Spin polarization dynamics in the Gubser-expanding background

Evolution of spin polarization in the Gubser-expanding conformal perfect-fluid hydrodynamic background is studied. The analysis of the conformal transformation properties of the conservation laws is extended to the case of the angular momentum conservation. The explicit forms of equations of motion for spin components are derived and analysed, and some special solutions are found.


I. INTRODUCTION
In the last decades relativistic hydrodynamics has proven to be very successful in describing the evolution of the strongly-interacting matter produced in relativistic heavy-ion collisions [1][2][3][4][5][6][7][8]. Recent measurements of spin polarization of Λ hyperons have shown that the spacetime evolution of quantum spin should also be included in this framework in order to properly capture effects seen in the experiment [9][10][11][12][13][14][15].
The first attempt to formulate relativistic hydrodynamics with spin has been done in Ref. [16]; see also the follow-up papers [17][18][19][20][21][22][23][24], as well as reviews [25,26]. In contrast to previous theoretical studies which considered generation of spin polarization of matter at freezeout due to spin-vorticity coupling  (see also related studies ), the spin hydrodynamic approach has been formulated entirely based on the conservation laws and assumption of local thermal equilibrium. In this case, enforcement of total angular momentum conservation leads to the need for introducing additional Lagrange multipliers which comprise the so-called spin polarization tensor ω µν that in the case of global equilibrium with rigid rotation reduces to the thermal vorticity tensor µν = − 1 2 (∂ µ β ν − ∂ ν β µ ) (here β µ = U µ /T is the ratio of the fluid flow vector U µ and the local temperature T ) [86][87][88]. The resulting spin hydrodynamic framework extends the standard one by adding additional dynamic equations which, in general, have to be solved numerically.
In the present paper we use the formalism of relativistic hydrodynamics with spin proposed in Refs. [22,25] to study evolution of spin polarization in the Gubserexpanding perfect-fluid hydrodynamic background [89,90]. The current work is an extension of the study presented in Ref. [22] where the space-time evolution of the spin polarization was considered in a boost-invariant and transversely homogeneous background, also known as Bjorken flow. Following other works [91][92][93][94][95][96][97][98][99][100] we first solve the perfect-fluid hydrodynamical equations using the Gubser symmetry arguments in the de Sitter coordinates and obtain analytical solutions for hydrody-namic variables. Subsequently, we extend the analysis of the properties of the conservation laws with respect to the conformal transformation to the case of the angular momentum conservation. We find that the latter is conformaly invariant only if the spin tensor is antisymmetric in all indices and the particles composing the fluid are massless. As the de Groot -van Leeuwen -van Weert (GLW) spin tensor [101] considered in this work is not, in general, satisfying these requirements, we relax the constraints related to the symmetry with respect to the special conformal transformations exploiting only the cylindrical symmetry and boost-invariance. Using expressions for temperature, chemical potential and velocity obtained for the Gubser-flowing background, we derive the equations of motion for spin polarization components which, as expected, exhibit non-trivial dynamics in both de Sitter time and angle, as well as a (small) sensitivity to the mass of the constituent particles. As in the formulation used herein we keep the assumption of small amplitude of polarization effects [22,25] the background equations expressing the baryon charge and energy-momentum conservation decouple completely from the angular momentum conservation, the background solutions are not spoiled by the breaking of the symmetry at the level of angular momentum conservation. For the special case of massless particles we find a set of special solutions which, in general, have a powerlike dependence on the temperature of the system. Finally, we also present some full numerical solutions with finite masses in Milne space-time. The solutions obtained in this work may be used for describing the collisions of systems with initial spin polarization as well as for testing the full 3+1D geometry codes.
The structure of the paper is as follows: We start by briefly reviewing the framework of relativistic hydrodynamics with spin for polarized systems of spin -1 /2 particles in Sec. III. In Sec. IV we introduce the boostinvariance and cylindrical symmetry, which is followed in Sec. V by the details about Gubser symmetry, Weyl rescaling of various thermodynamical and hydrodynamical quantities, and the information about the de Sitter basis vectors. Conformal transformation of the conservation equations is studied in Sec. VI. In Sec. VII we derive the Gubser-symmetric evolution equations for the background which we then use to finding the evolution equations of spin polarization components in the de Sitter spacetime. The resulting equations of motion are studied both analytically and numerically and transformed back to Milne spacetime. We conclude and summarize in Sec. VIII. Some additional information and details are given in the appendices.

