On the total mass of closed universes
arXiv:1212.0147 · doi:10.1007/s10714-013-1587-9
Abstract
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 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: 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 at least two, and a potentially viable gauge condition is suggested. The monotonicity properties of 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.
14 pages, plenary talk at the `Spanish Relativity Meeting in Portugal 2012', Guimarães 3-7 September; Final version, appearing in General Relativity and Gravitation