Analytic solutions of relativistic dissipative spin hydrodynamics with radial expansion in Gubser flow

We have derived the analytic solutions of dissipative relativistic spin hydrodynamics with Gubser expansion. Following the standard strategy of deriving the solutions in a Gubser flow, we take the Weyl rescaling and obtain the energy-momentum and angular momentum conversation equations in the $dS_{3}\times\mathbb{R}$ space-time. We then derive the analytic solutions of spin density, spin potential and other thermodynamic in $dS_{3}\times\mathbb{R}$ space-time and transform them back into Minkowski space-time $\mathbb{R}^{3,1}$. In the Minkowski space-time, the spin density and spin potential including the information of radial expansion decay as $\sim L^{-2}\tau^{-1}$ and $\sim L^{-2}\tau^{-1/3}$ in large $L$ limit, with $\tau$ being proper time and $L$ being the characteristic length of the system, respectively. Moreover, we observe the non-vanishing spin corrections to the energy density and other dissipative terms in the Belinfante form of dissipative spin hydrodynamics. Our results can also be used as test beds for future simulations of relativistic dissipative spin hydrodynamics.

One possible problem related to the sign problem is that the spin degrees of freedom may not reach the global equilibrium so that the Cooper-Frye formula [9,11] at global equilibrium fails to reproduce local spin polarization. It suggests that we need to add the spin degree of freedom to the current phenomenological frameworks and consider the off-equilibrium effects [22-24, 32, 73-77].
Very recently, the shear-induced polarization (SIP) as one of off-equilibrium effects [73][74][75][76][77] has been proposed and plays an important role to the spin polarization. The numerical results from relativistic hydrodynamics including SIP can give a correct sign in the comparison with the experimental data in the named the strange quark equilibrium scenarios [75], or in the isothermal equilibrium scenarios [77]. On the other hand, the total polarization in strange quark equilibrium scenarios is also found to be sensitive to the equation of state, freeze-out temperature and other parameters [78]. Similar studies on the parameter dependence at the √ s N N = 19.6GeV collisions are shown in Ref. [79] and one can also see Ref. [20,[80][81][82][83] for other related discussions. It indicates that the off-equilibrium effects need to be systematically studied in the future. Moreover, the modified Cooper-Frye formula including off-equilibrium effects, such as the effects of collisions [84] and spin [74] has been discussed.
On the other hand, there are two general ways to add the spin degree of freedom to the current phenomenological frames. As a macroscopic effective theory, the relativistic spin hydrodynamics is one possible way to consider the spin effects in the heavy ion collisions.
Moreover, in recent studies [63,74,110], a modified Cooper-Frye formula with spin potential at local equilibrium has been derived, in which the spin potential, just like the thermal vorticity and shear viscous tensor, contributes to spin polarization pseudo-vector.
Although there are intensive discussions on relativistic spin hydrodynamics, the codes for the numerical simulations have not been developed yet. It results in the lack of decaying behavior of spin density and spin potential. To see the influence of spin potential in the modified Cooper-Frye formula [74,110], we need to know the decaying behavior of these terms. Meanwhile, the numerical simulations also require some analytic solutions in special configurations as the test-beds.
To estimate the decay behavior of spin potential and find the suitable test-beds for the future numerical simulations, we search for the analytic solutions of relativistic spin hydrodynamics at some certain configurations. Based on the canonical form of relativistic spin hydrodynamics [58,64,70], we have already derived the analytic solutions in Bjorken expansion [71]. Our results show that the spin density and spin potential decay as τ −1 and τ −1/3 , respectively, where τ is the proper time. We also find that only one component of spin density, S xy , do not accelerate the Bjorken velocity. The transverse expansion of the medium and other components of spin density are not allowed in our previous study [71].
We emphasize that we derive the analytic solutions of relativistic spin hydrodynamics with radial expansion in a Gubser flow and it is different with some other approaches, in which the Bjorken or Gubser expansion is treated as the expanding background [129][130][131][132].
