Heavy Ion Collisions with Transverse Dynamics from Evolving AdS Geometries

Abstract: Currently there exists no known way to construct the Stress-Energy Tensor $(T_{\mu \nu})$ of the produced medium in heavy ion collisions at strong coupling from purely theoretical grounds. In this paper, some steps are taken in that direction. In particular, the evolution of $T_{\mu \nu}$ at strong coupling and at high energies is being studied for early proper times $(\tau)$. This is achieved in the context of the AdS/CFT duality by constructing the evolution of the dual geometry in an AdS$5$ background. Improving the earlier works in the literature, the two incident nuclei have an impact parameter $b$ and a non-trivial transverse profile. The nuclear matter is modeled by two shock waves corresponding to a non-zero five dimensional bulk Stress-Energy Tensor $J{MN}$. An analytic formula for $T_{\mu \nu}$ at small $\tau$ is derived and is used in order to calculate the momentum anisotropy and spatial eccentricity of the medium produced in the collision as a function of the ratio $\frac{\tau}{b}$. The result for eccentricity at intermediate $\frac{\tau}{b}$ agrees qualitatively with the results obtained in the context of perturbation theory and by using hydrodynamic simulations. Finally, the problem of the negative energy density and its natural connection to the eikonal approximation is discussed.