1]Physics Department and Center for Particle Physics and Field Theory, Fudan
University, Shanghai 200438, China

2]Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang
University, Hangzhou, Zhejiang 310027, China 3]Research Center for Nuclear
Physics, Osaka University, 10-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

4]Department of Physics, Niigata University, Niigata 950-2181, Japan 5]RIKEN
iTHEMS, RIKEN, Wako 351-0198, Japan

6]Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Fudan
University, Shanghai 200433, China 7]Shanghai Research Center for Theoretical
Nuclear Physics, NSFC and Fudan University, Shanghai 200438, China

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Gyrohydrodynamics:
Relativistic spinful fluid with strong vorticity

Zheng Cao  [  Koichi Hattori  [  [  Masaru Hongo  [  [  Xu-Guang Huang  [  [
Hidetoshi Taya  [  [  [  [

###### Abstract

We develop a relativistic (quasi-)hydrodynamic framework, dubbed the
gyrohydrodynamics, to describe fluid dynamics of many-body systems with spin
under strong vorticity based on entropy-current analysis. This framework
generalizes the recently-developed spin hydrodynamics to the regime where the
spin density is at the leading order in derivatives but suppressed by another
small parameter, the Planck constant $\hbar$, due to its quantum nature. Our
analysis shows that the complete first-order constitutive relations of
gyrohydrodynamics involve seventeen transport coefficients and are highly
anisotropic.

A52, D28, D31, E14, I55

Introduction.— Hydrodynamics is a low-frequency and long-wavelength effective
theory for interacting many-body systems Landau:Fluid . When incorporated with
relativity, it has been applied very successfully in astrophysics, cosmology,
and relativistic heavy-ion collisions. For example, relativistic hydrodynamics
provides an important foundation for the understanding of the core-collapse
supernova explosions, the structures and bulk properties of neutron stars and
white dwarfs, the dynamics of accretion discs around compact objects, and the
merging process of binary neutron stars and black holes; see, e.g., Refs.
Camenzind:book ; Rezzolla:book ; Baiotti:2016qnr ; Paschalidis:2016vmz ;
Muller:2020ard . In high-energy heavy-ion collisions, relativistic
hydrodynamics has become the ‘standard model’ to describe the evolution of
quark-gluon plasma created during the collisions; see e.g., Refs. Heinz:2013th
; Florkowski:2017olj ; Shen:2020mgh .

It is noteworthy that rotation of fluids, whether it is global or local (local
rotation is characterized by fluid vorticity), plays a crucial role in all of
the systems mentioned above. One of the critical consequences of rotation is
spin polarization when the constituents of fluids are spinful. In particular,
the strongest fluid vorticity has recently been detected in heavy-ion
collisions STAR:2017ckg . Measurements of hyperon polarization STAR:2017ckg ;
Adam:2018ivw ; Adam:2020pti  and vector-meson spin alignment Acharya:2019vpe ;
STAR:2022fan ; ALICE:2022sli  in heavy-ion collisions have indicated strong
spin polarization of quarks and/or gluons in the quark-gluon plasma,
supporting earlier theoretical ideas proposed in, e.g., Refs. Liang:2004ph ;
Liang:2004xn ; Betz:2007kg ; Becattini:2007sr ; Huang:2011ru ;
Becattini:2013fla . As such, it is desirable to develop a relativistic
hydrodynamic framework that deals appropriately with the spin degrees of
freedom and the vorticity generation. Such a framework may pave the way to
quantitative understanding of spin transports in quark-gluon plasma, supernova
explosions, and so on. Specifically, it may be used to resolve the so-called
sign problem of the local spin polarization, that is, the discrepancy between
the experimental data for azimuthal-angle dependence of hyperon polarization
and the theoretical calculations based on thermal vorticity (see below for
definition); see reviews Liu:2020ymh ; Gao:2020vbh ; Huang:2020xyr ;
Becattini:2020ngo ; Becattini:2022zvf  and references therein for more
details.

There have been great efforts on developing the framework of spin
hydrodynamics Montenegro:2017rbu ; Montenegro:2017lvf ; Florkowski:2017ruc ;
Florkowski:2018fap ; Montenegro:2018bcf ; Hattori:2019lfp ; Li:2019qkf ;
Fukushima:2020ucl ; Bhadury:2020puc ; Montenegro:2020paq ; Li:2020eon ;
Shi:2020htn ; Garbiso:2020puw ; Gallegos:2020otk ; She:2021lhe ; Hu:2021lnx ;
Gallegos:2021bzp ; Peng:2021ago ; Hongo:2021ona ; Hu:2021pwh ;
Cartwright:2021qpp ; Wang:2021wqq ; Hongo:2022izs ; Singh:2022ltu ;
Daher:2022xon ; Gallegos:2022jow ; Weickgenannt:2022zxs ; Bhadury:2022qxd . In
most of these developments, the spin and vorticity are considered to be of
similar magnitudes as the shear viscous correction to the hydrodynamics,
meaning that they are assumed to be $O(\partial^{1})$ in terms of the
derivative expansion. This is possible, since vorticity is the skew gradient
of velocity while the shear tenor is the symmetric gradient of velocity. Thus,
one expects that their strength may be comparable in magnitude. However, there
is a significant difference between the two: the shear tensor vanishes at the
global equilibrium, whereas vorticity does not. This implies that, on the
physical ground, these two can be regarded to be at different orders in
derivatives in and near equilibrium (cf. Ref. Li:2020eon  for another power
counting scheme based on the same consideration). Furthermore, the discovery
of strong spin polarization in the quark-gluon plasma also demands development
of a hydrodynamic theory with high spin density and vorticity.

Motivated by these observations, we will investigate a scenario in which
vorticity is counted as zeroth order in derivatives, whereas other types of
gradients of thermodynamic quantities are counted as first order in
derivatives. The spin polarization induced by the zeroth-order vorticity is
also regarded as the zeroth order in derivatives, but we assume it to be
suppressed by another small parameter, i.e., the Planck constant $\hbar$ due
to its quantum nature. We dub the resultant framework gyrohydrodynamics.

As we will demonstrate, the constitutive relations in gyrohydrodynamics have
much richer structure than those in spin hydrodynamics Hattori:2019lfp ;
Bhadury:2020puc ; Fukushima:2020ucl ; Gallegos:2021bzp ; Hongo:2021ona ;
Hu:2021pwh ; Gallegos:2022jow ; Weickgenannt:2022zxs ; Daher:2022xon .
Firstly, the leading-order constitutive relation describes the anisotropic
pressure induced by strong vorticity. Besides, we find seventeen anisotropic
transport coefficients: three bulk, four shear, three rotational viscosities,
four cross viscosities and three charge conductivities. Among these transport
coefficients, seven correspond to Hall-like transports, which do not induce
the entropy production. The appearance of the anisotropic transports is
analogous to magnetohydrodynamics, but there are no Hall transports in
magnetohydrodynamics in the strict hydrodynamic limit Grozdanov:2016tdf ;
Hongo:2020qpv .

We stress that spin is not a conserved quantity due to the inherent spin-orbit
coupling in relativistic systems. As a result, spin density is not a strict
hydrodynamic variable. Therefore, spin hydrodynamics and thus
gyrohydrodynamics should be regarded as quasi-hydrodynamics Grozdanov:2018fic
or a hydro+ theory Stephanov:2017ghc  valid in a regime where the relaxation
time of spin density towards its local equilibrium value is much longer than
that of other non-hydrodynamic modes Hongo:2021ona . To keep the notations
simple, however, we will continue calling spin density a hydrodynamic variable
and gyrohydrodynamics a hydrodynamic theory.

Throughout the paper, we use the mostly-plus metric
$\eta_{\mu\nu}=\mathrm{diag}(-1,+1,+1,+1)$, totally anti-symmetric tensor
$\epsilon^{\mu\nu\rho\sigma}$ with $\epsilon^{0123}=+1$, and notations
$X^{(\mu\nu)}=\frac{1}{2}(X^{\mu\nu}+X^{\nu\mu})$ and
$X^{[\mu\nu]}=\frac{1}{2}(X^{\mu\nu}-X^{\nu\mu})$ for an arbitrary tensor
$X^{\mu\nu}$.

Entropy current analysis for rotating spinful fluids.— The hydrodynamic
equations follow from Ward-Takahashi identities derived from symmetries of
underlying microscopic theories. In this Letter, we consider charged
relativistic fluids that enjoy the translational, Lorentz, and vector
$\text{U}(1)$ symmetries. These symmetries lead to the equations of motion,

$\partial_{\mu}\Theta^{\mu}_{~{}\nu}=0,\quad\partial_{\mu}\Sigma^{\mu}_{~{}\nu\rho}=-2\Theta_{[\nu\rho]},\quad\partial_{\mu}J^{\mu}=0,$
(1)

where $\Theta^{\mu\nu}$, $\Sigma^{\mu}_{~{}\rho\sigma}$, and $J^{\mu}$ are the
energy-momentum tensor, the spin current, and the charge current,
respectively. Note that the second equation follows from the total angular
momentum conservation $\partial_{\mu}J^{\mu}_{~{}\nu\rho}=0$ and a
decomposition of the total angular momentum tensor
$J^{\mu}_{~{}\nu\rho}=L^{\mu}_{~{}\nu\rho}+\Sigma^{\mu}_{~{}\nu\rho}$ with
$L^{\mu}_{~{}\nu\rho}=x_{\nu}\Theta^{\mu}_{~{}\rho}-x_{\rho}\Theta^{\mu}_{~{}\nu}$
being the orbital angular momentum. We choose the spin current
$\Sigma^{\mu}_{~{}\nu\rho}$ to be totally anti-symmetric, motivated by Dirac
fermions as the microscopic carrier of spin.

