Relativistic spin hydrodynamics revisited with general rotation by entropy-current analysis

Relativistic hydrodynamics, as an effective theory in terms of the IR variables, has proven highly successful in describing the macroscopic behavior of various many-body systems, spanning from astrophysics to relativistic heavy-ion collisions. Spin-orbit coupling plays a significant role in relativistic fluids with spinful constituents, leading to spin polarization, particularly in the presence of substantial angular momentum and/or strong vorticity. The polarization phenomena have been intensively studied in heavy-ion collisions since long Liang:2004ph ; Liang:2004xn ; Voloshin:2004ha ; Betz:2007kg ; Becattini:2007sr ; Huang:2011ru , and its occurrence has been confirmed through experimental measurements of hyperon polarization STAR:2017ckg ; STAR:2018gyt ; STAR:2020xbm and vector-meson spin alignment ALICE:2019aid ; STAR:2022fan ; ALICE:2022dyy . The global polarization of ${\Lambda}$ hyperons has been effectively captured by relativistic hydrodynamic models incorporating thermalized spin degrees of freedom Becattini:2007sr ; Becattini:2007nd ; Becattini:2013vja ; Becattini:2013fla ; Becattini:2015ska ; Becattini:2016gvu ; Karpenko:2016jyx ; Pang:2016igs ; Xie:2017upb . However, theoretical calculations Becattini:2017gcx ; Becattini:2020ngo concerning the azimuthal-angle dependence of hyperon polarization have yielded results with opposite signs compared to the experimental data Niida:2018hfw ; STAR:2019erd . This discrepancy, known as the“sign problem” in explaining local spin polarization, has also been addressed in related reviews Liu:2020ymh ; Gao:2020vbh ; Huang:2020xyr ; Becattini:2022zvf .

Relativistic spin hydrodynamics is a promising framework for understanding the sign problem, which has been advancing rapidly through various approaches in a significant body of research Montenegro:2017rbu ; Montenegro:2017lvf ; Florkowski:2017ruc ; Florkowski:2018fap ; Montenegro:2018bcf ; Hattori:2019lfp ; Li:2019qkf ; Montenegro:2020paq ; Garbiso:2020puw ; Gallegos:2020otk ; Fukushima:2020ucl ; Bhadury:2020puc ; Li:2020eon ; Shi:2020htn ; Hu:2021lnx ; Gallegos:2021bzp ; Peng:2021ago ; Hongo:2021ona ; Wang:2021ngp ; Wang:2021wqq ; She:2021lhe ; Hu:2021pwh ; Cartwright:2021qpp ; Hongo:2022izs ; Weickgenannt:2022zxs ; Bhadury:2022ulr ; Cao:2022aku ; Daher:2022xon ; Singh:2022ltu ; Gallegos:2022jow ; Biswas:2023qsw ; Xie:2023gbo ; Becattini:2023ouz . Essentially, relativistic spin hydrodynamics is constructed by introducing the spin modes linked to Lorentz symmetry as additional IR variables, alongside the conventional hydrodynamic variables related to relativistic translation invariance. The spin variables can typically be categorized into rotation and boost components based on the subgroups of the Lorentz transformation. The former characterizes the spin polarization in the fluid, while the role of the latter remains unclear. An entropy-current analysis approach to spin hydrodynamics was introduced in Hattori:2019lfp with the canonical spin current being antisymmetric only in its last two indices, referred to as the phenomenological formulation. The same approach is also implemented in the canonical formulations Hongo:2021ona ; Cao:2022aku where the canonical spin current of Dirac fermions is totally anti-symmetric. It is noteworthy that the canonical formulations in Hongo:2021ona ; Cao:2022aku are developed without considering the degree of freedom associated with boost symmetry. In contrast, another canonical formulation incorporating boost variables is presented in Daher:2022xon , which is connected to the phenomenological formulation through a pseudogauge transformation. It is meaningful to explore the inclusion of boost variables in a general relativistic framework, since boost symmetry, along with rotation, is a fundamental aspect of Lorentz covariant hydrodynamics. Furthermore, hydrodynamics with varying approaches to the treatment of boost modes may exhibit distinct behavior when subjected to spin-orbit coupling.

In this paper, we investigate the canonical formulation of spin hydrodynamics by considering its applicability to spinful fluids across a broad range of thermal vorticity intensities. In this scenario, the spin potential is presumed to be a general antisymmetric tensor in order to coincide with the thermal vorticity in equilibrium. Moreover, the magnitudes of both the thermal vorticity and the spin potential are assessed without regard to perturbative gradients. In the case of general spinful fluids, the entropy-current analysis suggests that the entropy principle cannot be fulfilled unless spin variables are included in the constitutive relations of the stress-energy tensor, with these spin variables needing to account for the degree of freedom linked to boost symmetry. The constitutive relations in the presence of boost variables have not yet been definitively determined. For simplilcity, we opt to connect the canonical formulation of spin hydrodynamics to the phenomenological approach through pseudogauge transformation. The linear-mode analysis using the resulting spin hydrodynamic equations reveals that the spin and hydrodynamic modes in this canonical formulation exhibit distinct dispersion relations compared to the phenomenological formulation up to the second order of gradients.

Throughout this paper, we adopt the mostly plus Minkowski metric ${\eta}^{{\mu}{\nu}}\equiv$diag$\left(-1,1,1,1\right)$. For the Levi-Civita symbol, we use the convention ${\epsilon}^{0123}=-{\epsilon}_{0123}=1$ and ${\epsilon}^{123}={\epsilon}_{123}=1$. We also define the notations $X^{({\mu}{\nu})}\equiv\frac{1}{2}\left(X^{{\mu}{\nu}}+X^{{\nu}{\mu}}\right)$ and $X^{[{\mu}{\nu}]}\equiv\frac{1}{2}\left(X^{{\mu}{\nu}}-X^{{\nu}{\mu}}\right)$.

The conservation equations of the Noether’s currents from the relativistic translation and Lorentz symmetry are

	$\displaystyle\partial_{\mu}{\Theta}^{{\mu}{\nu}}=0,$		(1)
	$\displaystyle\partial_{\alpha}J^{{\alpha}{\mu}{\nu}}=\partial_{\alpha}{\Sigma}^{{\alpha}{\mu}{\nu}}+{\Theta}^{{\mu}{\nu}}-{\Theta}^{{\nu}{\mu}}=0,$		(2)

where the total angular momentum is $J^{{\alpha}{\mu}{\nu}}\equiv{\Sigma}^{{\alpha}{\mu}{\nu}}+x^{\mu}{\Theta}^{{\alpha}{\nu}}-x^{\nu}{\Theta}^{{\alpha}{\mu}}$ with ${\Sigma}^{{\alpha}{\mu}{\nu}}$ being the spin tensor. The canonical stress-energy tensor ${\Theta}^{{\mu}{\nu}}$ is asymmetric in its two indices, comprising both symmetric and antisymmetric components, while the total angular momentum density $J^{{\alpha}{\mu}{\nu}}$ is antisymmetric in its last two indices.