II. CONVENTIONS
We use the following shorthand notation for the scalar product between two four-vectors a µ b µ ≡ a · b. For the Levi-Civita tensor αβγδ , the sign convention used throughout the paper is 0123 = − 0123 = 1. Throughout the text we use natural units with c = = k B = 1. The anti-symmetrization of arbitrary rank two tensor A is denoted as . Also, we are using "mostly plus" metric signature.

III. PERFECT FLUID HYDRODYNAMICS FOR SYSTEMS OF PARTICLES WITH SPIN 1/2
For reader's convenience in this section we briefly review the recently developed formalism of relativistic perfect fluid hydrodynamics for polarized systems of particles with spin 1 /2 [22]. We use the approximation of small polarization which, in the leading order, leads to decoupling of the background hydrodynamic equations for velocity, temperature and chemical potential (given by the standard net baryon number and energy-momentum conservation laws) from the dynamics of spin degrees of freedom (resulting from the conservation of angular momentum).

A. Conservation of net baryon number
The conservation law of net baryon number can be expressed in the following way where, in the case of no dissipative effects, the net baryon current N α reads with the net baryon density being given by The fluid four-velocity U α in Eq. (2) is normalized as U · U = −1 and d denotes the covariant derivative (see App. A for more information on the definitions of covariant derivative). The quantity α is the ratio of baryon chemical potential µ and temperature T , α ≡ µ/T . For an ideal relativistic gas of classical massive particles (and antiparticles) the auxiliary number density N (0) is given by the well known expression [1] where z ≡ m/T with m being the mass of the particles (and antiparticles), k ≡ 1/(2π 2 ) and K n denotes the n-th modified Bessel function of the second kind.

B. Conservation of energy and linear momentum
The conservation of energy and linear momentum is given by the expression where for perfect fluid the energy-momentum tensor T αβ takes the following standard form with ∆ αβ ≡ g αβ + U α U β being the spatial projection operator orthogonal to the fluid flow vector and the energy density and pressure being defined as respectively. In the case of ideal relativistic gas of spinless and neutral massive Boltzmann particles the auxiliary energy density and pressure is [1] respectively. The thermodynamic quantities defined above satisfy the standard thermodynamic relations, namely with S = ∂P ∂T µ , N = ∂P ∂µ T , and S (0) = ∂T .

C. Conservation of angular momentum
Total angular momentum may be expressed as a sum of orbital angular momentum part L α,βγ and spin angular momentum part S α,βγ as follows where in the second equality we expressed the orbital part in terms of the energy-momentum tensor. The conservation of total angular momentum is given by the expression Since the energy-momentum tensor used herein is symmetric T [βγ] = 0, see Eq. (6), the conservation of the total angular momentum implies separate conservation of its spin part.
The spin tensor in Eq. (14) is given by the following expression [22] S α,βγ = C U α ω βγ + S α,βγ where the auxiliary spin tensor is defined as with the quantity ω αβ (x) being the spin polarization tensor (to be discussed in the next section). The thermodynamic quantities A, B and C in Eqs. (15) and (16) are In what follows here on-wards we will refer Eqs. (15) and (16) as the de Groot -van Leeuwen -van Weert (GLW) spin tensor [101].

D. Spin polarization tensor
The spin polarization tensor ω αβ is an asymmetric rank-two tensor which can be decomposed with respect to the fluid four-velocity in the following way Any part of the auxiliary four-vectors κ α and ω α parallel to U α does not contribute to the right-hand side of Eq. (20) hence without the loss of generality we can assume that κ α and ω α satisfy the following orthogonality conditions Using above conditions, κ α and ω α can be expressed in terms of spin polarization tensor ω αβ as follows

IV. IMPLEMENTATION OF BOOST INVARIANCE AND CYLINDRICAL SYMMETRY
Let us consider central high-energy heavy-ion collisions which are boost-invariant and cylindrically-symmetric with respect to the beam (z) axis. Their dynamics is most conveniently described in the polar hyperbolic coordinates x µ = (τ, r, φ, η) where the line element reads with τ = √ t 2 − z 2 being the longitudinal proper time, η = tanh −1 (z/t) being the longitudinal spacetime rapidity, and r = x 2 +y 2 and φ = tan −1 (y/x) denoting the radial distance and the azimuthal angle, respectively, parameterizing the transverse plane perpendicular to the beam direction.
For convenience we introduce the following orthogonal four-vector basis in the laboratory frame which allows us to express the polar-hyperbolic metric tensor, g µν = diag(−1, 1, 1/r 2 , 1/τ 2 ), in the following form It is straightforward to check that the basis vectors (24) satisfy the following normalization conditions while all mixed scalar products vanish. One can convince himself that in the local rest frame U , R, Φ and Z are the unit vectors pointing in the τ , r, φ and η directions, respectively. Using the basis vectors (24) and orthogonality conditions (21) four-vectors κ α and ω α may be decomposed as follows where a i (τ, r) and b i (τ, r) are scalar coefficients characterizing the components of spin polarization tensor along the basis vectors.

