# Lorentz-invariant topological structures of the electromagnetic field
in a Fabry-Perot resonant slit-grating

Marina Yakovleva<EMAIL_ADDRESS>Université Paris-Saclay,
CNRS, Centre de Nanosciences et de Nanotechnologies, 91120, Palaiseau, France.
Jean-Luc Pelouard  Université Paris-Saclay, CNRS, Centre de Nanosciences et de
Nanotechnologies, 91120, Palaiseau, France.  Fabrice Pardo
<EMAIL_ADDRESS>Université Paris-Saclay, CNRS, Centre de
Nanosciences et de Nanotechnologies, 91120, Palaiseau, France.

###### Abstract

It is commonly assumed that the most correct description of the
electromagnetic world is the abstract one, and that topological constructs
such as lines of force are not covariant. In the present paper, we show that
for a $y$-invariant system with a $p$-polarized electromagnetic field, it is
possible to construct absolute (i.e. Lorentz invariant) lines, which we call
_electric spaghettis_ (ESs). The electromagnetic field is fully described by
the ES topology that transcend the limit between space and time, plus a new
invariant, the characteristic parameter $\eta$. In a Fabry-Perot resonant
slit-grating, three ES patterns can be distinguished, corresponding to three
regions: null straight lines in plane wave regions, rounded rhombuses in the
interference region inside the grating, and Bernoulli’s logarithmic spiral
patterns — the first ever described fractals — in the funneling region.

Faraday’s concept of lines of force [1, 2] was the first topology describing
the electromagnetic field. However, with our modern point of view, this
topological construct is not Lorentz invariant, and it is commonly assumed
that the most correct description of the electromagnetic field is the abstract
one [3, 4]. Nevertheless, few attempts to define a spacetime topology of the
electromagnetic field have been made. The idea of tubes of flux, a spacetime
generalization of Maxwell’s concept [2], was described for example by Misner,
Thorne and Wheeler in their book Gravitation [5]. Unfortunately it only leads
to a schematic presentation and does not construct well-defined spacetime
topological structures. A frequent approach of field 2-forms is to separate
space and time [6, 7], losing the Lorentz invariance. Absolute representations
of 2-forms were given only for simple cases, by Jancewicz for an electric
charge [8] and by Warnick and Russer for the case of a propagating plane wave
[9].

In an attempt to construct a Lorentz-invariant topology of the electromagnetic
field, Gratus [10] has suggested to consider fixed-coordinate space slices of
the 4D spacetime. A space slice, being arbitrary in the general case, is
canonical for a $y$-invariant $p$-polarized electromagnetic field. In this
case, the Maxwell’s 2-form becomes a simple 2-form, and its pullback onto any
arbitrary $y=\mathrm{Constant}$ plane defines lines in the 3D spacetime. There
is similarity between these lines and Faraday’s lines of force for a static
field in 3D space. The topology by itself defines at any point (or event) two
directional parameters, and a third parameter is necessary to completely
describe the field, either the value of the electrostatic potential, or the
field amplitude (that can be illustrated by the density of lines). However,
this field description by a topology plus a parameter cannot be
straightforwardly generalized from 3D space to 3D spacetime, mainly because of
the peculiarity of null-like directions in the latter. In the absence of
charge, it is evident that field lines never end in 3D space, but this is not
an established fact in 3D spacetime.

In this paper, we show that a $y$-invariant, $p$-polarized electromagnetic
field can be fully described by an absolute topology of lines in spacetime
(named here the _electric spaghettis_ (ESs)), with a new Lorentz-invariant
scalar measure on them, the $\eta$ parameter. This topology is studied in
detail for an infinite line of charge, and for a Fabry-Perot resonant slit
grating. The latter exhibits three specific topological ES structures: the
single plane wave region high above the grating, the two interfering plane
wave regions inside the grating, and the funnel region in the near field of
the grating. In the funneling region a fractal topological structure around
null field events is observed. These endless ES whirls, which seem to
contradict the absence of magnetic charge [10], are justified here by a
careful analysis.

For $y$-invariant $p$-polarized field, the 6 components of Faraday’s
$F=(\bm{-E},\bm{B})$ or Maxwell’s $\mathcal{G}=(\bm{H},\bm{D})$ 2-forms [11]
reduce to 3 components, with

$\mathcal{G}=H_{y}dt\wedge dy+D_{x}dy\wedge dz+D_{z}dx\wedge dy.$ (1)

This 2-form can be written as $\mathcal{G}=\phi_{e}\wedge dy$, with $\phi_{e}$
the 1-form

$\bm{\phi}_{e}=H_{y}dt+D_{z}dx-D_{x}dz.$ (2)

The dual of this 1-form is the vector:

$\bm{\tau}_{S}=H_{y}\partial_{t}-D_{z}\partial_{x}+D_{x}\partial_{z}.$ (3)

This vector defines integral curves, the ESs, that are absolute topological
structures in 3D spacetime: a given line connects the same set of events,
independently of any frame of reference. Starting from from any event
$\bm{s}_{0}$, the line $\mathrm{ES}(\bm{s}_{0})$ can be built using the
parametric equation

$\bm{s}(\eta)=\bm{s}_{0}+\int_{0}^{\eta}\bm{\tau}_{S}d\eta.$ (4)

The electric flux on a spaghetti segment $\mathrm{ES}_{e_{1}e_{2}}$ from the
event $\mathrm{e}_{1}=\bm{s}(\eta_{1})$ to the event
$\mathrm{e}_{2}=\bm{s}(\eta_{2})$ has the value

$\varphi_{e}(\mathrm{ES}_{e_{1}e_{2}})=\int_{\eta_{1}}^{\eta_{2}}(H^{2}-D^{2})d\eta.$
(5)

In this equation, the electric flux value $\varphi_{e}$ and the $H^{2}-D^{2}$
factor are both Lorentz-invariant. Hence
$\eta(\mathrm{ES}_{e_{1}e_{2}})=\eta_{2}-\eta_{1}$ is actually a new Lorentz
invariant.

From a topological point of view, a set of ESs can be constructed to cover any
region of the 3D spacetime. Furthermore, an oriented segment of ES can be
labeled with three quantities: 1) its interval length (but only if the segment
is purely timelike or spacelike), 2) its electric flux $\varphi_{e}$, 3) its
characteristic parameter $\eta$.

Unlike worldlines of physical objects, which are always timelike, ESs
transcend the boundary between time and space directions. Some parts of the
same ES can be timelike, spacelike, and even lightlike. The particular case
where ESs are contained in a light cone requires some attention, as
$H^{2}-D^{2}$, the interval and the electric flux are all equal to zero. In
contrast, the $\eta$ value of a non-degenerate (i.e. not reduced to a point)
ES segment is never zero, and has a sign fixed by the orientation of ES. If an
arbitrary inertial frame of reference is chosen, the components of the
electromagnetic field can be obtained from the ES topology and the $\eta$
value using Eq. (4):

$H_{y}=\frac{ds_{t}}{d\eta},D_{z}=-\frac{{ds_{x}}}{d\eta},D_{x}=\frac{{ds_{z}}}{d\eta},$
(6)

where $s_{t},s_{x},s_{z}$ are the ES coordinates. The smaller the field, the
larger the $\eta$ value. It may seem rather counter-intuitive, but the ES
characteristic parameter $\eta$ is an absolute, extensive physical property of
ESs.

Fig. 1(a) shows the case of a linear, constant density of charge $\lambda$
along the $y$-direction. The worldline of each charge is a straight line along
the $Ot$ axis, and this axis represents all charges in 3D spacetime $(t,x,z)$.
ESs appear as concentric green circles centered around the $Ot$ line. Every ES
circle contains the same electric flux value $\varphi_{e}=\lambda$. The pink
radial surfaces are Zero-Electric Flux Surfaces (ZEFSs). They are
perpendicular to the ESs and start on the $Ot$ charge worldline. The electric
flux on any line they contain is zero.

At first glance, Fig. 1(a) is a picture of equipotential lines (green circles,
which are indeed cylinders along $y$-axis in 4D spacetime) and electric field
strength lines, extended in the time direction (pink surfaces). However, the
concept of equipotential lines is frame-dependent: in a different frame of
reference, a magnetic field would appear, and both the electric field and the
equipotential lines would change. In contrast, ESs and ZEFSs are absolute,
connecting the same set of events, independently of any frame of reference.
The $\eta$ value calculated between two events of a given ES is absolute,
independent of the choice of a reference frame. The field components, which
are relative, can be simply derived from the ES topology and the parameter
$\eta$, using Eq. 6.

Figure 1: (a) A $t$-invariant (electrostatic) case: an $Oy$ charged line with
$Ot$ worldline. Green circles: ESs, pink surfaces: ZEFSs. (b) A
non-$t$-invariant case: ES (green line) with purely electric field part ($z$
direction, dark-blue frame), plane-wave part (null-like direction, light-blue
frame), and purely magnetic field ($t$ direction, purple frame). The ES
topology is absolute, the interval values $\delta s$, the flux values
$\delta\varphi_{e}$ and the characteristic parameter values $\delta\eta$ are
all absolute, independent of the reference frame. In contrast, $H_{y}$ and
$D_{x}$ depend on the axis choice ($t,z)$, with Eq. 3 and Eq. 6. When ES is
null-like, only the parameter $\eta$ has a non-zero value, able to describe
the electromagnetic field in this region.

In this electrostatic example, $\varphi_{e}$ has the same value $\lambda$ on
all circles. By contrast, $\eta=4\pi^{2}R^{2}/\lambda$ is proportional to the
square of the radius $R$. The usefulness of the parameter $\eta$, not obvious
in this example, is crucial when the ES part belongs to a light cone, as shown
in Fig. 1(b), where an ES is successively space-like, null-like, and time-
like, transcending the distinction between space and time.