The paper is organized as follows: In Sec. II, we review the basic idea of Gubser flow. In Sec. III, we introduce the canonical form of spin hydrodynamics, the conservation equations and constitutive relations in both Minkowski space R 3,1 and dS 3 × R space-time. In Sec.
IV, we simplify the differential equations in a Gubser flow and derive the analytic solutions.
We also discuss the results in the Belinfante form of spin hydrodynamics in Sec. IV D. We concludes and summarizes this work in Sec. V.

II. REVIEW ON GUBSER FLOW
In this section, following Refs. [117,118] we briefly review the main results in a Gubser flow. Besides the Bjorken boost invariance, Gubser flow can describe expansion with azimuthal symmetry in transverse plane [117,118].
Following [117,118], one can construct Gubser flow by imposing the "Gubser symmetry" We rewrite the metric in Minkowski space-time R 3,1 with coordinates (t, x, y, z) as where The coordinates τ, x ⊥ , ϕ and η denote longitudinal proper time, transverse plane radius, azimuthal angle and rapidity, respectively. We then introduce a time-like hyperbola embedding into the manifold R 3,1 . The radius of this hyperbola is normalized to be 1, i.e.
where X µ is the Cartesian coordinates in R 3,1 and can be parametrized as: Here, L is an adjustable parameter with dimension of length. The line element of dS 3 can be expressed as, Now, the metric of Minkowski space-time R 3,1 can be transformed into the one of dS 3 × R under Weyl rescaling with factor τ , where ds 2 is given by Eq. (1). Notice that Weyl rescaling is not a coordinate transformation [118]. Later, the metric (6) is rewritten by the Gubser coordinates (ρ, θ, ϕ, η), where Calculations in current work are performed mainly in this coordinates unless otherwise specified.
Now, we discuss the velocity profile in a Gubser flow. The Gubser symmetry requires that the normalized velocity must beû in dS 3 × R [117,118], [133]which means the Gubser flow is static in the (ρ, θ, ϕ, η) coordinate system. Under the Weyl rescaling u µ = τû µ and coordinate transformation, one can derive the fluid velocity in the (τ, x ⊥ , ϕ, η) coordinates in Minkowski space-time R 3,1 , where ρ is given by Eq.  The canonical form of energy momentum tensor T µν for dissipative spin hydrodynamics in Minkowski space-time reads [58,64,70], with energy density e, pressure p, heat flux h µ and viscosity tensor π µν . The antisymmetric part T [µν] is further decomposed as, where . The q µ and φ µν play a role of the source to produce or absorb the spin.
The main equations for dissipative spin hydrodynamics are where Σ αµν is the rank-three canonical spin components in the total angular momentum.
Note that we have replaced the ordinary derivative ∂ µ with the covariant derivative ∇ µ for the general space-time. The Eq.(14) comes from the total angular momentum conservation and describes the spin evolution. Furthermore, one can decompose Σ αµν as [58,64,70], where S µν = −S νµ is named as the spin density and Σ αµν (1) is perpendicular to the fluid velocity u α Σ αµν (1) = 0. The above decomposition for Σ αµν is called the a non-anti-symmetric gauge which has been used in Refs. [55,58,64,65,67,68,85] and also in spin hydrodynamics for massless fermions [135]. One can construct a total anti-symmetric tensor for the spin density, which has been used in Refs. [59,70].
We regard S µν as an independent variable and introduce the corresponding spin potential ω µν = −ω νµ , which is conjugate to spin density S µν . To add the effects of spin, the thermodynamic relations become, where T and s denote the temperature and entropy density, respectively. For simplicity, here we set the particle number density and the charge or baryon chemical potential are always zero throughout the current work. To highlight the spin effect, we also neglect the heat flux h µ in the energy-momentum tensor T µν in Eq. (11).
