On energy and its positivity in spacetimes with an expanding flat de Sitter background

TL;DR

This paper develops a new notion of energy in expanding de Sitter spacetimes with an umbilic second fundamental form, extending positive energy theorems to cosmological settings with horizons.

Contribution

It introduces a quasi-local energy definition based on Liu-Yau energy for initial data sets in de Sitter backgrounds with positive cosmological constant.

Findings

Established positivity of the new energy for certain cosmological constants.

Extended positive energy theorems to expanding universe models.

Provided a framework for energy in non-isolated, cosmological spacetimes.

Abstract

The positive energy theorems are a fundamental pillar in mathematical general relativity. Originally proved by Schoen-Yau and later Witten, these theorems were established for asymptotically flat manifolds where the metric tends to the standard Euclidean metric and whose second fundamental form decays to zero at infinity. This ansatz on the metric and second fundamental form is motivated by the desire to model an isolated gravitational system with a Minkowski space background for the spacetime. However, actual astrophysical massive objects are not truly isolated but rather exist within an expanding cosmological universe, where the second fundamental form is umbilic. With this in mind, we seek a notion of energy for initial data sets with an umbilic second fundamental form. In this work, we present a definition of energy in such an expanding cosmological setting. Instead of Minkowski space, we take de Sitter space as the background spacetime, which, when written in flat-expanding coordinates, is foliated by umbilic hypersurfaces each isometric to Euclidean 3-space. This cosmological setting necessitates a quasi-local energy definition, as the presence of a cosmological horizon in de Sitter space obstructs a global one. We define energy in this quasi-local setting by adapting the Liu-Yau energy to our framework and establish positivity of this energy for certain bounded values of the cosmological constant.