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 conservation equations in the $d{S}_3\times \mathbb{R}$ space-time. We then derive the analytic solutions of spin density, spin potential and other thermodynamic in $d{S}_3\times \mathbb{R}$ 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 nonvanishing 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.

Figure 1

We plot $d[\log (A/A_0)]/d[\log (\tau /{\tau }_0)]$ with $A={\mathrm{\Omega }}^{\tau x_{\perp }},{\xi }^{\tau x_{\perp }},{\omega }^{\tau x_{\perp }}$ as functions of proper time $\tau$ at $x_{\perp }=0.5\mathrm{fm}$ for (a) $L=4.3\mathrm{fm}$ and (b) $L=50\mathrm{fm}$. The parameters are chosen as ${\widehat{e}}_0=880$ at ${\rho }_0=0$ with ${\eta }_s=0$, $c_0=11$, and ${\widehat{T}}_0=({\widehat{e}}_0/c_0{)}^{1/4}$ [117, 118]. The red dotted, blue dashed, and green solid lines stand for the cases of $A={\mathrm{\Omega }}^{\tau x_{\perp }},{\xi }^{\tau x_{\perp }},{\omega }^{\tau x_{\perp }}$, respectively.