The power counting scheme can be assumed as [58,70]. The spin evolution equation (14) becomes, According to the second law of thermodynamics [58,64] or the effective theories [70], constitutive equations are given by, where η s and ζ are the shear viscosity and bulk viscosity, respectively and λ, γ are two new transport coefficients related to the spin. The four coefficients η s , ζ, λ, γ in the constitutive relations are all non-negative.
For simplicity, following the common strategy in a Gubser flow [117,118], we choose and set ζ = 0 from now on. The energy-momentum conservation (13) then reduces to, In general, the transformation rule for a physical variableÂ in dS 3 \timesR space-time to its corresponding quantity A in Minkowski space-time is [118,140,141,145], where ∆ A = [A] + m − n, and [A] is the mass dimension of A.
Next, we discuss the spin evolution equation (17). From Eq. (19), the spin evolution equation (17) is not covariant under Weyl rescaling. In dS 3 \timesR space-time, Eq. (17) becomes,∇ where we have used that Eq. (25) has several new terms proportional to τ −1∇ β τ . In this subsection, we extend the energy-momentum and angular momentum conservation equations to the dS 3 × R space-time. Unfortunately, these two kinds of conversation equations are not conformal invariant, i.e. we find that there are extra terms ∼∇ µ τ in Eqs. (23,25) under Weyl rescaling.
Here, we emphasize that we do NOT consider the conformal fluid in current study. In an ordinary fluid without spin, the anti-symmetric part of energy-momentum tensor T [µν] is zero and Eq. (23) reduces to the simplest expression∇ µT µν = 0. However, as discussed in Sec. III A, the anti-symmetric part of energy-momentum tensor T [µν] is non-vanishing in the spin hydrodynamics. We keep the general expression (23) for energy-momentum conservation here. Later, we will solve the conservation equations (23) in Sec. IV.
Applying the Eq. (19), the transformation of ∇ µ u ν is given by [141] It is straightforward to show that the bulk viscosity term ζ∆ µν ∇ α u α does not transform homogeneously. Based on Eq. (29), we can get [140,141,145] which lead to a compact form forπ µν andφ µν , Therefore,π µν andφ µν have the same structure as π µν and φ µν . Note that we deliberately writeη s andγ as (η s /ŝ)ŝ and (γ/ŝ)ŝ, respectively. Theη s /ŝ andγ/ŝ are dimensionless scalars which do not be modified when passing from Minkowski space-time R 3,1 to dS 3 × R space-time. We follow the standard strategy in Gubser flows and set Unfortunately,q µ becomeŝ whereλ = τ 3 λ. The last term in the bracket −2τ −1∆µα∇ α τ is generated by the Weyl rescaling. For simplicity, we need to set λ =λ = 0 in the current work, i.e. we set In fact, we have also checked that the nonzeroq µ breaks the Gubser symmetry and will change the velocityû µ = (−1, 0, 0, 0) due to the extra term −2τ −1∆µα∇ α τ . In this section, we extend the conservation equations and constitutive equations for the relativistic spin hydrodynamics from Minkowski space-time R 3,1 to the dS 3 × R space-time.
We find that neither the energy momentum conservation equation (23) nor the spin evolution equation (25) is covariant after Weyl rescaling. We also get the constitutive equations in the dS 3 × R space-time shown in Eqs. (27,31). We further set the bulk viscosity ζ = 0 and q µ = 0 for simplicity.

IV. ANALYTIC SOLUTIONS IN GUBSER FLOW
In this section, we derive the analytic solutions of dissipative spin hydrodynamics in a Gubser flow in high temperature limit.
We adopt the strategy similar to our previous works [71,[111][112][113][114][115][116]. and compare them with the our solutions in a Bjorken flow [71]. We also discuss the results for the spin hydrodynamics in the Belinfante form in Sec. IV D. Throughout this section, we use the Gubser coordinates (ρ, θ, φ, η) in dS 3 × R space-time if not specified.