V. GUBSER SYMMETRY AND CONFORMAL MAPPING TO DE SITTER SPACE
For boost-invariant and cylindrically symmetric systems one can construct a nontrivial four-velocity profile in Minkowski space R 3 ⊗R which is invariant with respect to the SO(3) q ⊗SO(1, 1)⊗Z 2 conformal symmetry group, also known as "Gubser's symmetry" [89,90]. The latter is composed of rotations in the r − φ plane coupled with two special conformal transformations parametrized by an arbitrary inverse length scale q (SO(3) q ), boosts in the η direction (SO(1, 1)) and reflections with respect to the r − φ plane (Z 2 ). The procedure of finding the flow pattern invariant with respect to this symmetry, while being intricate in Minkowski space, is quite transparent when considered in the curved spacetime formed by the product of the three-dimensional de Sitter space and a line, dS 3 ⊗R -hereafter we will shortly refer to it as "de Sitter space". The conformal map used to transform from one to the other constitutes of the Weyl rescaling of the line element where the conformal factor is Ω = τ , combined with the change from polar Milne coordinates x µ = (τ, r, φ, η) to de Sitter coordinatesx µ = (ρ, θ, φ, η) using The resulting rescaled line element of the de Sitter spacetime reads with the metricĝ µν = diag(−1, cosh 2 ρ, cosh 2 ρ sin 2 θ, 1) (hereafter all quantities defined in the de Sitter space will be denoted with a hat). Clearly, based on Eq. (33) we observe that the Weyl rescaling (30) combined with passing to the standard global coordinates on dS 3 (31)- (32) promotes the (SO(3) q ) conformal isometry to a manifest isometry (SO (3)) in (θ, φ).
Using rules for general coordinate transformations of tensors and Eq. (35) the relation between R 3 ⊗R metric and Using the information that ∆ gµν = −2 and the unit norm constraint g µν U µ U ν = −1, one obtains ∆ U µ = 1 which results in the transformation rulê Using Eq. (37) one can show that the four-velocity profile from Eqs. (24) in the de Sitter space is static, meaning it is invariant with respect to the Gubser symmetry group, provided the transverse rapidity profile has the form [89,90] ϑ(τ, r) = tanh −1 2 qτ qr 1 + (qτ ) 2 + (qr) 2 .
In the similar manner the remaining basis vectors (24) in the de Sitter space arê The metricĝ µν can be expressed aŝ while the determinant ofĝ µν iŝ Using the fact that the energy density Eq. (7) and pressure Eq. (8) In Eq. (13), x β have mass dimension [x β ] = −1 with one contravariant index, hence ∆ x β = 0, and using the information about the conformal weight of E and U α and Eq. (6), one can find that the stress energy tensor should have the conformal weight ∆ T αβ = 6. As a result one observes that the conformal weight of the spin tensor is ∆ S αβγ = 6, since each term in Eq. (13) should have the same conformal weight. Hence the GLW spin tensor in Eq. (15) [101] as well as canonical spin tensor [105], and the Hilgevoord-Wouthuysen (HW) spin tensor [106,107] should all have the same conformal weight. For instance, for a free Dirac field the canonical spin tensor S α,βγ C and the HW spin tensor S α,βγ HW are defined as [106,107] S α,βγ S α,βγ HW = S α,βγ respectively, with σ µν = i 2 [γ µ , γ ν ]. Since the conformal weight of both spinor ψ and dual spinorψ ≡ ψ † γ 0 is ∆ ψ = ∆ψ = 3 2 and conformal weight of Dirac gamma matrix is ∆ γ µ = 1 [108,109], therefore ∆ S αβγ = ∆ S αβγ C = ∆ S αβγ HW = 6. Similarly, using the information about the conformal weight of N in Eq. (2) we find that the conformal weight of the net baryon number current is ∆ N α = 4.
The conformal weight of the spin polarization tensor ω αβ is easily found if one notices the fact that the first term in Eq. (15) should have the same conformal weight as the spin tensor in Eq. (13). In this case for the ranktwo dimensionless spin polarization tensor with two contravariant indices one can find the conformal weight to be ∆ ω αβ = 2. Using this result in Eqs. (22) we see that ∆ κ α = 1 and ∆ ω α = 1. Here we used the fact that αβγδ has zero mass dimension and four contravariant indices, giving ∆ αβγδ = 4. From the information of how κ α , ω α and basis vectors (24) transform under Weyl rescaling and that all the spin polarization components a i and b i are dimensionless scalars, we see that ∆ ai = ∆ bi = 0. Hence, they are conformally invariant quantities.
Using Eq. (34) we summarize the transformation rules under Weyl rescaling (for four dimensional spacetime) as follows Finally, we note that the similar results may be found for the quantities with covariant indices. In particular one may find that ∆ T αβ = 2. The latter result is in agreement with the fact that raising (lowering) the Lorentz index with the metric tensor changes the conformal measure by a factor of 2 (−2).