In classical physics, the hydrodynamics of the Quark-Gluon Plasma (QGP) is formulated with IR variables while the details at a microscopic level are averaged out. For simplicity, we consider spin hydrodynamics without conserved charges. We follow Hongo:2021ona to assume that the coarse-grained spin tensor in hydrodynamics retains the entire antisymmetry of its corresponding quantum operator. Employing the fluid four-velocity $u^{\mu}$ with $u^{\mu}u_{\mu}=-1$, the spin density is introduced as ${\cal R}^{{\mu}{\nu}}\equiv-u_{\alpha}{\Sigma}^{{\alpha}{\mu}{\nu}}$ with ${\cal R}^{{\mu}{\nu}}=-{\cal R}^{{\nu}{\mu}}$. The resulting spin density satisfies ${\cal R}^{{\mu}{\nu}}u_{\nu}=0$ due to the total antisymmetry of ${\Sigma}^{{\mu}{\nu}{\alpha}}$. As a result, ${\cal R}^{{\mu}{\nu}}$ is fully saturated with only three independent components associated with the spatial rotation symmetry, while the remaining three attached to the boost symmetry are absent in the spin tensor ${\Sigma}^{{\mu}{\nu}{\alpha}}$¹¹1In general, a totally antisymmetric rank-3 tensor ${\Sigma}^{{\alpha}{\mu}{\nu}}$ can contain, at most, four independent components, of which only three can be present in ${\cal R}^{{\mu}{\nu}}$ by the definition ${\cal R}^{{\mu}{\nu}}\equiv-u_{\alpha}{\Sigma}^{{\alpha}{\mu}{\nu}}$. In any case, the totally antisymmetric spin tensor cannot encompass all six independent components associated with Lorentz symmetry.. In such condition, it is yet to be determined whether the spin variables, especially the boost ones, can be generally absent in the coarse-grained stress-energy tensor. To this end, we perform an entropy-current analysis to constrain the presence of spin variables in hydrodynamics with the entropy principle.

The totally antisymmetric spin tensor can be decomposed into longitudinal and transverse parts as

$\displaystyle{\Sigma}^{{\mu}{\nu}{\alpha}}=u^{{\mu}}{\cal S}^{{\nu}{\alpha}}+u^{{\nu}}{\cal S}^{{\alpha}{\mu}}+u^{{\alpha}}{\cal S}^{{\mu}{\nu}}+{\epsilon}^{{\mu}{\nu}{\alpha}{\sigma}}u_{\sigma}{\mathring{{\Sigma}}},$

where the antisymmetric component ${\cal S}^{{\mu}{\nu}}$ can be further decomposed as

$\displaystyle{\cal S}^{{\mu}{\nu}}={\epsilon}^{{\mu}{\nu}{\rho}{\sigma}}{\cal R}_{\rho}u_{\sigma}+2u^{[{\mu}}{\cal B}^{{\nu}]},$

with ${\cal R}_{\alpha}=\frac{1}{2}{\epsilon}_{{\alpha}{\mu}{\nu}{\sigma}}{\cal S}^{{\mu}{\nu}}u^{\sigma}$ and ${\cal B}^{\mu}={\cal S}^{{\mu}{\nu}}u_{\nu}$. Noting the identities

	$\displaystyle{\epsilon}^{{\mu}{\nu}{\alpha}{\sigma}}{\cal R}_{\sigma}$	$\displaystyle=\left(u^{{\mu}}{\epsilon}^{{\nu}{\alpha}{\rho}{\sigma}}+u^{\nu}{\epsilon}^{{\alpha}{\mu}{\rho}{\sigma}}+u^{\alpha}{\epsilon}^{{\mu}{\nu}{\rho}{\sigma}}\right){\cal R}_{\rho}u_{\sigma},$
	$\displaystyle 0$	$\displaystyle=u^{{\mu}}u^{[{\nu}}{\cal B}^{{\alpha}]}+u^{\nu}u^{[{\alpha}}{\cal B}^{{\mu}]}+u^{\alpha}u^{[{\mu}}{\cal B}^{{\nu}]},$		(3)

one readily writes the spin tensor into

$\displaystyle{\Sigma}^{{\mu}{\nu}{\alpha}}={\epsilon}^{{\mu}{\nu}{\alpha}{\sigma}}\left({\cal R}_{\sigma}+u_{\sigma}{\mathring{{\Sigma}}}\right),$

(4)

which immediately gives ${\cal R}^{{\mu}{\nu}}={\epsilon}^{{\mu}{\nu}{\rho}{\sigma}}{\cal R}_{\rho}u_{\sigma}$. Since ${\cal R}_{\rho}$ captures all the three independent components of ${\cal R}^{{\mu}{\nu}}$ in a covariant form, ${\mathring{{\Sigma}}}$ is automatically left as corrections out of equilibrium to be constrained by the entropy principle. Given ${\cal R}^{{\mu}{\nu}}$ representing the rotation components of the spin modes ${\cal S}^{{\mu}{\nu}}$, ${\cal B}^{{\mu}{\nu}}=2u^{[{\mu}}{\cal B}^{{\nu}]}$ as the rest part naturally denotes the boost modes where ${\cal B}^{\mu}$ contains all the three independent components related to the boost symmetry.

The local thermodynamic relations generalized with spin variables are

	$\displaystyle Ts={\varepsilon}+p-\frac{1}{2}{\omega}_{{\mu}{\nu}}{\cal S}^{{\mu}{\nu}},$		(5)
	$\displaystyle Tds=d{\varepsilon}-\frac{1}{2}{\omega}_{{\mu}{\nu}}d{\cal S}^{{\mu}{\nu}},$		(6)
	$\displaystyle sdT=dp-\frac{1}{2}{\cal S}^{{\mu}{\nu}}d{\omega}_{{\mu}{\nu}},$		(7)

where $T$, $s$, ${\varepsilon}$, $p$ and ${\omega}_{{\mu}{\nu}}=-{\omega}_{{\nu}{\mu}}$ denoting the local temperature, entropy density, energy density, pressure and spin potential respectively. An important point to note is that the local thermodynamic relations do not generally hold in the quantum-statistical description of a relativistic fluid Becattini:2023ouz , where the thermodynamic quantities, such as temperature, thermal velocity and spin potential can be unambiguously defined at the local thermodynamic equilibrium (LTE) Becattini:2014yxa . In this work, we adhere to the traditional hydrodynamics viewpoint Bhattacharya:2011tra ; Kovtun:2012rj and assume that it is always possible to establish the local thermodynamic relations with a proper redefinition of the thermodynamic quantities in a state near equilibrium. Although the thermodynamic quantities defined in the two frameworks may share different values, they should approach the same values as the fluid evolves to the global thermodynamic equilibrium (GTE).

Additionally, we take a general antisymmetric spin potential ${\omega}_{{\mu}{\nu}}$ without the requirement ${\omega}_{{\mu}{\nu}}u^{\nu}=0$ even when it is conjugate to ${\cal R}^{{\mu}{\nu}}$ in the local thermodynamic relations, as is case in the phenomenological formulation of spin hydrodynamicsHattori:2019lfp ; Fukushima:2020ucl ; Wang:2021ngp ; Wang:2021wqq ; Xie:2023gbo ; Daher:2022xon . One can also separate ${\omega}_{{\mu}{\nu}}$ into rotation and boost parts as ${\omega}_{{\mu}{\nu}}=r_{{\mu}{\nu}}+b_{{\mu}{\nu}}$ where

	$\displaystyle r_{{\mu}{\nu}}={\epsilon}_{{\mu}{\nu}{\rho}{\sigma}}r^{\rho}u^{\sigma},\quad b_{{\mu}{\nu}}=2b_{[{\mu}}u_{{\nu}]}$
	$\displaystyle r^{\sigma}=\frac{1}{2}{\epsilon}^{{\sigma}{\rho}{\mu}{\nu}}u_{\rho}r_{{\mu}{\nu}},\quad b_{\nu}=u^{\mu}b_{{\mu}{\nu}}.$