The Fabry-Perot resonant slit grating system studied in this paper consists of
a perfectly conducting metal with slits parallel to the $y$-direction (Fig.
2(a)), excited at normal incidence by a p-polarized plane wave (i.e. with the
magnetic field oriented along the $y$ direction). The grating period is
$L=\pi$, that is, half the wavelength $\Lambda=2\pi$ ($c=1$), so all the
diffracted waves are evanescent. The slit width is arbitrarily chosen as
$w=L/6$. This system is known for perfect transmission of incident radiation
[12], with a funneling mechanism (Fig. 2(b)) that can be understood by the
magneto-electric interference between the incident and evanescent fields [13].
The Fabry-Perot resonance condition corresponding to total transmission and no
reflection is obtained for a slit height $h$ equal to $h=h_{0}+n\,\Lambda/2$
with $h_{0}\approx 0.417\,\Lambda$, and $n$ is an arbitrary integer. The value
of $h_{0}$ is slightly smaller than $\Lambda/2$, as the phase of the internal
reflection is not exactly $\pi$. The value $h=0.917\,\Lambda$ was chosen for a
clear view of the interference pattern inside the slits.

Figure 2: (a) Slit grating, invariable in $y$-direction, made in a perfect
conductor. (b) Time-averaged Poynting vector lines in the resonsant case,
$p$-polarized normally incident monochromatic wave. (c) ESs in the central
plane of a slit. (d) ESs crossing the line z = - h/2, x =- 2 w/5.

Several numerical techniques based on Eq. 4 and Eq. 5 were used to compute ESs
and ZEFSs. Full descriptions of these and codebooks are presented in
Supplemental Material. Fig. 2(d) shows ESs crossing the line $z=-h/2,x=-2w/5$.
The median plane of a slit is a special place because the $D_{z}$ component is
here equal to zero, and the ESs remain localized in this plane (see Eq. 3 and
Fig. 2(c)). These ESs are also plotted in Fig. 3, along with ZEFS lines. As
there are no electric charge in the considered region, the electric flux is
constant between two neighboring bold ZEFS lines, and the density of these
lines in the $t$-direction corresponds to the value of the field component
$H_{y}$, and the density in the $z$-direction corresponds to the value of the
field component $D_{x}$.

Far from the slits, where a single plane wave propagates, $D_{x}=H_{y}$, the
ESs are straight null-like lines, according to Eq. 3. Inside a slit, where two
plane waves interfere, concentric patterns of rounded rhombi are observed. The
field is purely electric when the ESs (green lines) are parallel to the
$z$-direction, and purely magnetic when they are parallel to the
$t$-direction. Owing to the resonance inside the slit, the electric flux
density in this region is six times higher than in the plane-wave region. This
is clearly depicted in Fig. 3: in the slit region, there are six ZEFS lines
over a half-period in time, whereas in the upper part of the figure (plane-
wave region), there is only one bold ZEFS line. Compared with the interference
of two identical plane waves [14], the rounded squares are slightly distorted.
This is because the amplitude of the wave from bottom to top is smaller than
that from top to bottom.

Figure 3: Plot of the ES (green) and ZEFS (red) lines on the median plane for
one time period. The orange zone corresponds to the slit area. The electric
flux is constant between two thick red ZEFS lines, and also between two thin
red lines, with a factor $1/6$. Inset: zoom of the whirl zone, with very small
flux step between ZEFS lines.

In the funneling region, ESs form whirls that are centered on zero field
events. They seem to contradict the principle that tubes of flux with no
charge never end [5], so they require detailed analysis. The field in the
funneling region is the superposition of the incoming plane wave and of a
series of diffracted evanescent waves. For simplicity, let us restrict the
continuation of the study to only the first diffracted wave. In this model,
the incident plane wave has amplitude $\cos(z+t)$, and the evanescent wave has
amplitude $a\exp{(-\sqrt{3}z)}\sin(t)\cos(2x)$ where the amplitude $a=1.091$
and the phase $\pi/2$ are obtained numerically (see Supplemental Material).
The ES-ZEFS pattern of this interference is plotted as respectively green and
red lines in Fig. 4(a). The pattern of this simplified model is similar to the
one observed in Fig. 3. The whirls are centered on zero-field points
$H_{y}=D_{x}=0$.

Figure 4: (a) ESs (green) and ZEFS-lines (red) in the case of interference of
the plane wave and two evanescent ones (with equal amplitudes and opposite
propagation directions along $x$ axis). (b) Spacetime camembert-box with
incoming, never outgoing ESs at $x=0$. Other ESs are outgoing through one of
the sides.

Using a linear approximation of the field components around the zero-field
points $(t_{0}=\pi/3+p/2,\;z_{0}=0.7736)$, the parametric equation of an ES is
(see Supplemental Material):

$\displaystyle\begin{pmatrix}t(\eta)-t_{0}\\\ z(\eta)-z_{0}\end{pmatrix}$
$\displaystyle=z_{0}\begin{pmatrix}0&\beta\\\
1&-\alpha\end{pmatrix}\begin{pmatrix}\cos{\lambda_{i}\eta}\\\
\sin{\lambda_{i}\eta}\end{pmatrix}e^{-\lambda_{r}\eta}$ (7) $\displaystyle
x(\eta)$ $\displaystyle=x_{0}e^{2\lambda_{r}\eta}$ (8)

where
$\lambda_{i}=\sqrt{(3\sin{z_{0}}+\frac{\cos{z_{0}}}{\sqrt{3}})(\sin{z_{0}}-\frac{\cos{z_{0}}}{\sqrt{3}})}$,
and $\lambda_{r}=\sin{z_{0}}-\frac{\cos{z_{0}}}{\sqrt{3}}$. For $x_{0}=0$ this
is an affine transformation of a curve, named _spira mirabilis_ by Bernoulli
three centuries ago [15]. It is an endless spiral, the first fractal curve
ever studied. And this is the key of the paradox that ES should never end.
They never end indeed. Each loop of ES toward the center corresponds to
$2\pi/\lambda_{i}$ change of the parameter $\eta$ while the radius decreases
by a factor of $\exp(-2\pi\lambda_{r}/\lambda_{i})=0.12$.

The ES bundles make tubes of magnetic flux, corresponding to the Faraday
2-form $(-\bm{E},\bm{B})$, with $E_{x}dx\wedge dt+E_{z}dz\wedge
dt+B_{y}dx\wedge dz$. As there are no magnetic charges, any tube tube entering
on a given closed surface $\Sigma$ of spacetime $(t,x,z)$ must exit somewhere
on the same surface. The observed endless ESs do not contradict this law
because they are restricted to $x=0$ plane, having zero measure on the surface
$\Sigma$. If we consider a camembert-box as in Fig. 4(b) the incoming ESs at
the periphery exit through the sides of the disk thus defining tubes of
constant magnetic flux.

To conclude, we demonstrated that a $y$-invariant _p_ -polarized
electromagnetic field, which is usually represented by 3 field components (as
$H_{y},D_{z},D_{x}$, relative to a frame of reference), can be represented in
an absolute way. In a flat spacetime, the qualifier _absolute_ is synonymous
with _Lorentz invariant_. It should be noted that the content of this letter
can be extended to a curved spacetime, provided that there is translation
symmetry along $y$ direction. The electromagnetic field can be represented by
unique and absolute structures, the electric spaghettis (ESs), measured by the
absolute parameter $\eta$. The surfaces perpendicular to ESs (the Zero
Electric Flux Surfaces (ZEFSs)) are also absolute space-time structures.
Contrary to the worldlines of physical objects, which are always timelike, ESs
and ZEFSs transcend the limit between space and time. Any segment of ES is
characterized by two absolute electromagnetic quantities: its electric flux
$\varphi_{e}$, and its characteristic parameter $\eta$. Despite its counter-
intuitive nature (the larger it is, the smaller the field), $\eta$ is a
parameter that makes sense because it can describe the electromagnetic field
everywhere. In contrast, the flux and interval values, which are zero on null-
like segments of ESs, cannot describe the electromagnetic field everywhere.
ESs with $\eta$ milestones written on them, form a complete representation of
the electromagnetic world in the 3D $y$-invariant spacetime, giving in any
inertial frame of reference $(t,x,z)$, the field components
$H_{y},D_{z},D_{x}$.

We explored the ES topology of a Fabry-Perot resonant slit-grating. In the
plane wave region ESs are straight light-lines, with an electric field equal
to the magnetic field. Inside the slits, the ESs are mainly rounded rhombi,
corresponding to the interference of two plane waves, with purely $H_{y}$
zones and purely $D_{x}$ zones. The funneling region is of particular
interest, with ESs having the fractal behavior of logarithmic spirals. They
correspond to the interference of the incident wave with the evanescent waves
diffracted by the slit-grating.

ESs illustrate the profound unity of electric and magnetic fields, and give
them a topological structure that can be studied for itself, opening a new
field of research. Also, instead of considering field components, which are
related to an arbitrary frame of reference, the description of the field as an
absolute topology in spacetime with absolute measure $\eta$ is a radical
change of the conceptual approach to the electromagnetic world. This not only
has philosophical and educational consequences, but also suggests new
computational approaches.

## References

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# Equivalence of canonical and phenomenological formulations of spin
hydrodynamics

Asaad Daher<EMAIL_ADDRESS>Arpan Das<EMAIL_ADDRESS>Wojciech
Florkowski<EMAIL_ADDRESS>Radoslaw Ryblewski
<EMAIL_ADDRESS>
###### Abstract

Two formulations of relativistic hydrodynamics of particles with spin 1/2 are
compared. The first one uses expressions for the energy-momentum and spin
tensors that have properties that follow a direct application of Noether’s
theorem, including a totally antisymmetric spin tensor. The other one is based
on a simplified form of the spin tensor that is commonly used in the current
literature under the name of phenomenological pseudo-gauge. We show that these
two frameworks are equivalent, since they can be directly connected by a
suitably defined pseudogauge transformation. Our analysis uses arguments
related to the positivity of entropy production. The latter turns out to be
independent of the pseudo-gauge provided the canonical energy-momentum tensor
is properly improved.

###### keywords:

relativistic hydrodynamics with spin , dissipative spin hydrodynamics ,
pseudo-gauge , spin polarization

††journal: Physics Letters B

[inst1]Institute of Nuclear Physics Polish Academy of Sciences, PL-31-342
Krakow, Poland

[inst2]Institute of Theoretical Physics, Jagiellonian University, PL-30-348
Krakow, Poland

##  1 Introduction

Recent observations of spin polarization of weakly decaying hyperons produced
in relativistic heavy-ion experiments across various collision energies [1, 2,
3, 4, 5, 6, 7, 8, 9] have provided a unique probe to study polarization
phenomena in relativistic nuclear matter under rotation [10]. Motivated by the
earlier successes of relativistic fluid dynamics in heavy-ion phenomenology
[11], several extensions of relativistic hydrodynamics for the spin-polarized
fluids have been developed using quantum statistical density operators [12,
13, 14, 15, 16], relativistic kinetic theory [17, 18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29], effective Lagrangian approach [30, 31, 32, 33], entropy
current analysis [34, 35, 36, 37, 38, 39], holography [40, 41] and equilibrium
partition functions [42].

