A positive Bondi--type mass in asymptotically de Sitter spacetimes
László B Szabados, Paul Tod
TL;DR
The paper develops a rigorous framework for conserved quantities in asymptotically de Sitter spacetimes by contrasting Penrose-like curvature charges with a spinorial Nester–Witten construction. It demonstrates that Penrose’s 2-surface mass can vanish even with radiation, while a renormalized Nester–Witten energy is positive and rigid whenever the Witten equation admits a 2-surface twistor boundary data, tying finiteness to the twistor structure on non-contorted cuts. The authors formulate Bondi-type coordinates near 𝓘⁺, prove a positivity/rigidity theorem for the Nester–Witten energy, and show that zero energy implies a locally de Sitter domain of dependence. They further analyze the 2-surface twistor space ker 𝒯, its invariants, and under what extra structures a mass analogue can be canonically defined. The appendices develop the functional-analytic backbone (weighted Sobolev spaces and the isomorphism for the renormalized Sen–Witten operator) necessary to establish existence and uniqueness of solutions to the Witten equations in this setting.
Abstract
The general structure of the conformal boundary $\mathscr{I}^+$ of asymptotically de Sitter spacetimes is investigated. First we show that Penrose's quasi-local mass, associated with a cut ${\cal S}$ of the conformal boundary, can be zero even in the presence of outgoing gravitational radiation. On the other hand, following a Witten--type spinorial proof, we show that an analogous expression based on the Nester--Witten form is finite only if the Witten spinor field solves the 2-surface twistor equation on ${\cal S}$, and it yields a positive functional on the 2-surface twistor space on ${\cal S}$, provided the matter fields satisfy the dominant energy condition. Moreover, this functional is vanishing if and only if the domain of dependence of the spacelike hypersurface which intersects $\mathscr{I}^+$ in the cut ${\cal S}$ is locally isometric to the de Sitter spacetime. For non-contorted cuts this functional yields an invariant analogous to the Bondi mass.