On the total mass of closed universes

The total mass, the Witten type gauge conditions and the spectral properties of the Sen–Witten and the 3-surface twistor operators in closed universes are investigated. It has been proven that a recently suggested expression M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathtt{M}$$\end{document} for the total mass density of closed universes is vanishing if and only if the spacetime is flat with toroidal spatial topology; it coincides with the first eigenvalue of the Sen–Witten operator; and it is vanishing if and only if Witten’s gauge condition admits a non-trivial solution. Here we generalize slightly the result above on the zero-mass configurations: M=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathtt{M}=0$$\end{document} if and only if the spacetime is holonomically trivial with toroidal spatial topology. Also, we show that the multiplicity of the eigenvalues of the (square of the) Sen–Witten operator is even, and a potentially viable gauge condition is suggested. The monotonicity properties of M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathtt{M}$$\end{document} through the examples of closed Bianchi I and IX cosmological spacetimes are also discussed. A potential spectral characterization of these cosmological spacetimes, in terms of the spectrum of the Riemannian Dirac operator and the Sen–Witten and the 3-surface twistor operators, is also indicated.