Gravity dual of $1+1$ dimensional Bjorken expansion

We study the application of AdS/CFT duality to longitudinal boost-invariant Bjorken expansion of QCD matter produced in ultrarelativistic heavy ion collisions. As the exact $(1+4)$-dimensional bulk solutions for the $(1+3)$-dimensional boundary theory are not known, we investigate in detail the $(1+1)$-dimensional boundary theory, where the bulk is ${\mathrm{AdS}}_3$ gravity. We find an exact bulk solution, show that this solution describes part of the spinless Bañados-Teitelboim-Zanelli black hole with the angular dimension unwrapped, and use the thermodynamics of the Bañados-Teitelboim-Zanelli hole to recover the time-dependent temperature and entropy density on the boundary. After separating from the holographic energy-momentum tensor a vacuum contribution, given by the extremal black hole limit in the bulk, we find that the boundary fluid is an ideal gas in local thermal equilibrium. Including angular momentum in the bulk gives a boundary flow that is boost invariant but has a nonzero longitudinal velocity with respect to the Bjorken expansion.

Figure 1

The conformal diagram of the spinless nonextremal BTZ black-and-white hole (22) [17, 18]. The coordinate $\eta$ is suppressed, the coordinates of the diagram are $p=\arctan U$ and $q=\arctan V$ with $|p+q|<\pi /2$ and $|p-q|<\pi /2$, and the $p$ axis (respectively, $q$ axis) is tilted 45 degrees to the left (right) from the vertical. The Killing horizon of the Killing vector ${\partial }_t=(\sqrt{M}/\mathcal{L})(V{\partial }_V-U{\partial }_U)$ is at $pq=0$, where $r=r_{+}=\mathcal{L}\sqrt{M}$. The coordinate grid shows the curves of constant $\tau$ (solid curves) and the curves of constant $z$ (dotted curves) in the region $0<z<v\tau$, where $v$ is given by (25) and $M>1$ is assumed. The dashed line is the coordinate singularity $z=v\tau$, $r=\mathcal{L}\sqrt{M-1}$. On a given curve of constant $\tau$, $z$ takes the values $0<z<v\tau$, increasing from right to left, and on a given curve of constant $z$, $\tau$ takes the values $z/v<\tau <\infty$, increasing from bottom to top.

Figure 2

The conformal diagram of the nonextremal BTZ black-and-white hole spacetime with $J\ne 0$ [17, 18]. The solid (respectively, dotted) lines are curves of constant $\tau$ (constant $z$) in the region $0<z<v\tau$ of the metric (57), where $M>1$, $(M-1{)}^2-(J/\mathcal{L}{)}^2>0$, and $v$ is given by (53). $\tau$ increases upwards and $z$ increases to the left. The dashed line is the coordinate singularity $z=v\tau$.