De Sitter Light-Ray Operators

TL;DR

The paper addresses the challenge of defining meaningful detector-like observables in de Sitter space by constructing light-ray operators from null integrals of the stress tensor $T_{\mu\nu}$. It shows that the de Sitter analogs of Minkowski concepts—the energy flux, angular contribution to the Hamiltonian, ANEC, and the light transform—are generically distinct and observer-dependent in dS$_4$, unlike their Minkowski counterparts. Using the null embedding space formalism, the authors classify four such dS light-ray operators and compute their matrix elements in a free, conformally coupled scalar, demonstrating the expected symmetry and positivity properties. The work lays a foundation for extensions to generic masses and interactions, explores potential positivity bounds and cosmological applications, and points toward a broader, observer-centered paradigm for quantum fields in curved spacetimes and quantum gravity.

Abstract

In this work, we initiate the study of light-ray operators in four-dimensional de Sitter space focusing on null integrals of the stress tensor. In Minkowski space, the null integral of the stress tensor unifies several ostensibly different constructions, functioning simultaneously as the energy flux operator, the angular contribution to a conserved charge, the averaged null energy operator, and the light transform of the stress tensor. However, we show that the de Sitter analogs of these various interpretations do not necessarily coincide but rather lead to distinct, observer-dependent light-ray operators. We construct four such de Sitter analogs and analyze their matrix elements in a free, conformally coupled scalar theory, showing that they exhibit the expected symmetry and positivity properties.

Figures (14)

• Figure 1: The usual Penrose diagram for global dS$_4$ space. Each horizontal slice represents a 3-sphere $S^3$. At $\zeta = 0$, the spatial slice is a 3-sphere $S^3$ of unit radius (in units of $H^{-1}$), which grows hyperbolically toward $\mathscr{I}^\pm$, where it becomes infinitely large. Each point on such a slice represents a two-sphere $S^2$ that expands toward the center of the diagram and shrinks to a point at the left and right edges. The observer is conventionally placed on the left, at the South Pole.
• Figure 2: It will be more useful for our purposes to adopt the observer’s point of view. We therefore place the observer at the center and work with the doubled Penrose diagram, where the left and right edges are identified.
• Figure 3: The Penrose diagram of dS space covered by the expanding (EPP) and contracting (CPP) Poincaré patches, with the observers worldline at the center of the CPP. The boundary between the patches is the $\mathscr{H}^+$ of the observer at the center of the CPP and the $\mathscr{H}^-$ of the observer at the center of the EPP.
• Figure 4: Static patch (gold) seen as the overlap between two Poincaré patches (red and blue). This is the causal patch of our observer at the center sitting at $r=0$.
• Figure 5: The embedding of dS$_4$ as a hyperboloid in $\mathbb{R}^{1,4}$. The observer sits at the South Pole (denoted by the red line on the left). Each horizontal cross-section represents a spatial $S^3$ slice of the global geometry. Two overlapping Poincaré patches define a static patch. The static patch always has the same size (of unit radius) but its portion of the overall global dS slice varies with $\zeta$. At $\zeta = 0$ it coincides with the entire left half of the $S^3$. As the global $S^3$ slices grow away from $\zeta =0$, the static patch remains a fixed-radius disk that occupies a decreasing fraction of the full sphere.