Gravity dual of conformal matter collisions in $1+1$ dimensions

We find the three-dimensional gravity dual of a process in which two clouds of ($1+1$)-dimensional conformal matter moving in opposite directions collide. This gives the most general conformally invariant holographic flow in the ($1+1$)-dimensional boundary theory in terms of two arbitrary functions. With a suitable choice of the arbitrary functions the process can be interpreted as an opaque collision of two extended systems with central, fragmentation and interaction regions. Comparison with classical gluon field calculations relates the size of the system with the saturation scale.

Figure 1

Structure of the conformally invariant boundary flow following from the solution (8), assuming that $f$ and $g$ are vanishing at negative argument and are nonvanishing and have the same sign for positive argument. Within the past light cone of the origin the energy-momentum tensor vanishes. The left wedge $x^{+}<0<x^{-}$ contains null dust moving to the right, and the right wedge $x^{-}<0<x^{+}$ contains null dust moving to the left. The future light cone of the origin contains expanding matter with the flow given by (13) and energy density given by (14). The special solution (44) that gives rise to unregulated Bjorken similarity flow (20, 21) within the future light cone of the origin is also indicated; for this solution the energy density (21) diverges as $\tau \to 0$.

Figure 2

Flow lines (48) for $a=1$ and $x_i=1/4$, $1/3$, $1/2$, 1, 2, 3, 4 and 5, and the corresponding flow lines with $x^{+}$ and $x^{-}$ interchanged.

Figure 3

The boundary flow with the regularization (49). The various regions are described in the text. A curve of constant $\tau$ at $\tau >a\sqrt{2}$ is shown in the region $x^{\pm }>0$.

Figure 4

The energy density (53) divided by $\mathcal{L}(M-1)/(32\pi G_3)$ as a function of $\eta$ at constant $\tau$ in the regularization (49). The left panel shows the case $\tau >a\sqrt{2}$ and the right panel the case $\tau <a\sqrt{2}$. The parameter $b$ equals $a\sqrt{2}$.

Figure 5

Paths (55, 56) of the fluid elements moving initially to the left with $x^{+}=x_i^{+}$. In the Minkowski coordinates $(t,x)$, the fluid comes momentarily to rest on the light cone of the origin, and it stays at rest everywhere within the interaction diamond.