VI. CONFORMAL INVARIANCE OF CONSERVATION EQUATIONS
Considering four-dimensional conformal fluid dynamics, we aim at finding the conformal transformation of the conservation equations for net baryon number, energy and linear momentum, and spin, which, in general coordinates, are expressed as +Γ α αλ S λβγ + Γ β αλ S αλγ + Γ γ αλ S αβλ = 0, respectively, where Γ β αλ are Christoffel symbols defined in Eq. (A8).
One can show, that for the four-dimensional spacetime the conservation law for net baryon number (48) is already conformal-frame independent [89,90,[102][103][104], i.e. net baryon number is conserved in both Minkowski and de Sitter space-times. In this case, since the conformal weight of net baryon number is ∆ N α = 4, one can write Hence, we observe thatT αβ needs to be traceless in order to be conserved in de Sitter spacetime. Therefore, the breaking of conformal invariance is characterized only by the trace of the energy-momentum tensor 1 [112][113][114][115]. From Eq. (54) we find that the conformal invariance of the spin conservation law requires the spin tensor S αβγ to be totally antisymmetric. On the other hand, the GLW (15) and HW (44) forms, which are anti-symmetric only in their last two indices explicitly break the conformal invariance of Eq. (54). The consequences of this issue will be discussed in the next section.

VII. DYNAMICS IN DE SITTER COORDINATES
In this section we explore the dynamics of the spin polarization on top of the conformal Gubser-expanding hydrodynamic background in the de Sitter space-time. Our final results are obtained by transforming back to the Minkowski spacetime using Eqs. (42).

A. Conformal symmetry breaking and the equation of state
In Sec. VI we have shown that Eqs. (1) and (5) are conformally invariant, i.e. they transform homogeneously under Weyl transformation provided the constitutive relations do so and the energy-momentum tensor is traceless. As the presence of finite masses breaks the tracelessness condition (in our caseT λ λ = 3P −Ê ≈ O(m 2 )), in order to respect Gubser symmetry and keep the fourvelocity pattern invariant, the energy density (7), pressure (8) and net baryon density (3) have to be treated in the massless limit, meaninĝ respectively. Obviously, in this case one hasÊ = 3P.
On the other hand, for the conservation of spin to respect conformal invariance it is sufficient that the spin tensor in Eq. (54) is totally antisymmetric 2 . As noticed in Sec. VI the GLW form (15) of the spin tensor considered in this work does not satisfy this requirement. In general, if Eqs. (1), (5), and (14) were fully coupled this would lead to breaking of the Gubser symmetry and, in consequence, of the flow invariance. However, in the particular case studied herein, this is not the case since the spin dynamics given by Eq. (14) is treated only perturbatively [22], and hence decouples from the background hydrodynamic fields. This allows for a separate solution of background equations of motion (1) and (5) and subsequent solution of the spin part (14). As there is no backreaction to the background from the evolution of spin dynamics (polarization tensor does not enter Eqs. (1) and (5)), the conformal-breaking dynamics of spin polarization does not spoil the flow invariance (a similar issue was encountered also in other recent works, see for instance Ref. [116]). At this point, it is important to mention, that, despite this issue, we are still allowed to investigate the spin dynamics in the de Sitter coordinates, which we will do for the sake of convenience. Using the same arguments as given above we will also keep finite masses in expressions for A, B, and C defining the GLW spin tensor in Eqs. (15) and (16). 2 Note that this is not the case for the GLW and HW definitions whose additional terms ∼ 1/m make them ill-defined in the massless limit as well as break the conformal invariance of the spin conservation.

