Light-sheets and AdS/CFT

TL;DR

The paper addresses how to define the bulk holographic domain H(b) dual to a boundary region b in AdS/CFT. By bounding H(b) from below with the causally connected region $C(b)$ and from above with the light-sheet based region $L(b)$, and proving $C(b)=L(b)$, it establishes $H(b)=C(b)=L(b)$ in a covariant framework. It then develops a covariant RG flow along light-sheets, discusses nonlocal boundary operators and their bulk interpretation via regions $S$ and $\bar{H}$, and examines implications for black hole interiors. The work highlights both the power of covariant entropy bounds in constraining bulk reconstruction and the limitations of AdS/CFT in providing a nonperturbative description of spacetime behind horizons.

Abstract

One may ask whether the CFT restricted to a subset b of the AdS boundary has a well-defined dual restricted to a subset H(b) of the bulk geometry. The Poincare patch is an example, but more general choices of b can be considered. We propose a geometric construction of H. We argue that H should contain the set C of causal curves with both endpoints on b. Yet H should not reach so far from the boundary that the CFT has insufficient degrees of freedom to describe it. This can be guaranteed by constructing a superset of H from light-sheets off boundary slices and invoking the covariant entropy bound in the bulk. The simplest covariant choice is L, the intersection of L^+ and L^-, where L^+ (L^-) is the union of all future-directed (past-directed) light-sheets. We prove that C=L, so the holographic domain is completely determined by our assumptions: H=C=L. In situations where local bulk operators can be constructed on b, H is closely related to the set of bulk points where this construction remains unambiguous under modifications of the CFT Hamiltonian outside of b. Our construction leads to a covariant geometric RG flow. We comment on the description of black hole interiors and cosmological regions via AdS/CFT.

Figures (12)

• Figure 1: (a) The Poincaré patch of AdS, with the usual time slicing in the coordinates of Eq. (1.1). (b) Time slices of an arbitrary bulk coordinate system that covers the same near-boundary region as the Poincaré patch but a different region far from the boundary. This illustrates that there is no preferred coordinate system that would uniquely pick out a region described by the boundary, particularly if the bulk is not in the vacuum state.
• Figure 2: The boundary of AdS; the dashed lines should be identified. Examples of globally hyperbolic subsets $b$ are shown shaded. A causal diamond is a set of the form $I^-(q)\cup I^+(p)$, where $q$ is boundary event in the future of the boundary event $p$. Let $\tau$ be the time along a geodesic from $p$ to $q$ in the Einstein static universe of unit radius ($ds^2=-dt^2+d\Omega_{d-1}^2$). With $\tau=2\pi$, the causal diamond is the boundary of the Poincaré patch. A causal diamond with $\tau<2\pi$ ($\tau>2\pi$) is called "small" ("large"). An open interval $(t_1,t_2)$ with $t_2-t_1<\pi$ ($t_2-t_1>\pi$ is called "short strip" ("tall strip").
• Figure 3: The four null hypersurfaces orthogonal to a spherical surface $B$ in Minkowski space. The two cones $F_1$, $F_3$ have negative expansion and hence correspond to light-sheets. The covariant entropy bound states that the entropy of the matter on each light-sheet will not exceed the area of $B$. The other two families of light rays, $F_2$ and $F_4$, generate the skirts drawn in thin outline. Their cross-sectional area is increasing, so they are not light-sheets, and the entropy of matter on them is unrelated to the area of $B$.
• Figure 4: Penrose diagram of a collapsing star (shaded). At late times, the area of the star's surface becomes very small ($B$). The enclosed entropy in the spatial region $V$ stays finite, so that the spacelike entropy bound is violated. The covariant entropy bound avoids this difficulty because only future directed light-sheets are allowed by the nonexpansion condition. $L$ is truncated by the future singularity; it does not contain the entire star.
• Figure 5: (a) A square system in 2+1 dimensions, surrounded by a surface $B$ of almost vanishing length $A$. The entropy in the enclosed spatial volume can exceed $A$. (b) [Here the time dimension is projected out.] The light-sheet of $B$ intersects only with a negligible (shaded) fraction of the system, so the covariant entropy bound is satisfied.