An Uneventful Horizon in Two Dimensions

We investigate the possibility of firewalls in the Einstein-dilaton gravity model of CGHS. We use the results of the numerical simulation carried out by Ashtekar et al. to demonstrate that firewalls are absent and the horizon is drama free. We show that the lack of a firewall is consistent because the model does not satisfy one of the postulates of black hole complementarity. In particular, we show that the Hawking radiation is not pure, and is completely entangled with a long-lived remnant beyond the last ray.Comment: 28 pages, 4 figure

Holographic Reconstruction of General Bulk Surfaces

We propose a reconstruction of general bulk surfaces in any dimension in terms of the differential entropy in the boundary field theory. In particular, we extend the proof of Headrick et al. to calculate the area of a general class of surfaces, which have a 1-parameter foliation over a closed manifold. The area can be written in terms of extremal surfaces whose boundaries lie on ring-like regions in the field theory. We discuss when this construction has a description in terms of spatial entanglement entropy and suggest lessons for a more complete and covariant approach.Comment: 21 pages, 10 figures; v2: minor clarifications, references added, published versio

Spinning Geodesic Witten Diagrams

We present an expression for the four-point conformal blocks of symmetric traceless operators of arbitrary spin as an integral over a pair of geodesics in Anti-de Sitter space, generalizing the geodesic Witten diagram formalism of Hijano et al [arXiv:1508.00501] to arbitrary spin. As an intermediate step in the derivation, we identify a convenient basis of bulk three-point interaction vertices which give rise to all possible boundary three point structures. We highlight a direct connection between the representation of the conformal block as a geodesic Witten diagram and the shadow operator formalism.Comment: 28+6 pages, 8 figure

Integral Geometry and Holography

We present a mathematical framework which underlies the connection between information theory and the bulk spacetime in the AdS$_3$/CFT$_2$ correspondence. A key concept is kinematic space: an auxiliary Lorentzian geometry whose metric is defined in terms of conditional mutual informations and which organizes the entanglement pattern of a CFT state. When the field theory has a holographic dual obeying the Ryu-Takayanagi proposal, kinematic space has a direct geometric meaning: it is the space of bulk geodesics studied in integral geometry. Lengths of bulk curves are computed by kinematic volumes, giving a precise entropic interpretation of the length of any bulk curve. We explain how basic geometric concepts -- points, distances and angles -- are reflected in kinematic space, allowing one to reconstruct a large class of spatial bulk geometries from boundary entanglement entropies. In this way, kinematic space translates between information theoretic and geometric descriptions of a CFT state. As an example, we discuss in detail the static slice of AdS$_3$ whose kinematic space is two-dimensional de Sitter space.Comment: 23 pages + appendices, including 23 figures and an exercise sheet with solutions; a Mathematica visualization too

Bulk and Transhorizon Measurements in AdS/CFT

We discuss the construction of bulk operators in asymptotically AdS spacetimes, including the interiors of AdS black holes. We use this to address the question "If Schrodinger's cat were behind the horizon of an AdS black hole, could we determine its state by a measurement in the dual CFT?"Comment: 32 pages, 5 figures. v2: Author added, appendix on spacelike Green's function added, discussion of non-dependence on boundary Hamiltonian expande

Modular Berry Connection

The Berry connection describes transformations induced by adiabatically varying Hamiltonians. We study how zero modes of the modular Hamiltonian are affected by varying the region that supplies the modular Hamiltonian. In the vacuum of a 2d CFT, global conformal symmetry singles out a unique modular Berry connection, which we compute directly and in the dual AdS$_3$ picture. In certain cases, Wilson loops of the modular Berry connection compute lengths of curves in AdS$_3$, reproducing the differential entropy formula. Modular Berry transformations can be measured by bulk observers moving with varying accelerations.Comment: 5 pages, 2 figures. Some clarifications adde

Black Holes: Complementarity or Firewalls?

We argue that the following three statements cannot all be true: (i) Hawking radiation is in a pure state, (ii) the information carried by the radiation is emitted from the region near the horizon, with low energy effective field theory valid beyond some microscopic distance from the horizon, and (iii) the infalling observer encounters nothing unusual at the horizon. Perhaps the most conservative resolution is that the infalling observer burns up at the horizon. Alternatives would seem to require novel dynamics that nevertheless cause notable violations of semiclassical physics at macroscopic distances from the horizon.Comment: 22 pages, 1 figure. v2: We have not changed our minds. Various arguments are expanded and sharpened. v3: Relation to previous work clarified. v4: Additional reference to earlier wor

Tensor networks from kinematic space

We point out that the MERA network for the ground state of a 1+1-dimensional conformal field theory has the same structural features as kinematic space---the geometry of CFT intervals. In holographic theories kinematic space becomes identified with the space of bulk geodesics studied in integral geometry. We argue that in these settings MERA is best viewed as a discretization of the space of bulk geodesics rather than of the bulk geometry itself. As a test of this kinematic proposal, we compare the MERA representation of the thermofield-double state with the space of geodesics in the two-sided BTZ geometry, obtaining a detailed agreement which includes the entwinement sector. We discuss how the kinematic proposal can be extended to excited states by generalizing MERA to a broader class of compression networks.Comment: 35 pages, 17 figure