The bulk from the boundary; holography and AdS/CFT

Abstract

In this thesis we consider various methods by which one can extract (in detail) the metric structure of asymptotically anti-de Sitter spacetime, using only information from the boundary. This is motivated by the AdS/CFT correspondence, in particular the relation between geometrical properties of the bulk and certain field theory quantities such as "bulk-cone singularities" of two-point functions and entanglement entropy. These CFT quantities are directly related to endpoints of null bulk geodesies and regularised proper area of certain bulk minimal surfaces, respectively. Focussing initially on static, spherically symmetric spacetimes, we demonstrate how the endpoints of null geodesies, and the endpoints (along with the proper length) of zero-energy spacelike geodesies allow us to reconstruct the bulk spacetime metric, and detail explicit iterative algorithms by which the metric functions can be extracted numerically using this data, to an arbitrarily high degree of accuracy. The stability of the methods is demonstrated both via examples, and by an analytic consideration of the errors. Refinements of the algorithms are presented, and we consider the differences in how the two types of geodesic probe the bulk. We focus on a realistic application of our methods, namely extracting the physical properties of a "star" in AdS, which leads to an analysis of how their total mass varies with their core density in higher dimensions. We find the existence of a critical dimension (dc) separating two distinct regimes of behaviour; monotonic for d > d(_e), and oscillatory for d < d(_c). Finally, we consider how our iterative algorithms can be generalised to metrics with less symmetry, and discuss possible directions for future research.