We then introduce our dynamical variables — the energy density $e$, the fluid-
four velocity $u^{\mu}$, and the spin density $\sigma_{\nu\rho}$ — by

$\Theta^{(\mu\nu)}u_{\nu}=-eu^{\mu},\quad
u_{\mu}\Sigma^{\mu}_{~{}\nu\rho}=-\sigma_{\nu\rho},\quad J^{\mu}u_{\mu}=-n,$
(2)

where we employed the Landau frame in defining the fluid velocity and the
energy density Landau:Fluid  and normalize the fluid velocity as
$u^{\mu}u_{\mu}=-1$. The spin density $\sigma^{\mu\nu}$ only has three
dynamical degrees of freedom (rather than six), as the totally anti-symmetric
property of the spin current forces $\sigma_{\mu\nu}$ to satisfy the Frenkel
condition Frenkel:1926zz  $\sigma_{\mu\nu}u^{\nu}=0$ in addition to the anti-
symmetric condition $\sigma_{\nu\rho}=-\sigma_{\rho\nu}$. Consequently, three
components of the spin non-conservation law,

$u^{\rho}\partial_{\mu}\Sigma^{\mu}_{~{}\nu\rho}=-2\Theta_{[\nu\rho]}u^{\rho},$
(3)

give constraint equations rather than dynamical ones. This becomes manifest in
the fluid rest frame specified as $u^{\mu}=(1,\bm{0})$, where Eq. (3) does not
have any time derivative. In the following, for notational simplicity we also
use the dual variable
$\sigma^{\mu}\equiv\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}u_{\nu}\sigma_{\rho\sigma}$
(or $\sigma_{\mu\nu}=-\epsilon_{\mu\nu\rho\sigma}u^{\rho}\sigma^{\sigma}$),
which satisfies $\sigma^{\mu}u_{\mu}=0$.

We introduce entropy density $s$ as a function of our dynamical variables
$e,\sigma_{\mu\nu},n$, and define the conjugate variables — the inverse
temperature $\beta=1/T$, chemical potential $\mu$, and spin potential
$\mu_{\nu\rho}$ — as

$T\mathrm{d}s=\mathrm{d}e-\frac{1}{2}\mu^{\nu\rho}\mathrm{d}\sigma_{\nu\rho}-\mu\mathrm{d}n~{}\Leftrightarrow~{}\beta\equiv\frac{\partial
s}{\partial e},\quad\beta\mu^{\nu\rho}\equiv-2\frac{\partial
s}{\partial\sigma_{\nu\rho}},\quad\beta\mu\equiv-\frac{\partial s}{\partial
n}.$ (4)

Since the spin density $\sigma_{\nu\rho}$ is transverse to the fluid velocity
$u^{\mu}$, it is sufficient to assume that the spin potential
$\mu^{\nu\rho}(=-\mu^{\rho\nu})$ and its dual
$\mu_{\mu}\equiv\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}u^{\nu}\mu^{\rho\sigma}$
satisfy $\mu_{\mu}u^{\mu}=0=\mu^{\mu\nu}u_{\nu}$. We will also use the thermal
vector $\beta^{\mu}\equiv\beta u^{\mu}$ in the subsequent calculations for
brevity.

To find the constitutive relations of strongly rotating spinful fluids, we
rely on a power counting scheme different from that for the conventional
hydrodynamics. We count vorticity, the other gradients, and spin density all
differently: Spatial components of the (thermal) vorticity
$\omega_{\mu\nu}\equiv\partial_{[\mu}\beta_{\nu]}$ (or its dual vector
$\omega^{\mu}\equiv\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}u_{\nu}\omega_{\rho\sigma}$)
and the spin potential $\mu^{\nu\rho}$ are counted as $O(\partial^{0})$, while
their difference ($\omega_{\mu\nu}-\beta\mu_{\mu\nu}$) and the other gradients
are counted as $O(\partial^{1})$. Besides, we assume that the spin density is
suppressed by another small parameter $\hbar$ as $\sigma_{\nu\rho}=O(\hbar)$,
while the vorticity-induced part of the spin potential is not. This designed
power counting scheme is motivated by two observations: (i) vorticity belongs
to a family of non-dissipative gradients that does not produce entropy, and
(ii) the spin operator is accompanied by the Planck constant since spin is a
quantum object. As the spin potential is not completely fixed by the
vorticity, the three spin densities enjoy their intrinsic dynamics.

In our power counting scheme, the zeroth-order quantities that one can use for
the tensor decomposition are $u^{\mu}$,
$\hat{\omega}^{\mu}=\omega^{\mu}/\sqrt{\omega^{\nu}\omega_{\nu}}$,
$\epsilon^{\mu\nu}=\epsilon^{\mu\nu\rho\lambda}u_{\rho}\hat{\omega}_{\lambda}$,
and $\eta^{\mu\nu}$. Assuming the parity symmetry111For example, the parity-
odd term $u^{\mu}\hat{\omega}^{\nu}$ in $\Theta^{\mu\nu}$ is excluded in
parity-even systems since the fluid velocity $u^{\mu}$ and the vorticity
$\hat{\omega}^{\mu}$ are parity-odd and parity-even, respectively. and
noticing the totally anti-symmetric property of the spin current, one may
parametrize the (non-)conserved currents as

$\begin{split}\Theta^{\mu\nu}&=(e+p_{1})u^{\mu}u^{\nu}+p_{1}\eta^{\mu\nu}+p_{2}\hat{\omega}^{\mu}\hat{\omega}^{\nu}+p_{3}\epsilon^{\mu\nu}+u^{\mu}\delta
q^{\nu}-\delta q^{\mu}u^{\nu}+\delta\Theta^{\mu\nu},\\\
\Sigma^{\mu\nu\rho}&=\epsilon^{\mu\nu\rho\lambda}(\sigma_{\lambda}+u_{\lambda}\delta\sigma),\\\
J^{\mu}&=nu^{\mu}+\delta J^{\mu},\end{split}$ (5)

where $\delta\Theta^{\mu\nu}$, $\delta q^{\mu}$, and $\delta J^{\mu}$ satisfy
$\delta\Theta^{\mu\nu}u_{\nu}=0=u_{\mu}\delta\Theta^{\mu\nu}$ and $\delta
q^{\mu}u_{\mu}=0=\delta J^{\mu}u_{\mu}$. Note that the constraint equation (3)
with the above parametrization relates $\delta q^{\mu}$ with the spin density
$\sigma^{\mu\nu}$ as

$\displaystyle\delta
q^{\nu}=-\frac{1}{2}\epsilon^{\mu\nu\rho\lambda}u_{\rho}\partial_{\mu}(\sigma_{\lambda}+u_{\lambda}\delta\sigma).$
(6)

In the following, we will express the so-far undetermined quantities in Eq.
(5) in terms of our conjugate variables $\beta^{\mu},\mu^{\nu\rho}$, and
$\mu$. Those quantities are $p_{a}~{}(a=1,2,3)$ at $O(1)$ and
$\delta\Theta^{\mu\nu}$, $\delta\sigma$, and $\delta J^{\mu}$ at
$O(\partial)$. Then, the equations of motion (1) serve as a closed set of
partial differential equations.

To obtain the constitutive relations, we use the second law of local
thermodynamics: There exists an entropy current $s^{\mu}$ satisfying
$\partial_{\mu}s^{\mu}\geq 0$ for any configuration of $\beta^{\mu}$,
$\mu^{\nu\rho}$, and $\mu$. Expressing the entropy current as
$s^{\mu}=su^{\mu}+\delta s^{\mu}$ with the first-order correction $\delta
s^{\mu}$, one obtains constraints resulting from the condition
$\partial_{\mu}s^{\mu}\geq 0$ up to the second order in derivatives $\partial$
and spin $\hbar$. This enables us to relate $p_{a}~{}(a=1,2,3)$ to the
thermodynamic quantities and determine the tensor structures of
$\delta\Theta^{\mu\nu}$, $\delta\sigma$, and $\delta J^{\mu}$, and $\delta
s^{\mu}$. We first insert Eq. (5) into the equations of motion (1).
Contracting them with $\beta_{\nu}$, $\frac{1}{2}\beta\mu^{\nu\rho}$, and
$\beta\mu$, we have

$\displaystyle\beta De$ $\displaystyle=-\beta(e+p_{1})\theta-
p_{2}\hat{\omega}^{\mu}\hat{\omega}_{\nu}\partial_{\mu}\beta^{\nu}-p_{3}\epsilon^{\mu\nu}\partial_{\mu}\beta_{\nu}-\delta\Theta^{(\mu\nu)}\partial_{\mu}\beta_{\nu}-\delta\Theta^{[\mu\nu]}\partial_{\mu}\beta_{\nu}$
$\displaystyle\quad-\delta
q_{\nu}(D\beta^{\nu}+\partial^{\nu}\beta)+\partial_{\mu}(\beta\delta
q^{\mu}),$ $\displaystyle-\frac{1}{2}\beta\mu^{\nu\rho}D\sigma_{\nu\rho}$
$\displaystyle=\mu^{\mu}\sigma_{\nu}\partial_{\mu}\beta^{\nu}-\frac{1}{2}\delta\sigma\epsilon^{\mu\nu\rho\lambda}u_{\lambda}\partial_{\mu}(\beta\mu_{\nu\rho})+\partial_{\mu}(\beta\mu^{\mu}\delta\sigma)+\Big{[}p_{3}\beta\mu_{\nu\rho}\epsilon^{\nu\rho}+\beta\mu_{\mu\nu}\delta\Theta^{[\mu\nu]}\Big{]},$
$\displaystyle-\beta\mu Dn$ $\displaystyle=\beta\mu n\theta-\delta
J^{\mu}\partial_{\mu}(\beta\mu)+\partial_{\mu}(\beta\mu\delta J^{\mu}),$ (7)

where we introduced $D\equiv u^{\mu}\partial_{\mu}$ and
$\theta\equiv\partial_{\mu}u^{\mu}$ and eliminated $\delta q^{\mu}$ using the
constraint equation (6). Applying the chain rule for
$s(x)=s\big{(}e(x),\sigma_{\nu\rho}(x),n(x))$ and using Eq. (7), we rewrite
the entropy production rate as

