A positive Bondi-type mass in asymptotically de Sitter spacetimes

The general structure of the conformal boundary ${{\mathcal{I}}}^{+}$ of asymptotically de Sitter spacetimes is investigated. First we show that Penrose's quasi-local mass, associated with a cut ${\mathcal{S}}$ 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 two-surface twistor equation on ${\mathcal{S}},$ and it yields a positive functional on the two-surface twistor space on ${\mathcal{S}},$ 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 ${{\mathcal{I}}}^{+}$ in the cut ${\mathcal{S}}$ is locally isometric to the de Sitter spacetime. For non-contorted cuts this functional yields an invariant analogous to the Bondi mass