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A coarse-grained entropy for expanding cosmologies Leung, Hon Yin (Pompey)


The AdS/CFT correspondence relates quantum gravity in asymptotically anti-de Sitter (AdS) spacetimes to conformal field theories (CFT) living on its boundary. It is the most explicitly realised example of the holographic principle to date; holographic descriptions for more general spacetimes remain poorly understood. In seeking a microscopic description of cosmology through the holographic principle, this thesis explores the notion of a coarse-grained entropy over quantum gravitational degrees of freedom for expanding universes. The Engelhardt-Wall construction for the outer entropy is a coarse-grained generalization of the Hubeny-Rangamani-Takayanagi (HRT) formula for the von Neumann entropy. It admits a timelike thermodynamic second law for boundary operators with the bulk interpretation of coarse-graining over progressively larger regions to the interior of marginally trapped surfaces which foliate spacelike holographic screens. As holographic screens are generic objects in general relativity, a reasonable question to ask is whether we can define an analogous coarse-grained entropy for general spacetimes as a guide for probing the class of admissible underlying microscopic degrees of freedom. We show that expanding Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes admit everywhere timelike holographic screens and are thus prime candidates for a generalisation of the Engelhardt-Wall construction. After motivating the formulation of the future entropy as a cosmological coarse-grained entropy defined through a maximin prescription, we show by construction that it can always be expressed as the area of marginally anti-trapped surfaces foliating timelike holographic screens in expanding universes. In this manner, the future entropy enjoys an interpretation of quantifying the ignorance of the past subject to data fixed in the future. We then explicitly carry out our construction for spatially flat, expanding FLRW spacetimes and write an expression for the future entropy which increases with proper time as expected. Although the future entropy as we have defined is a purely geometric quantity, we entertain the possibility for a true statistical interpretation as a coarse-grained entropy which obeys a thermodynamic second law. Our hope is that the correct class of holographic theories for cosmological spacetimes would be able to reproduce our result given an appropriate coarse-graining procedure over microscopic degrees of freedom.

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