expressing $f*g$, we investigate $D*G$ with only $D$
being of the same particular class as in [28] and $G$ being an arbitrary Runge
domain. These considerations conclude in Theorem 3.4, which establishes a
relation, expressed in geometric terms, between $D*G$ and
$h_{\textnormal{{1}}_{N}}*G$. Here
$h_{\textnormal{{1}}_{N}}(z)=(1-z_{1}-\ldots-z_{N})^{-N}$ and the set
$h_{\textnormal{{1}}_{N}}*G$ is the largest domain containing the origin on
which every product $h_{\textnormal{{1}}_{N}}*g$, for $g\in\mathcal{O}_{0,G}$,
can be analytically continued. When $G$ is Runge domain, existence of
$h_{\textnormal{{1}}_{N}}*G$ is guaranteed by the aforementioned [28,
Proposition 4.1]. To calculate $D*G$ we must, however, face the problem of
computing $h_{\textnormal{{1}}_{N}}*G$. We deal with this topic in Section 4,
where, in Theorem 4.1, we derive a nice geometric formula for
$h_{\textnormal{{1}}_{N}}*G$, but, unfortunately, only for $N=2$. It is worthy
to emphasize that although obtaining a candidate for this set was quite
straightforward and relied mainly of certain integral formula, main
difficulties were met in demonstrating that this candidate is the largest one
on which all $h_{\textnormal{{1}}_{2}}*g$’s extend. Combining this result with
Theorem 3.4 announced above leads to a complete description of $D*G$ for two-
dimensional domains satisfying the listed assumptions. Nevertheless, the
question for higher dimensions remains open.

##  2\. Preliminaries

We begin by introducing basic concepts and notation setting grounds for this
study. We use the standard symbols $\mathbb{D}$, $\mathbb{T}$,
$\mathbb{C}_{*}$ and $\widehat{\mathbb{C}}$ to denote, respectively, the unit
disc in $\mathbb{C}$, its boundary, the punctured plane
$\mathbb{C}\setminus\\{0\\}$ and the Riemann sphere
$\mathbb{C}\cup\\{\infty\\}$. We assume that the set $\mathbb{N}$ of all
natural numbers contains $0$. By $\mathbb{P}_{N}(z,r)$ and
$\overline{\mathbb{P}}_{N}(z,r)$ we mean the open and closed polydiscs
centered at $z$ and having the radius $r$.

To shorten notation, we will use the word ’loop’ to denote a continuous map
defined on $\mathbb{T}$ and the word ’smooth’ to declare being of the
$\mathcal{C}^{\infty}$ class.

For points $z=(z_{1},\ldots,z_{N}),w=(w_{1},\ldots,w_{N})\in\mathbb{C}^{N}$,
by $z\bullet w$ we denote the product $z_{1}w_{1}+\ldots+z_{N}w_{N}$ and by
$z\cdot w$ or $zw$ we denote their coordinate-wise product, that is, the point
$(z_{1}w_{1},\ldots,z_{N}w_{N})$. The identity element of the latter
multiplication, $(1,1,\ldots,1)$ ($N$ times), is called
$\textnormal{{1}}_{N}$. Given two sets $A,B\subset\mathbb{C}^{N}$, by $AB$ or
$A\cdot B$ we mean their algebraic product, i.e. the set $\\{ab:a\in A,b\in
B\\}$. We use the classical notation of exponentiation, where for an
$\alpha=(\alpha_{1},\ldots,\alpha_{N})\in\mathbb{Z}^{N}$ the symbol
$z^{\alpha}$ denotes $z_{1}^{\alpha_{1}}\ldots z_{N}^{\alpha_{N}}$, holding
the convention that $z_{j}^{0}=1$. Moreover, $\alpha!$ and $|\alpha|$ will, as
usual, denote the product $\alpha_{1}!\ldots\alpha_{N}!$ and the sum
$\alpha_{1}+\ldots+\alpha_{N}$.

For compact sets $K\subset\mathbb{C}^{N}$ and $L\subset\Omega$ we use the
standard notation $\widehat{K}$ and $\widehat{L}_{\Omega}$ for the polynomial
hull of $K$ and the holomorphic hull of $L$ with respect to a domain $\Omega$.
The supremum of modulus of a complex-valued function over a set
$A\subset\mathbb{C}^{N}$ is denoted by $\|f\|_{A}$. Finally, if $0\in A$, then
by $\textnormal{cc}_{0}\,A$ we understand the connected component of $A$
containing $0$.

###  2.1. Integral formula for the $*$ product

Take two power series

$f(z)=\sum_{\alpha\in\mathbb{N}^{N}}f_{\alpha}z^{\alpha},\quad
g(z)=\sum_{\alpha\in\mathbb{N}^{N}}g_{\alpha}z^{\alpha}$

convergent in neighbourhoods of polydiscs $\overline{\mathbb{P}}_{N}(0,r)$ and
$\overline{\mathbb{P}}_{N}(0,\rho)$, respectively. A straightforward
calculation allows us to derive the equality

(1) $(f*g)(z)=\left(\frac{1}{2\pi
i}\right)^{N}\int_{\rho^{-1}\mathbb{T}^{N}}f(z\zeta)\,g\left(\zeta_{1}^{-1},\ldots,\zeta_{N}^{-1}\right)\frac{d\zeta_{1}}{\zeta_{1}}\ldots\frac{d\zeta_{N}}{\zeta_{N}}$

for $z=(z_{1},\ldots,z_{N})\in\overline{\mathbb{P}}_{N}(0,r\rho)$. Its one-
dimensional version was extensively used in research of Hadamard product in
one complex variable. Although in this paper we mostly rely on different
tools, the above fact will come in useful at some point in Section 4.

###  2.2. Integral formula for the $*$ product with special weights

Take a bounded smooth domain $0\in D\subset\mathbb{C}^{N}$, a neighbourhood
$V$ of $\partial D$ and a smooth map $\varphi:V\to\mathbb{C}^{N}$ such that
$\zeta\bullet\varphi(\zeta)=1$ for all $\zeta\in V$. If a function $f$ is
holomorphic in a neighbourhood of $\overline{D}$ and
$\sum_{\alpha\in\mathbb{N}^{N}}f_{\alpha}z^{\alpha}$ is its Taylor series
expansion at the origin, then from the considerations made in [28, Section
2.1] it follows that

$f_{\alpha}=c_{N}\frac{(N+|\alpha|-1)!}{\alpha!}\int_{\partial
D}f(\zeta)\varphi(\zeta)^{\alpha}\omega_{\varphi}(\zeta),$

where $c_{N}$ is a constant dependent only on $N$ and $\omega_{\varphi}$ is
certain smooth $(N,N-1)$ form on $V$ dependent only on $N$ and $\varphi$. Now,
if $g\in\mathcal{O}_{0}$ has the Taylor series expansion
$\sum_{\alpha\in\mathbb{N}^{N}}g_{\alpha}z^{\alpha}$, then for $z$ close to
$0$ one has that

(2)
$\sum_{\alpha\in\mathbb{N}^{N}}\frac{\alpha!(N-1)!}{(|\alpha|+N-1)!}f_{\alpha}g_{\alpha}z^{\alpha}=c_{N}(N-1)!\int_{\partial
D}f(\zeta)g(\varphi(\zeta)z)\omega_{\varphi}(\zeta).$

We will make use of this equality in Section 3.

##  3\. Description of $D*G$ for $D$ being of special class and $G$ being an
arbitrary Runge domain

Our goal in this part of the paper is to demonstrate Theorem 3.4. For the
Reader’s convenience, we begin by recalling definition and elementary facts
regarding the $*$-product of domains.

######  Assumption.

Throughout this section we assume that $N\geq 2$ and $D$, $G$ are domains in
$\mathbb{C}^{N}$ containing the origin.

######  Definition.

Assume that at least one of $D$, $G$ is Runge domain. Proposition 4.1 from
[28] establishes existence of the largest domain
$0\in\Omega\subset\mathbb{C}^{N}$ having the property that the image of
$\mathcal{O}_{0,D}\times\mathcal{O}_{0,G}$ by $*$ lies in
$\mathcal{O}_{0,\Omega}$. We define $D*G$ as this largest $\Omega$. The
referenced fact guarantees that $D*G$ itself is Runge domain.

For every pair of functions $f\in\mathcal{O}(D)$ and $g\in\mathcal{O}(G)$
there exists the only element of $\mathcal{O}(D*G)$ equal to $f*g$ near the
origin. Following [28], we denote it by $f*_{D,G}g$. We then obtain the
bilinear mapping

$*_{D,G}:\mathcal{O}(D)\times\mathcal{O}(G)\ni(f,g)\mapsto
f*_{D,G}g\in\mathcal{O}(D*G).$

From the closed graph theorem it follows that $*_{D,G}$ is separately
continuous. This, together with [26, page 88, Corollary 1], guarantees that it
is jointly continuous.

In a similar fashion we introduce the $*$-product of a germ from
$\mathcal{O}_{0}$ and a domain containing the origin.

######  Definition.

Assume that $G$ is Runge domain. If $\tau\in\mathcal{O}_{0}$, then, in virtue
of [28, Proposition 4.1], there exists the largest domain
$0\in\Omega\subset\mathbb{C}^{N}$ such that the image of $\mathcal{O}_{0,G}$
by the map $g\mapsto\tau*g$ lies in $\mathcal{O}_{0,\Omega}$. We denote this
largest $\Omega$ by $\tau*G$. As previously, $\tau*G$ is Runge domain.

For each $g\in\mathcal{O}(G)$ there exists the only function from
$\mathcal{O}(\tau*G)$ equal to $\tau*g$ near the origin. Denote this function
by $\tau*_{G}g$. The closed graph theorem yields that the linear operator

$\mathcal{O}(G)\ni g\mapsto\tau*_{G}g\in\mathcal{O}(\tau*G)$

is continuous.

######  Definition.

Similarly as in [28] we introduce the compact set

$D^{*}:=\left\\{\xi\in\mathbb{C}^{N}:\xi\bullet z\neq 1\text{ for all }z\in
D\right\\}.$

If $\xi\in D^{*}$, then the function

(3) $h_{\xi}(z):=\left(1-z\bullet\xi\right)^{-N}.$

belongs to $\mathcal{O}(D)$ and

(4)
$h_{\xi}(z)=\sum_{\alpha\in\mathbb{N}^{N}}\frac{(|\alpha|+N-1)!}{\alpha!(N-1)!}\xi^{\alpha}z^{\alpha}$

is its Taylor series expansion at the origin.

######  Lemma 3.1.

Assume that $G$ is Runge domain and $U\subset\mathbb{C}^{N}$ is an open set.
If a domain $\Omega\subset\mathbb{C}^{N}$ contains the origin and
$h_{\xi}*g\in\mathcal{O}_{0,\Omega}$ for all $\xi\in U$ and
$g\in\mathcal{O}_{0,G}$, then

$U\cdot\Omega\subset h_{\textnormal{{1}}_{N}}*G.$

###### Proof.

We need to show that $\xi\Omega\subset h_{\textnormal{{1}}_{N}}*G$ for each
$\xi\in U$. If $\xi\in U\cap(\mathbb{C}_{*})^{N}$, then, in view of (4), for
$g\in\mathcal{O}_{0,G}$ and $z$ close to $0$ we have

$(h_{\xi}*g)(z)=(h_{\textnormal{{1}}_{N}}*g)(\xi z),$

so the germ of the function on the right hand side belongs to
$\mathcal{O}_{0,\Omega}$. Hence,
$h_{\textnormal{{1}}_{N}}*g\in\mathcal{O}_{0,\xi\Omega}$. This holds for every
$g\in\mathcal{O}_{0,G}$, so the definition of $h_{\textnormal{{1}}_{N}}*G$
yields that it indeed contains $\xi\Omega$. On the other hand, if $\xi\in
U\setminus(\mathbb{C}_{*})^{N}$, then we can take a number $r>0$ so that
$\xi+r\mathbb{T}^{N}\subset U\cap(\mathbb{C}_{*})^{N}$. From the previous
considerations it follows that

$(\xi+r\mathbb{T}^{N})\cdot\Omega\subset h_{\textnormal{{1}}_{N}}*G.$

Since the set on the right hand side is Runge domain, for each $z\in\Omega$ it
contains the polynomial hull of $(\xi+r\mathbb{T}^{N})\cdot z$ and, in
particular, the point $\xi\cdot z$ itself. ∎

######  Lemma 3.2.

If $D$ is a pseudoconvex domain and $G$ is Runge domain, then

$D^{*}\cdot(D*G)\subset h_{\textnormal{{1}}_{N}}*G.$

Consequently,

$D*G\subset\textnormal{cc}_{0}\,\\{z\in\mathbb{C}^{N}:zD^{*}\subset
h_{\textnormal{{1}}_{N}}*G\\}.$

It is worthy to note that, as $D^{*}$ is compact, the set under
$\textnormal{cc}_{0}\,$ above is open.