The conjugations in (5)-(7) differ from the canonical formulations in Hongo:2021ona ; Cao:2022aku where the boost variables are absence and the spin potential is chosen as $r_{{\mu}{\nu}}$ with $r_{{\mu}{\nu}}u^{\nu}=0$. This difference is nontrivial. Although the Gibbs energy density $g$ from (5), i.e.,

$\displaystyle g={\varepsilon}+p-Ts=\frac{1}{2}{\omega}_{{\mu}{\nu}}{\cal S}^{{\mu}{\nu}}=\frac{1}{2}\left(r_{{\mu}{\nu}}{\cal R}^{{\mu}{\nu}}+b_{{\mu}{\nu}}{\cal B}^{{\mu}{\nu}}\right)=r_{{\mu}}{\cal R}^{{\mu}}+b_{{\mu}}{\cal B}^{{\mu}},$

(8)

gets no contributions from $b_{{\mu}{\nu}}{\cal R}^{{\mu}{\nu}}=r_{{\mu}{\nu}}{\cal B}^{{\mu}{\nu}}=0$, the conjugations in (6) and (7),

	$\displaystyle\frac{1}{2}{\omega}_{{\mu}{\nu}}d{\cal S}^{{\mu}{\nu}}=r_{{\mu}}d{\cal R}^{{\mu}}+b_{{\mu}}u_{{\nu}}{\epsilon}^{{\mu}{\nu}{\rho}{\sigma}}{\cal R}_{\rho}du_{{\sigma}}+b_{{\mu}}d{\cal B}^{{\mu}}+{\epsilon}_{{\mu}{\nu}{\rho}{\sigma}}r^{{\rho}}u^{{\nu}}{\cal B}^{\nu}du^{{\mu}},$
	$\displaystyle\frac{1}{2}{\cal S}^{{\mu}{\nu}}d{\omega}_{{\mu}{\nu}}={\cal R}^{{\mu}}dr_{{\mu}}-b_{{\mu}}u_{{\nu}}{\epsilon}^{{\mu}{\nu}{\rho}{\sigma}}{\cal R}_{\rho}du_{{\sigma}}+{\cal B}^{{\mu}}db_{{\mu}}-{\epsilon}_{{\mu}{\nu}{\rho}{\sigma}}r^{{\rho}}u^{{\nu}}{\cal B}^{\nu}du^{{\mu}},$		(9)

give $b$-${\cal R}$ and $r$-${\cal B}$ conjugations which are generally nonvanishing in the presence of vorticity where the non-inertial motion of fluid evolves along with the spin variables. It may seem that the non-inertial $b$-${\cal R}$ and $r$-${\cal B}$ terms are not well defined contributions to thermodynamic potentials. Actually, the velocity dependence is just an artifact from the introducing of rotation ${\cal R}^{\mu}$ and boost ${\cal B}^{\mu}$ vectors orthogonal to four-velocity as thermodynamic quantities into the generalized local thermodynamic relations. The general antisymmetric ${\omega}_{{\mu}{\nu}}$ is more physically appropriate than $r_{{\mu}{\nu}}$ with $r_{{\mu}{\nu}}u^{{\mu}}=0$, in the sense that ${\omega}_{{\mu}{\nu}}$ can smoothly transition into the GTE to coincide with the general thermal vorticity ${\varpi}_{{\mu}{\nu}}\equiv\partial_{[{\mu}}{\beta}_{{\nu}]}$ which may not necessarily be orthogonal to $u_{{\mu}}$. Especially, when the acceleration part $u^{\mu}{\varpi}_{{\mu}{\nu}}$ is as strong as the spatial part $\frac{1}{2}{\epsilon}^{{\sigma}{\rho}{\mu}{\nu}}u_{\rho}{\varpi}_{{\mu}{\nu}}$, we will see in the entropy-current analysis that the entropy principle with a general antisymmetric ${\omega}_{{\mu}{\nu}}$ necessarily requires the presence of ${\cal B}$ in the constitutive relations.

We start with a general tensor decomposition in spin hydrodynamics as follows,

	$\displaystyle{\Theta}^{{\mu}{\nu}}={\varepsilon}u^{{\mu}}u^{\nu}+p{\Delta}^{{\mu}{\nu}}+{\mathring{{\Theta}}}^{{\mu}{\nu}},$		(10)
	$\displaystyle{\Sigma}^{{\mu}{\nu}{\alpha}}={\epsilon}^{{\mu}{\nu}{\alpha}{\sigma}}\left({\cal R}_{\sigma}+{\mathring{{\Sigma}}}\,u_{{\sigma}}\right),$		(11)
	$\displaystyle s^{{\mu}}=s\,u^{{\mu}}+{\mathring{s}}^{{\mu}},$		(12)

where ${\Delta}^{{\mu}{\nu}}\equiv{\eta}^{{\mu}{\nu}}+u^{\mu}u^{\nu}$ is the transverse projection operator and the constitutive relations of the components with a circle are to be constrained by the entropy principle. To perform the entropy-current analysis, we derive the entropy production rate as follows. Taking the notations ${\beta}\equiv 1/T$, ${\beta}_{\mu}\equiv{\beta}u_{\mu}$, $D\equiv u^{{\nu}}\partial_{{\nu}}$ and ${\theta}\equiv\partial_{{\nu}}u^{{\nu}}$, we have

$\displaystyle\partial_{\mu}s^{{\mu}}=\partial_{\mu}\left(s\,u^{{\mu}}+{\mathring{s}}^{{\mu}}\right)=D\,s+s\,{\theta}+\partial_{\mu}{\mathring{s}}^{{\mu}}.$

(13)

We replace the first term in the above expression using (6) to get

$\displaystyle\partial_{\mu}s^{{\mu}}={\beta}\left[D{\varepsilon}-\frac{1}{2}{\omega}_{{\mu}{\nu}}D\left({\cal R}^{{\mu}{\nu}}+{\cal B}^{{\mu}{\nu}}\right)\right]+s\,{\theta}+\partial_{\mu}{\mathring{s}}^{{\mu}}.$

(14)

The two terms $D{\varepsilon}$ and $-\frac{1}{2}{\omega}_{{\mu}{\nu}}D{\cal R}^{{\mu}{\nu}}$ in the square brackets can be substituted by the components ${\mathring{{\Theta}}}^{{\mu}{\nu}}$ and ${\mathring{{\Sigma}}}$ that are to be determined by the entropy principle. To proceed, we contract (1) and (2) with $u_{\nu}$ and ${\omega}_{{\mu}{\nu}}$ respectively to get

	$\displaystyle D{\varepsilon}$	$\displaystyle=-\left({\varepsilon}+p\right){\theta}+u_{\nu}\partial_{\mu}{\mathring{{\Theta}}}^{{\mu}{\nu}},$
	$\displaystyle-\frac{1}{2}{\omega}_{{\mu}{\nu}}D{\cal R}^{{\mu}{\nu}}$	$\displaystyle={\omega}_{{\mu}{\nu}}\left[\frac{1}{2}{\theta}{\cal R}^{{\mu}{\nu}}+\partial_{\alpha}\left(u^{\nu}{\cal R}^{{\alpha}{\mu}}\right)\right]+\frac{1}{2}{\epsilon}^{{\alpha}{\mu}{\nu}{\sigma}}{\omega}_{{\mu}{\nu}}\partial_{\alpha}\left({\mathring{{\Sigma}}}u_{\sigma}\right)+{\omega}_{{\mu}{\nu}}{\mathring{{\Theta}}}^{[{\mu}{\nu}]}.$		(15)