At the macroscopic level, relativistic hydrodynamics with spin can be
formulated on the grounds of the conservation laws of conserved currents i.e.
energy-momentum tensor, baryon current, and the total angular momentum
current, with the last one composed of spin and orbital parts. In special
relativity, however, the decomposition of the total angular momentum current
obtained by choosing a particular form of the energy-momentum tensor and the
spin tensor is not unique. A given pair of these currents can always be
transformed to another one through the so-called pseudo-gauge [43, 24]. Of
particular importance in this context is the _Belinfante pseudo-gauge_ based
on Belinfante’s improved, i.e. symmetric, energy-momentum tensor [44, 45, 46],
which provides a natural connection to Einstein’s relativity theory. Another
choice, known as the _phenomenological pseudo-gauge_ , deals with an arbitrary
energy-momentum tensor, i.e. with symmetric and antisymmetric components, and
the spin tensor antisymmetric in the last two indices turns out to be
especially convenient for the construction of hydrodynamic frameworks. Last
but not least, at the microscopic level, the _canonical_ forms of conserved
currents with the energy-momentum tensor having both symmetric and
antisymmetric parts and the spin tensor being totally antisymmetric in all its
Lorentz indices, can be obtained. Such a choice of conserved currents is
motivated from the standpoint of the quantum field theory of Dirac fermions
when applying the Noether theorem [28].

While the change to a particular pseudo-gauge leaves the total charges and
dynamical equations intact, it modifies the densities of the macroscopic
quantities, hence introducing possible pseudo-gauge dependence of the
resulting hydrodynamic formalism. Moreover, in local thermal equilibrium, the
choice of decomposition of the angular momentum into an orbital and a spin
part can significantly affect the predictions of spin polarization [47]. To
explore such freedom, in Ref. [35] the dissipative spin hydrodynamics was
formulated using Belinfante pseudo-gauge, starting from the phenomenological
set of energy-momentum and spin tensors.

In the present work, we formulate the dissipative spin hydrodynamics using the
entropy current analysis and canonical forms of conserved currents. We find
that the totally antisymmetric nature of the spin tensor introduces the new
properties into the hydrodynamic description that are otherwise missing in the
phenomenological [34] or the Belinfante [35] frameworks. In particular, our
work shows that the naïve use of arbitrary forms of canonical currents
conflicts with the principle of entropy production. To resolve this problem,
we construct an _improved_ form of the canonical energy-momentum tensor
without affecting the conservation equations. We prove that the resulting
improved canonical definitions can be considered a suitable choice for a
first-order dissipative hydrodynamic framework. Using proper pseudo-gauge
transformation, we show an equivalence between the improved canonical
framework and the phenomenological framework of dissipative spin
hydrodynamics.

The paper is organized as follows. We begin our discussion of the canonical
formulation of first-order dissipative spin hydrodynamics in Sec. 2. In this
section, we argue that a naïve use of a general canonical currents is in
conflict with the principle of entropy production for a dissipative system. In
Sec. 3 we discuss an improved canonical framework in which one can uniquely
identify different dissipative currents using entropy current analysis. In
Sec. 4 we show that one can get the phenomenological framework of dissipative
spin hydrodynamics starting from the improved canonical framework, while in
Sec. 5 we justify the derivation of the improved canonical framework from the
phenomenological one. In Sec. 6 we summarize and conclude.

##  2 Entropy current analysis: canonical framework

For a spin-polarized fluid, the hydrodynamic framework can be constructed
based on the conservation of energy-momentum tensor $T^{\mu\nu}$ and total
angular momentum tensor $J^{\mu\alpha\beta}$111Here, for simplicity, we assume
there are no conserved charges in the system.

$\displaystyle\partial_{\mu}T^{\mu\nu}=0,$ (1)
$\displaystyle\partial_{\mu}J^{\mu\alpha\beta}=\partial_{\mu}S^{\mu\alpha\beta}+2T^{[\alpha\beta]}=0.$
(2)

For symmetric and antisymmetric part of arbitrary tensor $X^{\mu\nu}$ we use
the notation $X^{\mu\nu}_{(s)}\equiv X^{(\mu\nu)}=(X^{\mu\nu}+X^{\nu\mu})/2$
and $X^{\mu\nu}_{(a)}\equiv X^{[\mu\nu]}=(X^{\mu\nu}-X^{\nu\mu})/2$,
respectively. In Eq. (2) we have taken advantage of the fact that the total
angular momentum $J^{\mu\alpha\beta}$ can be decomposed into an orbital
angular momentum component $L^{\mu\alpha\beta}$, which can be expressed using
the energy-momentum tensor as $L^{\mu\alpha\beta}=2x^{[\alpha}T^{\mu\beta]}$,
and the intrinsic angular momentum (spin) component $S^{\mu\alpha\beta}$.

The orbital and the spin part, with the latter being given by the fully
antisymmetric tensor, can be argued to originate using the Noether theorem for
the Lorentz symmetry of a non-scalar field theory, namely, for our interest,
the Dirac field. Herein, we demand the Lorentz structure of the spin tensor to
be preserved at the macroscopic level. To this end, we define the constitutive
relations for the energy-momentum tensor and the totally antisymmetric spin
tensor up to the first order in the gradient expansion as

$\displaystyle
T^{\mu\nu}_{\text{can}}=T^{\mu\nu}_{(0)}+T^{\mu\nu}_{\text{can}(1)},$ (3)
$\displaystyle
S^{\mu\alpha\beta}_{\text{can}}=u^{\mu}S^{\alpha\beta}+u^{\beta}S^{\mu\alpha}+u^{\alpha}S^{\beta\mu}+S^{\mu\alpha\beta}_{\text{can}(1)}.$
(4)

The leading-order contribution to $T^{\mu\nu}_{\text{can}}$ in Eq. (3) has the
perfect-fluid form

$\displaystyle T^{\mu\nu}_{(0)}=\varepsilon u^{\mu}u^{\nu}-p\Delta^{\mu\nu},$
(5)

where $\varepsilon$ and $p$ are the energy density and pressure, respectively,
$u^{\mu}$ is the fluid four velocity satisfying the normalization condition
$u^{\mu}u_{\mu}=1$, $\Delta^{\mu\nu}\equiv g^{\mu\nu}-u^{\mu}u^{\nu}$ is the
symmetric operator projecting onto the space orthogonal to $u^{\mu}$, i.e.
$\Delta^{\mu\nu}u_{\mu}=0$, and $g_{\mu\nu}=\text{diag}(+1,-1,-1,-1)$ is the
Minkowski metric. The spin density tensor $S^{\mu\nu}$ is antisymmetric, i.e.
$S^{\mu\nu}=-S^{\nu\mu}$, therefore the leading-order part of the spin tensor
$S^{\mu\alpha\beta}_{\text{can}}$ is totally antisymmetric. The terms
$T^{\mu\nu}_{\text{can}(1)}$ and $S^{\mu\alpha\beta}_{\text{can}(1)}$ denote
the first-order derivative corrections to the energy-momentum tensor and the
spin tensor, respectively. In general, $T^{\mu\nu}_{\text{can}(1)}$ contains
both a symmetric $T^{\mu\nu}_{\text{can}(1s)}$ and an antisymmetric
$T^{\mu\nu}_{\text{can}(1a)}$ part. We can assume that, similarly to the
leading-order part, the $S^{\mu\alpha\beta}_{\text{can}(1)}$ part is totally
antisymmetric, although this does not affect our analysis. The tensor
$S^{\alpha\beta}$ plays a role analogous to the number density in the presence
of a conserved charge [34, 35]. Near local thermal equilibrium, it satisfies
the generalized laws of thermodynamics, namely

$\displaystyle\varepsilon+p$
$\displaystyle=Ts+\omega_{\alpha\beta}S^{\alpha\beta},$ $\displaystyle
d\varepsilon$ $\displaystyle=Tds+\omega_{\alpha\beta}dS^{\alpha\beta},$ (6)
$\displaystyle dp$ $\displaystyle=sdT+S^{\alpha\beta}d\omega_{\alpha\beta},$

where $s$ is the entropy density, $T$ is the temperature, and the
antisymmetric tensor $\omega^{\mu\nu}$ can be considered as the spin chemical
potential conjugate to the spin density $S^{\mu\nu}$ [34].

At this point we should comment on the gradient order of the quantities
associated with spin. Throughout the manuscript we consider
$\omega^{\mu\nu}\sim\mathcal{O}(\partial^{1})$, as argued in Refs. [34, 35],
taking into account that the chemical spin potential at global equilibrium can
be expressed by the thermal vorticity tensor
$\varpi_{\mu\nu}=-\partial_{[\mu}(u_{\nu]}/T)$. At the same time, $S^{\mu\nu}$
is considered to be $\mathcal{O}(\partial^{0})$, which is consistent with the
derivative counting of $\omega^{\mu\nu}$ in the high-temperature limit, where
one can assume that $S^{\mu\nu}\sim T^{2}\omega^{\mu\nu}$ [39]222This is not
necessarily the only choice, however. For example, in Ref. [37], dissipative
spin hydrodynamics was developed, where both the spin chemical potential and
the spin density tensor are assumed to be $\mathcal{O}(\partial^{0})$..

In the presence of the spin chemical potential and spin density, the
nonequilibrium entropy current can be generalized to first-order terms in the
gradient expansion as follows [34, 35]333Note that in addition to the
antisymmetry property of $S^{\mu\nu}$ one can also impose the condition that
$S^{\mu\nu}$ is orthogonal to $u^{\mu}$, i.e. $S^{\mu\nu}u_{\mu}=0$ [36]. This
is analogous to the Frenkel condition, but for hydrodynamics. This additional
condition reduces the number of degrees of freedom of the spin density tensor.
In the present study, we do not impose such conditions unless explicitly
stated..