$\displaystyle\partial_{\mu}s^{\mu}$ $\displaystyle=s\theta+\beta
De-\frac{1}{2}\beta\mu^{\nu\rho}D\sigma_{\nu\rho}-\beta\mu
Dn+\partial_{\mu}\delta s^{\mu}$ $\displaystyle=\left[s-\beta\left(e+p_{1}-\mu
n\right)\right]\theta-(p_{2}\hat{\omega}^{\mu}\hat{\omega}^{\nu}-\mu^{\mu}\sigma^{\nu})\partial_{\mu}\beta_{\nu}-p_{3}\epsilon^{\mu\nu}(\partial_{\mu}\beta_{\nu}-\beta\mu_{\mu\nu})$
$\displaystyle\quad-\delta\Theta^{(\mu\nu)}\partial_{(\mu}\beta_{\nu)}-\delta\Theta^{[\mu\nu]}(\partial_{[\mu}\beta_{\nu]}-\beta\mu_{\mu\nu})-\delta\sigma\partial_{\perp\mu}(\beta\mu^{\mu}-\omega^{\mu})-\delta
J^{\mu}\partial_{\mu}(\beta\mu)$
$\displaystyle\quad+\partial_{\mu}\left(\delta s^{\mu}+\beta\delta
q^{\mu}+\beta\mu^{\mu}\delta\sigma+\beta\mu\delta
J^{\mu}\right)+\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}u_{\rho}\partial_{\mu}\sigma_{\sigma}(\partial_{\nu}\beta+D\beta_{\nu}),$
(8)

where we used
$\omega^{\mu}Du_{\mu}=\partial_{\perp\mu}\omega^{\mu}-T\omega^{\mu}\partial_{\mu}\beta$,
following from a Bianchi-like identity
$\partial_{\mu}(\epsilon^{\mu\nu\rho\sigma}\omega_{\rho\sigma})=0$, and
introduced
$\partial_{\perp\mu}\equiv(\delta_{\mu}^{~{}\nu}+u_{\mu}u^{\nu})\partial_{\nu}$.
We also retained all the terms here, i.e., Eq. (8) is exact and does not
depend on the power counting scheme at this stage.

To implement our power counting scheme in a clear manner, we decompose
$\mu^{\mu}$ and $\sigma_{\nu}$ into the zeroth-order components along
$\hat{\omega}^{\mu}$ and the derivative corrections as

$\mu^{\mu}=\mu_{\parallel}\hat{\omega}^{\mu}+\mu_{\perp}^{\mu},\quad\sigma^{\nu}=\sigma_{\parallel}\hat{\omega}^{\nu}+\sigma_{\perp}^{\nu},$
(9)

where $\sigma_{\parallel}=O(\hbar^{1})$ and
$\sigma_{\perp}^{\mu}=O(\hbar^{1}\partial^{1})$ according to our power
counting scheme. On the other hand, the magnitude of the spin potential reads
$\mu_{\parallel}=O(1)$ and $\mu_{\perp}^{\mu}=O(\partial^{1})$ without the
suppression by $\hbar$ because the spin potential induced by vorticity is not
suppressed according to our assumption. The perpendicular components are
higher order in derivative because they vanish in the local equilibrium where
the entropy production terminates with a condition
$\omega_{\mu\nu}=\beta\mu_{\mu\nu}$. We then obtain the divergence of the
entropy current up to $O(\hbar^{1}\partial^{2},\partial^{3})$ as

$\displaystyle\partial_{\mu}s^{\mu}$
$\displaystyle=\left[s-\beta\left(e+p_{1}-\mu
n\right)\right]\theta-(p_{2}-\mu_{\parallel}\sigma_{\parallel})\hat{\omega}^{\mu}\hat{\omega}^{\nu}\partial_{\mu}\beta_{\nu}-p_{3}\epsilon^{\mu\nu}(\partial_{\mu}\beta_{\nu}-\beta\mu_{\mu\nu})$
$\displaystyle\quad-\delta\Theta^{(\mu\nu)}\partial_{(\mu}\beta_{\nu)}-\delta\Theta^{[\mu\nu]}(\partial_{[\mu}\beta_{\nu]}-\beta\mu_{\mu\nu})-\delta
J^{\mu}\partial_{\mu}(\beta\mu)$
$\displaystyle\quad+\partial_{\mu}\left(\delta
s^{\mu}+\beta\mu^{\mu}\delta\sigma+\beta\mu\delta
J^{\mu}\right)+O(\hbar^{1}\partial^{2},\partial^{3}).$ (10)

Note that here one can safely drop the term
$\delta\sigma\partial_{\perp\mu}(\beta\mu^{\mu}-\omega^{\mu})$ in Eq. (8),
which is higher-order in derivatives.

Constitutive relations of rotating spinful fluids.— We derive the constitutive
relations up to the first order in derivatives $\partial$ and spin $\hbar$ by
requiring that the divergence of the entropy current (10) respects the second
law of thermodynamics.

Let us first consider the leading-order constitutive relation for the energy-
momentum tensor. The first line of Eq. (10) results from the non-dissipative
terms, which we require to vanish so that $\partial_{\mu}s^{\mu}=0$ at the
leading order. We find

$e+p_{1}=Ts+\mu n,\quad p_{2}=\mu_{\parallel}\sigma_{\parallel},\quad
p_{3}=0,$ (11)

which relates the parameters $p_{a}~{}(a=1,2,3)$ with the thermodynamic
quantities. Equation (11) indicates that the leading-order energy-momentum
tensor has anistropic pressure induced by vorticity as

$\Theta^{\mu\nu}_{(0)}=eu^{\mu}u^{\nu}+p_{\perp}\Xi^{\mu\nu}+p_{\parallel}\hat{\omega}^{\mu}\hat{\omega}^{\nu},$
(12)

where we introduced $p_{\perp}\equiv p_{1}$ and $p_{\parallel}\equiv
p_{1}+\mu_{\parallel}\sigma_{\parallel}$ and the projection tensor
$\Xi^{\mu\nu}\equiv\eta^{\mu\nu}+u^{\mu}u^{\nu}-\hat{\omega}^{\mu}\hat{\omega}^{\nu}$.
Note that one can express the first relation of Eq. (11) as
$e+p_{\parallel}=Ts+\mu n+\mu_{\parallel}\sigma_{\parallel}$ that captures the
contribution of spin to the thermodynamic relation. The constitutive relation
(12) bears similarity with that for relativistic magnetohydrodynamics in which
the direction of magnetic field plays the role of $\hat{\omega}^{\mu}$ and
magnetization of the fluid induces difference between $p_{\perp}$ and
$p_{\parallel}$ Huang:2011dc ; Finazzo:2016mhm ; Grozdanov:2016tdf ;
Hernandez:2017mch ; Hongo:2020qpv .

We turn to the the first-order corrections to the constitutive relations.
Requiring the local second law of thermodynamics $\partial_{\mu}s^{\mu}\geq
0$, we find

$\begin{split}\delta\Theta^{(\mu\nu)}&=-T\eta^{\mu\nu\rho\sigma}\partial_{(\rho}\beta_{\sigma)}-T\xi^{\mu\nu\rho\sigma}(\partial_{[\rho}\beta_{\sigma]}-\beta\mu_{\rho\sigma}),\\\
\delta\Theta^{[\mu\nu]}&=-T\gamma^{\mu\nu\rho\sigma}(\partial_{[\rho}\beta_{\sigma]}-\beta\mu_{\rho\sigma})-T\xi^{\prime\,\mu\nu\rho\sigma}\partial_{(\rho}\beta_{\sigma)},\\\
\delta J^{\mu}&=-T\kappa^{\mu\nu}\partial_{\nu}(\beta\mu),\\\ \delta
s^{\mu}&=-\beta\mu^{\mu}\delta\sigma-\beta\mu\delta J^{\mu},\end{split}$ (13)

with the two viscous tensors $\eta^{\mu\nu\rho\sigma}$ and
$\gamma^{\mu\nu\rho\sigma}$, and conductivity tensor $\kappa^{\mu\nu}$. In
addition, there are “cross terms” given by $\xi^{\mu\nu\rho\sigma}$ and
$\xi^{\prime\,\mu\nu\rho\sigma}$ that convert the symmetric and antisymmetric
flow gradients to the antisymmetric and symmetric parts of
$\delta\Theta^{\mu\nu}$, respectively.222 The presence of the cross terms is
expected from the properties of the Coriolis force
${\bm{F}}\propto{\bm{v}}\times{\bm{\omega}}$ with the velocity ${\bm{v}}$ and
angular velocity ${\bm{\omega}}$. Operating derivatives on the Coriolis force,
one finds a nonzero divergence and rotation of the force,
$\nabla\cdot{\bm{F}}\propto{\bm{\omega}}\cdot(\nabla\times{\bm{v}})$ and
$\nabla\times{\bm{F}}\propto({\bm{\omega}}\cdot\nabla){\bm{v}}-{\bm{\omega}}(\nabla\cdot{\bm{v}})$,
induced by the rotation and gradient/divergence of ${\bm{v}}$, respectively.
Here, we assumed a constant ${\bm{\omega}}$ for demonstration.  Similar cross
terms have been discussed in the context of nematic liquid crystal
Parodi1970StressTF  (see also Ref. RevModPhys.46.617  for a review).