We then obtain the entropy production rate as

	$\displaystyle\partial_{\mu}s^{{\mu}}=$	$\displaystyle\left[s-{\beta}\left({\varepsilon}+p-\frac{1}{2}{\omega}_{{\mu}{\nu}}\left({\cal R}^{{\mu}{\nu}}+{\cal B}^{{\mu}{\nu}}\right)\right)\right]{\theta}+\partial_{{\mu}}\left({\mathring{s}}^{{\mu}}+{\mathring{{\Theta}}}^{{\mu}{\nu}}{\beta}_{\nu}+\frac{1}{2}{\mathring{{\Sigma}}}\,{\epsilon}^{{\mu}{\alpha}{\nu}{\sigma}}{\beta}{\omega}_{{\alpha}{\nu}}u_{\sigma}\right)$
		$\displaystyle-{\mathring{{\Theta}}}^{({\mu}{\nu})}\partial_{{\mu}}{\beta}_{{\nu}}-{\mathring{{\Theta}}}^{[{\mu}{\nu}]}\left(\partial_{{\mu}}{\beta}_{{\nu}}-{\beta}{\omega}_{{\mu}{\nu}}\right)-\frac{1}{2}{\mathring{{\Sigma}}}\,{\epsilon}^{{\alpha}{\mu}{\nu}{\sigma}}\partial_{\alpha}\left({\beta}{\omega}_{{\mu}{\nu}}\right)u_{\sigma}$
		$\displaystyle+\left[\partial_{\alpha}\left(u^{\nu}{\cal R}^{{\alpha}{\mu}}\right)-\frac{1}{2}\partial_{\alpha}\left(u^{{\alpha}}{\cal B}^{{\mu}{\nu}}\right)\right]{\beta}{\omega}_{{\mu}{\nu}}.$		(16)

In the absence of ${\cal B}$ and $b$, (16) agrees with Hongo:2021ona ; Cao:2022aku , irrespective of the specific power counting scheme. The entropy principle requires that (16) is not only semipositive in general, but also zero in the GTE where thermal vorticity ${\varpi}_{{\mu}{\nu}}$ becomes a constant anti-symmetric tensor and

$\displaystyle{\beta}_{\mu}=c_{\mu}+{\varpi}_{{\mu}{\nu}}x^{\nu},\quad{\beta}{\omega}_{{\mu}{\nu}}={\varpi}_{{\mu}{\nu}},$

(17)

with $c_{\mu}$ being a constant four-vector.

To explicitly seek the semipositivity of (16) to the second order of gradient, we adopt a general power counting scheme where perturbation expansion of spin variables are independent of the conventional hydrodynamic variables and their gradients,

	$\displaystyle{\varepsilon}\sim p\sim{\beta}\sim u^{\mu}\sim O\left(\partial^{0}\right),\quad{\cal S}^{{\mu}{\nu}}\sim O\left({\delta}\right),$
	$\displaystyle{\omega}_{{\mu}{\nu}}\sim{\varpi}_{{\mu}{\nu}}\sim O\left({\varpi}\right),\quad{\varpi}_{{\mu}{\nu}}-{\beta}{\omega}_{{\mu}{\nu}}\sim O\left(\partial\right),$		(18)

where $O\left({\delta}\right)$ could be $O\left({\hbar}{\varpi}\right)$ if the spin susceptibility is $O\left({\hbar}\partial^{0}\right)$. In general, $O\left({\varpi}\right)$ could range from $O\left(\partial\right)$ for hydrodynamics with an isotropy background to $O\left(\partial^{0}\right)$ for gyrohydrodynamics Cao:2022aku with an anisotropic background. We count $O\left({\delta}\right)$ and $O\left({\varpi}\right)$ independently of $O\left(\partial\right)$ so that the formulation of the spin hydrodynamics is applicable to a broad scale of the thermal vorticity instead of subject to a specific power counting scheme of it.

We aim to determine the constitutive relations of ${\mathring{{\Theta}}}^{{\mu}{\nu}}$, ${\mathring{{\Sigma}}}$ and ${\mathring{s}}^{\mu}$ to $O\left(\partial\right)$ where (16) should be semipositive to $O\left(\partial^{2}\right)$. In the precedent set by Hongo:2021ona , the entropy production rate is ensured to be semipositive to $O\left(\partial^{2}\right)$ under a specific power counting scheme with ${\cal R}^{{\mu}{\nu}}\sim{\omega}_{{\mu}{\nu}}\sim O\left(\partial\right)$ and without the presence of ${\cal B}^{{\mu}{\nu}}$, where the non-semipositive terms ${\mathring{{\Theta}}}^{[{\mu}{\nu}]}u_{\nu}\left(\partial_{{\mu}}{\beta}_{{\sigma}}-{\beta}{\omega}_{{\mu}{\sigma}}\right)u^{\sigma}$ and ${\beta}{\omega}_{{\mu}{\nu}}\partial_{\alpha}\left(u^{\nu}{\cal R}^{{\alpha}{\mu}}\right)$ in (16) can be neglected as $O\left(\partial^{3}\right)$. However, in a broad scale of the thermal vorticity, these non-semipositive terms are of $O\left(\partial{\varpi}{\delta}\right)$ which are generally non-ignorable and therefore have to be cancelled out. By noting (5), we drop the first term in the first line of (16). Taking the GTE limit (17) in (16), one has

	$\displaystyle 0=\partial_{\mu}s_{\mathrm{GTE}}^{{\mu}}=$	$\displaystyle\partial_{{\mu}}\left({\mathring{s}}_{\mathrm{GTE}}^{{\mu}}+{\mathring{{\Theta}}}_{\mathrm{GTE}}^{{\mu}{\nu}}{\beta}_{\nu}+\frac{1}{2}{\mathring{{\Sigma}}}_{\mathrm{GTE}}\,{\epsilon}^{{\mu}{\alpha}{\nu}{\sigma}}{\varpi}_{{\alpha}{\nu}}u_{\sigma}\right)$
		$\displaystyle+\left[\partial_{\alpha}\left(u^{\nu}{\cal R}^{{\alpha}{\mu}}\right)-\frac{1}{2}\partial_{\alpha}\left(u^{{\alpha}}{\cal B}^{{\mu}{\nu}}\right)\right]{\varpi}_{{\mu}{\nu}}.$		(19)

Noting the nonvanishing terms in the last brackets of (19), it is evident that ${\mathring{s}}^{{\mu}}\neq-{\mathring{{\Theta}}}^{{\mu}{\nu}}{\beta}_{\nu}-\frac{1}{2}{\mathring{{\Sigma}}}\,{\epsilon}^{{\mu}{\alpha}{\nu}{\sigma}}{\varpi}_{{\alpha}{\nu}}u_{\sigma}$ in general. Therefore, terms involving ${\cal R}$ and ${\cal B}$ must be present in either ${\mathring{{\Theta}}}^{{\mu}{\nu}}$, ${\mathring{s}}^{\mu}$ or ${\mathring{{\Sigma}}}$ to offset the above nonvanishing terms. Note that, in the vicinity of GTE, these nonvanishing terms arise from the leading-order term ${\epsilon}^{{\mu}{\nu}{\alpha}{\sigma}}{\cal R}_{\sigma}$ in the spin current (11). As pointed out in Hattori:2019lfp , the entropy production rate from the leading-order of the spin current is zero if spin and orbital angular momentum are separately conserved²²2Although there are no additional symmetries beyond the Lorentz group that would allow for individual conservation laws in field theory, and ideal spin hydrodynamics does not exist, we can establish an ad hoc criterion for formulating spin hydrodynamics, which suggests that a hydrodynamic framework should be non-dissipative at the leading-order in the conservation limit of the currents involved.. At the lowest order of the non-conservation equation (2) of the spin current, the dissipation of spin only stem from the source/absorption term ${\mathring{{\Theta}}}^{[{\mu}{\nu}]}$.

