Light-sheets and AdS/CFT

One may ask whether the conformal field theory (CFT) restricted to a subset $b$ of the anti-de Sitter (AdS) boundary has a well-defined dual restricted to a subset $H(b)$ of the bulk geometry. The Poincaré 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 $L$ of $H$ from light-sheets off boundary slices and invoking the covariant entropy bound in the bulk. The simplest covariant choice is $L=L^{+}\cap L^{-}$, 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 renormalization-group flow. We comment on the description of black hole interiors and cosmological regions via AdS/CFT.

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 ($d{s}^2=-d{t}^2+d{\Omega }_{d-1}^2$). 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.

Figure 6

An AdS-Schwarzschild black hole. A sphere on the regulated boundary encloses an infinitely large spatial region that extends all the way to the second, disconnected conformal boundary on the far side of the black hole.

Figure 7

The union of all future-directed light-sheets, $L^{+}(b)$, coming off the usual slicing of the boundary Minkowski space (left) covers precisely the Poincaré patch (the wedge-shaped region that lies both in the future of the boundary point $A$ and in the past of $D$). On the right, we show a different time slicing of the same boundary region. One of these slices is shown in blue in the bulk Penrose diagram (center); it curves up at $B$ and down at $C$. The future-directed light-sheet coming off the portion of the slice near $C$ is nearly the same as the future light cone of $C$ (shown in red/short-dashed), which reaches far beyond the Poincaré patch to the far side of AdS. The bulk region covered by $L^{+}(b)$ will thus be nearly two Poincaré patches, consisting of the points that lie in the future of $A$ but not in the future of $D$ (long-dashed).

Figure 8

Consider an arbitrary boundary region $b$ (enclosed by the solid line), and a point $P$ in the bulk region $C(b)$ (orthogonal to the page). The causal future of $P$, $J^{+}(P)$, intersected with $b$ is shown hatched. Roughly, the strategy of the proof is to demonstrate that there exists a time slice $s_{t_{*}(P)}$ on the boundary that is tangent to the lower boundary of future of $P$ in $b$. We show that $s_{t_{*}(P)}$ is the earliest time slice that has any intersection with the future of $P$, and that $P$ lies on the past light-sheet of $s_{t_{*}(P)}$.

Figure 9

The covariant bulk RG flow presented here reproduces the standard bulk RG flow in certain coordinate systems. Here we illustrate the construction in global coordinates of anti-de Sitter space. For a given coordinate time cutoff $\delta$, the union over $t$ of the intersection surfaces $s_t^{\delta }={\ell }_{t+\delta /2}^{-}\cap {\ell }_{t-\delta /2}^{+}$ form a timelike hypersurface $b^{\delta }$ in the bulk (left). The cross-sectional area of a given light-sheet will be greater on the surface $b^{\delta }$ than on $b^{{\delta }^{\prime }}$ (right). The difference $(A-A^{\prime })/4$ bounds the entropy on the red light-sheets going from $b^{\delta }$ to $b^{{\delta }^{\prime }}$, meaning that the bound applies to the entire darkly shaded wedge between them. The lightly shaded region between the hypersurfaces $b^{{\delta }^{\prime }}$ and $b^{\delta }$ is covered by such wedges.

Figure 10

A cross section of anti-de Sitter space, showing a short strip region $\mathcal{S}$ centered around $\tau =0$ on the boundary, and the bulk region $S(\mathcal{S})$ spacelike separated from $\mathcal{S}$. A local operator at the origin of the bulk can be written in terms of local operators on the boundary smeared over the boundary region spacelike related to the origin, within the green wedges. This region is much larger than $\mathcal{S}$ (red thick line), stretching from $\tau =-\pi /2$ to $\tau =+\pi /2$.

Figure 11

According to the Hamiltonian on the boundary strip $\mathcal{S}$, no source acts at the origin in the bulk, so the expectation value of $\varphi$ vanishes everywhere. At the time ${\tau }_0$ outside the strip, a source term for the nonlocal boundary operator dual to $\varphi (0,0)$ can be added to the boundary Hamiltonian. This causes the expectation value of $\varphi$ to be nonzero in the future of (0,0), in contradiction with the earlier conclusion about the same bulk points. Thus, unless we possess information about the exterior of $\mathcal{S}$ on the boundary guaranteeing that such operators do not act, the bulk interpretation of regions outside $\overline{H}(\mathcal{S})=C(\mathcal{S})$ is potentially ambiguous.

Figure 12

The shaded region shows bulk points spacelike related to a global boundary Cauchy surface $\sigma$. The union of all such sets over the collection of boundary Cauchy surfaces which do not intersect $\mathcal{S}$ has an ambiguous bulk interpretation when the boundary Hamiltonian is allowed to vary outside of $\mathcal{S}$. The unambiguous region, $\overline{H}(\mathcal{S})$, is the complement of this union. In this example, we see that $\overline{H}(\mathcal{S})=C(\mathcal{S})$.