###### Proof.

Fix an arbitrary domain $0\in\Omega\subset\subset D*G$. From the continuity of
$*_{D,G}$ it follows that we can find a constant $C>0$ and compact sets
$K\subset D$ and $L\subset G$ such that $K$ is holomorphically convex in $D$,
$0\in\textnormal{int}\,K$ and

(5) $\|f*_{D,G}g\|_{\Omega}\leq C\|f\|_{K}\|g\|_{L}$

for all $f\in\mathcal{O}(D)$ and $g\in\mathcal{O}(G)$.

Take a neighbourhood $U$ of $D^{*}$ such that $\eta\bullet z\neq 1$ for all
$z\in K$ and $\eta\in U$. If $\eta\in U$, then $h_{\eta}$ is holomorphic in a
neighbourhood of $K$, so, by the Oka-Weil theorem, it can be approximated
uniformly on $K$ by a sequence
$(f_{n})_{n\in\mathbb{N}}\subset\mathcal{O}(D)$. Hence, if
$g\in\mathcal{O}(G)$, then (5) gives that the functions $f_{n}*_{D,G}g$ form a
Cauchy sequence with respect to the supremum norm on $\Omega$. This means that
they converge in $\mathcal{O}(\Omega)$ and, thanks to the fact that
$0\in\textnormal{int}\,K$, the limit has to be equal to $h_{\eta}*g$ near the
origin. Consequently, $h_{\eta}*g\in\mathcal{O}_{0,\Omega}$ for every
$g\in\mathcal{O}_{0,G}$ and $\eta\in U$. Now, Lemma 3.1 allows us to conclude
that $D^{*}\cdot\Omega\subset U\cdot\Omega\subset h_{\textnormal{{1}}_{N}}*G$,
what completes the proof. ∎

######  Lemma 3.3.

Let $0\in W$ and $0\in D_{0}\subset D_{1}\subset D_{2}\subset\ldots$ be
domains in $\mathbb{C}^{N}$ such that $\bigcup_{n\in\mathbb{N}}D_{n}=D$. Set

$\Omega_{n}=\textnormal{cc}_{0}\,\\{z\in\mathbb{C}^{N}:zD_{n}^{*}\subset
W\\},\quad\Omega=\textnormal{cc}_{0}\,\\{z\in\mathbb{C}^{N}:zD^{*}\subset
W\\}.$

Then $\Omega_{0}\subset\Omega_{1}\subset\Omega_{2}\subset\ldots$ and
$\bigcup_{n\in\mathbb{N}}\Omega_{n}=\Omega$.

###### Proof.

Clearly, $D^{*}\subset D_{n+1}^{*}\subset D_{n}^{*}$, so
$\Omega_{n}\subset\Omega_{n+1}\subset\Omega$. To show that $\Omega$ is
contained in $\bigcup_{n\in\mathbb{N}}\Omega_{n}$, take an arbitrary connected
compact set $0\in K\subset\Omega$. One has that $K\cdot D^{*}\subset W$, so
$K\cdot U\subset W$ for a neighbourhood $U$ of $D^{*}$. The sequence
$(D_{n}^{*})_{n\in\mathbb{N}}$ decreases and it is straightforward to check
that $D^{*}=\bigcap_{n\in\mathbb{N}}D_{n}^{*}$. Therefore,
$D_{n_{0}}^{*}\subset U$ for some $n_{0}$, what gives that $K\cdot
D_{n_{0}}^{*}\subset W$. Hence, $K\subset\Omega_{n_{0}}$, because $K$ is
connected and contains the origin. From this we conclude that
$\Omega\subset\bigcup_{n\in\mathbb{N}}\Omega_{n}$. ∎

######  Definition.

Similarly as in [28] we define $\mathcal{D}_{N}$ as the family of all domains
in $\mathbb{C}^{N}$ containing the origin which are countable unions of non-
decreasing sequences of bounded smooth linearly convex domains.

Recall that $D$ is called _linearly convex_ if through every point of
$\mathbb{C}^{N}\setminus D$ one can pass an affine complex hyperplane disjoint
from $D$. As it is described in [28, Remark 4.11], each element of
$\mathcal{D}_{N}$, being a union of a non-decreasing sequence of Runge
domains, is also Runge domain.

######  Theorem 3.4.

If $D\in\mathcal{D}_{N}$ and $G$ is Runge domain, then

$D*G=\textnormal{cc}_{0}\,\\{z\in\mathbb{C}^{N}:zD^{*}\subset
h_{\textnormal{{1}}_{N}}*G\\}.$

###### Proof.

The left-to-right inclusion was established in Lemma 3.2, so it remains to
prove the opposite one. In virtue of Lemma 3.3 and [28, Proposition 4.5] it
suffices to restrict our considerations to the case when $D$ is bounded,
smooth and linearly convex. Then there exists a smooth map $\nu_{D}$ from a
neighbourhood of $\partial D$ to $\mathbb{C}^{N}$ such that at each point
$w\in\partial D$ its value $\nu_{D}(w)$ is the unit outward normal vector for
$D$ at $w$. Hence, for $w\in\partial D$ the equation
$(z-w)\bullet\overline{\nu}_{D}(w)=0$ describes the only complex hyperplane
passing through $w$ and disjoint from $D$. In particular,
$w\bullet\overline{\nu}_{D}(w)\neq 0$, as $0\in D$. This means that the
mapping

$\varphi:w\mapsto\overline{\nu}_{D}(w)\cdot(w\bullet\overline{\nu}_{D}(w))^{-1}$

is well-defined and smooth in a neighbourhood $V$ of $\partial D$. Moreover,
we have that $\varphi(\partial D)\subset D^{*}$ and $w\bullet\varphi(w)=1$ for
$w\in V$.

Denote by $\Omega$ the set on the right hand side of the conclusion and take a
domain $0\in U\subset\subset\Omega$. From the definition of $\Omega$ it
follows that $U\cdot\varphi(\partial D^{\prime})\subset
h_{\textnormal{{1}}_{N}}*G$ for a sufficiently large smooth domain $0\in
D^{\prime}\subset\subset D$ such that $\partial D^{\prime}\subset V$. If
$f\in\mathcal{O}(D)$ and $g\in\mathcal{O}(G)$, then, by (2), for $z$ lying
near the origin it holds that

$(f*g)(z)=c_{N}(N-1)!\int_{\partial
D^{\prime}}f(\zeta)(h_{\textnormal{{1}}_{N}}*_{G}g)(\varphi(\zeta)z)\omega_{\varphi}(\zeta).$

The integral on the right hand side defines a function of the variable $z$
which is holomorphic on $U$. Consequently, $f*g\in\mathcal{O}_{0,U}$. Since
$f$, $g$ and $U$ were taken arbitrarily, we conclude that $\Omega\subset D*G$.
∎

##  4\. Description of $h_{(1,1)}*D$ for Runge domains

This part is devoted to demonstration of Theorem 4.1, which completes,
although only in the two-dimensional case, the description of the star product
of domains established in Theorem 3.4. Recall that $h_{(1,1)}$ is the function
given by the formula (3), that is,

$h_{(1,1)}(z_{1},z_{2})=(1-z_{1}-z_{2})^{-2}=\sum_{\alpha_{1},\alpha_{2}\in\mathbb{N}}\frac{(\alpha_{1}+\alpha_{2}+1)!}{\alpha_{1}!\,\alpha_{2}!}z_{1}^{\alpha_{1}}z_{2}^{\alpha_{2}}.$

######  Assumption.

In this section we assume that $D$ is a domain in $\mathbb{C}^{2}$ containing
the origin.

######  Definition.

To simplify certain statements in this section, let us say that an open set
$U\subset\mathbb{C}_{*}$ _separates_ $0$ and $\infty$ if $U$ contains a loop
homotopic in $\mathbb{C}_{*}$ to the loop $\zeta\mapsto\zeta$. This is
equivalent to saying that $0$ and $\infty$ lie in different connected
components of $\widehat{\mathbb{C}}\setminus U$.

For a point $z=(z_{1},z_{2})\in\mathbb{C}^{2}$ introduce the mapping
$I_{z}:\mathbb{C}_{*}\to\mathbb{C}^{2}$ as

$I_{z}(\zeta):=(z_{1}(1+\zeta),z_{2}(1+\zeta^{-1})).$

One has that $I_{z}(-1)=(0,0)$, so $I_{z}^{-1}(D)$ is non-empty. It is also
important that $I_{z}$ is an injective proper map when
$z\in(\mathbb{C}_{*})^{2}$.

######  Theorem 4.1.

If $D$ is Runge domain, then

$h_{(1,1)}*D=\textnormal{cc}_{0}\,\left\\{z\in\mathbb{C}^{2}:\text{the set
}I_{z}^{-1}(D)\text{ separates }0\text{ and }\infty\right\\}.$

Directly from the definition of separating is follows that the set under
$\textnormal{cc}_{0}\,$ above is open. It also contains $(0,0)$, because
$I_{(0,0)}^{-1}(D)=\mathbb{C}_{*}$.

######  Remark 4.2.

Assume that $D$ is Runge domain and take $z\in\mathbb{C}^{2}$. The open set
$I_{z}^{-1}(D)$ is then $\mathcal{O}(\mathbb{C}_{*})$-convex in the sense that
$\widehat{L}_{\mathbb{C}_{*}}\subset I_{z}^{-1}(D)$ for every compact set
$L\subset I_{z}^{-1}(D)$. This means that every connected component of
$\widehat{\mathbb{C}}\setminus I_{z}^{-1}(D)$ contains $0$ or $\infty$
(possibly both of them). Consequently, the latter set is connected if and only
if $I_{z}^{-1}(D)$ does not separate $0$ and $\infty$.

######  Observation 4.3.

If an open set $\Omega\subset\mathbb{C}^{2}$ and a point
$z\in(\mathbb{C}_{*})^{2}$ are such that $I_{z}^{-1}(\Omega)$ does not
separate $0$ and $\infty$, then for every compact polynomially convex set
$K\subset\Omega$ the pre-image $I_{z}^{-1}(K)$ is either empty or polynomially
convex.

###### Proof.

First, note that the set $L:=I_{z}^{-1}(K)$, if non-empty, has to be compact
and holomorphically convex in $\mathbb{C}_{*}$, what means that every
connected component of $\widehat{\mathbb{C}}\setminus L$ contains $0$ or
$\infty$. But since $I_{z}^{-1}(\Omega)$ does not separate $0$ and $\infty$,
these points lie in the same connected component of
$\widehat{\mathbb{C}}\setminus L$. Hence, the set
$\widehat{\mathbb{C}}\setminus L$ is connected, so $L$ is polynomially convex.
∎

######  Lemma 4.4.

Let $K\subset\mathbb{C}^{2}$ be a compact polynomially convex set,
$z\in(\mathbb{C}_{*})^{2}$ and $\varrho\in(0,\infty)$. If $I_{z}^{-1}(K)$ is
empty or polynomially convex and

$I_{z}^{-1}(K)\cap\varrho\overline{\mathbb{D}}=\varnothing,$

then the union $K\cup I_{z}(\varrho\mathbb{T})$ is polynomially convex.

