de Sitter space, extremal surfaces and "time-entanglement"

TL;DR

The paper investigates extremal surfaces anchored at the de Sitter future boundary $I^+$ and shows that, in fully Lorentzian $dS$, these surfaces lack $I^+\rightarrow I^+$ turning points and extend into the past, leading to future-past timelike surfaces or no-boundary variants with complex areas. It introduces the notion of time-entanglement, exploring two quantum-mechanical realizations: a future-past thermofield-double state that preserves positive entanglement structure despite timelike separation, and a framework of reduced transition amplitudes derived from time evolution that generally yields complex entropies. By connecting these surface constructions to time contours and AdS rotations, the work provides a cohesive picture where de Sitter entropy and its holographic interpretation relate to timelike and interior boundary data, with no-boundary conditions yielding partial real contributions. The analysis combines explicit dS calculations across coordinate patches (Lorentzian, static, global) with 2D CFT toy models to illuminate how time-entanglement may manifest in holographic de Sitter setups and beyond.

Abstract

We refine previous investigations on de Sitter space and extremal surfaces anchored at the future boundary $I^+$. Since such surfaces do not return, they require extra data or boundary conditions in the past (interior). In entirely Lorentzian de Sitter spacetime, this leads to future-past timelike surfaces stretching between $I^\pm$. Apart from an overall $-i$ factor (relative to spacelike surfaces in $AdS$) their areas are real and positive. With a no-boundary type boundary condition, the top half of these timelike surfaces joins with a spacelike part on the hemisphere giving a complex-valued area. Motivated by these, we describe two aspects of "time-entanglement" in simple toy models in quantum mechanics. One is based on a future-past thermofield double type state entangling timelike separated states, which leads to entirely positive structures. Another is based on the time evolution operator and reduced transition amplitudes, which leads to complex-valued entropy.

Figures (3)

• Figure 1: $dS$ future-past extremal surfaces stretching between $I^\pm$ on an $S^{d-1}$ equatorial plane. The red curve is for generic subregion while the blue curve is a limiting curve as the subregion becomes the whole space.
• Figure 2: Global $dS$ future-past extremal surfaces stretching between $I^\pm$ on any $S^d$ equatorial plane in the IR limit ($\theta={\pi\over 2}$). This is also the picture for the $w=const$ slice in the static coordinates.
• Figure 3: Global $dS$ no-boundary extremal surfaces, with a top timelike part joining smoothly with a spatial part going around the hemisphere in the bottom half. The blue curve is the IR limit ($\theta={\pi\over 2}$) on some $S^d$ equatorial plane. This is also the picture for the $w=const$ slice in the static coordinates.