Again, for simplicity, we set the number density and chemical potential be zero. Following the standard Gubser flows [117,118], we can assume that the thermodynamic variableŝ e,p,T ,ŝ and transport coefficientsγ,η s are only functions of de Sitter time ρ. It suggests a natural assignment thatω µνŜ µν depends on ρ only. We emphasize that due to the nontrivial metricĝ µν = diag{−1, cosh 2 ρ, cosh 2 ρ sin 2 θ, 1} in the dS 3 × R space-time,Ŝ µν andω µν may be the functions of both ρ and θ.
To close the system, we need the equations of state besides Eq. (36). In III B, we assume which is a reasonable approximation in the ultra-relativistic or high temperature limits.
Here, c 2 s is the speed of sound and usually one can choose c 2 s = 1/3 for simplicity. We emphasize that EoS (37) does not imply the system is conformal invariant. In fact, there is no the conformal symmetry in our system. More discussion will be shown in the next subsection. On the other hand, inspired by the relation between particle number density and chemical potential, we assume another equation of state in high temperature limit [71], i.e.,Ŝ with dimensionless constant a. Eqs. (37,38) are regarded as two given conditions in the subsequent discussion.

B. Simplify the differential equations
In this subsection, our task is to find special configuration to hold the fluid velocity in a Gubser flow and simplify main differential equations (23,25).
Contactingû ν with both sides of Eq. (23) provides the conservation equation for energy, Using Eqs. (27,37,47), the evolution of energy density (52) reads Eq. (53) is the same as the one in ordinary relativistic hydrodynamics without spin effect in a Gubser flow [117,118] (also see Refs. [121,123] for extensions).
Third, we compute the evolution of spin following Eq. (25). After a long and tedious calculation, we eventually obtain six independent equations for the evolution of spin from Eq. (25), and, ∂ ρŜ ρθ + 3 tanh ρŜ ρθ = 0.
Remarkably, in the space-time dS 3 × R, there are extra terms proportional to∇ α τ in both energy-momentum conservation equation (23) and the evolution equations for spin (25). As mentioned in the previous subsection, these terms come from the Weyl rescaling and cannot be neglected in general. Fortunately, in the configuration for the Gubser flow, all of these terms vanish in Eqs. (53,58,59). It is of great help for us to derive the analytic solutions in the relativistic spin hydrodynamics in a Gubser flow. At last, we only have three independent differential equations, i.e. conservation equation for energy (53) and evolution equations for spin (58,59).
C. Analytic solutions in dS 3 × R and R 3,1 space-time In this subsection, we solve the differential equations (53,58,59) for the spin hydrodynamics in a Gubser flow. We then transform our solutions in dS 3 × R space-time to the Minkowski space-time R 3,1 .
We consider the high temperature limit and the spin chemical potential is much smaller than temperature in the relativistic heavy ion collisions, i.e. ω µν T , or We emphasize again thatη s /ŝ andγ/ŝ are small constants and we can assumeη s /ŝ,γ/ŝ 1.
Therefore, we can consider the ω 2 andη s /ŝ,γ/ŝ as small parameters and expand all the quantities in the power series of ω 2 andη s /ŝ,γ/ŝ.