B. Perfect-fluid background
Using Eq. (2) in Eq. (1) the conservation law for charge in the de Sitter coordinates can be written aŝ whereĝ is the determinant of de Sitter metric (41) and we employed definitions of covariant derivative from App. A. Identifying expansion scalar asθ ≡ 2 tanh ρ one may rewrite Eq. (58) in the following form In the similar way, transforming Eq. (5) to dS 3 ⊗R, using Eq. (6), and contracting it withÛ β , we get the conservation law for energy as follows One may check that all other projections of Eq. (5) are satisfied identically. The solutions of Eq. (59) and Eq. (60) can be found analytically giving [89,90], respectively, whereÊ 0 ≡Ê(ρ 0 ) andN 0 ≡N (ρ 0 ) are integration constants and ρ 0 is the initial de Sitter time. Using Eqs. (55)-(57) the corresponding solutions for temperature and baryon chemical potential in de Sitter space areT where, again,T 0 ≡T (ρ 0 ) andμ 0 ≡μ(ρ 0 ) are constants of integration. Based on above solutions we observe that α is a ρ independent quantity. Moreover, as shown in [89,90] one can see that the Gubser symmetry restricts the dynamics of the system in such a way that it depends only on the de Sitter time ρ. The space-time evolution of temperature as given by Eq. (63) and flow-vector components Fig. (1). For setting initial temperature parameters we followed Ref. [117] where it was assumed thatT 0 ≡T (ρ 0 ) = 1.2 at ρ 0 = 0 so that choosing q = 1 fm −1 results in T (τ 0 = 1 fm, r = 0) = 1.2 fm −1 . One can observe that the temperature and flow profiles are strongly correlated. (U τ , U r ) / (U τ ) 2 + (U r ) 2 (stream lines -the coloring of arrows is given by the rapidity ϑ); see the bar legends for the scalings.

C. Spin evolution
In this section we derive the spin evolution equations which we use to study spin dynamics in the Gubserflowing perfect-fluid background presented in the previous section. Herein, we assume that the spin polarization components, apart from ρ dependence, also exhibit θ dependence. Hence, following the findings in Sec. VI, we assume that, in general, spin equations of motion break the conformal invariance. In such a case ρ and θ coordinates in the de Sitter space serve just as an alternative parameterization of the directions (τ, r) in polar Milne coordinates. By substituting Eq. (15) with Eq. (16), Eq. (28) and Eq. (29) in Eq. (14), employing Eqs. (63)- (64), and subsequently projecting the resulting tensorial equation onÛ βRγ ,Û βΦγ ,Û βẐγ ,Φ βẐγ ,R βẐγ andR βΦγ we obtain the following equations of motion for the spin polarization componentŝ respectively, where( ) ≡ ∂ ρ ,( ) ≡ ∂ θ , andB andĈ are defined in Eq. (18) and Eq. (19). In the latter expressions, unlike in the case of hydrodynamic background, we use full expressions for energy density (7) and pressure (8) expressed in terms of finite particle masses.

Massless limit
From Eqs. (65)- (70) we observe that, unlike in the case of Bjorken expansion [22], only spin polarization components along R µ evolve independently when expressed in de Sitter coordinates. On the other handâ Φ (b Φ ) andb Z (â Z ), respectively, are coupled to each other. The cou-pling between the components emerge due to the conformal symmetry breaking, and manifests itself through the θ dependence of the latter.
In the massless case the solutions to Eqs. (65) and (68) may be found analytically givinĝ where,â 0 R ≡â R (ρ 0 ) andb 0 R ≡b R (ρ 0 ). The characteristic concave dependence ofb R on de Sitter time is qualitatively similar to that of temperature and baryon chemical potential; (in the case ofâ R one deals with the convex function of ρ).
The dynamics ofb Φ andâ Z components following from Eqs. (67) and (69) is more complicated, however, it shows certain characteristic features. In particular, ifb Φ is initially negligible, theâ Z component is approximately θindependent, yieldinĝ withâ 0 Z ≡â Z (ρ 0 ), i.e. givingâ Z (ρ) ∼ 1/ T (ρ). In the general case whereb Φ = 0, one may use the fact thatâ Z is a slowly varying function of θ and the second term on the left-hand side of Eq. (69) may be neglected. In this case the solution tob Φ component has the approximate form whereb 0 Φ ≡b Φ (ρ 0 ). Using exact numerical solutions one can show that, indeed,b Φ is a weakly-dependent function of θ and hence approximately proportional tob R .
In the case of the remainingâ Φ andb Z components simple solutions cannot be found and in general Eqs. (66) and (70) have to be solved numerically. However, due to specific structure of Eqs. (66) and (70) one may find some special solutions by requiring the θ terms to vanish, thus making theâ Φ andb Z independent of each other. The latter takes place for the following solutionŝ where, again,â 0 Φ ≡â Φ (ρ 0 ) andb 0 Z ≡b Z (ρ 0 ), and hencê a Φ ∼â R . Finally, we observe that, in the cases discussed above all components exhibit the universal dependence of the type (cosh ρ) c , with c being positive constant for a i and negative for b i .