We separate the non-dissipative parts from the dissipative parts by marking the former with subscript ${\delta}$ and the latter with tick, i.e., ${\mathring{{\Theta}}}^{{\mu}{\nu}}={\Theta}_{{\delta}}^{{\mu}{\nu}}+{\check{\Theta}}^{{\mu}{\nu}}$, ${\mathring{{\Sigma}}}={\Sigma}_{{\delta}}+{\check{\Sigma}}$ and ${\mathring{s}}^{\mu}=s_{{\delta}}^{\mu}+{\check{s}}^{\mu}$. At this stage, we manifest the assumption that the constitutive relations of ${\mathring{{\Theta}}}^{{\mu}{\nu}}$, ${\mathring{s}}^{\mu}$ and ${\mathring{{\Sigma}}}$, as expressions in terms of the spin hydrodynamic variables ${\beta}$, $u^{\mu}$, ${\omega}^{{\mu}{\nu}}$ and ${\cal S}^{{\mu}{\nu}}$, consistently satisfy the entropy principle, i.e.,

$\displaystyle\exists\,{\mathring{{\Theta}}}^{{\mu}{\nu}},{\mathring{s}}^{{\mu}},{\mathring{{\Sigma}}}\text{ as functions of }{\beta},u^{{\mu}},{\omega}^{{\mu}{\nu}}\text{ and }{\cal S}^{{\mu}{\nu}}\;\forall\,{\beta},u^{{\mu}},{\omega}^{{\mu}{\nu}}\text{ and }{\cal S}^{{\mu}{\nu}}:\partial_{\mu}s^{\mu}\geq 0.$

(20)

Here the ${\cal S}^{{\mu}{\nu}}$ dependent parts of the constitutive relations are to cancel out the non-semipositive terms in (16) where we take ${\cal S}^{{\mu}{\nu}}$ as extra free variables besides ${\beta},u^{{\mu}}$ and ${\omega}^{{\mu}{\nu}}$³³3Note that ${\cal S}^{{\mu}{\nu}}$ is free in the sense that the the dependence of ${\cal S}^{{\mu}{\nu}}$ on ${\beta},u^{{\mu}}$ and ${\omega}^{{\mu}{\nu}}$ could vary with specific type of fluid and physical regime while the constitutive relations should satisfy the entropy principle in general.. The entropy production rate is written as

	$\displaystyle\partial_{\mu}s^{{\mu}}$	$\displaystyle=\partial_{\mu}\left[{\check{s}}^{{\mu}}-{\check{\Theta}}^{{\mu}{\nu}}{\beta}_{{\nu}}+\frac{1}{2}{\check{\Sigma}}{\epsilon}^{{\mu}{\alpha}{\nu}{\sigma}}{\beta}{\omega}_{{\alpha}{\nu}}u_{\sigma}\right]$		(21)
		$\displaystyle-{\check{\Theta}}^{({\mu}{\nu})}\partial_{\mu}{\beta}_{{\nu}}-{\check{\Theta}}^{[{\mu}{\nu}]}\left(\partial_{{\mu}}{\beta}_{{\nu}}-{\beta}{\omega}_{{\mu}{\nu}}\right)-\frac{1}{2}{\check{\Sigma}}{\epsilon}^{{\alpha}{\mu}{\nu}{\sigma}}\partial_{\alpha}\left({\beta}{\omega}_{{\mu}{\nu}}\right)u_{\sigma}$
		$\displaystyle+\partial_{\mu}s_{{\delta}}^{\mu}+\partial_{\mu}{\Theta}_{{\delta}}^{{\mu}{\nu}}{\beta}_{\nu}+\left[{\Theta}_{{\delta}}^{{\mu}{\nu}}+\partial_{\alpha}\left(u^{\nu}{\cal R}^{{\alpha}{\mu}}\right)-\frac{1}{2}\partial_{\alpha}\left(u^{{\alpha}}{\cal B}^{{\mu}{\nu}}\right)+\frac{1}{2}{\epsilon}^{{\alpha}{\mu}{\nu}{\sigma}}\partial_{\alpha}\left({\Sigma}_{{\delta}}\,u_{\sigma}\right)\right]{\beta}{\omega}_{{\mu}{\nu}}.$

The dissipative part ${\check{\Theta}}^{{\mu}{\nu}}$ can be decomposed into the irreducible tensor basis Hattori:2019lfp ; Fukushima:2020ucl ; Daher:2022xon (see also Baier:2007ix ; Denicol:2012cn ; Molnar:2013lta ) as follows,

$\displaystyle{\check{\Theta}}^{({\mu}{\nu})}=2u^{({\mu}}h^{{\nu})}+{\tau}^{{\mu}{\nu}},\qquad{\check{\Theta}}^{[{\mu}{\nu}]}=2u^{[{\mu}}q^{{\nu}]}+{\phi}^{{\mu}{\nu}},$

(22)

where the dissipative currents satisfy ${\tau}^{{\mu}{\nu}}={\tau}^{{\nu}{\mu}},\,{\phi}^{{\mu}{\nu}}=-{\phi}^{{\nu}{\mu}},\,u_{\mu}h^{\mu}=u_{\mu}q^{\mu}=u_{\mu}{\tau}^{{\mu}{\nu}}=u_{\mu}{\phi}^{{\mu}{\nu}}=0$. We have $u_{\mu}{\check{\Theta}}^{{\mu}{\nu}}u_{\nu}={\varepsilon}$ while $u_{\mu}{\Theta}_{{\delta}}^{{\mu}{\nu}}u_{\nu}$ is not necessarily zero. Moreover, in contrast to Biswas:2023qsw , we do not require $u_{\mu}{\mathring{s}}^{\mu}\leq 0$. This is because we assume that the local thermodynamic relations (5)-(6) hold near LTE, where all the thermodynamic variables, including the entropy density $s$, are extended to be applicable out of equilibrium. Therefore, a general entropy density $s$ evolving towards equilibrium is not identically equal to the maximum value that is to be reached in equilibrium.

In addition to the dissipative parts in the entropy production rate, we have collected all the non-dissipative components into the last line of (21). Explicitly, the entropy principle requires that the sum of the non-dissipative terms gives zero entropy production rate

	$\displaystyle\partial_{\mu}s_{{\delta}}^{\mu}+\partial_{\mu}{\Theta}_{{\delta}}^{{\mu}{\nu}}{\beta}_{\nu}$
	$\displaystyle\quad+\left[{\Theta}_{{\delta}}^{{\mu}{\nu}}+\partial_{\alpha}\left(u^{\nu}{\cal R}^{{\alpha}{\mu}}\right)-\frac{1}{2}\partial_{\alpha}\left(u^{{\alpha}}{\cal B}^{{\mu}{\nu}}\right)+\frac{1}{2}{\epsilon}^{{\alpha}{\mu}{\nu}{\sigma}}\partial_{\alpha}\left({\Sigma}_{{\delta}}\,u_{\sigma}\right)\right]{\beta}{\omega}_{{\mu}{\nu}}=0.$		(23)

