de Sitter space, extremal surfaces, and time entanglement

Abstract

We develop further the investigations in arXiv:2210.12963 [hep-th] on de Sitter space, extremal surfaces and time entanglement. We discuss the no-boundary de Sitter extremal surface areas as certain analytic continuations from AdS while also amounting to space-time rotations. The structure of the extremal surfaces suggests a geometric picture of the time-entanglement or pseudo-entanglement wedge. The analytic continuation suggests a heuristic Lewkowycz-Maldacena formulation of the extremal surface areas. In the bulk, this is now a replica formulation on the Wavefunction which suggests interpretation as pseudo-entropy. Finally we also discuss aspects of future-past entangled states and time evolution.

Small Schwarzschild de Sitter black holes, quantum extremal surfaces and islands

“…E. The timelike separation implies that the on-shell generalized entropy becomes complex valued. While complex entropies are known in investigations in pure de Sitter space (which does not have a sufficiently wide Penrose diagram) and suggest new objects [118], [119], [120], [121], [122], it is consistent to ignore them in the Schwarzschild de Sitter context where spacelike separated quantum extremal surfaces do exist in accord with physical Page curve expectations for the black hole information paradox.…”

“…Our analysis has some parallels with the island studies in [87] for dS 2 arising under reductions from Nariai limits of higher dim Schwarzschild de Sitter. One might expect timelike separated quantum extremal surfaces for the future boundary resulting in complex-valued entropies as are known in pure de Sitter (see [68] for dS 2 , and [84] for reductions of higher dimensional Poincare dS; see also [118], [119], [120], [121], [122] for classical RT/HRT surfaces anchored at the future boundary). However Schwarzschild de Sitter has a "sufficiently wide" Penrose diagram so spacelike separated islands do exist here in accordance with physical expectations for the black hole Page curve (thus we discard timelike separated ones here).…”

Small Schwarzschild de Sitter black holes, the future boundary and islands

“…Finally, it is worth noting that there are also timelike separated quantum extremal surface solutions following from the extremization of the generalized entropy with respect to the future boundary observer: we discuss these solutions in appendix E. The timelike separation implies that the on-shell generalized entropy becomes complex valued. While complex entropies are known in investigations in pure de Sitter space (which does not have a sufficiently wide Penrose diagram) and suggest new objects [123][124][125][126][127], it is consistent to ignore them in the Schwarzschild de Sitter context where spacelike separated quantum extremal surfaces do exist in accord with physical Page curve expectations for the black hole information paradox.…”

“…Our analysis has some parallels with the island studies in [89] for dS 2 arising under reductions from Nariai limits of higher dim Schwarzschild de Sitter. One might expect timelike separated quantum extremal surfaces for the future boundary resulting in complex-valued entropies as are known in pure de Sitter (see [68] for dS 2 , and [86] for reductions of higher dimensional Poincare dS; see also [123][124][125][126][127] for classical RT/HRT surfaces anchored at the future boundary). However Schwarzschild de Sitter has a "sufficiently wide" Penrose diagram so spacelike separated islands do exist here in accordance with physical expectations for the black hole Page curve (thus we discard timelike separated ones here).…”

Modave lectures on de Sitter space & holography

“…Modave Lecture Notes on de Sitter Space & Holography Damián A. Galante holography, one should anchor extremal surfaces at the (stretched) horizon. See other recent proposals in [118][119][120][121][122]. Quantum extremal surfaces in the context of dS have also been actively studied recently [123][124][125][126][127][128][129], but it is fair to say that compared to the black hole case (see [130] for current state of affairs), the case of the cosmological horizon is still pretty much under development.…”