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A Positive Energy Theorem for Asymptotically deSitter Spacetimes

David Kastor, Jennie Traschen

TL;DR

The paper develops a positive energy framework for asymptotically de Sitter spacetimes by constructing conserved charges tied to asymptotic conformal Killing vectors, using a spinor-based (Nester/Witten) approach. It proves a positive conformal energy $Q_\psi$ associated with the CKV that generates conformal time translations, with time dependence arising from flux at infinity and a Newtonian-like limit $Q_\psi \approx a(t)\int \delta\rho$ for small perturbations. The authors relate this spinor charge to the Abbott-Deser charges for CKVs, show their equivalence in the de Sitter background, and discuss dynamical interpretations via Hamiltonian evolution, including the Schwarzschild–deSitter case where $Q_\psi=a(t)M$. These results provide a BPS-like positivity structure for de Sitter spacetimes, clarify mass notions in cosmological settings, and may inform connections to dS/CFT and perturbative mass definitions in expanding universes.

Abstract

We construct a set of conserved charges for asymptotically deSitter spacetimes that correspond to asymptotic conformal isometries. The charges are given by boundary integrals at spatial infinity in the flat cosmological slicing of deSitter. Using a spinor construction, we show that the charge associated with conformal time translations is necessarilly positive and hence may provide a useful definition of energy for these spacetimes. A similar spinor construction shows that the charge associated with the time translation Killing vector of deSitter in static coordinates has both positive and negative definite contributions. For Schwarzshild-deSitter the conformal energy we define is given by the mass parameter times the cosmological scale factor. The time dependence of the charge is a consequence of a non-zero flux of the corresponding conserved current at spatial infinity. For small perturbations of deSitter, the charge is given by the total comoving mass density.

A Positive Energy Theorem for Asymptotically deSitter Spacetimes

TL;DR

The paper develops a positive energy framework for asymptotically de Sitter spacetimes by constructing conserved charges tied to asymptotic conformal Killing vectors, using a spinor-based (Nester/Witten) approach. It proves a positive conformal energy associated with the CKV that generates conformal time translations, with time dependence arising from flux at infinity and a Newtonian-like limit for small perturbations. The authors relate this spinor charge to the Abbott-Deser charges for CKVs, show their equivalence in the de Sitter background, and discuss dynamical interpretations via Hamiltonian evolution, including the Schwarzschild–deSitter case where . These results provide a BPS-like positivity structure for de Sitter spacetimes, clarify mass notions in cosmological settings, and may inform connections to dS/CFT and perturbative mass definitions in expanding universes.

Abstract

We construct a set of conserved charges for asymptotically deSitter spacetimes that correspond to asymptotic conformal isometries. The charges are given by boundary integrals at spatial infinity in the flat cosmological slicing of deSitter. Using a spinor construction, we show that the charge associated with conformal time translations is necessarilly positive and hence may provide a useful definition of energy for these spacetimes. A similar spinor construction shows that the charge associated with the time translation Killing vector of deSitter in static coordinates has both positive and negative definite contributions. For Schwarzshild-deSitter the conformal energy we define is given by the mass parameter times the cosmological scale factor. The time dependence of the charge is a consequence of a non-zero flux of the corresponding conserved current at spatial infinity. For small perturbations of deSitter, the charge is given by the total comoving mass density.

Paper Structure

This paper contains 11 sections, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some DeSitter Basics
  3. A Positive Energy Theorem for DeSitter
  4. Derivation of the Spinor Identity
  5. Evaluation of $Q_\psi$ in the Asymptotic Limit
  6. CKV's and Conserved Charges
  7. General Construction
  8. Equivalence of $Q_\psi$ and $Q_\xi$ for CKV's
  9. Convergence of $Q_\psi$ and $Q_\xi$ for CKV's
  10. Dynamical Interpretation of $Q_\xi$ for CKV's
  11. Spinor Identity for deSitter KV's and anti-deSitter CKV's

Figures (1)

  • Figure 1: The conformal diagram for deSitter spacetime with the region covered by the flat cosmological coordinates of equations (\ref{['flatslicing']}) and (\ref{['transform']}) shaded and spatial slices at two different times are indicated.