Bulk-cone singularities & signatures of horizon formation in AdS/CFT

Abstract

We discuss the relation between singularities of correlation functions and causal properties of the bulk spacetime in the context of the AdS/CFT correspondence. In particular, we argue that the boundary field theory correlation functions are singular when the insertion points are connected causally by a bulk null geodesic. This implies the existence of "bulk-cone singularities" in boundary theory correlation functions which lie inside the boundary light-cone. We exhibit the pattern of singularities in various asymptotically AdS spacetimes and argue that this pattern can be used to probe the bulk geometry. We apply this correspondence to the specific case of shell collapse in AdS/CFT and indicate a sharp feature in the boundary observables corresponding to black hole event horizon formation.

Figures (21)

• Figure 1: Density and mass functions for the "star" geometry, where the central density is set to $\rho_0=10$.
• Figure 2: Metric functions for the "star" geometry with central density $\rho_0=10$ (red curves), and for comparison corresponding metric functions in the pure AdS geometry (blue curves; $f(r)$ is the higher and $h(r)$ is the lower of the two curves).
• Figure 3: Time delay for radial null geodesics through star in AdS, as a function of the star's internal density $\rho_0$.
• Figure 4: Null geodesics in star with $\rho_0=10$ in AdS, projected onto a constant $t$ slice and the $t-r$ plane, for varying angular momentum to energy ratio ($E = 10$ and $J= 0,1,\ldots,10$). On the left, the bold circle corresponds to the AdS boundary, whereas on the right, the dashed vertical line corresponds to the origin $r=0$ and the bold vertical line to the boundary at $\tan r = {\pi \over 2}$. The range of $t$ plotted is $(0, 1.1 \, \Delta t_0)$. (Analogous plots with larger $\rho_0$ appear in Fig. \ref{['starrphrho']} of Appendix \ref{['appstar']}.)
• Figure 5: Endpoints of null geodesics in the AdS geometry of star with (a)$\rho_0=1, 2,\ldots,10$ and (b)$\rho_0=10, 20,\ldots,100$. Each curve is plotted by varying ${\alpha}={E \over J}$. The top of the curves corresponds to big values of ${\alpha}$; in fact, one can easily show that the slope of each curve (as a function of ${\alpha}$) is given by $1/{\alpha}$.