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

Rodrigo Avalos, Eric Ling, Annachiara Piubello

TL;DR

The paper defines a de Sitter-adapted quasi-local energy $E_\\lambda$ for initial data sets with a cosmological constant, extending the Liu--Yau construction from flat Minkowski to the flat-expanding patch of de Sitter space. It proves positivity of $E_\\lambda$ for a broad class of data when the embedding lies inside the cosmological horizon, with explicit bounds on the expanding parameter $\\lambda$ expressed via $\\alpha_\\lambda$, and provides rigidity in the equality case. The results are illustrated on Schwarzschild--de Sitter and other $\\Lambda$-vacuum examples, showing $E_\\lambda>0$ under natural geometric and energy conditions. The work highlights open questions about rigidity, the $\\lambda$-dependence of the energy, and potential variational formulations in this cosmological setting, aiming to deepen the understanding of isolated systems in an expanding universe.

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.

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

TL;DR

The paper defines a de Sitter-adapted quasi-local energy for initial data sets with a cosmological constant, extending the Liu--Yau construction from flat Minkowski to the flat-expanding patch of de Sitter space. It proves positivity of for a broad class of data when the embedding lies inside the cosmological horizon, with explicit bounds on the expanding parameter expressed via , and provides rigidity in the equality case. The results are illustrated on Schwarzschild--de Sitter and other -vacuum examples, showing under natural geometric and energy conditions. The work highlights open questions about rigidity, the -dependence of the energy, and potential variational formulations in this cosmological setting, aiming to deepen the understanding of isolated systems in an expanding universe.

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.

Paper Structure

This paper contains 11 sections, 6 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main conjecture and theorem
  3. Positivity Results for Quasi-Local Energies with a Minkowski background
  4. Supporting lemmas and proof of main results
  5. Isometric embedding inside the cosmological horizon
  6. Positivity of energy
  7. Proof of main results.
  8. Examples
  9. Schwarzschild--de Sitter
  10. Other $\Lambda$-vacuum examples
  11. Discussion and conclusion

Key Result

Theorem 2.3

Let $(\Omega, g, k, \lambda)$ for $\lambda\geq 0$ be a compact initial data set with some boundary component $\Sigma= \partial \Omega$ of positive Gauss curvature $K > 0$ and $H > |{\rm tr}_\Sigma k|$. Assume that the dominant energy condition with respect to $\Lambda = 3\lambda^2$ holds. If $\lambd Moreover, if $0 < \lambda < \alpha_\lambda$, then $E_\lambda >0$; if $\lambda = \alpha_\lambda >0$

Figures (2)

  • Figure 1: One-half of de Sitter space is covered by the spatially flat FLRW model with an exponentially growing scale factor. In flat-expanding coordinates, each $t$-slice is isometric to the Euclidean space $\mathbb{R}^3$ and $\partial_t$ is Killing but timelike only for values within the cosmological horizon, i.e. for $r < 1/\lambda$. Compare with HE. For an in-depth analysis of how the flat-expanding coordinates relate to the global coordinates of de Sitter space, see Geshnizjani2023OnTI.
  • Figure 2: Left: the Penrose diagram of Minkowski spacetime. To compute the Liu--Yau energy, the surface $\Sigma$ is isometrically embedded in a constant $t$-slice of Minkowski spacetime. Right: the Penrose diagram of de Sitter spacetime. The surface is also isometrically embedded in a constant $t$-slice of the flat de Sitter spacetime and lies within the cosmological horizon (shown in red).

Theorems & Definitions (18)

  • Conjecture 2.1
  • Conjecture 2.2
  • Theorem 2.3
  • Remark
  • Remark
  • Corollary 2.4
  • Remark
  • Definition 3.1: Brown--York Energy
  • Theorem 3.1: ShiTam2002
  • Definition 3.2: Liu--Yau Energy
  • ...and 8 more