These tensors are decomposed by the use of the order-one spatial tensors
$\hat{\omega}^{\mu}$, $\Xi^{\mu\nu}$, and $\epsilon^{\mu\nu}$. The transverse,
longitudinal, and Hall-like components read

$\begin{split}\eta^{\mu\nu\rho\sigma}&=\zeta_{\perp}\Xi^{\mu\nu}\Xi^{\rho\sigma}+\zeta_{\parallel}\hat{\omega}^{\mu}\hat{\omega}^{\nu}\hat{\omega}^{\rho}\hat{\omega}^{\sigma}+\zeta_{\times}\big{(}\hat{\omega}^{\mu}\hat{\omega}^{\nu}\Xi^{\rho\sigma}+\Xi^{\mu\nu}\hat{\omega}^{\rho}\hat{\omega}^{\sigma}\big{)}\\\
&\quad+\eta_{\perp}\big{(}\Xi^{\mu\rho}\Xi^{\nu\sigma}+\Xi^{\mu\sigma}\Xi^{\nu\rho}-\Xi^{\mu\nu}\Xi^{\rho\sigma}\big{)}+2\eta_{\parallel}\big{(}\hat{\omega}^{\mu}\Xi^{\nu(\rho}\hat{\omega}^{\sigma)}+\hat{\omega}^{\nu}\Xi^{\mu(\rho}\hat{\omega}^{\sigma)}\big{)}\\\
&\quad+2\eta_{H_{\perp}}\big{(}\Xi^{\mu(\rho}\epsilon^{\sigma)\nu}+\Xi^{\nu(\rho}\epsilon^{\sigma)\mu}\big{)}+2\eta_{H_{\parallel}}\big{(}\hat{\omega}^{\mu}\epsilon^{\nu(\rho}\hat{\omega}^{\sigma)}+\hat{\omega}^{\nu}\epsilon^{\mu(\rho}\hat{\omega}^{\sigma)}\big{)},\\\
\gamma^{\mu\nu\rho\sigma}&=\gamma_{\perp}\big{(}\Xi^{\mu\rho}\Xi^{\nu\sigma}-\Xi^{\mu\sigma}\Xi^{\nu\rho}\big{)}+2\gamma_{\parallel}\big{(}\hat{\omega}^{\mu}\Xi^{\nu[\rho}\hat{\omega}^{\sigma]}-\hat{\omega}^{\nu}\Xi^{\mu[\rho}\hat{\omega}^{\sigma]}\big{)}\\\
&\quad+2\gamma_{H}\big{(}\hat{\omega}^{\mu}\epsilon^{\nu[\rho}\hat{\omega}^{\sigma]}-\hat{\omega}^{\nu}\epsilon^{\mu[\rho}\hat{\omega}^{\sigma]}\big{)},\\\
\xi^{\mu\nu\rho\sigma}&=2\xi_{\parallel}\big{(}\hat{\omega}^{\mu}\Xi^{\nu[\rho}\hat{\omega}^{\sigma]}+\hat{\omega}^{\nu}\Xi^{\mu[\rho}\hat{\omega}^{\sigma]}+\hat{\omega}^{\mu}\Xi^{\nu(\rho}\hat{\omega}^{\sigma)}-\hat{\omega}^{\nu}\Xi^{\mu(\rho}\hat{\omega}^{\sigma)}\big{)}\\\
&\quad+\zeta_{H_{\perp}}\big{(}\Xi^{\mu\nu}\epsilon^{\rho\sigma}-\Xi^{\rho\sigma}\epsilon^{\mu\nu}\big{)}+\zeta_{H\parallel}\big{(}\hat{\omega}^{\mu}\hat{\omega}^{\nu}\epsilon^{\rho\sigma}-\hat{\omega}^{\rho}\hat{\omega}^{\sigma}\epsilon^{\mu\nu}\big{)}\\\
&\quad+2\xi_{H}\big{(}\hat{\omega}^{\mu}\epsilon^{\nu[\rho}\hat{\omega}^{\sigma]}+\hat{\omega}^{\nu}\epsilon^{\mu[\rho}\hat{\omega}^{\sigma]}+\hat{\omega}^{\mu}\epsilon^{\nu(\rho}\hat{\omega}^{\sigma)}-\hat{\omega}^{\nu}\epsilon^{\mu(\rho}\hat{\omega}^{\sigma)}\big{)},\\\
\kappa^{\mu\nu}&=\kappa_{\perp}\Xi^{\mu\nu}+\kappa_{\parallel}\hat{\omega}^{\mu}\hat{\omega}^{\nu}+\kappa_{H}\epsilon^{\mu\nu}.\end{split}$
(14)

Those viscous tensor should satisfy the Onsager reciprocal relation stemming
from the time-reversal symmetry Onsager1931 ; de1984non . Note that
$\hat{\omega}$ is an odd quantity under the time-reversal transformation.
Then, the above tensors should satisfy the Onsager reciprocal relations
$\eta^{\mu\nu\rho\sigma}(\hat{\omega})=\eta^{\rho\sigma\mu\nu}(-\hat{\omega})$,
$\gamma^{\mu\nu\rho\sigma}(\hat{\omega})=\gamma^{\rho\sigma\mu\nu}(-\hat{\omega})$,
$\xi^{\mu\nu\rho\sigma}(\hat{\omega})=\xi^{\rho\sigma\mu\nu}(-\hat{\omega})$,
and
$\xi^{\prime\,\mu\nu\rho\sigma}(\hat{\omega})=\xi^{\rho\sigma\mu\nu}(-\hat{\omega})$.
The last relation provides $\xi^{\prime\,\mu\nu\rho\sigma}$ in terms of
$\xi^{\mu\nu\rho\sigma}$.

The terms appearing in the viscous and conductivity tensors can be classified
into two types according to their properties under the interchange between
(pairs of) Lorentz indices — $(\mu,\nu)\leftrightarrow(\rho,\sigma)$ for the
viscous tensors and $\mu\leftrightarrow\nu$ for the conductivity tensor —
without flipping the sign of $\hat{\omega}$. The first type, the terms having
the subscript $H$, is anti-symmetric under the index interchange. These are
non-dissipative Hall-like terms, which do not contribute to the entropy
production. The other terms, i.e., those without the subscript $H$, are
symmetric under the index interchange, meaning that they are dissipative and
contribute to the entropy production. Among these, the bulk viscosity
$\zeta_{\times}$ captures the non-equilibrium correction to the anisotropic
pressures $p_{\perp}$ and $p_{\parallel}$ in response to expansion or
compression in the parallel and perpendicular directions, respectively. Those
two processes are reciprocal to one another, and we required them to satisfy
the Onsager reciprocal relation. As a result, we find seventeen transport
coefficients: three bulk viscosities
$\zeta_{\parallel},\zeta_{\perp},\zeta_{\times}$, four shear viscosities
$\eta_{\parallel},\eta_{\perp},\eta_{H_{\parallel}},\eta_{H_{\perp}}$, three
rotational viscosities $\gamma_{\parallel},\gamma_{\perp},\gamma_{H}$, four
cross viscosities $\xi_{\parallel},\zeta_{H\parallel},\zeta_{H\perp},\xi_{H}$
and three conductivities $\kappa_{\perp},\kappa_{\parallel},\kappa_{H}$. Note
that we do not include a possible transverse structure,
$\Xi^{\mu[\rho}\epsilon^{\sigma]\nu}-\Xi^{\nu[\rho}\epsilon^{\sigma]\mu}$, in
$\gamma^{\mu\nu\rho\sigma}$ since it identically vanishes according to the so-
called Schouten identity
$\eta^{\mu\gamma}\epsilon^{\rho\nu\alpha\beta}-\eta^{\rho\gamma}\epsilon^{\mu\nu\alpha\beta}-\eta^{\nu\gamma}\epsilon^{\rho\mu\alpha\beta}-\eta^{\alpha\gamma}\epsilon^{\rho\nu\mu\beta}-\eta^{\beta\gamma}\epsilon^{\rho\nu\alpha\mu}=0$.

To respect the second law, the dissipative transport coefficients must satisfy
a set of inequalities,

$\begin{split}&\zeta_{\perp}\geq 0,\quad\zeta_{\parallel}\geq
0,\quad\zeta_{\perp}\zeta_{\parallel}-\zeta_{\times}^{2}\geq 0,\quad\\\
&\eta_{\perp}\geq 0,\quad\eta_{\parallel}\geq 0,\quad\gamma_{\perp}\geq
0,\quad\gamma_{\parallel}\geq
0,\quad\eta_{\parallel}\gamma_{\parallel}-\xi_{\parallel}^{2}\geq 0,\quad\\\
&\kappa_{\perp}\geq 0,\quad\kappa_{\parallel}\geq 0.\end{split}$ (15)

On the other hand, the signs of the seven coefficients
$\eta_{H_{\perp}},\eta_{H_{\parallel}},\gamma_{H},\zeta_{H\parallel},\zeta_{H\perp},\xi_{H},$
and $\kappa_{H}$ are not constrained by the second law, as they do not
contribute to the entropy production just like the Hall transports in
relativistic magnetohydrodynamics Huang:2011dc ; Finazzo:2016mhm ;
Hernandez:2017mch ; Grozdanov:2016tdf ; Hongo:2020qpv . Those non-dissipative
transports arise due to strong vorticity, which can be interpreted as an
analogue of strong magnetic fields Landau:Kinetics : The Coriolis force does
not do work, similarly to the Lorentz force by magnetic fields. One clear
difference from the Hall transports in magnetohydrodynamics is that those
transport coefficients in gyrohydrodynamics are even under charge conjugation
and can have non-vanishing values for neutral fluids.