$\displaystyle\mathcal{S}^{\mu}_{\text{can}}$
$\displaystyle=T^{\mu\nu}_{\text{can}}\,\beta_{\nu}+p\,\beta^{\mu}-\omega_{\alpha\beta}S^{\alpha\beta}\beta^{\mu}+\mathcal{O}(\partial^{2})$
$\displaystyle=\mathcal{S}^{\mu}_{(0)}+T^{\mu\nu}_{\text{can}(1)}\,\beta_{\nu}+\mathcal{O}(\partial^{2}).$
(7)

Here $\mathcal{S}^{\mu}_{(0)}\\!=\\!su^{\mu}$ is the equilibrium part of the
entropy current, $\beta\equiv 1/T$ is the inverse temperature, and
$\beta^{\mu}\equiv\beta u^{\mu}$. Note that up to the first order in the
gradient expansion, in the expression of the nonequilibrium entropy current
$S^{\mu\alpha\beta}_{\text{can}(1)}$ plays no role. Only the dissipative part
of the energy-momentum tensor, i.e. $T^{\mu\nu}_{\text{can}(1)}$ contributes
to the nonequilibrium entropy current.

So far, the explicit forms of the various dissipative currents in
$T^{\mu\nu}_{\text{can}(1)}$ are not uniquely defined. They can be uniquely
determined by imposing the second law of thermodynamics, i.e.
$\partial_{\mu}\mathcal{S}^{\mu}_{\text{can}}\geq 0$. Using Eq. (7), the
divergence of the entropy current can be expressed as

$\displaystyle\partial_{\mu}\mathcal{S}^{\mu}_{\text{can}}$
$\displaystyle=\beta\left[T\partial_{\mu}\mathcal{S}^{\mu}_{(0)}+u_{\nu}\partial_{\mu}T^{\mu\nu}_{\text{can}(1)}\right]+T^{\mu\nu}_{\text{can}(1)}\partial_{\mu}\beta_{\nu}+\mathcal{O}(\partial^{3}).$
(8)

Using the conservation equations of the canonical energy-momentum tensor and
the total angular momentum tensor, Eq. (8) can be simplified. The conservation
of the energy-momentum tensor (1) implies

$\displaystyle
u_{\nu}\partial_{\mu}T^{\mu\nu}_{\text{can}(1)}=-T\partial_{\mu}\mathcal{S}^{\mu}_{(0)}-\omega_{\alpha\beta}\partial_{\mu}(u^{\mu}S^{\alpha\beta}),$
(9)

which allows us to write Eq. (8) in the following way

$\displaystyle\partial_{\mu}\mathcal{S}^{\mu}_{\text{can}}$
$\displaystyle=-\beta\omega_{\alpha\beta}\partial_{\mu}(u^{\mu}S^{\alpha\beta})+T^{\mu\nu}_{\text{can}(1)}\partial_{\mu}\beta_{\nu}+\mathcal{O}(\partial^{3}).$
(10)

Furthermore, using the constitutive relation for the canonical spin tensor
given in Eq. (4), the conservation of total angular momentum (2) up to
$\mathcal{O}(\partial^{2})$ simplifies to

$\displaystyle\partial_{\mu}(u^{\mu}S^{\alpha\beta})=-2T^{\alpha\beta}_{\text{can}(1a)}-2\partial_{\mu}\Phi^{\mu\alpha\beta}_{\text{can(0)}},$
(11)

where for simplicity we have introduced the tensor
$\Phi^{\mu\alpha\beta}_{\text{can(0)}}\equiv u^{[\alpha}S^{\beta]\mu}$ which
is antisymmetric in the last two indices. Thus, if we substitute equation (11)
into Eq. (10), we find

$\displaystyle\partial_{\mu}\mathcal{S}^{\mu}_{\text{can}}$
$\displaystyle=T^{\alpha\beta}_{\text{can}(1s)}\partial_{\alpha}\beta_{\beta}+T^{\alpha\beta}_{\text{can}(1a)}\big{[}\partial_{\alpha}\beta_{\beta}+2\beta\omega_{\alpha\beta}\big{]}$
$\displaystyle+2\beta\omega_{\alpha\beta}\partial_{\mu}\Phi^{\mu\alpha\beta}_{\text{can(0)}},$
(12)

where we have decomposed $T^{\mu\nu}_{\text{can}(1)}$ into its symmetric and
antisymmetric parts. It should be noted that due to the totally antisymmetric
structure of the canonical spin tensor, the conservation of the total angular
momentum leads to mixing between the different components of the spin tensor,
as can be clearly seen in Eq. (11). This is a unique property of the canonical
framework of spin hydrodynamics.

To identify the transport coefficients in the canonical framework, we should
be able to write Eq. (12) in terms of the dissipative currents. The symmetric
and antisymmetric components of $T^{\alpha\beta}_{\text{can}(1)}$ can be
decomposed in terms of the irreducible tensor basis as follows [34, 35]

$\displaystyle
T^{\alpha\beta}_{\text{can}(1s)}=h^{\alpha}u^{\beta}+h^{\beta}u^{\alpha}+\tau^{\alpha\beta},$
(13) $\displaystyle
T^{\alpha\beta}_{\text{can}(1a)}=q^{\alpha}u^{\beta}-q^{\beta}u^{\alpha}+\phi^{\alpha\beta},$
(14)

respectively. Here, $h^{\mu}$, $\tau^{\mu\nu}$, $q^{\mu}$ and $\phi^{\mu\nu}$
can be identified as different dissipative currents and satisfy the following
conditions: $q\cdot u=0,h\cdot
u=0,\tau^{\mu\nu}u_{\mu}=0,\phi^{\mu\nu}u_{\mu}=0,\tau^{\mu\nu}=\tau^{\nu\mu},\phi^{\mu\nu}=-\phi^{\nu\mu}$.
For the symmetric energy-momentum tensor, the heat flux $h^{\mu}$ and the
viscous stress tensor $\tau^{\mu\nu}$ (which contains the shear and bulk
viscosity terms) are the only dissipative fluxes. However, in the case of a
non-vanishing antisymmetric part of the energy-momentum tensor, the new
dissipative currents $q^{\mu}$ and $\phi^{\mu\nu}$ arise.

Analogously to $T^{\mu\nu}_{\text{can}(1a)}$ in Eq. (14), we decompose the
antisymmetric term $\partial_{\mu}\Phi^{\mu\alpha\beta}_{\text{can(0)}}$ and
the spin chemical potential $\omega^{\alpha\beta}$ as

$\displaystyle\partial_{\mu}\Phi^{\mu\alpha\beta}_{\text{can(0)}}$
$\displaystyle=\delta q^{\alpha}u^{\beta}-\delta
q^{\beta}u^{\alpha}+\delta\phi^{\alpha\beta},$ (15)
$\displaystyle\omega^{\alpha\beta}$
$\displaystyle=k^{\alpha}u^{\beta}-k^{\beta}u^{\alpha}+\lambda^{\alpha\beta},$
(16)

where the components $\delta q$ and $k$ as well as $\delta\phi$ and $\lambda$
have the same properties as $q$ and $\phi$, respectively.

Moreover, using the decomposition $\partial_{\mu}\equiv\nabla_{\mu}+u_{\mu}D$,
with
$u^{\mu}\nabla_{\mu}=0,\nabla_{\mu}=\Delta^{\alpha}_{\mu}\partial_{\alpha}$
and $D=u^{\mu}\partial_{\mu}$, it can be shown that (for a detailed derivation
see A)

$\displaystyle\partial_{\mu}\mathcal{S}^{\mu}_{\text{can}}=$
$\displaystyle-\beta
h^{\mu}\left(\beta\nabla_{\mu}T-Du_{\mu}\right)+\beta\pi^{\mu\nu}\sigma_{\mu\nu}+\beta\Pi\theta$
$\displaystyle-\beta
q^{\mu}\left(\beta\nabla_{\mu}T+Du_{\mu}-4\omega_{\mu\nu}u^{\nu}\right)$
$\displaystyle+\phi^{\mu\nu}\left(\Omega_{\mu\nu}+2\beta\Delta^{\alpha}_{~{}\mu}\Delta^{\beta}_{~{}\nu}\omega_{\alpha\beta}\right)$
$\displaystyle+2\beta\left[2k_{\alpha}\delta
q^{\alpha}+\lambda_{\alpha\beta}\delta\phi^{\alpha\beta}\right].$ (17)

Here $\pi^{\mu\nu}$ is the traceless part of $\tau^{\mu\nu}$ and $\Pi$ is the
trace of $\tau^{\mu\nu}$, i.e.
$\tau^{\mu\nu}=\pi^{\mu\nu}+\Pi\Delta^{\mu\nu}$,
$\sigma_{\mu\nu}\equiv\nabla_{(\mu}u_{\nu)}-\frac{1}{3}\theta\Delta_{\mu\nu}$
is the shear tensor with $\theta=\nabla^{\alpha}u_{\alpha}$, and
$\Omega_{\mu\nu}\equiv\Delta^{\alpha}_{~{}\mu}\Delta^{\beta}_{~{}\nu}\partial_{[\alpha}\beta_{\beta]}=\beta\nabla_{[\mu}u_{\nu]}$
is the vorticity tensor.