###### Proof.

Write $z=(z_{1},z_{2})$ and define

$\mu(w_{1},w_{2}):=(w_{1}-z_{1})(w_{2}-z_{2})-z_{1}z_{2},\quad(w_{1},w_{2})\in\mathbb{C}^{2}.$

Clearly, $M:=\mu^{-1}(0)=I_{z}(\mathbb{C}_{*})$ is a complex submanifold of
$\mathbb{C}^{2}$ and the map $I_{z}:\mathbb{C}_{*}\to M$ is a biholomorphism.
By the assumptions, the union $I_{z}^{-1}(K)\cup\varrho\mathbb{T}$ is
holomorphically convex in $\mathbb{C}_{*}$. This implies that the set $(K\cap
M)\cup I_{z}(\varrho\mathbb{T})$, being its image by $I_{z}$, is
holomorphically convex in $M$ and thus polynomially convex in $\mathbb{C}^{2}$
(use e.g. [15, Theorem 7.4.8]). The conclusion now follows directly from the
subsequent general lemma. ∎

######  Lemma 4.5.

Let $V$ be an analytic subset of $\mathbb{C}^{N}$ and let
$K\subset\mathbb{C}^{N}$, $L\subset V$ be compact sets. If both $K$ and
$(K\cap V)\cup L$ are polynomially convex, then so is $K\cup L$.

Note that the sets $K$ and $L$ do not have to be disjoint.

###### Proof.

It is known (see e.g. [15, Theorems 6.5.2 and 7.1.5]) that the sheaf of germs
of holomorphic functions vanishing on $V$ is a coherent analytic sheaf on
$\mathbb{C}^{N}$. Therefore, if $X\subset\mathbb{C}^{N}$ is a compact,
polynomially convex set (intersecting $V$ or not), $U$ is a neighbourhood of
$X$, $F\in\mathcal{O}(U)$ and $F|_{U\cap V}\equiv 0$, then $F$ is a section of
this sheaf and it can be approximated uniformly on $X$ by global sections,
that is, by elements of $\mathcal{O}(\mathbb{C}^{N})$ vanishing on $V$. This
essential fact is a consequence of [15, Theorem 7.2.7].

Fix a point $z_{0}\in\mathbb{C}^{N}\setminus(K\cup L)$. We are going to show
that $z_{0}\not\in\widehat{K\cup L}$. If $z_{0}\not\in V$, then [15, Theorem
7.2.11] provides $f\in\mathcal{O}(\mathbb{C}^{N})$ vanishing on $V$ and such
that $f(z_{0})=1$. On the other hand, there exists
$g\in\mathcal{O}(\mathbb{C}^{N})$ having $g(z_{0})=1$ and $\|g\|_{K}<1$.
Hence, for sufficiently large number $n$ the function $g^{n}f$ maps $z_{0}$ to
$1$ and $K\cup L$ into $\mathbb{D}$.

It remains to consider the case when $z_{0}\in V$. Since $(K\cap V)\cup L$ is
polynomially convex, one can find another compact polynomially convex set
$A\subset\mathbb{C}^{N}$ such that

$(K\cap V)\cup L\subset\textnormal{int}\,A\text{ and }z_{0}\not\in A.$

Take a sequence $(p_{n})_{n\in\mathbb{N}}\subset\mathcal{O}(\mathbb{C}^{N})$
such that

$p_{n}(z_{0})=1\text{ and }\|p_{n}\|_{A}\to 0\text{ when }n\to\infty.$

The set $(\mathbb{C}^{N}\setminus V)\cup\textnormal{int}\,A$ is an open
neighbourhood of $K$, so it has a pseudoconvex open subset $\Omega$ containing
$K$. This means that $\Omega\cap V\subset\textnormal{int}\,A$ and hence
$p_{n}\to 0$ on $\Omega\cap V$. In virtue of [6, Theorem 13.1], there exists a
continuous linear extension operator from the Banach space of bounded
holomorphic functions on $\Omega\cap V$ into $\mathcal{O}(\Omega)$. Applying
it to $p_{n}$’s we obtain a sequence
$(g_{n})_{n\in\mathbb{N}}\subset\mathcal{O}(\Omega)$ convergent to $0$ in
$\mathcal{O}(\Omega)$ and such that $g_{n}-p_{n}=0$ on $\Omega\cap V$. As it
was described in the first paragraph of the proof, for every $n$ one can find
a function $q_{n}\in\mathcal{O}(\mathbb{C}^{N})$ so that $q_{n}|_{V}\equiv 0$
and $\|q_{n}+p_{n}-g_{n}\|_{K}<\frac{1}{n}$. Finally, set
$f_{n}:=p_{n}+q_{n}$. Every $f_{n}$ is an entire function and

$f_{n}=p_{n}\text{ on }V\text{ and }\|f_{n}-g_{n}\|_{K}<\frac{1}{n}.$

This implies that $f_{n}(z_{0})=1$ and $f_{n}\to 0$ on $K\cup L$ uniformly
when $n\to\infty$, so for large $n$ it holds that
$|f_{n}(z_{0})|>\|f_{n}\|_{K\cup L}$, as desired. ∎

For a holomorphic function $f$ of two variables define

$\Lambda(f)(z_{1},z_{2}):=f(z_{1},z_{2})+z_{1}\,\frac{\partial f}{\partial
z_{1}}(z_{1},z_{2}).$

For every domain $\Omega\subset\mathbb{C}^{2}$ the mapping
$f\mapsto\Lambda(f)$ defines a continuous linear operator on
$\mathcal{O}(\Omega)$.

######  Lemma 4.6.

If $\gamma$ is a loop in $\mathbb{C}_{*}$ homotopic to the loop
$\zeta\mapsto\zeta$ and $f$ is a polynomial in $\mathbb{C}^{2}$, then

$(h_{(1,1)}*_{\mathbb{C}^{2}}f)(z)=\frac{1}{2\pi
i}\int_{\gamma}(1+\zeta^{-1})\,\Lambda(f)(I_{z}(\zeta))d\zeta$

for all $z\in\mathbb{C}^{2}$.

###### Proof.

Fix a number $\rho\in(0,\frac{1}{2})$ and a polynomial $f$. From (1) it
follows that

$(h_{(1,1)}*_{\mathbb{C}^{2}}f)(z)=\left(\frac{1}{2\pi
i}\right)^{2}\int_{\rho^{-1}\mathbb{T}^{2}}\frac{f(z_{1}\zeta_{1},z_{2}\zeta_{2})\zeta_{1}\zeta_{2}}{(\zeta_{2}-1)^{2}(\zeta_{1}-\zeta_{2}(\zeta_{2}-1)^{-1})^{2}}d\zeta_{1}d\zeta_{2},$

when $z=(z_{1},z_{2})\in\mathbb{C}^{2}$. If $\zeta_{2}\in\rho^{-1}\mathbb{T}$,
then the point $\zeta_{2}(\zeta_{2}-1)^{-1}$ lies in $\rho^{-1}\mathbb{D}$,
so, in view of the Cauchy formula,

$\frac{1}{2\pi
i}\int_{\rho^{-1}\mathbb{T}}\frac{f(z_{1}\zeta_{1},z_{2}\zeta_{2})\zeta_{1}}{(\zeta_{1}-\zeta_{2}(\zeta_{2}-1)^{-1})^{2}}d\zeta_{1}\\\
=\left.\frac{d}{d\zeta_{1}}\Bigl{(}f(z_{1}\zeta_{1},z_{2}\zeta_{2})\zeta_{1}\Bigr{)}\right|_{\zeta_{1}=\zeta_{2}(\zeta_{2}-1)^{-1}}=\Lambda(f)\left(\frac{z_{1}\zeta_{2}}{\zeta_{2}-1},z_{2}\zeta_{2}\right).$

Therefore,

$(h_{(1,1)}*_{\mathbb{C}^{2}}f)(z)=\frac{1}{2\pi
i}\int_{\rho^{-1}\mathbb{T}}\frac{\zeta_{2}}{(\zeta_{2}-1)^{2}}\;\Lambda(f)\left(\frac{z_{1}\zeta_{2}}{\zeta_{2}-1},z_{2}\zeta_{2}\right)d\zeta_{2}.$

A homotopy argument allows us to integrate over $\rho^{-1}\mathbb{T}+1$
instead of $\rho^{-1}\mathbb{T}$. Then, after changing the variable via
$\zeta:=(\zeta_{2}-1)^{-1}$, we obtain the equality from the conclusion with
the integral over $\rho\mathbb{T}$. It is not affected by replacing
$\rho\mathbb{T}$ by $\gamma$, because the integrated function of $\zeta$ is
holomorphic on $\mathbb{C}_{*}$. ∎

###### Proof of Theorem 4.1.

Denote by $\Omega$ the set on the right hand side of the conclusion, that is,

$\Omega:=\textnormal{cc}_{0}\,\left\\{z\in\mathbb{C}^{2}:\text{the set
}I_{z}^{-1}(D)\text{ separates }0\text{ and }\infty\right\\}.$

The proof of the equality $h_{(1,1)}*D=\Omega$ is divided into a few steps.

Step 1: We show the inclusion $\Omega\subset h_{(1,1)}*D$.

Fix a function $f\in\mathcal{O}(D)$ and take a sequence
$(f_{n})_{n\in\mathbb{N}}$ of polynomials convergent to $f$ locally uniformly
on $D$. Then the functions $h_{(1,1)}*_{\mathbb{C}^{2}}f_{n}$ tend to
$h_{(1,1)}*_{D}f$ in the same manner on $h_{(1,1)}*D$. We claim that they form
a sequence convergent on $\Omega$ as well. Fix a point $a\in\Omega$ and take a
loop $\gamma:\mathbb{T}\to I_{a}^{-1}(D)$ homotopic in $\mathbb{C}_{*}$ to the
identity loop. Choose a closed ball $B\subset\Omega$ centered at $a$ so that
$I_{z}(\gamma(\mathbb{T}))\subset D$ for all $z\in B$. From Lemma 4.6 it
follows that

$(h_{(1,1)}*_{\mathbb{C}^{2}}f_{n})(z)=\frac{1}{2\pi
i}\int_{\gamma}(1+\zeta^{-1})\,\Lambda(f_{n})(I_{z}(\zeta))d\zeta$

for all $z\in\mathbb{C}^{2}$, $n\in\mathbb{N}$. Since
$\Lambda(f_{n})\to\Lambda(f)$ in $\mathcal{O}(D)$ as $n\to\infty$, the
integrals on the right hand side of the above equality converge uniformly with
respect to $z\in B$ to the identical integral with $f_{n}$ repalced by $f$.
Consequently, the sequence of polynomials $h_{(1,1)}*_{\mathbb{C}^{2}}f_{n}$
is uniformly convergent on $B$. Hence, it does converge locally uniformly on
$\Omega$, because $a\in\Omega$ was chosen arbitrarily. If
$g\in\mathcal{O}(\Omega)$ is the limit, then clearly $g=h_{(1,1)}*f$ near the
origin. This means that $h_{(1,1)}*f\in\mathcal{O}_{0,\Omega}$ for every
$f\in\mathcal{O}(D)$, so $\Omega\subset h_{(1,1)}*D$.

Step 2: We prove the inclusion
$(h_{(1,1)}*D)\cap(\mathbb{C}_{*})^{2}\subset\Omega$.

Suppose, to the contrary, that it is not valid, and choose a point
$z\in(h_{(1,1)}*D)\cap(\mathbb{C}_{*})^{2}\cap\partial\Omega$. There exist a
constant $C>0$ and a polynomially convex compact set $K\subset D$ such that

(6) $|(h_{(1,1)}*_{D}f)(z)|\leq C\|f\|_{K},\quad f\in\mathcal{O}(D).$

Take a number $\varrho>0$ so that
$I_{z}^{-1}(K)\cap\varrho\overline{\mathbb{D}}=\varnothing$. Since
$z\in\partial\Omega$, the set $I_{z}^{-1}(D)$ does not separate $0$ and
$\infty$, so from Observation 4.3 and Lemma 4.4 it follows that the union
$K\cup I_{z}(\varrho\mathbb{T})$ is polynomially convex. Hence, there exists a
compact set $L\subset\mathbb{C}^{2}$ such that
$I_{z}(\varrho\mathbb{T})\subset\textnormal{int}\,L$, $K\cap L=\varnothing$
and $K\cup L$ is polynomially convex (one can justify it making use, for
example, of the Kallin Lemma [27, Theorem 1.6.19]). This allows us to employ
the Oka-Weil theorem to get a sequence $(f_{n})_{n\in\mathbb{N}}$ of
polynomials in $\mathbb{C}^{2}$ uniformly convergent to $0$ on $K$ and to $1$
on $L$. Lemma 4.6 implies that

$(h_{(1,1)}*_{D}f_{n})(z)=\frac{1}{2\pi
i}\int_{\varrho\mathbb{T}}(1+\zeta^{-1})\,\Lambda(f_{n})(I_{z}(\zeta))d\zeta.$

If $n\to\infty$, then the left hand side goes to $0$, in view of (6). On the
other hand, the integrals converge to $1$, because $\Lambda(f_{n})\to 1$ on
$I_{z}(\varrho\mathbb{T})$. A contradiction.