Summary and Outlook.— In this Letter, we have reported the phenomenological
derivation of the constitutive relations for spinful fluids with strong
vorticity. It is formulated in the “spin hydrodynamic regime,” where spin
density enjoys its intrinsic dynamics as the energy-momentum and vector charge
densities do. We have found the leading and next-to-leading order constitutive
relations that capture the spin dynamics intertwined with rotating fluid
dynamics as well as the anisotropic pressure and transport phenomena induced
by the strong vorticity. We have obtained seventeen transport coefficients:
three bulk viscosities $\zeta_{\parallel},\zeta_{\perp},\zeta_{\times}$, four
shear viscosities
$\eta_{\parallel},\eta_{\perp},\eta_{H_{\parallel}},\eta_{H_{\perp}}$, three
rotational viscosities $\gamma_{\parallel},\gamma_{\perp},\gamma_{H}$, four
cross viscosities $\xi_{\parallel},\zeta_{H\parallel},\zeta_{H\perp},\xi_{H}$
and three conductivities $\kappa_{\perp},\kappa_{\parallel},\kappa_{H}$. A
subset of the transport coefficients should satisfy the positivity constraint
(15), while there are no such constraints on the Hall-like transport
coefficients.

The entropy-current analysis leaves the seventeen transport coefficients as
phenomenological parameters. The theoretical evaluation of these parameters
can be carried out on the basis of either the linear-mode analysis or the
Green-Kubo formula Green ; Nakano ; Kubo ; Kadanoff-Martin1963 ;
Luttinger:1964zz , both of which are regarded as the matching conditions for
the (quasi-)hydrodynamic description. Deriving such matching conditions for
the seventeen transport coefficients and evaluating them with microscopic
theories (such as quantum chromodynamics) are important outlooks in line with
the present Letter.

We investigated the formulation with the totally anti-symmetric spin current
that is motivated by the canonical spin current of Dirac fermions, while
previous works do not necessarily assume this tensor structure (e.g., Ref.
Hattori:2019lfp ). One may always redefine the spin current and energy-
momentum tensor operators — or change what one calls spin current and energy-
momentum tensor operators — without modifying the form of their equations of
motion via the so-called pseudo-gauge transformations Belinfante1939 ;
Belinfante1940 ; Rosenfeld1940  (see, e.g., Refs. Becattini:2018duy ;
Li:2020eon ; Speranza:2020ilk  for recent discussion). This may lead to
another definition of a spin current operator which is not necessarily totally
anti-symmetric. It may be worth investigating what the resulting hydrodynamic
equation looks like with a different choice of the spin and energy-momentum
tensor. Besides, strong vorticity can give rise to intriguing phenomena in
spinless case as well. Interested readers can find discussions in Ref.
Li:2020eon  for possible anisotropies in constitutive relations and in, e.g.,
Ref. Jensen:2011xb  for the modification of thermodynamics in a $2+1$
dimensional fluid. We leave all these issues as future works.

## Acknowledgment

The authors thank Matthias Kaminski, Yu-Chen Liu, Kazuya Mameda, Misha
Stephanov, Xiao-Liang Xia and Ho-Ung Yee for fruitful discussions. This work
is partially supported by JSPS KAKENHI under grant Nos. 20K03948, 22H02316,
and 22K14045 and RIKEN iTHEMS Program (in particular iTHEMS Non-Equilibrium
Working group and iTHEMS Mathematical Physics Working group). Z. C. and X.-G.
H. are supported by NSFC under Grant No. 12075061 and Shanghai NSF under Grant
No. 20ZR1404100.

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# Updated kinematics of the Radcliffe Wave: non-synchronous, dipole-like
vertical oscillations

Zhi-Kai Zhu Guangxi Key Laboratory for Relativistic Astrophysics, School of
Physical Science and Technology,
Guangxi University, Nanning 530004, China  Min Fang Purple Mountain
Observatory, Chinese Academy of Sciences, 10 Yuanhua Road, Nanjing 210023,
China  Zu-Jia Lu Guangxi Key Laboratory for Relativistic Astrophysics, School
of Physical Science and Technology,
Guangxi University, Nanning 530004, China  Junzhi Wang Guangxi Key Laboratory
for Relativistic Astrophysics, School of Physical Science and Technology,
Guangxi University, Nanning 530004, China  Guang-Xing Li South-Western
Institute for Astronomy Research, Yunnan University, Kunming 650500, China
Shiyu Zhang Purple Mountain Observatory, Chinese Academy of Sciences, 10
Yuanhua Road, Nanjing 210023, China  Veli-Matti Pelkonen Institut de Ciències
del Cosmos, Universitat de Barcelona, IEEC-UB, Martí i Franquès 1, E08028
Barcelona, Spain  Paolo Padoan Institut de Ciències del Cosmos, Universitat de
Barcelona, IEEC-UB, Martí i Franquès 1, E08028 Barcelona, Spain  Department of
Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH
03755, USA  En-Wei Liang Guangxi Key Laboratory for Relativistic Astrophysics,
School of Physical Science and Technology,
Guangxi University, Nanning 530004, China

###### Abstract

The kinematic structure of the Radcliffe Wave (RW) is crucial for
understanding its origin and evolution. In this work, we present an accurate
measurement of the vertical velocity $V_{Z}$ by where the radial velocity (RV)
measures are taken into consideration. This is achieved in two ways. First,
the velocities are measured towards Young Stellar Objects (YSOs), using their
RV and proper motion measurements from APOGEE-2 and Gaia DR3. Second, we
combine RV measurements toward clouds with proper motion measurements of
associated YSOs to determine the vertical velocities of the clouds. The
results reveal that the oscillations in $V_{Z}$ are not synchronous with the
vertical coordinate. The difference is caused by a combination of the effect
of the radial velocity which we include in this paper, and the difference in
models. By supplementing our analysis with additional young star samples, we
find a consistent dipole pattern in $V_{Z}$. The fact that no significant
amplitude differences are found among the analyzed samples indicates that
there is no apparent age gradient within the dipole. We propose that RW
evolves at a relatively slow rate. The fact that it will take a much longer
time for RW to complete a full period compared to the cloud lifetimes
challenges its classification as a traditional “wave”. This age discrepancy
should explain the phase difference, and non-synchronous oscillation found in
kinematic studies.

ISM: clouds; ISM: structure; ISM: kinematics and dynamics; stars: kinematics
and dynamics

††thanks<EMAIL_ADDRESS>
##  1 Introduction

The recent discovery of the Radcliffe Wave (RW) has sparked significant
interest as it reveals a novel distribution pattern of molecular clouds and
dense gas in the solar neighborhood. The The RW was first identified by [1],
which belongs to a class of Glactic-scale gas filament reported since [30].
[52] that had obtained the accurate distances of molecular clouds. It is a
coherent molecular cloud structure spanning 2.7 kiloparsecs in length and
exhibiting vertical oscillations in the Z direction of the Galactic coordinate
system. They described the undulating behavior as a damped sinusoidal mode.
Besides, they argued that a part of the well-known Gould Belt, first described
by Benjamin Gould in 1879 and after whom the Belt is named [24], is a
projection effect of RW.

The nature of the Gould’s belt has been studied for decades after it was first
pointed out. [27] and [20] used observation data of O and B stars to separate
the Gould’s Belt from the Galactic Belt and to determine the parameters of the
velocity field in both belts. They found that within the Gould’s Belt, the
young stars form a local irregularity of the velocity field that arises from
the superposition of the motion of different local groups of young stars. The
kinematics of the Gould’s Belt has been studied considering the spatial
distribution and the motion of the stars [e.g. 8, 7, 36, 16]. [53] used Gaia
space telescope observation data to analyze the kinematics of the Gould’s
Belt. Their results show that nearly all of the star-forming complexes in the
solar vicinity lie on the surface of the Local Bubble and that the velocities
of the young stars show outward expansion mainly perpendicular to the bubble’s
surface. So their results supported the picture that supernovae created the
expansion of the Local Bubble.

The kinematics of RW is crucial for understanding its nature. The spatial
existence of RW has been well-confirmed, but its kinematics as a wave-like
structure remains uncertain. Recent studies by [29] and [44] have investigated
the vertical oscillations of RW using tracers such as young stars and young
cluster members. [29] found that the vertical velocity ($V_{Z}$) and vertical
coordinate of RW oscillate synchronously, which means they exhibit a constant
phase difference. The oscillations are consistent with the structure behaving
either like a standing wave, or a wave traveling towards the left. Conversely
[44] found a dipole in $V_{Z}$ along RW, which means the vertical velocities
on the two sides of RW are comparable in magnitude but opposite in direction.
However, both papers made an approximation ignoring the radial velocity (RV)
when calculating $V_{Z}$, and obtained oscillation results with only
relatively weak amplitudes of approximately 4-7 km/s. These weak amplitudes
indicate that the approximation they used may lead to a significant impact on
the results, emphasizing the importance of considering RV when studying the
kinematic structure of RW. While [46] used RV from Gaia DR2 [22] to calculate
$V_{Z}$, they considered only a very limited region of the clouds. In
addition, Gaia DR3 [23] contains approximately four times as many sources with
RV measurements than Gaia DR2.