In leading order of ω 2 , the Eq. (53) becomes, whose solution is given by, with initial valueT 0 ≡T (ρ 0 ). Here, the auxiliary function B(ρ) is where F (a, b; c; z) is the hyper-geometric function. Substituting Eq. (64) to Eq. (41), we obtain the expression for the energy densityê, With the EoS (38), the solutions of evolution equations for spin (58,59) are, where is a constant determined by the initial condition and The f (θ) in Eq. (68) is a function of θ. As explained in Sec. IV A, although the Gubser flow requires the scalarsω µνŜ µν and bothω µνω µν andŜ µνŜ µν depend on ρ only, EoS (38) implies thatω µν orŜ µν could also depend on θ due to the metricĝ µν = diag −1, cosh 2 ρ, cosh 2 ρ sin 2 θ, 1 . Using Eq. (67), we find that Unless with a constant determined by initial value ofŜ ϕη 0 =Ŝ ϕη (ρ 0 , θ 0 ), theŜ µνŜ µν would not be independent on θ. Finally, the expression for the spin densityŜ ϕη becomeŝ If the dimensionless quantityγ/ŝ can be regarded as a small constant, i.e.γ/ŝ 1, we can obtain Next, we transform the analytic solutions (64,66,67,74) where we introduce the with an adjustable parameter L defined in Eq. (4), and the τ 0 and x ⊥0 stands for the initial proper time and the transverse position x ⊥ . Here, we have used the identitŷ sinceη s /ŝ is a scalar under the Weyl rescaling. Using the same method, it is straightforward to get the expression for temperature T from Eq. (64). The T as a function of τ, x ⊥ is similar to the e(τ, x ⊥ ).
Next, we take the Weyl rescaling and the coordinate transformation to the spin density.
By using Eq. (8,24), the nonzero spin density S µν in the (τ, x ⊥ , ϕ, η) coordinate system of Minkowski space-time R 3,1 is given by We find that the exponential factor A(ρ) in Eq. (70) is always less than 1. It means that dissipative effects ∝γ accelerate decaying. Furthermore, we express the spin density in Cartesian coordinates (t, x, y, z) by coordinate transformation Eq. (2), i.e., where we introduce that and η, ϕ, ρ are the functions of (t, x, y, z) given by Eqs. (2,8). The other two components, S 0z and S xy vanish.
Let us comment on our results here. In large L limit, i.e., x ⊥ , τ L, we have G(L, τ, x ⊥ ) ∼ L 4 and the energy density and temperature become, The spin density in Eqs. (80)(81)(82)(83) reduces to, From EoS (38), the spin chemical potential ω µν decays like The decay behavior of e, T, in large L limit is the same as those in the spin hydrodynamics in a Bjorken expansion [71]. Meanwhile, due to the dissipative effects, the spin density and spin potential can decay more rapidly. We find that the exponential factor A(ρ) in Eq. (70) is always less than 1. It means that dissipative effects ∝γ accelerate decaying.
Notably, in a Bjorken flow, we only get the nonzero solutions for the spin component S xy [71]. Here, there are four non-vanishing spin density components found in the current work due to the radial expansion in a Gubser flow.
Before end this section, let us discuss the terms in the modified Cooper-Frye formula. As discussed in recent works for shear induced polarization [73][74][75][76][77] (also see Refs. [74,84,110] for other terms related to spin density), the thermal vortical Ω µν and thermal shear tensor and spin potential ω µν = S µν /(aT 2 ) appear in modified Cooper-Frye formula.
From Eqs. (10,24,64), we find the nonzero components of Ω µν are Ω τ x ⊥ and Ω x ⊥ τ . Note that, in dS 3 × R space-time, the nonzero thermal vortical tensorsΩ τ x ⊥ andΩ x ⊥ τ come from the space-time derivatives of the temperature and the extra terms proportional to∇ µ τ shown in Eq. (29) from Weyl rescaling. In the R 3,1 space-time, the ∇ µ u ν with u µ given by Eq. (10) is obviously nonzero and can contribute to Ω τ x ⊥ and Ω x ⊥ τ .
We now analyze the evolution behavior of thermal vortical and shear tensors.
In large L but small η/s, γ/s limits, by using Eqs. (10,38,64,79), we notice that and obtain the evolution behavior of nonzero components of Ω µν and ξ µν listed in Tab I.
We find that the only nonzero component of thermal vortical tensor, Ω τ x ⊥ , is much smaller than the maximum component of thermal shear tensor, ξ τ τ , but it has the same order of magnitude as the spin potential in Eq. (88). Table I. Evolution behavior of nonzero components for thermal vortical tensor Ω µν and shear tensor ξ µν and spin potential ω µν in R 3,1 space-time in large L but small η/s, γ/s limits.