Step 3: We show that if $(z_{1},0)\in h_{(1,1)}*D$, then $(z_{1},0)\in D$.
Thanks to evident symmetry, it will also mean that $(0,z_{2})\in D$ when
$(0,z_{2})\in h_{(1,1)}*D$.

Suppose, to the contrary, that $(z_{1},0)\not\in D$. One can find a constant
$C>0$ and a compact polynomially convex set $K\subset D$ satisfying

(7) $|(h_{(1,1)}*_{D}f)(z_{1},0)|\leq C\|f\|_{K},\quad f\in\mathcal{O}(D).$

Take a closed ball $B$ centered at $(z_{1},0)$ so small that $K\cap
B=\varnothing$ and $K\cup B$ is polynomially convex. In view of the Oka-Weil
theorem, there exists a sequence $(f_{n})_{n\in\mathbb{N}}$ of polynomials in
$\mathbb{C}^{2}$ uniformly convergent to $0$ on $K$ and to $1$ on $B$.

From Lemma 4.6 is follows that

(8)
$(h_{(1,1)}*_{\mathbb{C}^{2}}f_{n})(z_{1},0)=\Lambda(f_{n})(z_{1},0)=f_{n}(z_{1},0)+z_{1}\,\frac{\partial
f_{n}}{\partial z_{1}}(z_{1},0).$

Now, if $n\to\infty$, then (7) implies that the left hand side of (8) goes to
$0$, while the right hand side converges to $1$. This contradiction completes
the proof of this step.

Step 4: We obtain the inclusion $h_{(1,1)}*D\subset\Omega$.

It remains to prove that if $(z_{1},0)\in h_{(1,1)}*D$, then
$(z_{1},0)\in\Omega$ (the same statement for $(0,z_{2})$ can be demonstrated
identically). Take such a point $(z_{1},0)$ and choose a curve
$\gamma:[0,1]\to h_{(1,1)}*D$ so that $\gamma(0)=(0,0)$, $\gamma(1)=(z_{1},0)$
and $\gamma(t)\in(\mathbb{C}_{*})^{2}$ for $t\in(0,1)$. The conclusion of Step
2 guarantees that $\gamma(t)\in\Omega$ every $t\in(0,1)$, so
$(z_{1},0)\in\overline{\Omega}$. Moreover, from Step 3 we know that
$(z_{1},0)\in D$, so $\epsilon\mathbb{D}_{*}\subset I_{(z_{1},0)}^{-1}(D)$ for
a small number $\epsilon>0$. Hence, the latter set separates $0$ and $\infty$,
what gives that $(z_{1},0)\in\Omega$. The proof is complete. ∎

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# Entropy current and entropy production in relativistic spin hydrodynamics

Francesco Becattini<EMAIL_ADDRESS>Department of Physics, University of
Florence and INFN Via G. Sansone 1, I-50019, Sesto F.no (Firenze), Italy
Asaad Daher<EMAIL_ADDRESS>Institute of Nuclear Physics Polish
Academy of Sciences, PL-31-342 Kraków, Poland  Xin-Li Sheng<EMAIL_ADDRESS>INFN Sezione di Firenze, Via G. Sansone 1, I-50019, Sesto F.no (Firenze),
Italy

###### Abstract

We use a first-principle quantum-statistical method to derive the expression
of the entropy production rate in relativistic spin hydrodynamics. We show
that the entropy current is not uniquely defined and can be changed by means
of entropy-gauge transformations, much the same way as the stress-energy
tensor and the spin tensor can be changed with pseudo-gauge transformations.
We show that the local thermodynamic relations, which are admittedly educated
guesses in relativistic spin hydrodynamics inspired by those at global
thermodynamic equilibrium, do not hold in general and they are also non-
invariant under entropy-gauge transformations. Notwithstanding, we show that
the entropy production rate is independent of those transformations and we
provide a universally applicable expression, extending that known in
literature, from which one can infer the dissipative parts of the energy
momentum and spin tensors.

##  I Introduction

Motivated by the evidence of spin polarization of particles produced in
relativistic heavy ion collisions [1, 2], there is a growing interest in the
so-called relativistic spin hydrodynamics [3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19]. Relativistic spin hydrodynamics stipulates that
the description of a relativistic fluid requires the addition of a spin
tensor, that is the mean value of a rank 3 tensor operator ${\widehat{\cal
S}}^{\lambda\mu\nu}$ (the last two indices anti-symmetric) contributing to the
overall angular momentum current:

$\widehat{\cal
J}^{\lambda\mu\nu}=x^{\mu}{\widehat{T}}^{\lambda\nu}-x^{\nu}{\widehat{T}}^{\lambda\mu}+{\widehat{\cal
S}}^{\lambda\mu\nu}\,,$

where ${\widehat{T}}^{\mu\nu}$ is the stress-energy tensor operator. This
current is conserved, which implies that the spin tensor fulfills the
continuity equation:

$\partial_{\lambda}{\widehat{\cal
S}}^{\lambda\mu\nu}={\widehat{T}}^{\nu\mu}-{\widehat{T}}^{\mu\nu}\,.$ (1)

It is important to point out that the spin tensor - and the stress-energy
tensor as well - are not uniquely defined. Indeed, they can be changed with a
so-called pseudo-gauge transformation [20, 21] to a new couple of tensors
fulfilling the same dynamical equations and providing the same integrated
conserved charges. Since the spin tensor can be made vanishing with a suitable
pseudo-gauge transformation, its dynamical meaning has been questioned, yet it
was observed in ref. [22] that for a fluid not in global thermodynamic
equilibrium (such as the QGP throughout its lifetime) the quantum state of the
system (i.e. the density operator describing initial local equilibrium) is not
invariant under pseudo-gauge transformations. Thus, in principle, the physical
measurements depend on the pseudo-gauge if the initial quantum state is not
invariant, and particularly on the intensive quantity which is
thermodynamically conjugated to the spin tensor, the spin potential. For
instance, if the spin tensor does not vanish, the spin polarization of final
particles depends on the difference between spin potential and thermal
vorticity [23]. The microscopic conditions underpinning spin hydrodynamics
have been studied and elucidated in ref. [11], where it was made clear that
spin hydrodynamics regime occurs under a specific hierarchy of the interaction
time scales in the system.

A key problem in the relativistic hydrodynamics with spin is the derivation of
the constitutive equations of the spin tensor as well as of the anti-symmetric
part of the stress-energy tensor. This problem has drawn significant attention
over the past few years, with several derivations of constitutive equations
[7, 8, 9, 10, 11, 12, 13] based on the requirement of the positivity of local
entropy production rate. However, like in the traditional approach to
relativistic hydrodynamics, the entropy current is not really derived, but it
is obtained from an educated guess of a thermodynamic relations between the
proper densities of entropy, energy, charge and “spin density”
$S^{\mu\nu}\equiv u_{\lambda}{\cal S}^{\lambda,\mu\nu}$ as follows:

$\begin{split}&Ts+\mu n=\rho+p-\frac{1}{2}\omega_{\mu\nu}S^{\mu\nu}\\\ &{\rm
d}p=s\,{\rm d}T+n\,{\rm d}\mu+\frac{1}{2}S^{\mu\nu}{\rm
d}\omega_{\mu\nu}\end{split}$ (2)

where $T$ is temperature, $\mu$ a chemical potential, $\rho$ the proper energy
density, $p$ the pressure, $n$ the charge density and $\omega_{\mu\nu}$ is the
spin potential111This is related to $\Omega$ defined in ref. [22] by the
relation $\omega=T\Omega$..

In this work, we apply the quantum-statistical approach to relativistic
hydrodynamics [24, 25, 26] by including spin tensor. The quantum statistical
method based on local equilibrium density operator has several advantages over
other approaches in that it makes it possible to derive from first principles
a form of the entropy current and entropy production rate rather than
constructing it assuming a particular form of the local thermodynamic relation
such as the equation (2). We will use a recent result on the extensivity of
the logarithm of the partition function to obtain an exact form of the entropy
current [27]. We will be able to show that the relation (2) is incomplete and
that the entropy density has, in general, additional terms involving the spin
tensor. Furthermore, we will extend the derivation of ref. [24] of the entropy
production rate to include the spin tensor. Such general relation is the
starting point to derive the constitutive relations for the anti-symmetric
part of the stress-energy tensor and the spin tensor.

##  II Entropy current and local equilibrium

In the quantum-statistical description of a relativistic fluid, the local
equilibrium density operator denoted as ${\widehat{\rho}}_{LE}$ is obtained by
maximizing entropy $S=-\Tr({\widehat{\rho}}\log{\widehat{\rho}})$ over some
preset space-like hypersurface by constraining the mean values of the energy,
momentum, charge and spin densities to be equal to their actual values [22]:

$\displaystyle{\widehat{\rho}}_{\rm LE}=\frac{1}{Z_{\rm
LE}}\exp\left[-\int_{\Sigma}{\rm
d}\,\Sigma_{\mu}\,\left({\widehat{T}}^{\mu\nu}\beta_{\nu}-\zeta{\widehat{j}}^{\mu}-\frac{1}{2}\Omega_{\lambda\nu}{\widehat{\cal
S}}^{\mu\lambda\nu}\right)\right]\,,$ (3)

where ${\rm d}\Sigma_{\mu}\equiv{\rm d}\Sigma\,n_{\mu}$, $n$ being the unit
vector perpendicular to the hypersurface $\Sigma$; the function $Z_{\rm LE}$
is the partition function, and the operators ${\widehat{T}}^{\mu\nu}$,
${\widehat{\cal S}}^{\mu\lambda\nu}$ are the energy-momentum and spin tensor
operators, a particular couple amongst all the possible couples connected by
pseudo-gauge transformations. The constraints read:

$n_{\mu}T^{\mu\nu}=n_{\mu}T^{\mu\nu}_{\rm LE}\,,\qquad
n_{\mu}j^{\mu}=n_{\mu}j^{\mu}_{\rm LE}\,,\qquad n_{\mu}{\cal
S}^{\mu\lambda\nu}=n_{\mu}{\cal S}^{\mu\lambda\nu}_{\rm LE}\,,$ (4)

where the local equilibrium values are defined as:

$X_{\rm LE}\equiv\Tr({\widehat{\rho}}_{\rm
LE}\widehat{X})-\bra{0}\widehat{X}\ket{0}\,,$ (5)

with $\ket{0}$ being the supposedly non-degenerate lowest lying eigenvector of
the operator in the exponent of (3). In the equation (3), the fields
$\beta_{\nu}$, $\zeta$ and $\Omega_{\lambda\nu}$ are the Lagrange multipliers
related to this problem, and they are the thermal velocity four-vector, the
chemical potential to temperature ratio, and the spin potential to temperature
ratio respectively, that is:

$\beta=\frac{u}{T}\,,\qquad\zeta=\frac{\mu}{T}\,,\qquad\Omega=\frac{\omega}{T}\,.$
(6)

It is worth pointing out that they can be obtained as solutions of the
constraint equations (4) [26], if the exact values of the stress-energy tensor
and other currents is known. In relativistic hydrodynamics, since they are not
known a priori, they are solutions of the hydrodynamic partial differential
equations with initial conditions expressed by the equations (4) over the
initial Cauchy space-like hypersurface. It should also be stressed that
$\beta$ thereby defines a so-called hydrodynamic frame in its own (the so-
called thermodynamic or thermometric or $\beta$ frame), which does not
coincide with the better known Landau or Eckart frames. At global equilibrium
one has:

$\beta_{\mu}=b_{\mu}+\varpi_{\mu\nu}x^{\nu}\,,\;\;{\rm with}\;\;b,\varpi={\rm
const}\,,\qquad\Omega=\varpi\,,\qquad\zeta={\rm const}\,,$ (7)

where $\varpi$ is a constant anti-symmetric tensor, the thermal vorticity.