In order to obtain precise measurements of the vertical oscillations of RW, we
focus on studying its kinematic structure by calculating the velocities of the
clouds using two approaches that incorporate the RV component. We emphasize
the crucial role that RV plays in accurately determining the cloud velocities.
In the first approach, we utilize multiple samples of young stars as tracers,
incorporate the APOGEE-2 dataset [33] and the Gaia DR3 dataset to extract
their RV information. By combining RV with the proper motion, parallax, right
ascension, and declination data from Gaia DR3, we can derive the three-
dimensional velocities of objects in Galactic coordinates, in particular
$V_{Z}$, and the vertical distance (Z) to the Galactic disk.

In the second approach, we directly study molecular clouds to derive
measurements of their radial velocities and locations. This is accomplished by
utilizing the whole-Galaxy CO survey conducted by [12] and the compendium
provided by [52]. For more precise measurements of RV and distance for Cygnus
X and North America region, we also rely on the data (51) from the Milky Way
Imaging Scroll Painting (MWISP) project, which is a multi-line survey in
12CO/13CO/C18O along the northern Galactic plane with PMO-13.7m telescope
[43]. We then estimate the proper motions of the molecular clouds based on
those of associated young stars. This approach enables us to calculate the
approximate $V_{Z}$ for the majority of molecular clouds in RW.

By utilizing these two methods, we aim to obtain more accurate measurements of
the vertical oscillations of RW, allowing for a comprehensive understanding of
its kinematics and origin. The structure of the paper is as follows. In § 2 we
describe the data samples used in our analysis, followed by our results in §
3. We discuss our main results in § 4 and summarise our conclusions in § 5.

##  2 data and method

###  2.1 Young Star Samples

In the first method, we select a sample of Class I/II Young Stellar Objects
(YSOs) from the catalogue provided by [34] to serve as a major tracer for
studying RW. Those YSOs are the youngest stars that are more likely to retain
the locations and kinematics inherited from their parent clouds. The sample is
constructed by cross-matching Gaia DR2 and AllWISE datasets by a machine
learning method. We select Class I/II YSOs by cross-matching with Gaia within
1 arcsec error in an almost rectangular area along RW for the proper motion
and parallax measurements. The selected regions cover the height range
extending in the $Z$-direction from -300 pc to 250 pc. It also has a width of
500 pc and a length of 3.7 kpc, centered around RW. Through this process, we
obtained a sample of 5194 Class I/II YSOs that are relevant to our study of
RW. We name it YSO sample I.

In addition, as a supplement and for comparison, we incorporate three
additional catalogs: the Class III YSO catalogue by [34], the Alma catalogue
of OB stars II by [38] and the open cluster catalogue by [5]. All these
catalogues are based on Gaia DR2 data. To ensure consistency, we update the
proper motion and parallax measurements using the more recent Gaia DR3 data.
Subsequently, we perform a cross-match for RV in the same manner as we do for
the Class I/II YSOs, within the same rectangular area encompassing RW. This
cross-matching process allow us to obtain three distinct samples: Class III
YSOs, OB stars, and young open cluster members (members of open clusters with
ages less than 100 Myr).

In the second method, when approximating the proper motion of molecular
clouds, for a more comprehensive coverage, we also refer to another YSO
catalogue from [35] and combine it with the Class I/II YSO catalogue from
[34]. After selecting from the same rectangular area, we get YSO sample II.

By combining all the aforementioned samples of young stars—YSO sample I, YSO
sample II, Class III YSO sample, OB star sample, and young open cluster member
sample—we aim to explore RW from multiple perspectives and gain a
comprehensive understanding of its characteristics and dynamics.

###  2.2 Radial Velocity

In the first method, young stars are utilized as tracers. To derive RV of
young star samples, we perform a cross-match within 1 arcsec tolerance between
the young stars and two datasets: Sloan Digital Sky Survey IV (SDSS-IV; 4)
APOGEE-2 [33] and Gaia Data Release 3 (Gaia DR3; 21, 23), utilizing the
“VHELIO AVG” and “radial velocity” measurements as the RV values respectively.
Gaia and APOGEE are two independent astronomical projects that collect RV data
for stars. Gaia utilizes spectroscopic observations to measure RVs of stars by
analyzing the Doppler shift in stellar spectra obtained from multiple
observations. APOGEE measures RVs of stars by analyzing the absorption
features in their spectra. APOGEE primarily operates in the H-band, which
covers the wavelength range of 1.51-1.70 $\mu$m in the near-infrared spectrum.
This range allows for the observation of sources that are difficult to detect
in optical bands due to the advantage of infrared light in penetrating dust
and gas. According to the findings reported in [25], however, RV measurements
obtained from Gaia are not as accurate when compared to the high-resolution
RVs derived from APOGEE observations of spectroscopically stable young stars.
This discrepancy is attributed to the presence of unique spectral features
exhibited by YSOs resulting from both accretion and activity processes.
Specifically, these features are prominent in the vicinity of the Ca II
triplet at approximately 8500 Å, which falls within the spectral range covered
by Gaia’s Radial Velocity Spectrometer spectrograph, as highlighted by [39].
So when taking young stars as tracers, we mainly focus on APOGEE’s RVs to
correct $V_{Z}$, and Gaia’s RVs as an auxiliary supplement. Then by
incorporating the proper motion, parallax, right ascension, and declination
data from Gaia DR3, we are able to calculate the three-dimensional velocities
of young stars along RW. In the YSO sample I including 5194 YSOs mentioned in
section 2.1, 1196 YSOs have RVs obtained from APOGEE, and 893 YSOs from Gaia
DR3. Notably, 224 YSOs have RV measurements available from both sources, and
in Appendix B we discuss the relationship of RV values from these two sources.

Figure 1: The distribution of the vertical component of radial velocity
(${\mathrm{RV}}\cdot\sin b$) obtained from APOGEE; $b$ is the galactic
latitude.

We want to emphasize the significance of considering RV in our analysis. As
illustrated in Figure 1, when we take into account the YSO sample I and
calculate the vertical component of RV using the latitude information of YSOs
along with RV data from APOGEE-2, it becomes evident that a substantial
portion of the absolute values of the vertical component exceeds the amplitude
of $V_{Z}$ oscillations reported in previous studies. Specifically, 62.5$\%$
of the sample exhibit vertical components below $-$5 km/s, which is comparable
to the typical value of the weak-amplitude oscillations reported in [29]. This
finding underscores the necessity of considering RV.

Figure 2: $V_{Z}$ distribution of YSOs along the Radcliffe Wave. The upper
panel shows the result using RV data from APOGEE, while the lower panel shows
the result using RV data from Gaia. The red points represent more accurate
$V_{Z}$ corrected by RV, while the blue points indicate approximate $V_{Z}$
values calculated without considering RV. $T$ represents the direction that
the Radcliffe Wave extends, with the zero point on the $T$ axis indicating the
closest point to the sun along the Radcliffe Wave. The blue lines represent
the median values of blue crosses within 200 pc bins along the $T$ axis, with
each bin containing more than 30 points. The black dashed lines represent the
median values of the red circles within the same bin.

In the second method, we study molecular clouds directly and extract RV
measurements for clouds from the whole-Galaxy CO surveys by [12], which cover
the majority of molecular clouds within RW. These comprehensive CO surveys
were conducted using the 1.2-meter Millimeter-Wave Telescope at the CfA and
its counterpart on Cerro Tololo in Chile. These two instruments have provided
the most extensive, uniform, and widely utilized survey of interstellar carbon
monoxide (CO) in the Milky Way, serving as the best indicator of the
predominantly invisible molecular hydrogen that constitutes the bulk of mass
within molecular clouds [42, 11]. From the CO surveys, we get RV measurements
of the molecular clouds. However, calculating $V_{Z}$ requires information on
the proper motions of the clouds, which are challenging to measure directly
and still lacking. Fortunately, there are strong kinematic correlations
between the young stellar populations and associated molecular gas and clouds,
as reported in [18], [45] and [17]. By considering CO and its isotope
molecules as tracers for clouds, there are pieces of evidence supporting that
young stars inherit RV from the molecular clouds, as reported in [9, 10]. This
suggests a strong relationship between the kinematics of molecular clouds and
associated young stars, which implies there is also a similarity in their
proper motions. In this approach, we perform a spatial cross-match between the
clouds and the associated YSOs in YSO sample II. By doing this, we acquire
approximate proper motions for certain molecular clouds. This approximation
allows us to calculate $V_{Z}$ for the majority of clouds within RW. To obtain
more precise RV and distance measurements for the Cygnus X and North America
regions, we also use the data from the MWISP project, referring to 51 for
comprehensive information.

By combining RV with the proper motion and location data from Gaia DR3, along
with the known component of Solar motion
$(U_{\odot},V_{\odot},W_{\odot})=(11.1,12.24,7.25)\,\rm km\,\rm s^{-1}$ [41],
we are able to derive the three-dimensional velocities of objects. We also
calculate the vertical distance to the Galactic disk based on the precise
measurement of the Sun’s position, $Z_{\odot}=20.8\,\rm pc$, from [3].

