Black Holes, Shock Waves, and Causality in the AdS/CFT Correspondence

TL;DR

The paper develops a concrete link between bulk causality in AdS and its CFT description by relating boosted AdS black holes to gravitational shock waves and to light-cone localized CFT states. Symmetry fixes the CFT energy-momentum tensor for Schwarzschild AdS, and boosting corresponds to a dilation that localizes energy near the light cone, illustrating the UV/IR connection. By matching shock-wave dynamics with light-cone states, the authors reproduce bulk causal relations and show there are no alpha' corrections to the shock wave, supporting robustness of the AdS/CFT correspondence for causal structure. The work also outlines extensions to other backgrounds and implications for understanding black hole formation within the CFT framework.

Abstract

We find the expectation value of the energy-momentum tensor in the CFT corresponding to a moving black hole in AdS. Boosting the black hole to the speed of light, keeping the total energy fixed, yields a gravitational shock wave in AdS. The analogous procedure on the field theory side leads to ``light cone'' states, i.e., states with energy-momentum tensor localized on the light cone. The correspondence between the gravitational shock wave and these light cone states provides a useful tool for testing causality. We show, in several examples, how the CFT reproduces the causal relations in AdS.

Figures (4)

• Figure 1: The UV/IR relation at work: (a) On the supergravity side a black hole is moving along a radial geodesic in AdS with maximal radial direction $\sim 1/\Delta r$. (b) On the SYM side, the energy of the SYM state associated with the black hole is concentrated around the light cone with minimal size of $\Delta r$.
• Figure 2: Regions of $AdS_d$ covered by different coordinates: The coordinates (\ref{['lim']}) cover the shaded region in (a). While the coordinates (\ref{['bn']}) cover the shaded region in (b). The matching between the coordinates is such that the line $U=0, t=\infty$ in (b) is the same as $y_-=0$ in (a).
• Figure 3: On the supergravity side (a) we have two shock waves which do not interact. On the SYM side (b) the states associated with these shock waves evolve on light cones which do not cross.
• Figure 4: (a) The arrows indicate the trajectories of the massless particles. The dashed lines indicate the shock wave of the particle whose trajectory initiates at $x_1 =c$. The shock wave of the other particle coincides (in the figure) with its trajectory. The shock waves cross each other at $t=c/2$ which is in agreement with SYM (b).