Starting from the equation (3), it is possible to prove [27] that if the
operator:

$\widehat{\Upsilon}\equiv\int_{\Sigma}{\rm
d}\,\Sigma_{\mu}\,\left({\widehat{T}}^{\mu\nu}\beta_{\nu}-\zeta{\widehat{j}}^{\mu}-\frac{1}{2}\Omega_{\lambda\nu}{\widehat{\cal
S}}^{\mu\lambda\nu}\right)$

is bounded from below and the lowest lying eigenvalue $\ket{0}$ is non-
degenerate, the logarithm of $Z_{\rm LE}$ is extensive, namely it can be
written as an integral over $\Sigma$:

$\log Z_{\rm LE}=\int_{\Sigma}{\rm
d}\Sigma_{\mu}\;\phi^{\mu}-\bra{0}\widehat{\Upsilon}\ket{0}=\int_{\Sigma}{\rm
d}\Sigma_{\mu}\;\left[\phi^{\mu}-\bra{0}\left({\widehat{T}}^{\mu\nu}\beta_{\nu}-\zeta{\widehat{j}}^{\mu}-\frac{1}{2}\Omega_{\lambda\nu}{\widehat{\cal
S}}^{\mu\lambda\nu}\right)\ket{0}\right]$ (8)

where

$\phi^{\mu}=\int_{1}^{\infty}{\rm d}\lambda\;\left(T^{\mu\nu}_{\rm
LE}(\lambda)\beta_{\nu}-\zeta j^{\mu}_{\rm
LE}(\lambda)-\frac{1}{2}\Omega_{\lambda\nu}{\cal S}^{\mu\lambda\nu}_{\rm
LE}(\lambda)\right)$ (9)

is defined as the thermodynamic potential current. In the equation (9), the
integration variable $\lambda$ is a dimensionless parameter which multiplies
the exponent of the local equilibrium density operator (3), that is:

${\widehat{\rho}}_{\rm LE}(\lambda)=\frac{1}{Z_{\rm
LE}(\lambda)}\exp\left[-\lambda\int_{\Sigma}{\rm
d}\,\Sigma_{\mu}\,\left({\widehat{T}}^{\mu\nu}\beta_{\nu}-\zeta{\widehat{j}}^{\mu}-\frac{1}{2}\Omega_{\lambda\nu}{\widehat{\cal
S}}^{\mu\lambda\nu}\right)\right]\,,$ (10)

and $T^{\mu\nu}_{\rm LE}(\lambda),j^{\mu}_{\rm LE}(\lambda),{\cal
S}^{\mu\lambda\nu}_{\rm LE}(\lambda)$ are calculated with the equation (5)
with the modified density operator just defined in the eq. (10). As $\lambda$
multiplies $\beta$, $\zeta$ and $\Omega$, this coefficient plays the role of a
rescaled inverse temperature, so it possible to change the integration
variable in (9) from $\lambda$ to $T^{\prime}(x)=T(x)/\lambda$ and rewrite the
thermodynamic potential current:

$\phi^{\mu}(x)=\int_{0}^{T(x)}\frac{{\rm d}T^{\prime}}{T^{\prime
2}}\;\left(T^{\mu\nu}_{\rm
LE}(x)[T^{\prime},\mu,\omega]u_{\nu}(x)-\mu(x)j^{\mu}_{\rm
LE}(x)[T^{\prime},\mu,\omega]-\frac{1}{2}\omega_{\lambda\nu}(x){\cal
S}^{\mu\lambda\nu}_{\rm LE}(x)[T^{\prime},\mu,\omega]\right)\,,$ (11)

where we used the eq. (6). The equation (11) shows that the thermodynamic
potential current can be calculated by integrating in temperature the mean
values at local thermodynamic equilibrium of the various involved currents. It
is important to stress the meaning of the square brackets, which denote a
functional dependence on the arguments. Indeed, the local equilibrium values
of the currents at some point $x$ depend not just on the value of
$T^{\prime},\mu,\omega$ at the same point $x$, but on the whole functions
$T^{\prime}(y),\mu(y),\omega(y)$; tantamount, assuming analiticity, on the
value of the functions and all their gradients at the point $x$.

Once the thermodynamic potential current $\phi^{\mu}$ is determined, an
entropy current can be defined. By using the definition (5) and the equations
(3),(8) we have:

$\begin{split}S=&-\Tr({\widehat{\rho}}_{\rm LE}\log{\widehat{\rho}}_{\rm
LE})=\log Z_{\rm LE}+\int_{\Sigma}{\rm
d}\Sigma_{\mu}\;\left(\Tr({\widehat{\rho}_{\rm
LE}}{\widehat{T}}^{\mu\nu})\beta_{\nu}-\zeta\Tr({\widehat{\rho}_{\rm
LE}}{\widehat{j}}^{\mu})-\frac{1}{2}\Omega_{\lambda\nu}\Tr({\widehat{\rho}_{\rm
LE}}{\widehat{\cal S}}^{\mu\lambda\nu})\right)\\\ =&\int_{\Sigma}{\rm
d}\Sigma_{\mu}\;\left(\phi^{\mu}+T^{\mu\nu}_{\rm LE}\beta_{\nu}-\zeta
j^{\mu}_{\rm LE}-\frac{1}{2}\Omega_{\lambda\nu}{\cal S}^{\mu\lambda\nu}_{\rm
LE}\right)\,,\end{split}$ (12)

which implies that we can define an entropy current as:

$s^{\mu}=\phi^{\mu}+T^{\mu\nu}_{\rm LE}\beta_{\nu}-\zeta j^{\mu}_{\rm
LE}-\frac{1}{2}\Omega_{\lambda\nu}{\cal S}^{\mu\lambda\nu}_{\rm LE}\,.$ (13)

##  III Entropy current: quasi-objective form and entropy-gauge
transformations

The equations (11) and (13) define the fields $\phi^{\mu}$ and $s^{\mu}$.
However, they depend not just on $x$ but also on the space-like hypersurface
employed to define the local equilibrium mean values of the currents through
the density operator (3) ${\widehat{\rho}_{\rm LE}}$. More specifically, to
each point $x$ there must be a corresponding hypersurface $\Sigma$ needed to
define local thermodynamic equilibrium through the constraints (4).
Altogether, to define the thermodynamic potential and entropy current at each
point $x$ one needs to specify in advance a family of 3D space-like
hypersurfaces, a so-called foliation of the space-time.

Figure 1: An example of a family of 3D space-like hypersurfaces (solid lines)
defining a foliation parametrized by the real variable $\tau$, which is
necessary to define local thermodynamic equilibrium. The 3D space-like
hypersurface $\Sigma$ does not belong to the foliation.

The dependence of the currents (11) and (13) on the foliation involves a
problem in that, if we are to calculate the total entropy by integrating the
entropy current (13) on some $\Sigma$ which does not belong to the foliation
(see figure 1), the result is in general different from the total entropy
which would be obtained from the Von Neumann formula imposing the constraints
of local equilibrium (4) over this particular $\Sigma$. In symbols:

$\int_{\Sigma}{\rm d}\Sigma_{\mu}\;s^{\mu}\neq-\Tr({\widehat{\rho}_{\rm
LE}}(\Sigma)\log{\widehat{\rho}_{\rm LE}}(\Sigma))\,,$ (14)

with equality applying only if $\Sigma$ belongs to the foliation. Such a
situation is quite disturbing, as one of the requested features of the entropy
current field is to provide the actual value of the total entropy.

To settle the issue, one can define the entropy current more in general as:

$s^{\mu}=\phi^{\mu}+T^{\mu\nu}\beta_{\nu}-\zeta
j^{\mu}-\frac{1}{2}\Omega_{\lambda\nu}{\cal S}^{\mu\lambda\nu}\,,$ (15)

with (omitting most arguments to make the expression more compact):

$\phi^{\mu}=\int_{0}^{T}\frac{{\rm d}T^{\prime}}{T^{\prime
2}}\;\left(T^{\mu\nu}[T^{\prime}]u_{\nu}-\mu
j^{\mu}[T^{\prime}]-\frac{1}{2}\omega_{\lambda\nu}{\cal
S}^{\mu\lambda\nu}[T^{\prime}]\right)\,.$ (16)

Indeed, in the equations (15) and (16) we have used the actual values of the
conserved currents. By doing so, whenever we integrate the currents over some
hypersurface $\Sigma$, not necessarily belonging to the original foliation,
the result is the same that we would have obtained by enforcing the
constraints (4) on $\Sigma$ itself. Since:

$\displaystyle\int_{\Sigma}{\rm d}\Sigma_{\mu}s^{\mu}$
$\displaystyle=\int_{\Sigma}{\rm
d}\Sigma\;n_{\mu}\left(\phi^{\mu}+T^{\mu\nu}\beta_{\nu}-\zeta
j^{\mu}-\frac{1}{2}\Omega_{\lambda\nu}{\cal S}^{\mu\lambda\nu}\right)$
$\displaystyle=\int_{\Sigma}{\rm d}\Sigma\;n_{\mu}\left(\int_{0}^{T}\frac{{\rm
d}T^{\prime}}{T^{\prime 2}}\;\left(T^{\mu\nu}[T^{\prime}]u_{\nu}-\mu
j^{\mu}[T^{\prime}]-\frac{1}{2}\omega_{\lambda\nu}{\cal
S}^{\mu\lambda\nu}[T^{\prime}]\right)+T^{\mu\nu}\beta_{\nu}-\zeta
j^{\mu}-\frac{1}{2}\Omega_{\lambda\nu}{\cal S}^{\mu\lambda\nu}\right)$

a local equilibrium density operator like in equation (3) can be built on
$\Sigma$ by enforcing the constraints (4) therein. Therefore, the above
expression becomes, by using the eqs. (9) and (12):

$\displaystyle\int_{\Sigma}{\rm d}\Sigma\;n_{\mu}\left(\int_{0}^{T}\frac{{\rm
d}T^{\prime}}{T^{\prime 2}}\;\left(T^{\mu\nu}_{\rm LE}[T^{\prime}]u_{\nu}-\mu
j^{\mu}_{\rm LE}[T^{\prime}]-\frac{1}{2}\omega_{\lambda\nu}{\cal
S}^{\mu\lambda\nu}_{\rm LE}[T^{\prime}]\right)+T^{\mu\nu}_{\rm
LE}\beta_{\nu}-\zeta j^{\mu}_{\rm LE}-\frac{1}{2}\Omega_{\lambda\nu}{\cal
S}^{\mu\lambda\nu}_{\rm LE}\right)$ $\displaystyle=-\Tr({\widehat{\rho}_{\rm
LE}}(\Sigma)\log{\widehat{\rho}_{\rm LE}}(\Sigma))\,.$

Similarly, the equation (8) can be extended to a relation which applies to any
space-like hypersurface $\Sigma$:

$\int{\rm d}\Sigma_{\mu}\phi^{\mu}=\log Z_{\rm
LE}(\Sigma)+\bra{0}\widehat{\Upsilon}\ket{0}$ (17)

The two equations (15) and (16) are the final expressions defining the entropy
current for a system which is close to local thermodynamic equilibrium. It is
worth remarking that those equations imply that the entropy current depends on
the actual mean values of the conserved currents ensuing from the quantum
field Lagrangian. The question is whether, with the definitions (16) and (15)
the thermodynamic potential and entropy current fields are objective, namely
independent of a predefined foliation. For this purpose, in the first place
the Lagrange multiplier fields $\beta,\zeta,\Omega$ which also appear in those
definitions should be independent thereof, a condition which is achieved in
relativistic hydrodynamics if they are obtained as solutions of partial
differential equations from given initial conditions.

