The AdS/CFT correspondence and topological censorship

Abstract

In [1] it was shown that (n + 1)-dimensional asymptotically anti-de-Sitter spacetimes obeying natural causality conditions exhibit topological censorship. We use this fact in this paper to derive in arbitrary dimension relations between the topology of the timelike boundary-at-infinity, I, and that of the spacetime interior to this boundary. We prove as a simple corollary of topological censorship that any asymptotically anti-de Sitter spacetime with a disconnected boundary-at-infinity necessarily contains black hole horizons which screen the boundary components from each other. This corollary may be viewed as a Lorentzian analog of the Witten and Yau result [2], but is independent of the scalar curvature of I. Furthermore, as shown in [1], the topology of V ′ , the Cauchy surface (as defined for asymptotically anti-de Sitter spacetime with boundary-at-infinity) for regions exterior to event horizons, is constrained by that of I; the homomorphism Π 1 (Σ 0 ) → Π 1 (V ′ ) induced by the inclusion map is onto where Σ 0 is the intersection of V ′ with I. In 3 + 1 dimensions, the homology of V ′ can be completely determined from this as shown in [1]. In this paper, we prove in arbitrary dimension that H n−1 (V ; Z) = Z k where V is the closure of V ′ and k is the number of boundaries Σ i interior to Σ 0 . As a consequence, V does not contain any wormholes or other compact, non-simply connected topological structures. Finally, for the case of n = 2, we show that these constraints and the onto homomorphism of the fundamental groups from which they follow are sufficient to limit the topology of interior of V to either B 2 or I × S 1 .

Topological Censorship for Kaluza–Klein Space-Times

“…this effect have been established in the literature under various conditions [10,13,14,17,18,21]. Of particular relevance to our work is [14], where topological censorship is reduced to a null convexity condition of timelike boundaries.…”

“…As noted in [17], topological censorship can be viewed as the statement that any curve in T with endpoints in T is homotopic to a curve in T , or equivalently that the map j * : π 1 (T ) → π 1 ( T ) is surjective. We reproduce here the argument for completeness.…”

Entanglement between two interacting CFTs and generalized holographic entanglement entropy

“…This suggests that its gravity dual looks like the upper right picture in figure 5, where the minimal surface γ does not reach the AdS boundary. However, we should be careful that if we naively consider such a geometry with two boundaries which are causally connected, it contradicts with the topological censorship [75]. Thus we need to take into account quantum corrections or complicated structures of internal manifolds.…”

A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric

“…However, by a theorem of Nomizu [31] (see also [4]), if the underlying domain is simply connected, then analytic continuation does give rise to a single-valued extension. By the topological censorship theorem [10,11], the domain of outer communication has this property. Consequently, there exists a unique, single valued extension of K a to the domain of outer communication, i.e., the exterior of the black hole (with respect to a given end of infinity).…”