Imposing the second law of thermodynamics, i.e.
$\partial_{\mu}\mathcal{S}^{\mu}_{\text{can}}\geq 0$ we obtain the conditions

$\displaystyle h^{\mu}$
$\displaystyle=-\kappa\left(Du^{\mu}-\beta\nabla^{\mu}T\right),$ (18)
$\displaystyle\pi^{\mu\nu}$ $\displaystyle=2\eta\sigma^{\mu\nu},$ (19)
$\displaystyle\Pi$ $\displaystyle=\zeta\theta,$ (20) $\displaystyle q^{\mu}$
$\displaystyle=\lambda\left(\beta\nabla^{\mu}T+Du^{\mu}-4\omega^{\mu\nu}u_{\nu}\right),$
(21) $\displaystyle\phi^{\mu\nu}$
$\displaystyle=\gamma\left(\Omega^{\mu\nu}+2\beta\Delta^{\mu}_{~{}\alpha}\Delta^{\nu}_{~{}\beta}\omega^{\alpha\beta}\right),$
(22)

together with the constraint

$\displaystyle 2k_{\alpha}\delta
q^{\alpha}+\lambda_{\alpha\beta}\delta\phi^{\alpha\beta}\geq 0.$ (23)

Here various transport coefficients can be identified with the conditions:
$\kappa\geq 0$, $\eta\geq 0$, $\zeta\geq 0$, $\lambda\geq 0,\gamma\geq 0$.
Note that $k^{\alpha}$, $\lambda^{\alpha\beta}$ as well as

$\displaystyle\delta q^{\alpha}=$
$\displaystyle\Delta^{\alpha}_{~{}\mu}u_{\nu}\partial_{\lambda}\Phi^{\lambda\mu\nu}_{\text{can(0)}},$
(24) $\displaystyle\delta\phi^{\alpha\beta}=$
$\displaystyle\Delta^{\alpha}_{~{}[\mu}\Delta^{\beta}_{~{}\nu]}\partial_{\lambda}\Phi^{\lambda\mu\nu}_{\text{can(0)}},$
(25)

originate from completely different terms. Therefore, Eq. (23) is not
satisfied for arbitrary initial conditions in general. The presence of such an
additional condition implies that the canonical framework with the energy-
momentum tensor and the spin tensor, as given in Eqs. (3) and (4), make the
resulting hydrodynamic framework not a well-defined initial value problem for
an arbitrary set of hydrodynamic variables, $T$, $u^{\mu}$ and
$\omega^{\mu\nu}$. However, such a problem can be solved by the proper
modification of the energy-momentum tensor, as we discuss in the next section.

##  3 Improved canonical framework

The constitutive relation of the energy-momentum tensor, as given in Eq. (3),
is not unique. In particular, one can add to it an additional totally
divergent term, namely

$\displaystyle\widetilde{T}^{\mu\nu}_{\text{can}}=T^{\mu\nu}_{(0)}+T^{\mu\nu}_{\text{can}(1)}+\partial_{\lambda}\left(u^{\nu}S^{\mu\lambda}\right).$
(26)

Since the additional term is $\mathcal{O}(\partial^{1})$, it does not affect
the conservation law of the energy-momentum tensor (1), but it does affect the
evolution of the spin tensor (2). Moreover, in the presence of the spin
density tensor $S^{\mu\nu}$, at the level of the first-order gradient
expansion one can always add such a term to the energy-momentum tensor.

For the choice of the energy-momentum tensor in Eq. (26) and the spin tensor
in Eq. (4), the conservation of total angular momentum (2) implies

$\displaystyle\partial_{\mu}S^{\mu\alpha\beta}_{\text{can}}$
$\displaystyle=-2T^{\alpha\beta}_{\text{can}(1a)}-\partial_{\mu}(u^{\beta}S^{\alpha\mu})+\partial_{\mu}(u^{\alpha}S^{\beta\mu})$
$\displaystyle\implies\partial_{\mu}(u^{\mu}S^{\alpha\beta})$
$\displaystyle=-2T^{\alpha\beta}_{\text{can}(1a)}.$ (27)

Similarly to the previous section, in this case we can also define the
nonequilibrium entropy current up to first order terms in the gradient
expansion as follows

$\displaystyle\widetilde{\mathcal{S}}^{\mu}_{\text{can}}$
$\displaystyle=\widetilde{T}^{\mu\nu}_{\text{can}}\,\beta_{\nu}+p\,\beta^{\mu}-\omega_{\alpha\beta}S^{\alpha\beta}\beta^{\mu}+\mathcal{O}(\partial^{2})$
$\displaystyle=\mathcal{S}^{\mu}_{(0)}+\widetilde{T}^{\mu\nu}_{\text{can}(1)}\,\beta_{\nu}+\mathcal{O}(\partial^{2}).$
(28)

However, compared to Eq. (7), the first-order correction to the equilibrium
energy-momentum tensor here includes an additional term
$\partial_{\lambda}(u^{\nu}S^{\mu\lambda})$, see Eq. (26).

The divergence of the entropy current (28) can now be expressed as

$\displaystyle\partial_{\mu}\widetilde{\mathcal{S}}^{\mu}_{\text{can}}=\widetilde{T}^{\mu\nu}_{\text{can}(1)}\partial_{\mu}\beta_{\nu}-\beta\omega_{\alpha\beta}\partial_{\mu}(u^{\mu}S^{\alpha\beta})+\mathcal{O}(\partial^{3}),$
(29)

which, using Eqs. (26) and (27), can be further rewritten as

$\displaystyle\partial_{\mu}\widetilde{\mathcal{S}}^{\mu}_{\text{can}}=T^{\alpha\beta}_{\text{can}(1s)}\partial_{\alpha}\beta_{\beta}+T^{\alpha\beta}_{\text{can}(1a)}\left[\partial_{\alpha}\beta_{\beta}+2\beta\omega_{\alpha\beta}\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\partial_{\mu}\beta_{\nu}\partial_{\lambda}(u^{\nu}S^{\mu\lambda}).$
(30)

Note that the last term in the above equation equals a total derivative term
$\partial_{\mu}\left[\beta_{\nu}\partial_{\lambda}(u^{\nu}S^{\mu\lambda})\right]$
which can be absorbed in the divergence of the entropy current and gives the
following

$\displaystyle\partial_{\mu}\widetilde{\mathcal{S}}^{\prime\mu}_{\text{can}}=T^{\alpha\beta}_{\text{can}(1s)}\partial_{\alpha}\beta_{\beta}+T^{\alpha\beta}_{\text{can}(1a)}\left[\partial_{\alpha}\beta_{\beta}+2\beta\omega_{\alpha\beta}\right],$
(31)

where

$\displaystyle\widetilde{\mathcal{S}}^{\prime\mu}_{\text{can}}$
$\displaystyle=\widetilde{\mathcal{S}}^{\mu}_{\text{can}}-\beta_{\nu}\partial_{\lambda}\left(u^{\nu}S^{\mu\lambda}\right)$
$\displaystyle=\mathcal{S}^{\mu}_{(0)}+\beta_{\nu}T^{\mu\nu}_{\text{can}(1)}+\mathcal{O}(\partial^{2}).$
(32)

The right-hand side of Eq. (31) can be easily shown to be positive definite by
properly identifying the dissipative currents. Indeed, after decomposing
$T^{\mu\nu}_{\text{can}(1)}$ into the irreducible tensor basis, as discussed
in the previous section, we obtain

$\displaystyle\partial_{\mu}\widetilde{\mathcal{S}}^{\prime\mu}_{\text{can}}=$
$\displaystyle-\beta
h^{\mu}\left(\beta\nabla_{\mu}T-Du_{\mu}\right)+\beta\pi^{\mu\nu}\sigma_{\mu\nu}+\beta\Pi\theta$
$\displaystyle-\beta
q^{\mu}\left(\beta\nabla_{\mu}T+Du_{\mu}-4\omega_{\mu\nu}u^{\nu}\right)$
$\displaystyle+\phi^{\mu\nu}\left(\Omega_{\mu\nu}+2\beta\Delta^{\alpha}_{~{}\mu}\Delta^{\beta}_{~{}\nu}\omega_{\alpha\beta}\right).$
(33)

Note that, unlike Eq. (17), Eq. (33) is free of the potentially problematic
terms due to the use of an improved energy-momentum tensor from (26). If we
impose the second law of thermodynamics, we can identify the dissipative
currents as in Eqs. (18)-(22), but without the constraint (23). Without the
additional condition (23), the improved version of the canonical framework is
well defined and the dissipative currents can be uniquely expressed by the
hydrodynamic variables, i.e. $T$, $u^{\mu}$ and $\omega^{\mu\nu}$.

Some remarks on Eqs. (18)-(22) are in order here. Note that the expressions of
$h^{\mu}$, $\pi^{\mu\nu}$, and $\Pi$ in terms of the transport coefficients,
i.e. heat conductivity $(\kappa)$, shear viscosity ($\eta$), and bulk
viscosity ($\zeta$), are expected from relativistic Navier-Stokes
hydrodynamics. The new coefficients $\lambda$ and $\gamma$ appear in the
dissipative formulation of hydrodynamic with spin and were also found earlier
[34, 35]. It should be emphasized that the transport coefficients $\kappa$,
$\eta$, $\zeta$, $\lambda$ and $\gamma$ enter directly into the expression of
the energy-momentum tensor and are expected to play an important role in the
equilibration of the system and in the fluctuation-dissipation relations.
Since the choice of the canonical energy-momentum tensor (26) together with
the totally antisymmetric spin tensor (4) leads to a well-defined theory of
first-order dissipative spin hydrodynamics, we will refer to the set of
tensors (4) and (26) as canonical currents in the remainder of the manuscript.

##  4 Transition from canonical to phenomenological formulation

In the previous section, we discussed in detail the canonical framework of
dissipative spin hydrodynamics with dissipative currents uniquely determined
by the positivity of the divergence of the entropy current. The canonical
framework uses the energy-momentum tensor, which contains both symmetric and
antisymmetric parts, and the canonical spin tensor is totally antisymmetric.
The latter property follows directly from a microscopic theory [28], but does
not provide a natural identification of the tensor $S^{\mu\nu}$ in Eq. (4) as
the spin density. 444If one additionally uses the Frenkel condition
$S^{\mu\nu}u_{\nu}=0$, then $S^{\mu\nu}$ can indeed be identified as the spin
density, which in this case follows from the definition of the canonical spin
tensor.. Therefore, many formulations of spin hydrodynamics use a simpler form
of the spin tensor with
$S^{\mu\alpha\beta}_{\text{ph}(0)}=u^{\mu}S^{\alpha\beta}$. In this paper we
refer to such formulations as to phenomenological ones and use the subscript
”ph” in this context.