We consider the non-dissipative constitutive relations of ${\Theta}_{{\delta}}^{{\mu}{\nu}}$, $s_{{\delta}}^{\mu}$ and ${\Sigma}_{{\delta}}$ to all orders as solutions to (23). For this purpose, we explicitly write ${\Theta}_{{\delta}}^{{\mu}{\nu}}$ and $s_{{\delta}}^{\mu}$ into the terms of $O\left(\partial^{0}{\omega}^{0}{\delta}\right)$, $O\left(\partial^{0}{\omega}{\delta}\right)$, $O\left(\partial{\omega}^{0}{\delta}\right)$ and higher orders in a general form as

	$\displaystyle{\Theta}_{{\delta}}^{{\alpha}{\sigma}}={\Theta}_{0}^{{\alpha}{\sigma}}+{\Theta}_{{\omega}}^{{\alpha}{\sigma}{\mu}{\nu}}{\omega}_{{\mu}{\nu}}+{\Theta}_{\partial}^{{\alpha}{\sigma}}-\frac{1}{2}{\epsilon}^{{\mu}{\alpha}{\sigma}{\nu}}\partial_{\mu}\left({\Sigma}_{{\delta}}\,u_{\nu}\right)+O\left(\partial{\omega}{\delta}\right),$
	$\displaystyle s_{{\delta}}^{\alpha}=s_{0}^{{\alpha}}+s_{{\omega}}^{{\alpha}{\mu}{\nu}}{\beta}{\omega}_{{\mu}{\nu}}+s_{\partial}^{\alpha}+O\left(\partial{\omega}{\delta}\right),$		(24)

where the $O\left(\partial^{0}{\omega}^{0}{\delta}\right)$ components ${\Theta}_{0,{\omega}},s_{0,{\omega}}$ and $O\left(\partial{\omega}^{0}{\delta}\right)$ components ${\Theta}_{\partial},s_{\partial}$ are expressions in terms of ${\beta},u^{\mu}$ and ${\cal S}^{{\mu}{\nu}}$. Now we collect the terms involving ${\beta}{\omega}_{{\mu}{\nu}}$, $\partial_{\alpha}\left({\beta}{\omega}_{{\mu}{\nu}}\right)$ and ${\beta}{\omega}_{{\alpha}{\sigma}}{\beta}{\omega}_{{\mu}{\nu}}$ into

$\displaystyle X^{{\mu}{\nu}}{\beta}{\omega}_{{\mu}{\nu}}+Y^{{\alpha}{\mu}{\nu}}\partial_{\alpha}\left({\beta}{\omega}_{{\mu}{\nu}}\right)+{\Theta}_{{\omega}}^{{\alpha}{\sigma}{\mu}{\nu}}T{\beta}{\omega}_{{\alpha}{\sigma}}{\beta}{\omega}_{{\mu}{\nu}},$

(25)

where $X^{{\mu}{\nu}}$ and $Y^{{\alpha}{\mu}{\nu}}$ are defined as

	$\displaystyle X^{{\mu}{\nu}}\equiv\partial_{\alpha}s_{{\omega}}^{{\alpha}{\mu}{\nu}}+\partial_{\alpha}\left({\Theta}_{{\omega}}^{{\alpha}{\sigma}{\mu}{\nu}}T\right){\beta}_{\sigma}+{\Theta}_{\partial}^{{\mu}{\nu}}+\partial_{\alpha}\left(u^{\nu}{\cal R}^{{\alpha}{\mu}}-\frac{1}{2}u^{{\alpha}}{\cal B}^{{\mu}{\nu}}\right),$
	$\displaystyle Y^{{\alpha}{\mu}{\nu}}\equiv s_{{\omega}}^{{\alpha}{\mu}{\nu}}+{\Theta}_{{\omega}}^{{\alpha}{\sigma}{\mu}{\nu}}u_{\sigma}.$		(26)

The two parts must vanish for any values of ${\beta}{\omega}_{{\mu}{\nu}}$ and $\partial_{\alpha}\left({\beta}{\omega}_{{\mu}{\nu}}\right)$, resulting in the constraints

$\displaystyle X^{[{\mu}{\nu}]}=0\;\text{ and }\;Y^{{\alpha}[{\mu}{\nu}]}=0.$

(27)

Likewise, to ensure the term ${\Theta}_{{\omega}}^{{\alpha}{\sigma}{\mu}{\nu}}{\beta}{\omega}_{{\alpha}{\sigma}}{\beta}{\omega}_{{\mu}{\nu}}$ vanishes for arbitrary values of ${\beta}{\omega}_{{\alpha}{\sigma}}{\beta}{\omega}_{{\mu}{\nu}}$ in (25), ${\Theta}_{{\omega}}^{{\alpha}{\sigma}{\mu}{\nu}}$ can run through several switches as follows

$\displaystyle{\Theta}_{{\omega}}^{{\alpha}{\sigma}[{\mu}{\nu}]}=0\;\text{ or }\;{\Theta}_{{\omega}}^{[{\alpha}{\sigma}]{\mu}{\nu}}=0\;\text{ or }\;{\Theta}_{{\omega}}^{{\alpha}{\sigma}{\mu}{\nu}}=-{\Theta}_{{\omega}}^{{\mu}{\nu}{\alpha}{\sigma}}\;\text{ or }\;{\Theta}_{{\omega}}^{{\alpha}{\sigma}{\mu}{\nu}}=-{\Theta}_{{\omega}}^{{\nu}{\mu}{\sigma}{\alpha}}.$

(28)

We then combine (27) and (28) to constrain the solutions to (23). As a straightforward application of these constraints, one can readily confirm that

$\displaystyle{\Theta}_{\partial}^{{\mu}{\nu}}=0\;\text{ and }\;{\Theta}_{{\omega}}^{{\alpha}{\sigma}{\mu}{\nu}}=0\;\text{ and }\;\eqref{cnstrXY}\;\to\;s_{{\omega}}^{{\alpha}[{\mu}{\nu}]}=0\;\to\;\partial_{\alpha}\left({\cal R}^{{\alpha}[{\mu}}u^{{\nu}]}-\frac{1}{2}u^{{\alpha}}{\cal B}^{{\mu}{\nu}}\right)=0,$

(29)

leading to a contradiction as the left-hand side of the final equation is not identically zero. This implies that the stress-energy tensor must depend on $S^{{\mu}{\nu}}$ at $O\left(\partial{\delta}\right)$ or $O\left({\omega}{\delta}\right)$. In general, one can analyze the terms to all orders in (23) to obtain a complete constraint for the solution. Nevertheless, given that (23) must hold order by order, we concentrate exclusively on the constraints related to the $O\left(\partial{\delta}\right)$ and $O\left({\omega}{\delta}\right)$ terms. It will become apparent in the next section that the components dependent on ${\omega}$ within these orders are sufficient to illustrate the difficulties in upholding the entropy principle in the absence of boost variables.