Numerical results
Let us turn to some numerical solutions of the spin equations of motion.
In Fig. (2) we present the exact numerical solutions of Eqs. (65) and (68) for a R and b R components as functions of proper time τ and radial distance r. The initial values of thê a R andb R components are chosen in such a way so thatâ 0 R = 0.1 andb 0 R = 0.1, which implies a R (τ 0 = 1fm, r = 0) = b R (τ 0 = 1fm, r = 0) = 0.1.
The mass in the calculations is set to m = 0.5T 0 . We find that the dynamics of a R and b R components is qualitatively different with a R having minimum and b R having a maximum at ρ 0 (the black dotted lines). Moreover, by comparing with the massless case we checked (see dashed lines) that the mass is weakly affecting the polarization dynamics as long as the mass is small, i.e. z < 1).
In Fig. (3) we present analogous plots to Fig. (2) but for a Z and b Φ components using the numerical solutions of Eqs. (67) and (69). In this case we initialize the components in the similar way as previously assumingâ 0 Z = 0.1 andb 0 Φ = 0.1. We observe a weak θ dependence of the solutions which is due to mixing of the two components. A comparison to massless case (dashed lines) confirms a small sensitivity of the solution to mass of the particles. Moreover we find that the increase of mass in general increases the values of the spin polarization components within the considered region. A similar conclusions hold for the remaining a Φ and b Z components.

VIII. SUMMARY AND CONCLUSIONS
In this paper we have used the formalism of hydrodynamics with spin [22,25] to derive the equations of motion for spin polarization in the Gubser-expanding conformal hydrodynamic background [89,90]. Follow-ing previous works on this topic we have first solved the perfect-fluid hydrodynamical equations using the Gubser symmetry arguments in the de Sitter coordinates and obtained analytical solutions for hydrodynamic variables, see Eqs. (61)- (64). Subsequently, we have extended the analysis of the properties of the conservation laws with respect to the conformal transformation to the case of the angular momentum conservation. We have found that the latter is conformaly invariant provided the spin tensor is completely antisymmetric and the particles are massless. As, in general, the currents in the GLW (Groot -van Leeuwen -van Weert) framework studied here break these requirements we have solved the spin dynamics in the Gubser expanding background in the de Sitter coordinates (for the sake of convenience) allowing, however, for both, ρ and θ dependence, of the spin polarization components. As there is no back reaction of the latter on the background we performed this analysis in the general massive case. The solutions obtained in this way are mapped back to the Milne coordinates, giving us the complete spatio-temporal evolution for the system with boost-invariance and cylindrical symmetry. In the massless limit we have found certain special solutions which, to a large extent, exhibit power-law type dependence on temperature. We have found that, unlike in the Bjorken case, only radial components behave independently.
The framework used herein might be used, for instance, for the description of head-on collisions of initially polarized particles/ions at high energies [118], describing the mixing between polarization components along beam and in the azimuthal direction. Moreover, the formalism worked out here can be used for cross checking the numerical simulations within the full 3+1D geometry framework, which is now being developed. need the covariant derivative operator which is an extension of partial derivative to arbitrary manifolds, and which reduces to partial derivative operator in flat spacetime in Cartesian coordinates. Its action on an arbitrary scalar V , rank-1 V µ and rank-2 V µν tensors is where d denotes covariant derivative, ∂ is partial derivative, g = det{g µν } and Γ ν µλ are Christoffel symbols which are expressed as Γ ν µλ ≡ Γ ν λµ = 1 2 g νσ (∂ µ g σλ + ∂ λ g σµ − ∂ σ g µλ ) . (A8)