Yet, a complete independence cannot be achieved. Looking carefully at the
thermodynamic potential current in the eq. (16), it appears that its
definition involves the knowledge of the conserved currents as functionals of
the temperature. However, such functionals can be constructed only if the
local equilibrium operator is introduced, hence a separation between the local
equilibrium term and the dissipative term, which does require the introduction
of a foliation. We can signify this limitation by saying that the
thermodynamic potential current, and the entropy current as well, can be made
quasi-objective. The quasi-objective nature of the entropy current also shows
up in the entropy production rate (32), as will be discussed in Section V.

A further issue is that the thermodynamic potential current and the entropy
current fields are not unique. It is quite clear that a transformation of the
thermodynamic potential current:

$\phi^{\prime\mu}=\phi^{\mu}+\nabla_{\lambda}A^{\lambda\mu}\,,$ (18)

where $A^{\lambda\mu}$ is an arbitrary anti-symmetric tensor, implying:

$s^{\prime\mu}=s^{\mu}+\nabla_{\lambda}A^{\lambda\mu}$ (19)

will leave the total entropy

$S=\int_{\Sigma}{\rm d}\Sigma_{\mu}\;s^{\mu}$

invariant because of the relativistic Stokes theorem, provided that the tensor
$A$ fulfills suitable boundary conditions. Therefore, just like $T^{\mu\nu}$
and ${\cal S}^{\mu\lambda\nu}$, the entropy current $s^{\mu}$ is not uniquely
defined and can be changed with transformations (19), henceforth defined as
entropy-gauge transformations. Such a freedom in defining the entropy current
affects the local thermodynamic relations, as we will see. Nevertheless, the
divergence of the entropy current is invariant under pseudo-gauge
transformations because:

$\nabla_{\mu}s^{\prime\mu}=\nabla_{\mu}s^{\mu}+\nabla_{\mu}\nabla_{\lambda}A^{\lambda\mu}=\nabla_{\mu}s^{\mu}\,.$

The entropy production rate will be discussed in Section V.

##  IV Discussion on local thermodynamic relations

The local thermodynamic relation between proper densities can be obtained by
contracting the entropy current with a suitable four-velocity vector. For
instance, one can contract the (15) with the four-velocity defined by the
direction of $\beta$ that is
$u^{\mu}=\beta^{\mu}/\sqrt{\beta^{2}}=T\beta^{\mu}$ 222Note that if one
contracts the (15) with the normalized time-like eigenvector of the stress-
energy tensor $u_{L}$, which defines with the Landau frame, the obtained LTR
reads:  $s_{L}=s^{\mu}u_{L\mu}=\phi\cdot u_{L}+u_{L}\cdot\beta\rho_{L}-\zeta
n_{L}-\frac{1}{2}\Omega_{\lambda\nu}u_{L\mu}{\cal
S}^{\mu\lambda\nu}\equiv\phi\cdot u_{L}+u_{L}\cdot\beta\rho_{L}-\zeta
n_{L}-\frac{1}{2}\Omega_{\lambda\nu}S^{\lambda\nu}_{L}$ (20) Since
$u_{L}\cdot\beta\neq\sqrt{\beta^{2}}=1/T$, it turns out that, even if the
entropy current was quasi-objective, the LTR is frame-dependent [26] and much
care should be taken when using it to derive constitutive equations.:

$s\equiv s^{\mu}u_{\mu}=\phi\cdot u+\frac{1}{T}\rho-\zeta
n-\frac{1}{2}\Omega_{\lambda\nu}u_{\mu}{\cal S}^{\mu\lambda\nu}\equiv\phi\cdot
u+\frac{1}{T}\rho-\frac{\mu}{T}n-\frac{1}{2}\Omega_{\lambda\nu}S^{\lambda\nu}\,,$
(21)

where $\rho=u_{\mu}u_{\nu}T^{\mu\nu}$ and $n=u_{\mu}j^{\mu}$. Defining the
pressure as:

$p\equiv T\phi\cdot u\,,$

the eq. (21) coincides with the first thermodynamic relation in eq. (2). It
should be pointed out though, that only at global equilibrium with $\beta={\rm
const}$ this quantity coincides with the hydrostatic pressure, that is the
diagonal spatial component of the mean value of the stress-energy tensor (see
Appendix A); in all other cases, it does not need to. By contracting the eq.
(16) with $u=\beta/\sqrt{\beta^{2}}$ we obtain:

$\begin{split}p=T\phi\cdot u&=T\int_{0}^{T}\frac{{\rm d}T^{\prime}}{T^{\prime
2}}\;\left(u_{\mu}T^{\mu\nu}[T^{\prime}]u_{\nu}-\mu
u_{\mu}j^{\mu}[T^{\prime}]-\frac{1}{2}\omega_{\lambda\nu}u_{\mu}{\cal
S}^{\mu\lambda\nu}[T^{\prime}]\right)\\\ &=T\int_{0}^{T}\frac{{\rm
d}T^{\prime}}{T^{\prime 2}}\;\left(\rho[T^{\prime}]-\mu
n_{[}T^{\prime}]-\frac{1}{2}\omega_{\lambda\nu}S^{\lambda\nu}[T^{\prime}]\right)\,,\end{split}$
(22)

whence the following relation can be readily obtained:

$\frac{\partial p}{\partial T}\Big{|}_{\mu,\omega}=s$ (23)

by using the (21). This equation is the first step in proving the second
relation (2), but in fact the remaining two partial derivative of the pressure
function do not need to coincide with the charge density and the spin density
and in general:

$\frac{\partial p}{\partial\mu}\Big{|}_{T,\omega}\neq
n\,,\qquad\qquad\frac{\partial
p}{\partial\omega_{\lambda\nu}}\Big{|}_{T,\mu}\neq S^{\lambda\nu}\,.$

Indeed, for the equality to apply, one would need the following thermodynamic
relation to hold:

$T{\rm d}s={\rm d}\rho-\mu\,{\rm d}n-\frac{1}{2}\omega_{\lambda\nu}{\rm
d}S^{\lambda\nu}\,,$ (24)

and yet this cannot be obtained from the definitions (21) and (16).

Furthermore, the relation (23) is not invariant under entropy-gauge
transformations. The thermodynamic potential current can be redefined
according to the (18) and, contracting with the four-velocity we get:

$p^{\prime}=T\phi^{\prime}\cdot u=p+Tu_{\mu}\nabla_{\lambda}A^{\lambda\mu}\,,$

where the transformed quantities are denoted with a prime. It is then easy to
show that:

$\frac{\partial p^{\prime}}{\partial
T}\Big{|}_{\mu\omega}=s^{\prime}+u_{\mu}T\frac{\partial}{\partial
T}\nabla_{\lambda}A^{\lambda\mu}\Big{|}_{\mu\omega}\,.$ (25)

If the second term on the right hand side is non-vanishing, even the relation
(23) is broken. An example of an entropy-gauge transformation which breaks the
(23) is shown in Appendix C for the global equilibrium with rotation.

In conclusion, the local thermodynamic relations (2) are not fully appropriate
in the derivation of the divergence of the entropy current. On one hand, it
turns out that the differential relation in (2) cannot be proved in general
and on the other hand, perhaps most importantly, because they are both non-
invariant under entropy-gauge transformations.

##  V Entropy production rate

The entropy production rate, which is important to obtain the constitutive
equations of relativistic hydrodynamics, is determined by taking the
divergence of the equation (15). By using the continuity equations of the
stress-energy tensor, the number current and the spin tensor, that is:

$\nabla_{\mu}T^{\mu\nu}=0\,,\qquad\nabla_{\mu}j^{\mu}=0\,,\qquad\nabla_{\mu}{\cal
S}^{\mu\lambda\nu}=T^{\nu\lambda}-T^{\lambda\nu}\,,$ (26)

we obtain:

$\displaystyle\begin{split}\nabla_{\mu}s^{\mu}&=\nabla_{\mu}\phi^{\mu}+T^{\mu\nu}\nabla_{\mu}\beta_{\nu}-j^{\mu}\nabla_{\mu}\zeta-\frac{1}{2}{\cal
S}^{\mu\lambda\nu}\nabla_{\mu}\Omega_{\lambda\nu}-\frac{1}{2}\Omega_{\lambda\nu}\nabla_{\mu}{\cal
S}^{\mu\lambda\nu}\\\
&=\nabla_{\mu}\phi^{\mu}+T_{S}^{\mu\nu}\xi_{\mu\nu}-j^{\mu}\nabla_{\mu}\zeta+T_{A}^{\mu\nu}(\Omega_{\mu\nu}-\varpi_{\mu\nu})-\frac{1}{2}{\cal
S}^{\mu\lambda\nu}\nabla_{\mu}\Omega_{\lambda\nu}\,,\end{split}$ (27)

where $T_{S}$ and $T_{A}$ are the symmetric and anti-symmetric parts of the
stress-energy tensor and

$\xi_{\mu\nu}=\frac{1}{2}\left(\nabla_{\mu}\beta_{\nu}+\nabla_{\nu}\beta_{\mu}\right)\,,\qquad\varpi_{\mu\nu}=\frac{1}{2}\left(\nabla_{\nu}\beta_{\mu}-\nabla_{\mu}\beta_{\nu}\right)$

are the thermal shear and thermal vorticity tensor respectively.

Figure 2: The hypersurface $\Sigma$ is mapped to $\Sigma_{\epsilon}$ by an
infinitesimal diffeomorphism $\epsilon$. The vector field $\delta(x)$ is
defined as ${\rm d}x^{\prime}(x,\epsilon)/{\rm d}\epsilon|_{\epsilon=0}$.

The next step, as it appears from the equation (27), is the calculation of the
divergence of the thermodynamic potential current, $\nabla_{\mu}\phi^{\mu}$.
To derive it, it is convenient to study the change of $\log Z_{\rm LE}$ under
an infinitesimal change of the integration 3D hypersurface (see figure 2). An
infinitesimal change of hypersurface may be seen, in simple terms, as the
result of locally moving every point $x\in\Sigma$ to a point
$x^{\prime}(x,\epsilon)\in\Sigma_{\epsilon}$, $\epsilon$ being a finite real
parameter. Setting $x^{\prime}(x,0)=x$, we define:

$\frac{{\rm d}x^{\prime\mu}(x,\epsilon)}{{\rm
d}\epsilon}\Big{|}_{\epsilon=0}=\delta^{\mu}(x)\,.$

For a small $\epsilon$, the vector field $\delta$ loosely represents the
direction in which the hypersurface is locally modified and the parameter
$\epsilon$ describes how far along the vector field $\delta$ we move the
hypersurface. Formally, these definitions are those of a one-parameter group
of diffeomorphisms, which are a prerequisite to define the Lie derivative. For
the special case of the integration of a vector field $V^{\mu}$ over a
3D-hypersurface, one has (see Appendix B):

$\displaystyle\lim_{\epsilon\to\
0}\frac{1}{\epsilon}\left(\int_{\Sigma_{\epsilon}}{\rm
d}\Sigma_{\mu}V^{\mu}-\int_{\Sigma}{\rm
d}\Sigma_{\mu}\,V^{\mu}\right)=\int_{\partial\Sigma}{\rm
d}\tilde{S}_{\mu\nu}\,\delta^{\mu}V^{\nu}+\int_{\Sigma}{\rm
d}\Sigma\cdot\delta\,\nabla_{\mu}V^{\mu}\,,$ (28)

where $\partial\Sigma$ is the 2-D boundary surface. We can apply this equation
to the (17) to obtain the infinitesimal change of $\log Z_{\rm LE}$ by a
change of the hypersurface:

