We study the collision of planar shock waves in
AdS

Article funded by SCOAP3

e^{i {\bm k} \cdot \bm x_\perp} \, \widetilde H_\pm({\bm k},z_\mp) \, \frac{8I_2(k/r)}{k^2 r^2} \,. \label{eq:h} \end{equation} The AdS curvature scale has been set to unity. The boundary of the asymptotically AdS spacetime lies at radial coordinate $r = \infty$. The single-shock metric~(\ref{eq:FG}) is an exact solution to Einstein's equations for any choice of $\widetilde H_\pm$~\cite{Gubser-ml-2008pc}. This geometry represents a state in the dual SYM theory with stress tensor, \begin{equation} \label{eq:hatvsnohat} T^{\mu \nu} = \frac{N_{\rm c}^2}{2 \pi^2} \widehat T^{\mu \nu}, \end{equation} with non-zero components \begin{equation} \label{eq:singleshock} \widehat T^{00} = \widehat T^{zz} = \pm \widehat T^{0z} = H_\pm(\bm x_\perp,z_\mp), \end{equation} where $H_\pm$ is the 2D transverse Fourier transform of $\widetilde H_\pm$. Note that here and in what follows we used hats to denote quantities normalized by $\frac{N_{\rm c}^2}{2 \pi^2}$. That is, for any function $F$ we define $\widehat F$ by $F = \frac{N_{\rm c}^2}{2 \pi^2} \widehat F$. We choose \begin{equation} \label{eq:initialH} H_\pm(\bm x_\perp,z_\mp) = \mu^3 \delta_w(z_\mp), \end{equation} where $\mu$ is an energy scale and $\delta_w(z_\mp)$ is a smeared $\delta$-function which satisfies the normalization and variance conditions in eq.~(\ref{eq:normandvar}). We employ two different shock profiles \begin{equation} \label{eq:smearing} \delta_w(z) = \frac{1}{\sqrt{2 \pi w^2}} e^{-\frac 12 z^2/w^2} \ \ \ \ {\rm and} \ \ \ \ \delta_w(z) = \frac{M}{e^{\frac 12 z^2/W^2} + 1}, \end{equation} with \begin{equation} M = \textstyle \frac{1}{w} \sqrt{\textstyle-\frac{(4 + 3 \sqrt{2}) \zeta(\frac 32)}{ 4 \pi \zeta(\frac 12)^3} } \approx \frac{0.74}{w}, \qquad W = \textstyle w \sqrt{-\frac{\sqrt{2} \zeta(\frac 12)}{\zeta( \frac 32)}} \approx 0.89 w. \end{equation} We refer to the first shock profile in~(\ref{eq:smearing}) as the Gaussian profile and the second as the non-Gaussian profile. These shock profiles are plotted in figure~\ref{fig:smearingfunctions}. \begin{figure}[t] \centering \includegraphics[scale = 0.45]{smearingfunctions.pdf} ]]>