##  3 results

###  3.1 Non-synchronous $V_{Z}$ oscillations

Initially, we utilize RV measurements from APOGEE and Gaia in the first method
and obtain more accurate vertical velocities of YSO sample I. These refined
$V_{Z}$ are shown with the red data points in Figure 2. In Figure 2, $T$ is
the direction RW extends and the zero point on the $T$ axis represents the
closest point to the sun along RW (corresponding to the $Y_{\mathrm{prime}}$
axis in the 1). To facilitate comparison, we use blue points to represent the
approximate $V_{Z}$ values that were calculated without taking RV measurements
into consideration. Additionally, we mark black lines that represent the
median values of the blue points within bins of 200 pc along the $T$ axis. We
only include bins that contain more than 10 stars in order to ensure
statistical significance in our analysis. The upper panel displays the results
using RV from APOGEE, while the lower panel shows the results using RV from
Gaia. In both panels, the peak of the approximate oscillation (-500 pc¡$T$¡0
pc, corresponding to 0.75 kpc¡$X^{\prime}$¡1.25 kpc in Figure 3 in 29) tends
to flatten when RV is taken into account. This suggests that the synchronous
oscillation reported in previous studies may arise from the neglect of RV, or
at least it should be corrected by incorporating RV measurements.

Figure 3: The spatial and $V_{Z}$ distributions of the cross-matched molecular
clouds along the Radcliffe Wave. The matched parts of clouds are depicted as
red circles, where the radii represent the relative sizes of the matched parts
based on the $T$ axis. The upper panel provides a side view of the spatial
distribution, with the red circles indicating the cloud locations and the pink
arrows indicating the directions and magnitudes of the $V_{Z}$ component. The
length of each arrow corresponds to the value of $V_{Z}$ (in km/s) multiplied
by 15. The orange curve marks the fit to the position of the Radcliffe Wave
model suggested by [1]. In contrast, locations of the YSOs in YSO sample II
are represented by grey points, and the purple ones indicate the cross-matched
YSOs. The lower panel displays the distribution of $V_{Z}$. The grey circles
and red circles respectively represent the $V_{Z}$ result obtained without and
with RV information, accompanied by error bars indicating 1$\sigma$
dispersions.

However, when considering the RV of YSOs, only approximately 20% of the
objects have available RV measurements. The coverage of these measurements is
primarily on nearby regions such as the Orion, Taurus, and Perseus clouds.
However, the coverage of other regions, such as Cygnus X, North America, and
the Mon R2 cloud, is inadequate. To gain a more comprehensive understanding,
we directly study the molecular clouds as a complementary approach. We utilize
the widely recognized whole-galaxy CO surveys conducted by [12] and the data
from the MWISP project to obtain RV measurements of these molecular clouds.
Subsequently, we combine the proper motions of associated YSOs in YSO sample
II, which are highly likely to inherit and preserve the kinematics of their
parent clouds. More details of the method is described in Appendix A. As a
result, we obtain the spatial distribution and $V_{Z}$ distribution of the
crossed-matched clouds, as illustrated in Figure 3. In Figure 3, the matched
parts of molecular clouds are shown with red circles. The size of the circles
represents the relative sizes of the matched parts based on the $T$ axis. The
upper panel shows a side view of the spatial distribution, where the red
circles indicate the locations of the clouds. The pink arrows represent the
values of the $V_{Z}$ components. The lower panel displays the distribution of
$V_{Z}$ for the clouds, in which the red circles represent the $V_{Z}$ values
of clouds obtained by combining RV from the CO surveys and proper motion from
associated YSOs, and the grey circles illustrate the results obtained without
considering RVs. Our analysis reveals the crucial role of RV measurements in
understanding the $V_{Z}$ distribution of RW, particularly in the $T$¡300pc
region. Incorporating the RV components leads to a general decrease in the
values of $V_{Z}$ in this region.

Figure 4: Fitting results of the spatial and $V_{Z}$ distribution of clouds in
the $T$¡300 pc region. In the upper panel, the grey line and red line are the
fitting results of vertical velocities without and with RV measurements,
respectively. The orange line is the fitting result of vertical coordinates
(right $y$-axis). The lower panel shows the phase difference ($\delta\phi$)
between vertical coordinates and vertical velocities, with the grey and red
curves representing the distribution of $\delta\phi$ without and with RV
measurements, respectively. Figure 5: A comparison of RV between clouds and
their associated YSOs. Each data point represents a cloud, where the x-axis
represents the median RV obtained from APOGGE for the associated YSOs
(${\mathrm{RV}}_{\mathrm{APO}}$), and the y-axis represents the residual
between the average RV of clouds (${\mathrm{RV}}_{\mathrm{MC}}$) and
${\mathrm{RV}}_{\mathrm{APO}}$. Only the clouds, of which the associated YSOs
have RV measurements from APOGEE, are shown in this figure.

Figure 6:  The $V_{Z}$ distribution of clouds (calculated using RV from CO
surveys) and young star samples (calculated using RV from APOGEE), with error
bars indicating 1$\sigma$ dispersions. The upper panel presents the results of
clouds and various young star samples for comparison, including YSO sample II,
OB stars, and young open cluster members. For further comparison, the lower
panel displays the results for YSOs with and without disks.

In order to investigate the potential presence of synchronous oscillations
between $V_{Z}$ and vertical coordinates, we perform separate fitting analyses
on $V_{Z}$ without RV and $V_{Z}$ with RV in the $T$¡300 pc region,
corresponding to the $X^{\prime}$¡1.55 kpc region as depicted in Figure 3 of
[29], where they previously suggested there are synchronous oscillations. In
our fitting, we adopt the damped sinusoidal function as proposed by [1] in
their equation (1). Figure 4 displays the fitting results for the spatial and
$V_{Z}$ distributions of clouds within the $T<300$ pc region. The upper panel
of Figure 4 displays curves of different colors representing the fitting
results of both the vertical coordinates and the vertical velocities. The grey
curve represents the fitting result of the $V_{Z}$ distribution obtained
without considering RV measurements. This fitting curve resembles the yellow
line shown in Figure 3 of [29]. However, when the RV components are taken into
account, as indicated by the red fitting curve, the values of $V_{Z}$
generally decrease in this region. Although a peak still exists in the -500
pc¡$T$¡0 pc region, the red curve exhibits a shorter period and a faster-
changing phase compared to the grey curve. In the lower panel of the figure,
the phase difference ($\delta\phi$) between the vertical coordinates and the
vertical velocities is depicted. When calculated without RV measurements,
$\delta\phi$ appears to be constant, similar to the findings reported in [29].
However, when a more precise calculation is performed considering RV
measurements, $\delta\phi$ decreases as $T$ increases. These results strongly
indicate that the previously reported synchronous oscillation arises due to
the neglect of the RV component in previous studies, and there are no
synchronous oscillations between vertical coordinates and vertical velocities.
Furthermore, separate K-S tests are conducted to evaluate the goodness of fit
for the two groups of $V_{Z}$ data, yielding p-values of 0.94 for both tests,
which are very close to 1. This indicates excellent fitting performance.

For evaluating the feasibility of this method, we assess the consistency in
RVs between the clouds and their associated YSOs. For each successfully
matched cloud, we compare its average RV with the median RV of the associated
YSOs, which is derived from APOGEE data, as demonstrated in Figure 5.
Remarkably, we observe a strong agreement between the two velocities, as
evidenced by a maximum absolute residual of only 2.14 km/s and a high
correlation coefficient of 0.98. This strong correlation suggests that the
method is indeed feasible. It should be noted that our analysis only considers
clouds with a minimum of 10 associated YSOs, each having APOGEE RV
measurements. This criterion helps mitigate potential random errors that could
arise from small sample sizes.

###  3.2 A $V_{Z}$ dipole pattern without age gradient

Additionally, Figure 3 reveals that the revised vertical velocities
demonstrate an overall upward trend as $T$ increases, showing as a dipole in
the $V_{Z}$. This trend is similar to the results presented in [44], but with
a larger maximum amplitude of approximately 8 km/s.

Moreover, we extend our investigation to explore the $V_{Z}$ distribution of
additional star samples along RW, including the OB star sample, the young open
cluster member sample, and the Class III YSO sample. In contrast to the study
by [44], we incorporate RV measurements by utilizing APOGEE data.
Additionally, we use the YSO catalogue from [35] and the catalogues of Class
I/II YSOs, and Class III YSOs from [34] for comparative analysis. As a result,
Figure 6 depicts the $V_{Z}$ distribution of clouds and young star samples. In
Figure 6, we analyze the median vertical velocities of different young samples
in bins of 200 pc along the $T$ axis, keeping the bins containing more than 10
stars. In the upper panel, we can observe a consistent overall upward trend in
vertical velocities with increasing coordinate $T$, although there are some
small oscillations in the intermediate range. This provides a piece of robust
evidence for the presence of dipoles in the $V_{Z}$ distribution. For a more
in-depth analysis, we incorporate the Class III YSO catalog with YSO sample II
in the lower panel and divide this combined sample into two subgroups: YSOs
with disks and YSOs without disks. This division is based on the methodology
proposed by [48] and [47]. In the lower panel, we also observe the presence of
$V_{Z}$ dipoles.

Our findings diverge from those of [44], which observed an age gradient in the
oscillation of stars, with the youngest sample exhibiting a maximum dipole. In
contrast, our analysis, as depicted in Figure 6, does not provide significant
evidence indicating a substantial difference in the size of the dipole between
the different samples. Particularly in the lower panel, the younger YSOs that
have disks do not exhibit a larger dipole than the older YSOs that do not have
disks. These findings suggest that age may not be the primary factor
influencing the size of the dipole in $V_{Z}$ among these samples.

