Shear viscosity in late time of hydrodynamic evolution in AdS/CFT duality

We investigate the shear viscosity $\eta$ and the entropy density $s$ of strongly coupled $\mathcal{N}=4$ super Yang-Mills (SYM) plasma in late time of hydrodynamic evolution with AdS/CFT duality and Bjorken scaling. We use correlation function method proposed by Kovtun, Son and Starinets. We obtain the metric $g_{\mu\nu}$ in a proper time dependent $AdS_{5}$ space through holographic renormalization, whose boundary condition is given by energy-momentum tensor of the plasma in 2+1 dimension with transverse expansion or radial flow. With the metric we compute $\eta$ and $s$ of fluids in 1+1 and 2+1 dimension without and with radial flow. We find the ratio $\eta/s=1/(4\pi)$ in 1+1 dimension consistent with the Kovtun-Son-Starinets bound if next-to-leading terms in proper time are included in the equation of motion for metric perturbations. For 2+1 dimension the result is unchanged in the leading order of transverse rapidity.