It is well known that one can always generate a new set of energy-momentum and
spin tensors from a given set of such tensors. This is achieved by performing
the so-called pseudogauge transformation [20, 43, 24, 44, 45, 46]. In this
section we show that it is possible to construct explicitly a pseudogauge
leading directly from the (improved) canonical formulation given by the set of
tensors (4) and (26) to the phenomenological one. The starting point is

$\displaystyle
T^{\mu\nu}_{\text{ph}}=\widetilde{T}^{\mu\nu}_{\text{can}}+\frac{1}{2}\partial_{\lambda}\left(\Sigma^{\lambda\mu\nu}-\Sigma^{\mu\lambda\nu}-\Sigma^{\nu\lambda\mu}\right),$
(34) $\displaystyle
S^{\lambda\mu\nu}_{\text{ph}}=S^{\lambda\mu\nu}_{\text{can}}-\Sigma^{\lambda\mu\nu},$
(35)

where the pseudogauge potential is of the form

$\displaystyle\Sigma^{\lambda\mu\nu}=2\Phi^{\lambda\mu\nu}_{\text{can(0)}}+\Sigma_{(1)}^{\lambda\mu\nu},$
(36)

where we introduced $\Sigma_{(1)}^{\lambda\mu\nu}$ to include all higher order
correction terms in the spin current. Note that here both
$\Sigma^{\lambda\mu\nu}$ and $\Sigma^{\lambda\mu\nu}_{(1)}$ are antisymmetric
only in the last two indices.

Using the pseudogauge transformation we obtain (up to terms of order
$\mathcal{O}(\partial)$)

$\displaystyle S^{\lambda\mu\nu}_{\text{ph}}$
$\displaystyle=u^{\lambda}S^{\mu\nu}+S_{\text{ph(1)}}^{\lambda\mu\nu}$ (37)

and

$\displaystyle T^{\mu\nu}_{\text{ph}}$
$\displaystyle=\widetilde{T}^{\mu\nu}_{\text{can}}-\partial_{\lambda}(u^{\nu}S^{\mu\lambda})$
$\displaystyle=T^{\mu\nu}_{(0)}+T^{\mu\nu}_{\text{can}(1)}$
$\displaystyle=T^{\mu\nu}_{(0)}+T^{\mu\nu}_{\text{ph}(1)},$ (38)

where
$S_{\text{ph(1)}}^{\lambda\mu\nu}=S^{\lambda\mu\nu}_{\text{can}(1)}-\Sigma_{(1)}^{\lambda\mu\nu}$
is first order in gradient expansion.

Having determined the energy-momentum tensor and the spin tensor for the
phenomenological framework, we can study the nonequilibrium entropy current,
which can be expressed as follows

$\displaystyle\mathcal{S}^{\mu}_{\text{ph}}$
$\displaystyle=\mathcal{S}^{\mu}_{(0)}+\beta_{\nu}T^{\mu\nu}_{\text{ph}(1)}+\mathcal{O}(\partial^{2}).$
(39)

Using the conservation equation of the phenomenological energy-momentum
tensor, the divergence of the phenomenological entropy current can be
simplified as

$\displaystyle\partial_{\mu}\mathcal{S}^{\mu}_{\text{ph}}=T^{\mu\nu}_{\text{ph}(1)}\partial_{\mu}\beta_{\nu}-\beta\omega_{\alpha\beta}\partial_{\mu}(u^{\mu}S^{\alpha\beta})+\mathcal{O}(\partial^{3}).$
(40)

From Eqs. (29) and (40), we can see that the divergence of the canonical
entropy current and the divergence of the phenomenological entropy current are
not the same. However,
$\partial_{\mu}\widetilde{\mathcal{S}}^{\mu}_{\text{can}}$ and
$\partial_{\mu}\mathcal{S}^{\mu}_{\text{ph}}$ differ only by a total
divergence term, see Eq. (30). Using the expression of
$T^{\mu\nu}_{\text{ph}(1)}$ (38) in Eq. (40) we find,

$\displaystyle\partial_{\mu}\mathcal{S}^{\mu}_{\text{ph}}$
$\displaystyle=\partial_{\mu}\beta_{\nu}T^{\mu\nu}_{\text{can}(1s)}+\bigg{[}\partial_{\alpha}\beta_{\beta}+2\beta\omega_{\alpha\beta}\bigg{]}T^{\alpha\beta}_{\text{can}(1a)}.$
(41)

In the last equation, we used conservation of total angular momentum for the
phenomenological currents. From the equations (32) and (39), we see that
$\widetilde{\mathcal{S}}^{\prime\mu}_{\text{can}}=\mathcal{S}^{\mu}_{\text{ph}}$.
Moreover, from Eqs. (31) and (41), we can conclude that the entropy
constraints are the same in the canonical and phenomenological formalisms.
Using the tensor decomposition of $T^{\alpha\beta}_{\text{can}(1s)}$ and
$T^{\alpha\beta}_{\text{can}(1a)}$ together with the condition
$\partial_{\mu}\mathcal{S}^{\mu}_{\text{ph}}\geq 0$ we get back the
dissipative currents as given in Eqs. (18)-(22). Note that the expressions for
the dissipative currents as given in Eqs. (18)-(22) correspond exactly to the
dissipative currents obtained in Ref. [34]. Thus, we conclude that starting
from the improved canonical framework, it is indeed possible to recover the
phenomenological formalism of first-order dissipative spin hydrodynamics.

##  5 Canonical framework starting from the phenomenological formalism

Till now we have derived the dissipative spin hydrodynamics framework starting
from the canonical formalism with a general structure of the energy-momentum
tensor and the totally antisymmetric spin tensor. In such a framework we
uniquely determined the dissipative currents using the entropy current
analysis. Furthermore, using the pseudogauge transformation we have obtained
the phenomenological energy-momentum and spin tensor and also studied the
dissipative spin hydrodynamics in this framework. The energy-momentum tensor
in the phenomenological as well as in the canonical framework contains
symmetric and antisymmetric parts. However, contrary to the phenomenological
spin tensor, in the canonical framework the spin tensor is always totally
antisymmetric.

Note that to achieve a well defined description of dissipative spin
hydrodynamics we have improved the energy-momentum tensor adding to the
standard dissipative part a totally divergent term. We have argued that in the
presence of a spin density tensor $S^{\mu\nu}$ such a term is very natural. In
this section we will explicitly show that such a term automatically arises if
one considers the pseudogauge transformation. We also argue that additional
term which originates from the pseudogauge transformation also modifies
various dissipative currents. Appearance of the pseudogauge corrections to
various dissipative currents is straightforward to understand: as the
pseudogauge transformation enters into the expression of the energy-momentum
tensor at $\mathcal{O}(\partial)$, it effectively introduces new dissipative
terms in the original energy-momentum tensor. To put the above statement on
the firm mathematical ground here we will first discuss the framework of
dissipative spin hydrodynamics using the phenomenological energy-momentum
tensor and spin tensor as introduced in Ref. [34]. After that, we will derive
the canonical framework using pseudogauge transformation.

In the phenomenological framework we start with the constitutive relations for
$T^{\mu\nu}_{\text{ph}}$ and $S^{\mu\alpha\beta}_{\text{ph}}$ up to first
order in the gradient expansion 555Note that the calculations discussed in
this section are completely independent of the previous discussions. However,
without introducing new notations for the energy-momentum tensor and the spin
tensor we use the same as introduced earlier in the manuscript. [34],

$\displaystyle
T^{\mu\nu}_{\text{ph}}=T^{\mu\nu}_{(0)}+T^{\mu\nu}_{\text{ph}(1)},$ (42)
$\displaystyle
S^{\mu\alpha\beta}_{\text{ph}}=u^{\mu}S^{\alpha\beta}+S^{\mu\alpha\beta}_{\text{ph}(1)}.$
(43)

In general, $T^{\mu\nu}_{\text{ph}(1)}$ contains symmetric as well as
antisymmetric components, while $S^{\mu\alpha\beta}_{\text{ph}(1)}$ is a
tensor antisymmetric in last two indices only. This also implies that the spin
tensor in the phenomenological framework is only antisymmetric in last two
indices.

Analogous to the previous sections, we define the nonequilibrium entropy
current up to the first-order term in the gradient expansion as

$\displaystyle\mathcal{S}^{\mu}_{\text{ph}}$
$\displaystyle=\beta_{\nu}T^{\mu\nu}_{\text{ph}}+p\beta^{\mu}-\beta\omega_{\alpha\beta}S^{\alpha\beta}u^{\mu}+\mathcal{O}(\partial^{2}).$
(44)

The divergence of Eq. (44) can then be expressed in the form

$\displaystyle\partial_{\mu}\mathcal{S}^{\mu}_{\text{ph}}$
$\displaystyle=T^{\mu\nu}_{\text{ph}(1s)}\partial_{\mu}\beta_{\nu}+T^{\mu\nu}_{\text{ph}(1a)}(\partial_{\mu}\beta_{\nu}+2\beta\omega_{\mu\nu})+\mathcal{O}(\partial^{3}).$
(45)

Decomposing the symmetric and antisymmetric components of the phenomenological
energy-momentum tensor as
$T^{\mu\nu}_{\text{ph}(1s)}=\overline{h}^{\mu}u^{\nu}+\overline{h}^{\nu}u^{\mu}+\overline{\tau}^{\mu\nu}$
and
$T^{\mu\nu}_{\text{ph}(1a)}=\overline{q}^{\mu}u^{\nu}-\overline{q}^{\nu}u^{\mu}+\overline{\phi}^{\mu\nu}$,
we obtain the following constraint on the divergence of the phenomenological
entropy current

$\displaystyle\partial_{\mu}\mathcal{S}^{\mu}_{\text{ph}}=$
$\displaystyle-\beta\overline{h}^{\mu}\bigg{(}\beta\nabla_{\mu}T-Du_{\mu}\bigg{)}+\beta\overline{\pi}^{\mu\nu}\sigma_{\mu\nu}+\beta\overline{\Pi}\theta$
$\displaystyle-\beta\overline{q}^{\mu}\bigg{(}\beta\nabla_{\mu}T+Du_{\mu}-4u^{\nu}\omega_{\mu\nu}\bigg{)}$
$\displaystyle+\overline{\phi}^{\mu\nu}\left(\Omega_{\mu\nu}+2\beta\Delta^{\alpha}_{\mu}\Delta^{\beta}_{\nu}\omega_{\alpha\beta}\right)\geq
0$ (46)

with dissipative currents
$\overline{h},\overline{\pi},\overline{\Pi},\overline{q}$ and
$\overline{\phi}$ satisfying relations similar to those shown in Eqs.
(18)-(22).

