Wednesday, December 5, 2012

1212.0147 (László B Szabados)

On total masses in GR    [PDF]

László B Szabados
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. A non-negative expression, built from the norm of the 3-surface twistor operator and the energy-momentum tensor of the matter fields on a spacelike hypersurface, is found which, in the asymptotically flat/hyperboloidal case, provides a lower bound for the ADM/Bondi--Sachs mass. In closed universes the analogous expression coincides with the first eigenvalue of the Sen--Witten operator, and it is vanishing if and only if the closed data set is in a flat spacetime with spatial topology $S^1\times S^1\times S^1$. Moreover, its vanishing is equivalent to the existence of non-trivial solutions of Witten's gauge condition. Thus, it provides a positive definite measure of the strength of the gravitational field (with physical dimension mass) on closed hypersurfaces, i.e. some sort of the total mass of closed universes. Its monotonicity properties are discussed through the examples of closed Bianchi I and IX cosmological spacetimes. A spectral characterization of these cosmological spacetimes is also given.
View original: http://arxiv.org/abs/1212.0147

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