##  4 discussion

###  4.1 Origin of the non-synchronous oscillation

Different from previous conclusion, we belive that the vertical oscillation in
the Radcliffe Wave is non-synchronous. There are two reason. The first is the
inclusion of RV measurements. After correcting $V_{Z}$ for RW using two
methods, we get a more accurate and comprehensive perspective of the $V_{Z}$
distribution. In contrast to the results that neglect RV components, we
observe a general decrease in the values of $V_{Z}$ in the $T$¡300 pc region,
while there is little change in other regions. As a result, the phase
difference ($\delta\phi$) between the vertical coordinates and the vertical
velocities is not consistently constant in this region, contrary to the
results reported in [29]. Instead, there is a noticeable decrease in the phase
difference along the direction of RW extension ($T$ axis). This suggests that
the previous finding of synchronous oscillations in RW was not valid. Part of
the reason is our incorporation of the RV component, which has been neglected
in previous studies. However, we also note that different authors have adopted
different models. It is a combination of these factors which leads to the
different conclusions.

###  4.2 Nature of the dipole pattern

Additionally, the revised $V_{Z}$ consistently demonstrates an upward trend as
$T$ increases, revealing dipole patterns in the distribution of $V_{Z}$.
Notably, we observe similar dipole patterns when using several young star
samples as tracers, which resembles the findings reported in the study by
[44], but with a larger amplitude. However, our approach improves upon their
results by considering RV measurements from APOGEE-2, while they neglected
them. Moreover, contrary to their results, we find no age gradient within the
dipole patterns, particularly when comparing the younger YSOs with disks with
older YSOs without disks. This suggests that the origin of RW may have
comparable effects on both molecular clouds and stars, raising doubts about
some formation hypotheses that have a stronger impact on the clouds, such as
the Kelvin-Helmholtz instability proposed by [19] and the satellite galaxy
perturbations hypothesis proposed by [44]. Currently, the origin of the RW
remains a mystery, and our precise and comprehensive analysis of the $V_{Z}$
patterns will contribute to a better understanding and identification of the
underlying origin and physical mechanisms involved.

Furthermore, we gain insights into the vertical evolution of RW by considering
the influence of the gravitational potential. We approximate the gravitational
force in the vertical direction as $F_{Z}$(kg$\cdot$km2
s-2pc-1)$=-0.003Z$(pc)$\cdot M$(kg), referring to Figure 2 in [40] and [2],
where $Z$ represents the vertical distance and $M$ denotes the mass of the
cloud. As depicted in Figure 7, RW behaves as an oscillation propagating
slowly to the left. It oscillates on the Galactic disk, resembling a harmonic
oscillator. The precise $V_{Z}$ information in this article can help to
determine the exact amplitude and phase of the oscillation for each cloud. The
clouds within RW exhibit similar periods, which can be estimated as
t=$2\pi$$\sqrt{\frac{M\times Z}{-F_{Z}}}\approx 110$ Myr. This implies that RW
oscillates at such a slow rate that it cannot complete a full period within a
cloud’s lifetime, which is estimated to be less than 40 Myr according to [37]
and [6]. Additionally, we also investigate the evolution in the X and Y
directions and obtain similar results as reported in [31]. RW undergoes
significant stretching during the cloud’s lifetime. Combining consideration of
the lack of synchronous oscillations between the vertical coordinates and
$V_{Z}$, the Radcliffe Wave can hardly be classified as a “wave” in the strict
sense.

Figure 7: Evolution of the Radcliffe Wave in the vertical direction at
different moments: Nowadays, 10 Myr later, 20 Myr later and 40 Myr later. The
orange curve marks the current position of RW. The direction of $V_{Z}$
component is represented by pink arrows, with their lengths corresponding to
the values of $V_{Z}$ (in km/s) multiplied by 15.

###  4.3 Future of kinematics studies

To understand the kinematics of RW, it is crucial to have precise measurements
of both RV and the proper motion of its gas and cloud members. However, the
latter measurements are very uncertain and challenging to define and determine
accurately. As a result, using YSOs as tracers may still be a more viable
approach. However, the current RV measurements of YSOs are limited to a
relatively nearby region and are not comprehensive enough. The upcoming Gaia
DR4 and DR5 releases, as well as future RV measurements, will help corroborate
the pattern of the vertical oscillations depicted in Figure 3 and Figure 6. By
inferring $V_{Z}$ maps of YSOs located near the galactic disk and examining
their oscillation patterns, we can obtain valuable insights into the nature of
RW.

##  5 conclusion

We analyze the kinematics of the Radcliffe Wave and find that considering
radial velocity measurements is crucial for accurately determining the $V_{Z}$
pattern. By incorporating radial velocity measurements using two methods, we
gain a more precise and comprehensive understanding of the distribution of
$V_{Z}$. We observe that the oscillations of vertical coordinates and $V_{Z}$
are not synchronized, but exhibit a decreasing phase difference in the $T<300$
pc region (where $T$ corresponds to the $Y_{\mathrm{prime}}$ axis in 1).
Furthermore, in the direction along the Radcliffe Wave, $V_{Z}$ shows an
upward trend and displays a dipole-like pattern with a maximum amplitude of
around 8 km/s. Furthermore, there is no apparent age gradient in the
oscillations. Noting that the vertical evolution of the Radcliffe Wave
progresses slowly, making it unable to complete a full oscillation within the
lifetime of the wave itself, which is around 30 Myr [31], we propose that the
Radcliffe Wave does not exhibit typical characteristics of a traditional
“wave” in the strict sense.

##  6 Acknowledgements

This research made use of the data from the Milky Way Imaging Scroll Painting
(MWISP) project, which is a multi-line survey in 12CO/13CO/C18O along the
northern galactic plane with PMO-13.7m telescope. We are grateful to all the
members of the MWISP working group, particularly the staff members at
PMO-13.7m telescope, for their long-term support. MWISP was sponsored by
National Key R$\&$D Program of China with grants 2023YFA1608000 $\&$
2017YFA0402701 and by CAS Key Research Program of Frontier Sciences with grant
QYZDJ-SSW-SLH047. ZJL acknowledges support by the grant AD23026127, funded by
Guangxi Science and Technology Project. GXL acknowledges support from NSFC
grant No. 12273032 and 12033005. VMP and PP acknowledge financial support by
the grant PID2020-115892GB-I00, funded by MCIN/AEI/10.13039/501100011033. This
work is supported by the National Natural Science Foundation of China (Grant
No.12133003).

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##  Appendix A cross match with CO surveys

Figure A1: Velocity-integrated spatial CO Map of the Perseus area. YSOs are
indicated by black points, marking their respective coordinates within the
map. The relatively associated region is marked by the enclosing blue folding
frame.

When combining YSO sample II with RV measurements of molecular clouds from the
whole-Galaxy CO surveys conducted by [12], our focus is on the regions where
YSOs and clouds with higher density (corresponding to larger velocity-
integrated intensity of CO—$W_{\mathrm{CO}}$) are associated.

For the spatial cross-matching process, we initially match associated areas in
the velocity-integrated spatial ($l$-$b$) CO map based on galactic latitude
($b$) and galactic longitude ($l$). This enables us to identify relatively
associated regions, such as the area enclosed by the blue frame depicted in
Figure A1 for the Perseus molecular cloud region.

To obtain more accurate RV measurements, we eliminate redundant peaks that may
be attributed to overlapping clouds with similar latitude and longitude but at
different distances, with reference to previous works, such as [13] for the
Perseus molecular cloud area. For other clouds, we also refer to several
articles including [32, 28, 14, 26, 49, 50, 10, 15].

Additionally, we determine the distances of the clouds by referring to the
compendium provided by [52]. For more accurate RV and distance measurements of
Cygnus X and the North America region, we also refer to the data (51) from the
MWISP project. Thereafter, we utilize the parallax measurements of YSOs
obtained from Gaia DR3. This allows us to refine the distance estimation and
effectively eliminate YSOs that are not spatially correlated with the
molecular clouds in the radial direction.

Subsequently, we combine RV measurements of the associated cloud areas with
the median value of the proper motion measurements of YSOs within the same
areas. This combination enables us to calculate the vertical velocities of the
clouds.

##  Appendix B APOGEE vs Gaia

Figure B1: Comparison of radial velocities obtained from Gaia DR3
(${\mathrm{RV}}_{\mathrm{Gaia}}$) and radial velocities obtained from APOGEE-2
(${\mathrm{RV}}_{\mathrm{APO}}$). The black dashed line is the one-to-one
line.

In the YSO sample I mentioned in section 2.1, which contains Class I and Class
II YSOs along RW, there are 224 YSO members with available RV information from
both Gaia DR3 and APOGEE datasets. In Figure B1, we compare the RV
measurements from Gaia and APOGEE, where each blue violet point represents a
YSO member. The Pearson correlation coefficient ($r$) of the RV values between
these two datasets is only 0.10, indicating that the correlation is weak for
the sources. In this sample, there is a substantial deviation between RVs
obtained from Gaia and APOGEE. Besides, as depicted in Figure B1, RVs from
Gaia DR3 exhibit a larger dispersion compared to those from APOGEE. Those two
databases exhibit notable differences, emphasizing the need to prioritize the
more accurate one for the RVs of young stars. Based on the findings in [25]
and [39], which indicate that APOGEE-2 provides more accurate RV measurements
for young stars compared to Gaia DR3, we prioritize the results derived from
APOGEE in the paper.