Let us now discuss the improved canonical framework that results from applying
the pseudogauge transformation to the phenomenological framework. In this
case, using the pseudogauge transformation as

$\displaystyle
T^{\mu\nu}=T^{\mu\nu}_{\text{ph}}+\frac{1}{2}\partial_{\lambda}\bigg{(}\Psi^{\lambda\mu\nu}-\Psi^{\mu\lambda\nu}-\Psi^{\nu\lambda\mu}\bigg{)},$
(47) $\displaystyle
S^{\mu\alpha\beta}=S^{\mu\alpha\beta}_{\text{ph}}-\Psi^{\mu\alpha\beta},$ (48)

and choosing the pseudogauge potential $\Psi^{\lambda\mu\nu}$ in the form

$\displaystyle\Psi^{\mu\alpha\beta}=S_{\text{ph}}^{\alpha\mu\beta}-S_{\text{ph}}^{\beta\mu\alpha},$
(49)

we obtain the following constitutive relations

$\displaystyle T^{\mu\nu}=\widetilde{T}^{\mu\nu}_{\text{can}}$
$\displaystyle=T^{\mu\nu}_{\text{ph}}+\delta T^{\mu\nu}_{\text{ph(1)}},$ (50)
$\displaystyle S^{\mu\alpha\beta}=S_{\text{can}}^{\mu\alpha\beta}$
$\displaystyle=S_{\text{ph}}^{\mu\alpha\beta}+S_{\text{ph}}^{\beta\mu\alpha}+S_{\text{ph}}^{\alpha\beta\mu},$
(51)

where $S_{\text{can}}^{\mu\alpha\beta}$ is totally antisymmetric and we have
used in Eq. (50) the expression of $S^{\nu\mu\lambda}_{\text{ph}}$ from Eq.
(43).

It should be emphasized that the term $\delta
T^{\mu\nu}_{\text{ph(1)}}=\partial_{\lambda}\left(u^{\nu}S^{\mu\lambda}\right)$
was introduced in the previous section in a heuristic way to obtain a well-
defined description within the canonical formalism. Interestingly, such a term
is obtained here in a natural way applying the pseudogauge transformation to
the phenomenological framework.

One can observe that the additional term $\delta T^{\mu\nu}_{\text{ph(1)}}$
gives rise to the first-order corrections for various components of the
initial energy-momentum tensor, which are additional to those arising from
$T^{\mu\nu}_{\text{ph}(1)}$. Thus, the resulting energy-momentum tensor can be
written in the following form

$\displaystyle\widetilde{T}^{\mu\nu}_{\text{can}}$
$\displaystyle=T^{\mu\nu}_{(0)}+T^{\mu\nu}_{\text{ph}(1)}+\delta
T^{\mu\nu}_{\text{ph(1)}},$ (52)
$\displaystyle=\widetilde{\varepsilon}u^{\mu}u^{\nu}-p\Delta^{\mu\nu}+\widetilde{h}^{\mu}u^{\nu}+\widetilde{h}^{\nu}u^{\mu}+\widetilde{\tau}^{\mu\nu}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\widetilde{q}^{\mu}u^{\nu}-\widetilde{q}^{\nu}u^{\mu}+\widetilde{\phi}^{\mu\nu},$
(53)

where

$\displaystyle\widetilde{\varepsilon}=\varepsilon+\delta\overline{\varepsilon},$
(54)
$\displaystyle\widetilde{h}^{\mu}=\overline{h}^{\mu}+\delta\overline{h}^{\mu},$
(55)
$\displaystyle\widetilde{\tau}^{\mu\nu}=\overline{\tau}^{\mu\nu}+\delta\overline{\tau}^{\mu\nu},$
(56)
$\displaystyle\widetilde{q}^{\mu}=\overline{q}^{\mu}+\delta\overline{q}^{\mu},$
(57)
$\displaystyle\widetilde{\phi}^{\mu\nu}=\overline{\phi}^{\mu\nu}+\delta\overline{\phi}^{\mu\nu}.$
(58)

For the details of the decomposition in Eq. (53) and the definitions of the
additional $\delta$ contributions in Eqs. (54)-(58) see B.

As mentioned above, the nonequilibrium entropy current can be defined as

$\displaystyle\mathcal{S}^{\mu}_{\text{can}}$
$\displaystyle=\beta_{\nu}\widetilde{T}^{\mu\nu}_{\text{can}}+p\beta^{\mu}-\beta^{\mu}\omega_{\alpha\beta}S^{\alpha\beta}+\mathcal{O}(\partial^{2})$
$\displaystyle={\cal
S}_{(0)}^{\mu}+\beta_{\nu}\widetilde{T}^{\mu\nu}_{\text{can}(1)}+\mathcal{O}(\partial^{2}).$
(59)

Using the conservation of canonical energy-momentum tensor, the divergence of
the canonical entropy current can be written as follows

$\displaystyle\partial_{\mu}\mathcal{S}^{\mu}_{\text{can}}$
$\displaystyle=\partial_{\mu}\beta_{\nu}\widetilde{T}^{\mu\nu}_{\text{can}(1)}-\beta\omega_{\alpha\beta}\partial_{\mu}(u^{\mu}S^{\alpha\beta})+\mathcal{O}(\partial^{3})$
$\displaystyle=\partial_{\mu}\beta_{\nu}T^{\mu\nu}_{\text{ph}(1)}+\partial_{\mu}\bigg{[}\beta_{\nu}\partial_{\lambda}(u^{\nu}S^{\mu\lambda})\bigg{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\beta\omega_{\alpha\beta}\partial_{\mu}(u^{\mu}S^{\alpha\beta})+\mathcal{O}(\partial^{3}).$
(60)

Absorbing that total derivative term into the definition of the entropy
current, we get

$\displaystyle\partial_{\mu}\mathcal{S}^{\mu}_{\text{ph}}=\partial_{\mu}\beta_{\nu}T^{\mu\nu}_{\text{ph}(1)}-\beta\omega_{\alpha\beta}\partial_{\mu}(u^{\mu}S^{\alpha\beta})+\mathcal{O}(\partial^{3})$
(61)

where
$\mathcal{S}^{\mu}_{\text{ph}}=\mathcal{S}^{\mu}_{\text{can}}-\beta_{\nu}\partial_{\lambda}(u^{\nu}S^{\mu\lambda})$.
Thus, we conclude that the entropy current constraints in these two frameworks
are equivalent. Although the various dissipative currents are different due to
the contribution coming from the pseudogauge transformation.

##  6 Summary and conclusions

We formulate the first-order dissipative spin hydrodynamics for the canonical
framework with a totally antisymmetric spin tensor. We argue that the naive
use of a general energy-momentum tensor containing both a symmetric and an
antisymmetric part, together with a totally antisymmetric spin tensor, is in
conflict with the principle of entropy production. We show that this problem
can be solved by an improved form of the canonical energy-momentum tensor,
obtained by modifying the canonical energy-momentum tensor with a suitable
totally divergent term that does not affect the conservation of the energy-
momentum tensor. The origin of such a term can also be justified using the
concept of the pseudogauge transformation. Using such an improved form, we
show that the framework of dissipative spin hydrodynamics is well defined
within the canonical framework. We argue that one can always recover the
phenomenological framework of spin hydrodynamics from such an improved
canonical framework using an appropriate pseudogauge transformation, and vice
versa. Therefore, the improved canonical framework and the phenomenological
framework of first-order dissipative spin hydrodynamics are equivalent since
they are connected by a suitable pseudogauge transformation. Although a
pseudogauge transformation can lead to modified dissipative currents, entropy
current analysis shows that the entropy production condition for a dissipative
system remains unchanged in these two frameworks, i.e., the entropy current
constraint is independent of a pseudogauge transformation.

Acknowledgements. This work was supported in part by the Polish National
Science Centre Grants No. 2016/23/B/ST2/00717 and No. 2018/30/E/ST2/00432.
A.D. would like to thank Shi Pu for some important discussions and
clarifications on the formulation of dissipative spin hydrodynamics.

## Appendix A

Here we present the detailed derivation of Eq. (17). We start with Eq. (12)
which using the decomposition of symmetric and antisymmetric components of
$T^{\mu\nu}_{\text{can}(1)}$ in terms of the dissipative currents as given in
Eqs. (13) and (14) can be rewritten as

$\displaystyle\partial_{\mu}\mathcal{S}^{\mu}_{\text{can}}$
$\displaystyle=2\beta\omega_{\alpha\beta}\left[T^{\alpha\beta}_{\text{can}(1a)}+\partial_{\mu}\Phi^{\mu\alpha\beta}_{\text{can(0)}}\right]$
$\displaystyle~{}~{}~{}~{}+h^{\mu}u^{\nu}(\partial_{\mu}\beta_{\nu}+\partial_{\nu}\beta_{\mu})+\tau^{\mu\nu}\partial_{\mu}\beta_{\nu}$
$\displaystyle~{}~{}~{}~{}+q^{\mu}u^{\nu}(\partial_{\mu}\beta_{\nu}-\partial_{\nu}\beta_{\mu})+\phi^{\mu\nu}\partial_{\mu}\beta_{\nu}.$
(62)

Using the decomposition $\partial_{\mu}\equiv\nabla_{\mu}+u_{\mu}D$, with
$u^{\mu}\nabla_{\mu}=0,\nabla_{\mu}=\Delta^{\alpha}_{\mu}\partial_{\alpha}$,
and $D=u^{\mu}\partial_{\mu}$, it can be shown that

$\displaystyle\partial_{\mu}\beta_{\nu}=u_{\nu}\nabla_{\mu}\beta+u_{\nu}u_{\mu}D\beta+\beta\nabla_{\mu}u_{\nu}+\beta
u_{\mu}Du_{\nu}.$ (63)

This implies

$\displaystyle
h^{\mu}u^{\nu}\left(\partial_{\mu}\beta_{\nu}+\partial_{\nu}\beta_{\mu}\right)=$
$\displaystyle-\beta h^{\mu}\left(\beta\nabla_{\mu}T-Du_{\mu}\right),$ (64)

and

$\displaystyle
q^{\mu}u^{\nu}\left(\partial_{\mu}\beta_{\nu}-\partial_{\nu}\beta_{\mu}\right)=-\beta
q^{\mu}\left(\beta\nabla_{\mu}T+Du_{\mu}\right).$ (65)