0)$ eigenvector of the stress tensor, \begin{equation} T^{\mu}_{\ \nu} \, u^\nu = -\epsilon \, u^\mu \,, \label{eq:veldef} \end{equation} with $\epsilon$ the associated eigenvalue. In terms of $\epsilon$ and $u^\mu$ the constitutive relations of fluid/gravity read \begin{equation} \label{eq:hydroconstituative} T^{\mu \nu}_{\rm hydro} = p g^{\mu \nu} + (\epsilon + p) u^\mu u^\nu + \Pi^{\mu \nu}, \end{equation} where $p = \frac{\epsilon}{3}$ is the pressure and $\Pi^{\mu \nu}$ is the viscous stress. The viscous stress satisfies $u_\mu \Pi^{\mu \nu} = 0$ and $g_{\mu \nu} \Pi^{\mu \nu} = 0$ and at first order in gradients is given by $\Pi^{\mu \nu} = - \eta \sigma^{\mu \nu}$ with $\eta$ the shear viscosity and \begin{equation} \sigma_{\mu \nu} = \partial_{(\mu} u_{\nu )} + u_{(\mu} u^\rho \partial_\rho u_{\nu)} - {\textstyle \frac{1}{3}} \partial_\alpha u^\alpha \left [ \eta_{\mu \nu} + u_\mu u_\nu \right], \end{equation} the shear tensor. The shear viscosity may be expressed in terms of the proper energy via~\cite{Kovtun-ml-2004de,Policastro-ml-2001yc} \begin{equation} \label{eq:viscosity} \eta = \frac{1}{3 \pi T} \epsilon, \end{equation} with the temperature $T$ given by \begin{equation} \label{eq:Tdef} T = \left (\frac{8 \epsilon}{ 3 \pi^2 N_{\rm c}^2} \right )^{1/4}. \end{equation} The hydrodynamic equations of motion are given by the energy-momentum conservation equation $\partial_\mu T^{\mu \nu}_{\rm hydro} = 0$. Note that the hydrodynamic stress tensor is completely determined by four functions. In contrast, in general the exact (traceless) stress tensor contains nine independent function. Instead of solving the hydrodynamic equations of motion for the evolution of $\epsilon$ and $u^\mu$, a simple way to compare the gravitational evolution to hydrodynamics is to extract the exact $\epsilon$ and $u^\mu$ from the eigenvalue equation~(\ref{eq:veldef}) with the exact stress tensor. In the domain where hydrodynamics is a good description this should yield the same time evolution for proper energy density and fluid velocity as hydrodynamics. With the exact proper energy and fluid velocity known, we can then construct $T^{\mu \nu}_{\rm hydro}$ from eqs.~(\ref{eq:hydroconstituative})--(\ref{eq:Tdef}) and compare $T^{\mu \nu}$ and $T^{\mu \nu}_{\rm hydro}$. To quantify the domain in which hydrodynamics is applicable we then define the residual measure \begin{equation} \label{eq:deltadef} \Delta \equiv \frac{1}{ p}\sqrt{\Delta T_{\mu \nu}\Delta T^{\mu \nu}}, \qquad \Delta T^{\mu \nu} \equiv T^{\mu \nu} - T^{\mu \nu}_{\rm hydro}. \end{equation} The quantity $\Delta$, evaluated in the local fluid rest frame, measures the relative difference between the spatial stress in $T^{\mu \nu}$ and $T^{\mu \nu}_{\rm hydro}$. Regions of spacetime with $\Delta \ll 1$ are evolving hydrodynamically. Let us first focus on the hydrodynamic evolution produced by Gaussian shock collisions. In figure~\ref{fig:hydroresidual} we plot the hydrodynamic residual $\Delta$ for Gaussian shock profiles with widths $w = 5 w_o$ (top) and $w = w_o$ (bottom). Note that we only plot $\Delta$ in the region $\mathcal R$ defined to be the largest connected region in spacetime where $\Delta \leq 0.15$. Outside of $\mathcal R$ hydrodynamics is not a good description and the fluid velocity need not even be well defined~\cite{Arnold-ml-2014jva} (i.e.\ the stress need not have a time-like eigenvector). The dashed line in the figure, which bounds the region $\mathcal R$, is given by \begin{equation} \label{eq:hydrodomain} \tau_* = \sqrt{(t - \Delta t)^2 -z^2}, \end{equation} with $\mu \tau_* = 1.5$ and $\mu \Delta t = 0.58$. We therefore conclude that the domain of applicability of hydrodynamics is approximately the same for both shock thicknesses. Figure~\ref{fig:hydroresidual} clearly shows that our planar shock collisions result in the formation of an expanding volume of fluid which is well described by hydrodynamics everywhere except near the light cone, where non-hydrodynamic effects become important. At mid-rapidity viscous hydrodynamics becomes a good approximation at time \begin{equation} \mu t_{\rm hydro} \approx 2. \end{equation} \begin{figure}[t] \centering \includegraphics[scale = 0.4]{hydroresidual2.pdf} ]]>