A positive Bondi--type mass in asymptotically de Sitter spacetimes

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.