The symmetric tensor $\tau^{\mu\nu}$ can be decomposed into trace and
traceless part, $\tau^{\mu\nu}=\pi^{\mu\nu}+\Pi\Delta^{\mu\nu}$. Using Eq.
(63) and the fact that $\tau^{\mu\nu}$ is orthogonal to the fluid four
velocity we obtain

$\displaystyle\tau^{\mu\nu}\partial_{\mu}\beta_{\nu}=\tau^{\mu\nu}\beta\nabla_{\mu}u_{\nu}=\beta\left(\pi^{\mu\nu}+\Pi\Delta^{\mu\nu}\right)\nabla_{\mu}u_{\nu}$
$\displaystyle=\beta\pi^{\mu\nu}\bigg{[}\frac{1}{2}\left(\nabla_{\mu}u_{\nu}+\nabla_{\nu}u_{\mu}\right)-\frac{1}{3}\Delta_{\mu\nu}\nabla^{\alpha}u_{\alpha}+\frac{1}{3}\Delta_{\mu\nu}\nabla^{\alpha}u_{\alpha}\bigg{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\beta\Pi\Delta^{\mu\nu}\nabla_{\mu}u_{\nu}$
$\displaystyle=\beta\pi^{\mu\nu}\bigg{[}\nabla_{(\mu}u_{\nu)}-\frac{1}{3}\theta\Delta_{\mu\nu}\bigg{]}+\beta\Pi\theta$
$\displaystyle=\beta\pi^{\mu\nu}\sigma_{\mu\nu}+\beta\Pi\theta,$ (66)

where $\theta=\nabla^{\alpha}u_{\alpha}$. In the third line of the above
equation we have used that $\pi^{\mu\nu}$ is traceless, i.e.
$\pi^{\mu}_{~{}\mu}=0$ and $\pi^{\mu\nu}u_{\nu}=0=\pi^{\mu\nu}u_{\mu}$. It is
also important to note that
$\sigma_{\mu\nu}=\nabla_{(\mu}u_{\nu)}-\frac{1}{3}\theta\Delta_{\mu\nu}$ is
traceless, i.e. $\sigma^{\mu}_{~{}\mu}=0$, and orthogonal to the fluid four
velocity, i.e. $\sigma^{\mu\nu}u_{\mu}=0=\sigma^{\mu\nu}u_{\nu}$.

Moreover it can be shown that

$\displaystyle\phi^{\mu\nu}\partial_{\mu}\beta_{\nu}=\frac{1}{2}\phi^{\mu\nu}\left(\partial_{\mu}\beta_{\nu}-\partial_{\nu}\beta_{\mu}\right)=\phi^{\mu\nu}\partial_{[\mu}\beta_{\nu]}$
$\displaystyle=\beta\phi^{\mu\nu}\nabla_{[\mu}u_{\nu]}$
$\displaystyle=\frac{\beta}{2}\phi^{\mu\nu}\left(\Delta^{\alpha}_{\mu}\partial_{\alpha}u_{\nu}-\Delta^{\alpha}_{\nu}\partial_{\alpha}u_{\mu}\right)$
$\displaystyle=\phi^{\mu\nu}\Delta^{\alpha}_{~{}\mu}\Delta^{\beta}_{~{}\nu}\partial_{[\alpha}\left(\beta
u_{\beta]}\right)=\phi^{\mu\nu}\Omega_{\mu\nu},$ (67)

where we defined
$\Omega_{\mu\nu}\equiv\Delta^{\alpha}_{~{}\mu}\Delta^{\beta}_{~{}\nu}\partial_{[\alpha}\beta_{\beta]}=\beta\nabla_{[\mu}u_{\nu]}$.

Using Eqs. (64)-(67), the divergence of the entropy current as given in Eq.
(62) can be expressed as

$\displaystyle\partial_{\mu}\mathcal{S}^{\mu}_{\text{can}}$
$\displaystyle=2\beta\omega_{\alpha\beta}\partial_{\mu}\Phi^{\mu\alpha\beta}_{\text{can(0)}}$
$\displaystyle~{}-\beta
h^{\mu}\bigg{(}\beta\nabla_{\mu}T-Du_{\mu}\bigg{)}+\beta\pi^{\mu\nu}\sigma_{\mu\nu}+\beta\Pi\partial^{\alpha}u_{\alpha}$
$\displaystyle-\beta
q^{\mu}\left(\beta\nabla_{\mu}T+Du_{\mu}-4\omega_{\mu\nu}u^{\nu}\right)$
$\displaystyle+\phi^{\mu\nu}\left(\Omega_{\mu\nu}+2\beta\Delta^{\alpha}_{~{}\mu}\Delta^{\beta}_{~{}\nu}\omega_{\alpha\beta}\right)$
(68)

Using in the above equation decompositions (15)-(16) we obtain Eq. (17).

## Appendix B

Here we provide details on the decomposition of the first-order gradient term,
the term $\delta
T^{\mu\nu}_{\text{ph(1)}}=\partial_{\lambda}\left(u^{\nu}S^{\mu\lambda}\right)$
into different irreducible tensor structures. This can be done by first
decomposing $\delta T^{\mu\nu}_{\text{ph(1)}}$ into symmetric and
antisymmetric parts

$\displaystyle\delta T^{\mu\nu}_{\text{ph(1)}}$ $\displaystyle=\delta
T^{(\mu\nu)}_{\text{ph(1)}}+\delta T^{[\mu\nu]}_{\text{ph(1)}},$ (69)

which can be decomposed further in the following way

$\displaystyle\delta
T^{(\mu\nu)}_{\text{ph(1)}}=\delta\overline{\varepsilon}u^{\mu}u^{\nu}+\delta\overline{h}^{\mu}u^{\nu}+\delta\overline{h}^{\nu}u^{\mu}+\delta\overline{\tau}^{\mu\nu}$
(70) $\displaystyle\delta
T^{[\mu\nu]}_{\text{ph(1)}}=\delta\overline{q}^{\mu}u^{\nu}-\delta\overline{q}^{\nu}u^{\mu}+\delta\overline{\phi}^{\mu\nu},$
(71)

along with the conditions
$\delta\overline{h}^{\mu}u_{\mu}=0,~{}\delta\overline{\tau}^{\mu\nu}u_{\mu}=0,~{}\delta\overline{q}^{\mu}u_{\mu}=0,~{}\delta\overline{\phi}^{\mu\nu}u_{\mu}=0,~{}\delta\overline{\tau}^{\mu\nu}=\delta\overline{\tau}^{\nu\mu},~{}\delta\overline{\phi}^{\mu\nu}=-\delta\overline{\phi}^{\nu\mu}$.

The symmetric tensor $\delta\overline{\tau}^{\mu\nu}$ can be further
decomposed into a traceless part and a trace

$\displaystyle\delta\overline{\tau}^{\mu\nu}$
$\displaystyle=\bigg{(}\Delta^{\mu}_{(\alpha}\Delta^{\nu}_{\beta)}-\frac{1}{3}\Delta^{\mu\nu}\Delta_{\alpha\beta}\bigg{)}\delta\overline{\tau}^{\alpha\beta}+\frac{1}{3}\Delta^{\mu\nu}\Delta_{\alpha\beta}\delta\overline{\tau}^{\alpha\beta}$
$\displaystyle=\delta\overline{\tau}^{\langle\mu\nu\rangle}+\delta\overline{\tau}\Delta^{\mu\nu}.$
(72)

Note that $\delta\overline{\phi}^{\mu\nu}$ is antisymmetric, therefore it is
traceless.

The unknown quantities in Eqs. (70) and (71) can be obtained by taking the
following projections

$\displaystyle\delta\overline{\varepsilon}=u_{\mu}u_{\nu}\partial_{\lambda}(u^{(\nu}S^{\mu)\lambda})=u_{\mu}\partial_{\lambda}S^{\mu\lambda},$
(73)
$\displaystyle\delta\overline{h}^{\beta}=\Delta^{\beta}_{~{}\nu}u_{\mu}\partial_{\lambda}(u^{(\nu}S^{\mu)\lambda})=\frac{1}{2}u_{\mu}(\partial_{\lambda}u^{\beta})S^{\mu\lambda}+\frac{1}{2}\Delta^{\beta}_{~{}\nu}\partial_{\lambda}S^{\nu\lambda},$
(74) $\displaystyle\delta\overline{\tau}^{\langle\mu\nu\rangle}=$
$\displaystyle\bigg{(}\Delta^{\mu}_{(\alpha}\Delta^{\nu}_{\beta)}-\frac{1}{3}\Delta^{\mu\nu}\Delta_{\alpha\beta}\bigg{)}\partial_{\lambda}(u^{(\beta}S^{\alpha)\lambda})$
$\displaystyle=$
$\displaystyle\bigg{(}\Delta^{\mu}_{(\alpha}\Delta^{\nu}_{\beta)}-\frac{1}{3}\Delta^{\mu\nu}\Delta_{\alpha\beta}\bigg{)}\partial_{\lambda}(u^{\alpha}S^{\beta\lambda}),$
(75)
$\displaystyle\delta\overline{\tau}=\frac{1}{3}\Delta_{\mu\nu}\partial_{\lambda}(u^{(\nu}S^{\mu)\lambda})=\frac{1}{3}\Delta_{\mu\nu}\partial_{\lambda}(u^{\nu}S^{\mu\lambda}),$
(76)
$\displaystyle\delta\overline{q}^{\beta}=u_{\nu}\Delta^{\beta}_{~{}\mu}\partial_{\lambda}(u^{[\nu}S^{\mu]\lambda}),$
(77)
$\displaystyle\delta\overline{\phi}^{\alpha\beta}=\Delta^{[\alpha}_{~{}\mu}\Delta^{\beta]}_{~{}\nu}\partial_{\lambda}(u^{[\nu}S^{\mu]\lambda})=\Delta^{[\alpha}_{~{}\mu}\Delta^{\beta]}_{~{}\nu}\partial_{\lambda}(u^{\nu}S^{\mu\lambda}).$
(78)

Therefore, using Eqs. (69), (70) and (71), the canonical energy-momentum
tensor takes the form (53) where $\delta$ contributions in Eqs. (54)-(58) are
calculable from Eqs. (73)-(78).

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