Evolution behavior in large L and small η/s, γ/s limits From Eq. (18), in the global equilibrium, one of the most important conclusion for spin should be zero [58,64,88]. Here, we compare the evolution behavior of the following quantities, We conclude that in large L but small η/s, γ/s limits the thermal shear tensor is more important than than spin potential and thermal vortical tensor.
We also discuss the evolution behavior at finite L case. We follow the in-viscid case of Gubser flow in Ref. [117,118]  as a function of τ . Here, we choose A = Ω τ x ⊥ , ξ τ x ⊥ , ω τ x ⊥ and A 0 stands for the value of these quantities at initial proper time τ 0 . In Fig. 1, we choose x ⊥ = 0.5fm for simplicity.
We have checked numerically that power law behavior of these quantities for other fixed x ⊥ ( 4fm) almost the same. The maximum proper time is chosen as 4fm/c. After 4fm/c, the temperature will be less than the typical freeze out one [117,118].
In Fig. 1(a), we find that within 4fm/c the Ω τ x ⊥ and ω τ x ⊥ always increases or decreases, respectively. While, the thermal shear tensor ξ τ x ⊥ increases as ∝ τ 1/3 at early time but decrease rapidly as ∝ τ −1 after 2.3fm/c. Surprisingly, the thermal shear tensor decays much faster than the spin potential in this model. As a comparison, when L = 50fm in Fig. 1(b), we observe a consistent results as expected in Tab I.
We conclude that in finite L case, the evolution behavior of the quantities mentioned above depends on the parameters L. Therefore, we can not naively drop any one of them in where and Σ λµν is given by Eq. (15). The Belinfante total angular momentum reads, It is obviously that both T µν and J αµν are conserved.
After a short calculation, up to O(∂ 1 ), one get [64], where are the spin corrections to the energy density, heat flow, shear viscous tensor and bulk viscous pressure, respectively.
In our previous work [71], all of these spin corrections vanish in a Bjorken flow. Inserting our results (79) with Gubser velocity (10) into Eq. (96) yields, the spin corrections to the energy density, where G and constant c 1 are given by Eqs. (77) and (69), respectively. We get the other non-vanishing spin corrections in the (τ, x ⊥ , φ, η) coordinates, One may wonder why the spin correction to the bulk pressure δΠ s in Eq. (98) is nonzero and may break the conformal invariance. Again, we comment that EoS (37) is the leading order one in the ultra high temperature limits and is not related to the conformal invariance directly. Therefore, there is no inconsistency between the finite δΠ s and EoS (37 Now, we turn to estimate how large these spin corrections will be. In the large L limit, we find energy density e ∼ τ −4/3 and its spin corrections δe s ∼ 1/(τ x ⊥ L 2 ). It gives that lim L→∞ δe s /e ∝ τ 1/3 /L 2 → 0 at late proper time, i.e. the spin correction to the energy density δe s is a small correction to the energy density e.  [117,118].
Note that, the due to the differences of notations, η s /s in this work is twice as much as that in Ref. [117,118]. Since the proper time is larger than 4.0fm/c, the temperature is less than the typical freeze out temperature T ≤ 150MeV. We choose the range of proper time as 0.5 − 4.0fm/c similar to the standard Gubser flow [117,118]. Although total energy To avoid the divergent behavior of δe, we have imposed the constraint x ⊥ > 0.5fm in Eq. (99). The parameter c 1 defined in Eq. (69) should not be too large due to our power counting scheme in Eq. (60) in Sec. IV C. Here, we choose |c 1 | ≤ 2 as a reasonable test. Using these parameters and Eqs. (10,18,37,76,97,98), we obtain δe s /e < 0.1, at x ⊥ ∈ [0.5, 4.0]fm, τ ∈ [0.5, 4.0]fm/c, δπ µν s /π µν , δΠ s /p < 0.1, at x ⊥ ∈ [0.0, 4.0]fm, τ ∈ [0.5, 4.0]fm/c.