$\displaystyle\lim_{\epsilon\to\ 0}\frac{1}{\epsilon}\left[\log Z_{\rm
LE}(\Sigma_{\epsilon})-\log Z_{\rm LE}(\Sigma)\right]=\int_{\Sigma}{\rm
d}\Sigma\cdot\delta\,\nabla_{\mu}\left(\phi^{\mu}-\bra{0}{\widehat{T}}^{\mu\nu}\beta_{\nu}-\zeta{\widehat{j}}^{\mu}-\frac{1}{2}\Omega_{\lambda\nu}{\widehat{\cal
S}}^{\mu\lambda\nu}\ket{0}\right)$ $\displaystyle=\int_{\Sigma}{\rm
d}\Sigma\cdot\delta\,\nabla_{\mu}\phi^{\mu}-\int_{\Sigma}{\rm
d}\Sigma\cdot\delta\,\bra{0}{\widehat{T}}_{S}^{\mu\nu}\xi_{\mu\nu}-{\widehat{j}}^{\mu}\nabla_{\mu}\zeta+{\widehat{T}}_{A}^{\mu\nu}(\Omega_{\mu\nu}-\varpi_{\mu\nu})-\frac{1}{2}{\widehat{\cal
S}}^{\mu\lambda\nu}\nabla_{\mu}\Omega_{\lambda\nu}\ket{0}\,,$ (29)

where, in the last step, we have used the continuity equations (26), holding
at operator level. On the other hand, the logarithm of the partition function
can be calculated by means of its definition as a trace. For an infinitesimal
$\epsilon$ one has:

$\displaystyle Z_{\rm
LE}(\Sigma_{\epsilon})=\Tr\left(\exp\left[-\int_{\Sigma_{\epsilon}}{\rm
d}\Sigma_{\mu}\left({\widehat{T}}^{\mu\nu}\beta_{\nu}-\zeta{\widehat{j}}^{\mu}-\frac{1}{2}\Omega_{\lambda\nu}{\widehat{\cal
S}}^{\mu\lambda\nu}\right)\right]\right)$
$\displaystyle\simeq\Tr\left(\exp\left[-\int_{\Sigma}{\rm
d}\Sigma_{\mu}\left({\widehat{T}}^{\mu\nu}\beta_{\nu}-\zeta{\widehat{j}}^{\mu}-\frac{1}{2}\Omega_{\lambda\nu}{\widehat{\cal
S}}^{\mu\lambda\nu}\right)-\epsilon\int_{\Sigma}{\rm
d}\Sigma\cdot\delta\nabla_{\mu}\left({\widehat{T}}^{\mu\nu}\beta_{\nu}-\zeta{\widehat{j}}^{\mu}-\frac{1}{2}\Omega_{\lambda\nu}{\widehat{\cal
S}}^{\mu\lambda\nu}\right)\right]\right)$
$\displaystyle=\Tr\left(\exp\left[-\int_{\Sigma}{\rm
d}\Sigma_{\mu}\left({\widehat{T}}^{\mu\nu}\beta_{\nu}-\zeta{\widehat{j}}^{\mu}-\frac{1}{2}\Omega_{\lambda\nu}{\widehat{\cal
S}}^{\mu\lambda\nu}\right)\right.\right.$
$\displaystyle-\left.\left.\epsilon\int_{\Sigma}{\rm
d}\Sigma\cdot\delta\;\left({\widehat{T}}_{S}^{\mu\nu}\xi_{\mu\nu}-{\widehat{j}}^{\mu}\nabla_{\mu}\zeta+{\widehat{T}}_{A}^{\mu\nu}(\Omega_{\mu\nu}-\varpi_{\mu\nu})-\frac{1}{2}{\widehat{\cal
S}}^{\mu\lambda\nu}\nabla_{\mu}\Omega_{\lambda\nu}\right)\right]\right)\,,$

where we have used the equation (28) - assuming that the boundary term
vanishes - and, again, the continuity equations (26) at operator level. By
expanding the trace in the small parameter $\epsilon$, and keeping in mind the
equation (3), we obtain:

$\displaystyle Z_{\rm LE}(\Sigma_{\epsilon})$ $\displaystyle\simeq Z_{\rm
LE}(\Sigma)-\epsilon Z_{\rm LE}(\Sigma)$ $\displaystyle\times\int_{\Sigma}{\rm
d}\Sigma\cdot\delta\;\left(\Tr({\widehat{\rho}_{\rm
LE}}{\widehat{T}}_{S}^{\mu\nu})\xi_{\mu\nu}-\Tr({\widehat{\rho}_{\rm
LE}}{\widehat{j}}^{\mu})\nabla_{\mu}\zeta+\Tr({\widehat{\rho}_{\rm
LE}}{\widehat{T}}_{A}^{\mu\nu})(\Omega_{\mu\nu}-\varpi_{\mu\nu})-\frac{1}{2}\Tr({\widehat{\rho}_{\rm
LE}}{\widehat{\cal
S}}^{\mu\lambda\nu})\nabla_{\mu}\Omega_{\lambda\nu}\right)\,,$

whence:

$\displaystyle\lim_{\epsilon\to\ 0}\frac{1}{\epsilon}\left[\log Z_{\rm
LE}(\Sigma_{\epsilon})-\log Z_{\rm LE}(\Sigma)\right]$ (30)
$\displaystyle=-\int_{\Sigma}{\rm
d}\Sigma\cdot\delta\;\left(\Tr({\widehat{\rho}_{\rm
LE}}{\widehat{T}}_{S}^{\mu\nu})\xi_{\mu\nu}-\Tr({\widehat{\rho}_{\rm
LE}}{\widehat{j}}^{\mu})\nabla_{\mu}\zeta+\Tr({\widehat{\rho}_{\rm
LE}}{\widehat{T}}_{A}^{\mu\nu})(\Omega_{\mu\nu}-\varpi_{\mu\nu})-\frac{1}{2}\Tr({\widehat{\rho}_{\rm
LE}}{\widehat{\cal
S}}^{\mu\lambda\nu})\nabla_{\mu}\Omega_{\lambda\nu}\right)\,.$

Therefore, by comparing the equation (V) with the equation (30), taking into
account that both $\Sigma$ and the field $\delta$ are arbitrary, we can infer
that:

$\displaystyle\nabla_{\mu}\phi^{\mu}$
$\displaystyle=-\left[\left(\Tr({\widehat{\rho}_{\rm
LE}}{\widehat{T}}_{S}^{\mu\nu})-\bra{0}{\widehat{T}}_{S}^{\mu\nu}\ket{0}\right)\xi_{\mu\nu}-\left(\Tr({\widehat{\rho}_{\rm
LE}}{\widehat{j}}^{\mu})-\bra{0}{\widehat{j}}^{\mu}\ket{0}\right)\nabla_{\mu}\zeta\right.$
$\displaystyle+\left.\left(\Tr({\widehat{\rho}_{\rm
LE}}{\widehat{T}}_{A}^{\mu\nu})-\bra{0}{\widehat{T}}_{A}^{\mu\nu}\ket{0}\right)(\Omega_{\mu\nu}-\varpi_{\mu\nu})-\frac{1}{2}\left(\Tr({\widehat{\rho}_{\rm
LE}}{\widehat{\cal S}}^{\mu\lambda\nu})-\bra{0}{\widehat{\cal
S}}^{\mu,\lambda\nu}\ket{0}\right)\nabla_{\mu}\Omega_{\lambda\nu}\right]$
$\displaystyle=-\left(T^{\mu\nu}_{S\rm(LE)}\xi_{\mu\nu}-j^{\mu}_{\rm
LE}\nabla_{\mu}\zeta+T^{\mu\nu}_{A\rm(LE)}(\Omega_{\mu\nu}-\varpi_{\mu\nu})-\frac{1}{2}{\cal
S}^{\mu\lambda\nu}_{\rm LE}\nabla_{\mu}\Omega_{\lambda\nu}\right)\,,$ (31)

where, in the last step, we have used the definition of local equilibrium
values.

Now, substituting back the eq. (V) into the eq. (27), we obtain the evolution
of entropy current:

$\nabla_{\mu}s^{\mu}=\left(T_{S}^{\mu\nu}-T^{\mu\nu}_{S(\rm
LE)}\right)\xi_{\mu\nu}-\left(j^{\mu}-j^{\mu}_{\rm
LE}\right)\nabla_{\mu}\zeta+\left(T^{\mu\nu}_{A}-T^{\mu\nu}_{A(\rm
LE)}\right)(\Omega_{\mu\nu}-\varpi_{\mu\nu})-\frac{1}{2}\left({\cal
S}^{\mu\lambda\nu}-{\cal S}^{\mu\lambda\nu}_{\rm
LE}\right)\nabla_{\mu}\Omega_{\lambda\nu}\,.$ (32)

The equation (32) is the main result of this work and it is the starting point
to derive the constitutive equations of dissipative spin hydrodynamics, which
relate the anti-symmetric part of the stress-energy tensor and the spin tensor
to the gradients of the spin potential and the difference between spin
potential and thermal vorticity, besides the (thermal) shear tensor and the
gradient of $\zeta=\mu/T$. In the above form, it is in fact a generalization
of the one found by Van Weert and Zubarev [24, 25], with the addition of the
last two terms involving spin tensor and the spin potential. We stress that
the formula (32) is exact and not an approximation at some order of a gradient
expansion. Indeed, with respect to all previous assessments of dissipative
spin hydrodynamics based on different approaches [7, 8, 9, 10, 11, 12, 13], a
novel feature is apparently the simultaneous appearance of the last two terms
of the right hand side. While the last term is neglected in almost all
derivations, it was actually obtained in ref. [10]. However, it should be
pointed out that some terms in previous derivations may have been omitted
because of a gradient power counting method. A complete analysis of the
constitutive equations implied by the (32) will be presented in a forthcoming
study.

##  VI Discussion and conclusions

The formula (32) shows that entropy production rate, in general, is non-
vanishing whenever there is a difference between the actual value of the
conserved (or conserved-related) currents and the corresponding values at
local thermodynamic equilibrium, such as $T_{S(\rm LE)}$, $j_{\rm LE}$, etc.
As we have emphasized in this paper, local equilibrium depends on the choice
of a family of 3D space-like hypersurfaces, i.e. a foliation. In relativistic
hydrodynamics, this freedom ultimately corresponds to the choice of a four-
velocity vector, so-called hydrodynamic frame. The dependence on the foliation
shows up in the divergence of the entropy current (32), which is manifestly
dependent on local equilibrium values (see the discussion at the end of
Section III).

We emphasize that the formula (32) is exact, not an approximation at some
order of a gradient expansion. In other words, fixing the order in a gradient
expansion of hydrodynamic quantities is not required to obtain it. However,
for future work, once constitutive equations are determined, a gradient
ordering can be made based on the involved scales in the physical problem.

In conclusion, in this work we have employed a quantum-statistical approach to
derive the entropy current and entropy production rate without assuming the
traditional local thermodynamic relations (2). In fact, we have shown that the
local thermodynamic relations do not hold in general and that they are also
non-invariant under allowed transformations of the entropy current, that we
have defined as entropy-gauge transformations. We have obtained an expression
of the entropy production rate (32) which extends to spin hydrodynamics
previous expression obtained in refs. [24, 25]. This form is especially well-
suited to derive the constitutive equations of dissipative spin hydrodynamics,
what will be the subject of a forthcoming work.

## Acknowledgements

Part of this work was carried out in the workshop ”The many faces of
relativistic fluid dynamics” held in the Kavli institute in Santa Barbara (CA)
USA, supported in part by the National Science Foundation under Grants No. NSF
PHY-1748958 and PHY-2309135. F.B. gratefully acknowledges fruitful discussions
with the participants in the workshop, especially J. Armas, G. Denicol, P.
Kovtun and M. Hippert Teixeira. Interesting discussions with R. Ryblewski, E.
Grossi and A. Giachino are also acknowledged. A.D. thanks the Department of
Physics, University of Florence and INFN for the hospitality. A.D.
acknowledges the financial support provided by the Polish National Agency for
Academic Exchange NAWA under the Programme STER–Internationalisation of
doctoral schools, Project no. PPI/STE/2020/1/00020 and the Polish National
Science Centre Grants No.2018/30/E/ST2/00432.

