Optical functions in de Sitter

TL;DR

This paper develops a gauge framework for de Sitter spacetimes, constructing solutions to the eikonal equation and analyzing null foliations, with implications for stability analyses of Schwarzschild de Sitter spacetime.

Contribution

It introduces a method to construct global solutions to the eikonal equation in de Sitter space and analyzes the structure of null foliations relevant for spacetime stability.

Findings

Small perturbations cause horizon spheres to pinch off at infinity.

Well-behaved null foliations can be constructed from infinity.

Final gauge choices are necessary for stability analysis.

Abstract

This paper addresses pure gauge questions in the study of (asymptotically) de Sitter spacetimes. We construct global solutions to the eikonal equation on de Sitter, whose level sets give rise to double null foliations, and give detailed estimates for the structure coefficients in this gauge. We show two results which are relevant for the foliations used in the stability problem of the expanding region of Schwarzschild de Sitter spacetimes [Schlue (2016)]: (i) Small perturbations of round spheres on the cosmological horizons lead to spheres that pinch off at infinity. (ii) Globally well behaved double null foliations can be constructed from infinity using a choice of spheres related to the level sets of a mass aspect function. While (i) shows that in the above stability problem a final gauge choice is necessary, the proof of (ii) already outlines a strategy for the case of spacetimes with decaying, instead of vanishing, conformal Weyl curvature.