We now investigate the framework in the abscence of the degree of freedom related to the boost symmetry where there are only seven independent dynamical variables with four from relativistic translation symmetry and three from rotation symmetry. In such circumstances, it is necessary to select three out of the ten equations in (1)-(2) as redundant in order to avoid overdetermination. As pointed out in Hongo:2021ona , the physically meaningful choice is to consider the three equations ensuing from the boost symmetry in (2) as redundant identities since the boost variables are vanishing. The identities in the local rest frame are obtained by setting ${\mu}=0,{\nu}=i$ or ${\mu}=i,{\nu}=0$ in (2), while the covariant form is manifested by projecting (2) onto $u_{\nu}$ as

	$\displaystyle\left(\partial_{\alpha}{\Sigma}^{{\alpha}{\mu}{\nu}}+2{\Theta}^{[{\mu}{\nu}]}\right)u_{\nu}=0$
	$\displaystyle\qquad\Rightarrow\quad\frac{1}{2}{\epsilon}^{{\alpha}{\mu}{\nu}{\sigma}}u_{\nu}\partial_{\alpha}\left({\cal R}_{\sigma}+{\Sigma}_{{\delta}}u_{\sigma}\right)+{\Theta}_{{\delta}}^{[{\mu}{\nu}]}u_{\nu}+\frac{1}{2}{\epsilon}^{{\alpha}{\mu}{\nu}{\sigma}}u_{\nu}{\check{\Sigma}}\,\partial_{\alpha}u_{\sigma}+q^{\mu}=0.$		(30)

Noting that $q^{\mu}$ at $O\left(\partial{\delta}\right)$ and ${\check{\Sigma}}$ at $O\left({\delta}\right)$ are both zero to ensure the semipositivity of the dissipative parts in (21), we isolate the $O\left(\partial{\delta}\right)$ terms from the other parts in the above identity,

	$\displaystyle q^{\mu}$	$\displaystyle=-\frac{1}{2}{\epsilon}^{{\alpha}{\mu}{\nu}{\sigma}}u_{\nu}{\check{\Sigma}}\,\partial_{\alpha}u_{\sigma},$		(31)
	$\displaystyle{\Theta}_{{\delta}}^{[{\mu}{\nu}]}u_{\nu}$	$\displaystyle=-\frac{1}{2}{\epsilon}^{{\alpha}{\mu}{\nu}{\sigma}}u_{\nu}\partial_{\alpha}\left({\cal R}_{\sigma}+{\Sigma}_{{\delta}}u_{\sigma}\right),$		(32)

where the identity at $O\left(\partial{\delta}\right)$ in (32) should hold for arbitrary ${\cal R}$. Utilizing the identities (31)-(32) as the result of the vanishing boost variables, we can demonstrate that it is not possible to cancel out the non-semipositive term $\partial_{\alpha}\left(u^{\nu}{\cal R}^{{\alpha}{\mu}}\right){\beta}{\omega}_{{\mu}{\nu}}$ in (23).

For the ${\cal R}$ dependent parts, we further collect the $O\left(\partial{\omega}^{0}{\delta}\right)$ and $O\left(\partial^{0}{\omega}{\delta}\right)$ terms in (32) to obtain the extra constraints from the vanshing of ${\cal B}$ as

	$\displaystyle{\Theta}_{\partial}^{[{\mu}{\nu}]}u_{\nu}=-\frac{1}{2}{\epsilon}^{{\alpha}{\mu}{\nu}{\sigma}}u_{\nu}\partial_{\alpha}{\cal R}_{\sigma},$		(33)
	$\displaystyle{\Theta}_{{\omega}}^{[{\mu}{\nu}]{\alpha}{\sigma}}u_{\nu}{\omega}_{{\alpha}{\sigma}}=0\quad\to\quad{\Theta}_{{\omega}}^{[{\mu}{\nu}][{\alpha}{\sigma}]}u_{\nu}=0.$		(34)

We now examine the combined constraints on ${\Theta}_{{\omega}}$ in (27)-(28) and (33)-(34). For the first switch in (28), one has

$\displaystyle{\Theta}_{{\omega}}^{{\alpha}{\sigma}[{\mu}{\nu}]}=0\;\text{ and }\;Y^{{\alpha}[{\mu}{\nu}]}=0\,\to\,s_{{\omega}}^{{\alpha}[{\mu}{\nu}]}=0\;\to\;X^{[{\mu}{\nu}]}={\Theta}_{\partial}^{[{\mu}{\nu}]}+\partial_{\alpha}\left({\cal R}^{{\alpha}[{\mu}}u^{{\nu}]}\right)=0,$

(35)

which obviously contradicts the identity (33). Thus, we get ${\Theta}_{{\omega}}^{{\alpha}{\sigma}[{\mu}{\nu}]}\neq 0$.

For the rest three switches in (28), using (33) in (27) while noting $u\cdot u=-1$ and $u\cdot{\cal R}=0$, we get

	$\displaystyle{\Theta}_{{\omega}}^{{\alpha}{\sigma}[{\mu}{\nu}]}u_{\nu}u_{\sigma}\partial_{\alpha}\ln{\beta}+{\Theta}_{{\omega}}^{{\alpha}{\sigma}[{\mu}{\nu}]}u_{\nu}\partial_{\alpha}u_{\sigma}-W_{1}^{{\alpha}{\mu}{\nu}}{\cal R}_{\nu}\partial_{\alpha}\left(u\cdot u\right)-W_{2}^{{\alpha}{\mu}}\partial_{\alpha}\left(u\cdot{\cal R}\right)$
	$\displaystyle=\partial_{\alpha}{\cal R}^{{\alpha}[{\mu}}u^{{\nu}]}u_{\nu}-\frac{1}{2}{\epsilon}^{{\alpha}{\mu}{\nu}{\sigma}}u_{\nu}\partial_{\alpha}{\cal R}_{\sigma}=\frac{1}{2}\partial_{\alpha}\left(u^{\alpha}{\cal R}^{{\mu}{\nu}}\right)u_{\nu}=-\frac{1}{2}u^{\alpha}{\epsilon}^{{\mu}{\nu}{\sigma}{\lambda}}{\cal R}_{\lambda}u_{\nu}\partial_{\alpha}u_{\sigma},$		(36)

where $W_{1}$ and $W_{2}$ could be any dimensionless tensors as expressions in terms of ${\beta},u^{\mu}$ and ${\cal R}^{{\mu}}$. We have used the first identity of (3) in the second equality. Given that the above equation holds for arbitrary $\partial_{\alpha}\ln{\beta}$, $\partial_{\alpha}{\cal R}_{\sigma}$ and $\partial_{\alpha}u_{\sigma}$, we have the constraints

	$\displaystyle{\Theta}_{{\omega}}^{{\alpha}{\sigma}[{\mu}{\nu}]}u_{\nu}u_{\sigma}=0\;\text{ and }\;W_{2}^{{\alpha}{\mu}}u^{\sigma}=0$
	$\displaystyle\;\text{ and }\;{\Theta}_{{\omega}}^{{\alpha}{\sigma}[{\mu}{\nu}]}u_{\nu}-W_{1}^{{\alpha}{\mu}{\nu}}{\cal R}_{\nu}u^{\sigma}-W_{2}^{{\alpha}{\mu}}{\cal R}^{\sigma}-W_{3}^{{\alpha}{\mu}}a^{\sigma}=-\frac{1}{2}u^{\alpha}{\epsilon}^{{\mu}{\nu}{\sigma}{\lambda}}{\cal R}_{\lambda}u_{\nu},$		(37)

which renders

	$\displaystyle 0={\Theta}_{{\omega}}^{{\alpha}{\sigma}[{\mu}{\nu}]}u_{\nu}u_{\sigma}=-W_{1}^{{\alpha}{\mu}{\nu}}{\cal R}_{\nu}\;\to\;W_{1}^{{\alpha}{\mu}{\nu}}=0$
	$\displaystyle\text{ and }\;W_{2}^{{\alpha}{\mu}}u^{\sigma}=0\;\to\;W_{2}^{{\alpha}{\mu}}=0\;\to\;{\Theta}_{{\omega}}^{{\alpha}{\sigma}[{\mu}{\nu}]}u_{\nu}=-\frac{1}{2}u^{\alpha}{\epsilon}^{{\mu}{\nu}{\sigma}{\lambda}}{\cal R}_{\lambda}u_{\nu}$
	$\displaystyle\to{\Theta}_{{\omega}}^{[{\alpha}{\sigma}][{\mu}{\nu}]}u_{\nu}=-\frac{1}{2}u^{[{\alpha}}{\epsilon}^{{\sigma}]{\mu}{\nu}{\lambda}}{\cal R}_{\lambda}u_{\nu}\neq 0.$		(38)

This excludes the second switch in (28), i.e., ${\Theta}_{{\omega}}^{[{\alpha}{\sigma}]{\mu}{\nu}}\neq 0$.

