| Summary: | In this thesis we find constraints to asymptotically anti de-Sitter space dual to holographic conformal field theory states using the holographic duality. A recent conjecture involving the causal holographic information surface propsed that for smooth asymptotically anti de-Sitter spacetimes that obey the null energy condition, the area of the Ryu-Takayanagi surface will always be less than or equal to the area of the causal holographic information surface. This conjecture is explored in three dimensional spacetimes that are dual to translation invariant states on the boundary conformal field theory in two dimensions. A series expansion of the Ryu-Takayanagi surface and causal holographic information surface is derived, and is used to translate the constraint between the ar- eas of the two surfaces into a constraint on the asymptotic structure of such geometries near the conformal boundary. The translated constraints are compared to the constraints given by the null energy condition - and it is found that the first two leading order constraints are the same. We then outline some preliminary results of an ongoing project whose goal is to understand the dual of relative entropy of holographic states defined on null cone regions on the conformal boundary. We derive the modular Hamiltonian for vacuum states defined on null cone regions in a conformal field theory using known results for modular Hamiltonians on null planes. We also derive the Ryu- Takayanagi surface associated with such null cone regions. Using these results, it is argued that, for null cones whose base is cut by a constant time cut, will not give new constraints beyond what is already known for ball shaped regions. === Science, Faculty of === Physics and Astronomy, Department of === Graduate
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