Abstract
We examine the holographic complexity conjectures in the context of holographic theories of FRW spacetimes. Analyzing first the complexity-action conjecture for a flat FRW universe with one component, we find that the complexity grows as $t^2$, regardless of the value of $w$. In addition, we examine the holographic complexity for a flat universe sourced by a scalar field that is undergoing a transition. We find that the complexity decreases when the holographic entanglement entropy decreases for this universe. Moreover, the calculations show that, while the entanglement entropy decreases only slightly, the magnitudes of the corresponding fractional decreases in complexity are much larger. This presumably reflects the fact that entanglement is computationally expensive. Interestingly, we find that the gravitational action behaves like a complexity, while the total action is negative, and is thus ill-suited as a measure of complexity, in contrast to the conjectures in AdS settings. Finally, the implications of the complexity-volume conjecture are examined. The results are qualitatively similar to the complexity-action conjecture.
Highlights
Recent work has revealed deep connections between gravity, spacetime, and information
The AdS/CFT correspondence posits that any theory of quantum gravity in d þ 1-dimensional anti-de Sitter space (AdS) is equivalent to a conformal field theory (CFT) in d dimensions [1,2,3]. This correspondence is a concrete realization of the holographic principle, which conjectures that the degrees of freedom in a theory of quantum gravity are encoded in one fewer dimension
The celebrated Ryu-Takayanagi formula [4] of the AdS/CFT correspondence posits an equivalence between entanglement entropy in a holographic CFT and minimal surfaces in the bulk
Summary
Recent work has revealed deep connections between gravity, spacetime, and information. The celebrated Ryu-Takayanagi formula [4] (and its covariant generalization, the Hubeny-Rangamani-Takayanagi formula [5]) of the AdS/CFT correspondence posits an equivalence between entanglement entropy in a holographic CFT and minimal surfaces in the bulk This has led to an improved understanding of the rich interplay between the spacetime in the bulk theory, and information in the boundary theory. The qualitative behavior does not depend on these details: the complexity, in general, grows linearly with time for a large amount of time after the entanglement entropy saturates This line of reasoning led to the complexity-volume conjecture, which says that the CFT quantity dual to the maximum-volume slice is the complexity of the CFT state. The complexityaction conjecture posits that the complexity of the state CFT state jψðtL; tRÞi is given by [6]
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