The last two switches in (28) combined with (34) reduce to

	$\displaystyle\left({\Theta}_{{\omega}}^{{\alpha}{\sigma}{\mu}{\nu}}=-{\Theta}_{{\omega}}^{{\mu}{\nu}{\alpha}{\sigma}}\;\text{ or }\;{\Theta}_{{\omega}}^{{\alpha}{\sigma}{\mu}{\nu}}=-{\Theta}_{{\omega}}^{{\nu}{\mu}{\sigma}{\alpha}}\right)\;\text{ and }\;{\Theta}_{{\omega}}^{[{\alpha}{\sigma}][{\mu}{\nu}]}u_{\sigma}=0$
	$\displaystyle\,\to\,\left({\Theta}_{{\omega}}^{{\alpha}{\sigma}[{\mu}{\nu}]}=-{\Theta}_{{\omega}}^{[{\mu}{\nu}]{\alpha}{\sigma}}\;\text{ or }\;{\Theta}_{{\omega}}^{{\alpha}{\sigma}[{\mu}{\nu}]}=-{\Theta}_{{\omega}}^{[{\nu}{\mu}]{\sigma}{\alpha}}\right)\;\text{ and }\;{\Theta}_{{\omega}}^{[{\alpha}{\sigma}][{\mu}{\nu}]}u_{\sigma}=0$
	$\displaystyle\,\to\,{\Theta}_{{\omega}}^{[{\alpha}{\sigma}][{\mu}{\nu}]}u_{\nu}=-{\Theta}_{{\omega}}^{[{\mu}{\nu}][{\alpha}{\sigma}]}u_{\nu}=0\;\text{ or }\;{\Theta}_{{\omega}}^{[{\alpha}{\sigma}][{\mu}{\nu}]}u_{\nu}=-{\Theta}_{{\omega}}^{[{\nu}{\mu}][{\sigma}{\alpha}]}u_{\nu}=0,$		(39)

which is also ruled out by (38). Hence, there is no consistent result for ${\Theta}_{{\omega}}$ to ensure the vanishing of the $O\left(\partial{\omega}{\delta}\right)$ and $O\left(\partial^{0}{\omega}^{2}{\delta}\right)$ parts in (23). In other words, with a general antisymmetric spin potential ${\omega}_{{\mu}{\nu}}$ and vanishing boost variables ${\cal B}^{{\mu}{\nu}}$, it is generally not possible for the constitutive relations of spin hydrodynamics to satisfy the entropy principle. Note that (27)-(28) and (34) are constraints resulting from a general antisymmetric spin potential ${\omega}_{{\mu}{\nu}}$. A meaningful complement would be to apply the entropy-current analysis presented in this study to the case of a special spin potential $r_{{\mu}{\nu}}$ with $r_{{\mu}{\nu}}u^{\nu}=0$, as utilized in Hongo:2021ona ; Cao:2022aku , to investigate the existance of a solution for ${\Theta}_{{\delta}}$ and $s_{{\delta}}$. We will not attempt to address this issue here.

The challenge in adhering to the entropy principle arises from the lack of degree of freedom associated with boost symmetry. Therefore, it is inevitable to activate the boost variables so that the canonical formulation of spin hydrodynamics aligns with the entropy principle. In this scenario, the boost components of the conservation law (2) are independent equations, rather than being fixed as identities like in (31)-(32). The semipositivity of the dissipative parts is ensured by adopting the constitutive relations that are basically the same as those in Hattori:2019lfp ; Fukushima:2020ucl ; Daher:2022xon ,

	$\displaystyle h^{\mu}$	$\displaystyle=-Th^{{\mu}{\nu}{\alpha}}\partial_{\alpha}{\beta}_{\nu},$	$\displaystyle q^{\mu}$	$\displaystyle=-Tq^{{\mu}{\nu}{\alpha}}\left(\partial_{\alpha}{\beta}_{\nu}-{\beta}{\omega}_{{\alpha}{\nu}}\right),$
	$\displaystyle{\tau}^{{\mu}{\nu}}$	$\displaystyle=-T{\tau}^{{\mu}{\nu}{\alpha}{\sigma}}\partial_{\alpha}{\beta}_{{\sigma}},$	$\displaystyle{\phi}^{{\mu}{\nu}}$	$\displaystyle=-T{\phi}^{{\mu}{\nu}{\alpha}{\sigma}}\left(\partial_{\alpha}{\beta}_{{\sigma}}-{\beta}{\omega}_{{\alpha}{\sigma}}\right),$		(40)
	$\displaystyle{\check{\Sigma}}$	$\displaystyle=-\frac{1}{2}T{\xi}{\epsilon}^{{\mu}{\nu}{\alpha}{\sigma}}u_{\sigma}\partial_{\alpha}\left({\beta}{\omega}_{{\mu}{\nu}}\right),$	$\displaystyle{\check{s}}^{{\mu}}$	$\displaystyle={\beta}h^{{\mu}}-{\beta}q^{{\mu}}-\frac{1}{2}{\check{\Sigma}}{\epsilon}^{{\mu}{\alpha}{\nu}{\sigma}}{\beta}{\omega}_{{\alpha}{\nu}}u_{\sigma},$

where

	$\displaystyle h^{{\mu}{\nu}{\alpha}}$	$\displaystyle\equiv{\kappa}{\Delta}^{{\mu}({\nu}}u^{{\alpha})},\hskip 40.00006pt\ignorespaces q^{{\mu}{\nu}{\alpha}}\equiv{\kappa}_{s}{\Delta}^{{\mu}[{\nu}}u^{{\alpha}]},$
	$\displaystyle{\tau}^{{\mu}{\nu}{\alpha}{\beta}}$	$\displaystyle\equiv 2{\eta}\left[\frac{1}{2}\left({\Delta}^{{\mu}{\alpha}}{\Delta}^{{\nu}{\beta}}+{\Delta}^{{\mu}{\beta}}{\Delta}^{{\mu}{\nu}}\right)-\frac{1}{3}{\Delta}^{{\mu}{\nu}}{\Delta}^{{\alpha}{\beta}}\right]+{\zeta}{\Delta}^{{\mu}{\nu}}{\Delta}^{{\alpha}{\beta}},$
	$\displaystyle{\phi}^{{\mu}{\nu}{\alpha}{\beta}}$	$\displaystyle\equiv\frac{1}{2}{\eta}_{s}\left({\Delta}^{{\mu}{\alpha}}{\Delta}^{{\nu}{\beta}}-{\Delta}^{{\mu}{\beta}}{\Delta}^{{\nu}{\alpha}}\right),$		(41)

with positive coefficients ${\kappa},\,{\kappa}_{s},\,{\eta},\,{\zeta},\,{\eta}_{s}$ and ${\xi}$. In the case $O\left({\varpi}\right)\sim O\left(\partial^{0}\right)$, the dissipative currents in (40) can be further decomposed according to the anisotropy in gyrohydrodynamics Cao:2022aku .