We comment that Eqs. (97) and (98) are the evidence to show that spin corrections in the Belinfante form of spin hydrodynamic exist. These spin corrections are expected in Ref. [64].

V. CONCLUSION
In this work we have obtained the analytical solutions for the dissipative spin hydrodynamics with radial expansion in a Gubser flow.
After a short review on the standard Gubser flow, we briefly discuss the relativistic dissipative spin hydrodynamics in the Minkowski space-time R 3,1 and extend the main equations to the dS 3 × R space-time under Weyl rescaling. Unfortunately, we find that there are ex-tra contributions ∝∇ µ τ from Weyl rescaling to both the energy momentum conservation Eq. (23) and angular momentum conservation Eq. (25). We emphasize that the energymomentum conservation equations is no longer conformal invariant in the current work due to its anti-symmetric components. For simplicity, we drop the bulk viscous pressure andq µ .
We further assume the transport coefficientsη s /ŝ andγ/ŝ are small constants similar to the ordinary Gubser flow.
We then discuss the thermodynamic relations (36) and the equations of state (37,38) in the dS 3 × R space-time. For convenience, we introduce a dimensionless scalar ω 2 in Eq.
(39). Next, we derive the special configuration for the fluid, in which the fluid velocity in a Gubser flow holds. Fortunately, the extra terms from Weyl rescaling ∼∇ µ τ in energy momentum and angular momentum conservation equations vanish in this configuration. In the power series expansion of small ω 2 ,η s /ŝ,γ/ŝ, we have derived the analytic solutions for the dissipative spin hydrodynamics in the dS 3 × R space-time. The evolution of energy density and spin densityŜ ρθ ,Ŝ ϕη are shown in Eqs. (66,67,68) in dS 3 × R space-time. We also transform these physical quantities back to the Minkowski space-time R 3,1 .
Our main results for the energy density e and spin density S 0x , S 0y , S xz , S yz in Minkowski space-time R 3,1 are given by Eqs. (76,(80)(81)(82)(83). There are two remarkable difference between the solutions found here and in a Bjorken flow [71]. The first thing is that the solutions in a Gubser flow provide the additional information for transverse expansion of the systems, which is missing in a Bjorken flow [71]. Meanwhile, now we have derived four nonzero components S 0x , S 0y , S xz , S yz in spin density tensor in current work, while we only have one nonzero component S xy in Bjorken flow [71]. It indicates that our current findings are not a simple extension of Bjorken flow.
In large L and small η/s, γ/s limits we find that e ∝ τ −4/3 , T ∝ τ −1/3 , S µν ∝ L −2 τ −1 , ω µν ∝ L −2 τ −1/3 , which are similar to the behavior in Bjorken expansion [71]. Moreover, S µν decay much faster in the cases of a finite L and with nonzero dissipative effects than in large L and prefect fluid limit.
In our model, we find that in large L and small η/s, γ/s limits the thermal shear tensor ξ µν may be more important than spin potential ω µν and thermal vortical tensor Ω µν , but in finite L case their evolution behavior depends on the parameters L strongly. We can not naively drop any one of them in the modified Cooper-Frye formula. To clarify it, we need further studies based on spin hydrodynamics in the future.
At the end, after the pseudogauge transformation, we discuss the results for spin hydrodynamics in the Belinfante form. We observe that the spin corrections to the energy density and other dissipative terms do not vanish in a Gubser flow, which is quite different with the results in a Bjorken flow.
Our analytic solutions also provide the test beds for the future numerical simulations of relativistic dissipative spin hydrodynamics .

ACKNOWLEDGMENTS
This work is partly supported by National Natural Science Foundation of China (NSFC) under Grants No. 12075235 and 12135011.