A Bound on Holographic Entanglement Entropy from Inverse Mean Curvature Flow

Entanglement entropies are notoriously difficult to compute. Large-N strongly-coupled holographic CFTs are an important exception, where the AdS/CFT dictionary gives the entanglement entropy of a CFT region in terms of the area of an extremal bulk surface anchored to the AdS boundary. Using this prescription, we show -- for quite general states of (2+1)-dimensional such CFTs -- that the renormalized entanglement entropy of any region of the CFT is bounded from above by a weighted local energy density. The key ingredient in this construction is the inverse mean curvature (IMC) flow, which we suitably generalize to flows of surfaces anchored to the AdS boundary. Our bound can then be thought of as a "subregion" Penrose inequality in asymptotically locally AdS spacetimes, similar to the Penrose inequalities obtained from IMC flows in asymptotically flat spacetimes. Combining the result with positivity of relative entropy, we argue that our bound is valid perturbatively in 1/N, and conjecture that a restricted version of it holds in any CFT.Comment: 33+7 pages, 7 figures. v2: addressed referee comment

Complex Entangling Surfaces for AdS and Lifshitz Black Holes?

We discuss the possible relevance of complex codimension-two extremal surfaces to the the Ryu-Takayanagi holographic entanglement proposal and its covariant Hubeny-Rangamani-Takayanagi (HRT) generalization. Such surfaces live in a complexified bulk spacetime defined by analytic continuation. We identify surfaces of this type for BTZ, Schwarzschild-AdS, and Schwarzschild-Lifshitz planar black holes. Since the dual CFT interpretation for the imaginary part of their areas is unclear, we focus on a straw man proposal relating CFT entropy to the real part of the area alone. For Schwarzschild-AdS and Schwarzschild-Lifshitz, we identify families where the real part of the area agrees with qualitative physical expectations for the appropriate CFT entropy and, in addition, where it is smaller than the area of corresponding real extremal surfaces. It is thus plausible that the CFT entropy is controlled by these complex extremal surfaces.Comment: 28+5 pages. v2: Addressed referee comment

Flowing Funnels: Heat sources for field theories and the AdS_3 dual of CFT_2 Hawking radiation

We construct the general 2+1 dimensional asymptotically AdS_3 solution dual to a stationary 1+1 CFT state on a black hole background. These states involve heat transport by the CFT between the 1+1 black hole and infinity (or between two 1+1 black holes), and so describe the AdS dual of CFT Hawking radiation. Although the CFT stress tensor is typically singular at the past horizon of the 1+1 black hole, the bulk 2+1-dimensional solutions are everywhere smooth, and in fact are diffeomorphic to AdS_3. In particular, we find that Unruh states of the CFT on any finite-temperature 1+1 black hole background are described by extreme horizons in the bulk.Comment: 24 page

Locality from Quantum Gravity: All or Nothing

In a full theory of quantum gravity, local physics is expected to be approximate rather than innate. It is therefore important to understand how approximate locality emerges in the semiclassical limit. Here we show that any notion of locality emergent from a holographic theory of quantum gravity is "all or nothing": local data is not obtained gradually from subregions of the boundary, but is rather obtained all at once when enough of the boundary is accessed. Our assumptions are mild and thus this feature is quite general; in the special case of AdS/CFT, a slightly different manifestation follows from well-known and familiar properties.Comment: 7 pages; 4 figures. v2: added references, minor edit

Natural‐language processing applied to an ITS interface

The aim of this paper is to show that with a subset of a natural language, simple systems running on PCs can be developed that can nevertheless be an effective tool for interfacing purposes in the building of an Intelligent Tutoring System (ITS). After presenting the special characteristics of the Smalltalk/V language, which provides an appropriate environment for the development of an interface, the overall architecture of the interface module is discussed. We then show how sentences are parsed by the interface, and how interaction takes place with the user. The knowledge‐acquisition phase is subsequently described. Finally, some excerpts from a tutoring session concerned with elementary geometry are discussed, and some of the problems and limitations of the approach are illustrated

Conserved Charges in Asymptotically (Locally) AdS Spacetimes

We review issues related to conservation laws for gravity with a negative cosmological constant subject to asymptotically (locally) anti-de Sitter boundary conditions. Beginning with the empty AdS spacetime, we introduce asymptotically (locally) AdS (AlAdS) boundary conditions, important properties of the boundary metric, the notion of conformal frames, and the Fefferman-Graham expansion. These tools are used to construct variational principles for AlAdS gravity, to more properly define the notion of asymptotic symmetry, and to construct the associated boundary stress tensor. The resulting conserved charges are shown to agree (up to possible choices of zero-point) with those built using Hamiltonian methods. Brief comments are included on AdS positive energy theorems and the appearance of a central extension of the AdS$_3$ asymptotic symmetry algebra. We also describe the algebra of boundary observables and introduce the anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence using only tools from gravitational physics (and without other input from string theory). Our review focuses on motivations, current status, and open issues as opposed to calculational details. We emphasize the relativist (as opposed to particle physics) perspective and assume as background a standard graduate course in general relativity.Comment: To appear as Ch 19 of "The Springer Handbook of Spacetime," edited by A. Ashtekar and V. Petkov. (Springer-Verlag, expected 2013). v2: Amended references. v3: Corrected assorted minus sign convention