##  Appendix A Thermodynamic potential current at homogeneous global
equilibrium

Homogeneous global equilibrium is defined by the condition
$\beta=\mbox{const}$ i.e. vanishing thermal vorticity in the equation (7).
Plugging this form in the equation (3), the density operator takes on the
familiar form (for simplicity, we assume there are no charges in the system):

$\displaystyle{\widehat{\rho}}_{\rm
GE}=\frac{1}{Z}\exp\left[-\beta\cdot{\widehat{P}}\right]\,.$ (33)

Due to the symmetries of the above operator, the mean value of the stress-
energy tensor operator has the ideal form:

$T^{\mu\nu}=\Tr({\widehat{\rho}}_{\rm
GE}{\widehat{T}}^{\mu\nu})-\bra{0}{\widehat{T}}^{\mu\nu}\ket{0}=(\rho+p)u^{\mu}u^{\nu}-pg^{\mu\nu}\,,$
(34)

where $u\equiv\beta/{\sqrt{\beta^{2}}}$. According to eq. (16), the
thermodynamic potential current is:

$\displaystyle\phi^{\mu}=\int_{0}^{T}\frac{{\rm d}T^{\prime}}{T^{\prime
2}}\;T^{\mu\nu}[T^{\prime}]u_{\nu}=\int_{0}^{T}\frac{{\rm
d}T^{\prime}}{T^{\prime 2}}\;\rho[T^{\prime}]u^{\mu}\,,$ (35)

where $T=1/\sqrt{\beta^{2}}$. The above expression confirms the expectation
that, at the homogeneous global equilibrium, any vector field should be
parallel to $\beta$ with a coefficient depending on $\beta^{2}$ or,
equivalently, the temperature $T$. Therefore:

$\phi^{\mu}=\beta^{\mu}\phi(\beta^{2})$ (36)

and the goal is now to show that such scalar coefficient $\phi(\beta^{2})$ is
just the pressure, as defined by the equation (34).

By taking the derivative with respect to $\beta$ of the partition function, we
have:

$-\frac{\partial\log Z}{\partial\beta_{\nu}}=\Tr({\widehat{\rho}}_{\rm
GE}{\widehat{P}}^{\nu})=\int_{\Sigma}{\rm
d}\Sigma_{\mu}\;\Tr({\widehat{\rho}}_{\rm GE}{\widehat{T}}^{\mu\nu})\,.$ (37)

Since:

$\log Z=\int{\rm
d}\Sigma_{\mu}\;\phi^{\mu}-\beta_{\nu}\bra{0}{\widehat{P}}^{\nu}\ket{0}$

from the (8), from the (37) we can obtain the following equality

$\displaystyle\int_{\Sigma}{\rm
d}\Sigma_{\mu}\left(\frac{\partial\phi^{\mu}}{\partial\beta_{\nu}}\right)=-\int_{\Sigma}{\rm
d}\Sigma_{\mu}T^{\mu\nu}\,,$ (38)

where we have used the (34). Since the integration hypersurface $\Sigma$ is
arbitrary, being at global equilibrium, we can infer the following relation:

$T^{\mu\nu}=-\frac{\partial\phi^{\mu}}{\partial\beta_{\nu}}+\partial_{\lambda}A^{\lambda\mu\nu}\,,$

where the rank 3 tensor $A^{\lambda\mu\nu}$ is anti-symmetric in the indices
$\lambda\mu$. Such a gradient term is allowed by the Stokes theorem in
Minkowski space-time if suitable boundary conditions are fulfilled. Yet, since
the equilibrium is homogeneous, it must vanish due to traslational invariance
as all mean values ought to be constant and uniform. This implies that, by
using the equation (36):

$\displaystyle
T^{\mu\nu}=-\frac{\partial\phi^{\mu}}{\partial\beta_{\nu}}=-\frac{\partial}{\partial\beta_{\nu}}\phi\beta^{\mu}=-\phi
g^{\mu\nu}+T\frac{\partial\phi}{\partial T}u^{\mu}u^{\nu}\,.$ (39)

We can now compare (34) with (39) and infer that $\phi=p$ and consequently
$\phi^{\mu}=p\beta^{\mu}$. By plugging the latter equation in the (35) and
taking the derivative with respect to $T$ we obtain:

$T\frac{\partial p}{\partial T}=\rho+p\,,$

which makes also the second term on the right hand side of equation (39)
consistent with the identification $\phi=p$.

##  Appendix B Lie derivatives and integration

Suppose we have a one-parameter group of diffeomorphisms
$x^{\prime}(x,\epsilon)$ with $\epsilon$ a real number. Let $\omega$ be a rank
3 differential form which is to be integrated over a 3D hypersurface embedded
in the 4D space-time. We denote with $\omega^{\prime}_{\epsilon}$ the
differential form which is obtained from $\omega$ through the diffeomorphism,
that is:

$\omega^{\prime}_{\epsilon}(x)_{\mu_{1}\mu_{2}\mu_{3}}=J_{\mu_{1}}^{\nu_{1}}J_{\mu_{2}}^{\nu_{2}}J_{\mu_{3}}^{\nu_{3}}\;\omega(x^{\prime}(x,\epsilon))_{\nu_{1},\nu_{2},\nu_{3}}$

where $J_{\mu}^{\nu}=\partial x^{\prime}(x,\epsilon)^{\nu}/\partial x^{\mu}$
is the jacobian matrix element of the diffeomorphism. Let $\Sigma_{\epsilon}$
be the image of the hypersurface $\Sigma$ through the diffeomorphism. Then we
have:

$\int_{\Sigma_{\epsilon}}\omega=\int_{\Sigma}\omega^{\prime}_{\epsilon}$

whence:

$\displaystyle\lim_{\epsilon\to
0}\frac{1}{\epsilon}\left(\int_{\Sigma_{\epsilon}}\omega(x)-\int_{\Sigma}\omega(x)\right)=\lim_{\epsilon\to
0}\frac{1}{\epsilon}\int_{\Sigma}\omega^{\prime}_{\epsilon}(x)-\omega(x)=\int_{\Sigma}{\cal
L}_{\delta}(\omega(x))\,,$

where ${\cal L}_{\delta}$ stands for the Lie derivative along the vector field
$\delta(x)={\rm d}x^{\prime}(x,\epsilon)/{\rm d}\epsilon|_{\epsilon=0}$.

The so-called Cartan magic formula can now be used in the last expression,
leading to:

$\int_{\Sigma}{\cal L}_{\delta}(\omega)=\int_{\Sigma}i_{\delta}{\rm
d}\omega+{\rm d}(i_{\delta}\omega)=\int_{\Sigma}i_{\delta}{\rm
d}\omega+\int_{\partial\Sigma}i_{\delta}\omega\,,$ (40)

where $i_{\delta}$ stands for the interior product of the form with the vector
field $\delta$ and ${\rm d}$ stands for the exterior derivative. The second
term on the right hand side of (40) is an integral of an exterior derivative
and it has been turned into a 2D boundary integral of $i_{\delta}\omega$ by
using the generalized Stokes theorem for differential forms.

We can apply the above formulae to the differential form which is the dual of
a vector field $V$ in a 4D space-time, namely:

$\omega_{\mu\nu\rho}=\frac{1}{6}E_{\mu\nu\rho\sigma}V^{\sigma}=\frac{1}{6}\sqrt{|g|}\epsilon_{\mu\nu\rho\sigma}V^{\sigma}\,.$
(41)

With this form, it can be shown that:

$\int_{\Sigma}\omega(x)=\int_{\Sigma}{\rm d}\Sigma_{\mu}V^{\mu}\,.$ (42)

The exterior derivative can be readily worked out by using the definition:

$({\rm
d}\omega)_{\lambda\mu\nu\rho}=-\frac{1}{24}E_{\lambda\mu\nu\rho}\nabla\cdot
V\,,$

which leads, by using the definition of interior product, to:

$(i_{\delta}{\rm
d}\omega)_{\mu\nu\rho}=\frac{1}{6}E_{\mu\nu\rho\sigma}\delta^{\sigma}\nabla\cdot
V\,.$

Therefore, by using the above expression and the (42) we get:

$\int_{\Sigma}i_{\delta}{\rm d}\omega=\int_{\Sigma}{\rm
d}\Sigma_{\mu}\,\delta^{\mu}\nabla\cdot V\,.$

The second integral in the (40) can be similarly worked out and one eventually
obtains the equation (28).

##  Appendix C Non-invariance of the local thermodynamic relations: an example

We are going to show that the local thermodynamic relation (23) is not
invariant under entropy-gauge transformation (18), namely that the equation
(25) applies with a non-trivial second term in the right hand side.

We consider, as specific example, global equilibrium with non-vanishing
thermal vorticity in the equation (7). Let:

$A^{\lambda\mu}=f(\kappa^{2})\varpi^{\lambda\mu}\,,$

where $\kappa^{\mu}=\varpi^{\mu\nu}\beta_{\nu}$ and
$f(\kappa^{2})=g(\kappa^{2})/\kappa^{2}$ with $g(\kappa^{2})$ an adimensional
differentiable function (this form of $f(\kappa^{2})$ ensures that
$A^{\lambda\mu}$ has the correct dimension for the entropy gauge
transformation (18)). One has, in Cartesian coordinates:

$\nabla_{\lambda}A^{\lambda\mu}=\partial_{\lambda}A^{\lambda\mu}=f^{\prime}(\kappa^{2})\varpi^{\lambda\mu}\partial_{\lambda}\kappa^{2}=f^{\prime}(\kappa^{2})\varpi^{\lambda\mu}2\kappa^{\nu}\partial_{\lambda}\kappa_{\nu}=f^{\prime}(\kappa^{2})\varpi^{\lambda\mu}2\kappa^{\nu}\partial_{\lambda}(\varpi_{\nu\rho}\beta^{\rho})=f^{\prime}(\kappa^{2})\varpi^{\lambda\mu}2\kappa^{\nu}\varpi_{\nu\rho}\varpi^{\rho}_{\;\lambda}\,,$

where, in the last step, we have used the relation
$\varpi_{\mu\nu}=\partial_{\nu}\beta_{\mu}$ which applies at global
equilibrium where $\partial_{\mu}\beta_{\nu}+\partial_{\nu}\beta_{\mu}=0$.

Now let $\gamma^{\rho}=\varpi^{\rho\nu}\kappa_{\nu}$ so that:

$\partial_{\lambda}A^{\lambda\mu}=-2f^{\prime}(\kappa^{2})\gamma^{\rho}\varpi_{\rho\lambda}\varpi^{\lambda\mu}\,.$
(43)

Contracting the equation (43) with $u_{\mu}$ we get:

$u_{\mu}\partial_{\lambda}A^{\lambda\mu}=T\beta_{\mu}\partial_{\lambda}A^{\lambda\mu}=-2f^{\prime}(\kappa^{2})T\gamma^{\rho}\varpi_{\rho\lambda}\varpi^{\lambda\mu}\beta_{\mu}=-2f^{\prime}(\kappa^{2})T\gamma^{\rho}\varpi_{\rho\lambda}\kappa^{\lambda}=-2f^{\prime}(\kappa^{2})T\gamma^{2}\,.$
(44)

The derivative in (25) must be taken by keeping $\omega=T\varpi$ constant.
Therefore, being:

$\kappa^{\mu}=\varpi^{\mu\nu}\beta_{\nu}=\frac{1}{T^{2}}\omega^{\mu\nu}u_{\nu}\,,\qquad\qquad\gamma^{\rho}=\varpi^{\rho\nu}\kappa_{\nu}=\frac{1}{T^{3}}\omega^{\rho\nu}\omega_{\nu\alpha}u^{\alpha}\,,$

and choosing $g(\kappa^{2})=1$, we have that the expression in the equation
(44) is proportional to $T^{3}$,

$T\frac{\partial}{\partial
T}\left(u_{\mu}\partial_{\lambda}A^{\lambda\mu}\right)=T\frac{2}{(\kappa^{2})^{2}}\gamma^{2}\,,$
(45)

which is non-vanishing. Therefore, using the (45) in the equation (25) we get:

$\frac{\partial p^{\prime}}{\partial
T}\Big{|}_{\mu\omega}=s^{\prime}+T\frac{2}{(\kappa^{2})^{2}}\gamma^{2}\,,$

which proves the non-invariance of the local thermodynamic relation.

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