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The eigenvalue problem for the Sen--Witten operator on closed spacelikehypersurfaces is investigated. The (square of its) eigenvalues are shown to begiven exactly by the 3-surface integral appearing in the expression of thetotal energy-momentum of the matter+gravity systems in Witten's energypositivity proof. A sharp lower bound for the eigenvalues, given in terms ofthe constraint parts of the spacetime Einstein tensor, i.e. the energy andmomentum densities of the matter fields, is given.

A lower bound for the eigenvalues of the Sen-Witten operator on closed spacelike hypersurfaces