A New Type of Singularity Theorem

A new type of singularity theorem, based on spatial averages of physical quantities, is presented and discussed. Alternatively, the results inform us of when a spacetime can be singularity-free. This theorem provides a decisive observational difference between singular and non-singular, globally hyperbolic, open cosmological models.Comment: 6 pages, no figures. Contribution to appear in the Proceedings of the Spanish Relativity Meeting ERE-07, "Relativistic Astrophysics and Cosmology

Equations for general shells

The complete set of (field) equations for shells of arbitrary, even changing, causal character are derived in arbitrary dimension. New equations that seem to have never been considered in the literature emerge, even in the traditional cases of everywhere non-null, or everywhere null, shells. In the latter case there arise field equations for some degrees of freedom encoded exclusively in the distributional part of the Weyl tensor. For non-null shells the standard Israel equations are recovered but not only, the additional relations containing also relevant information. The results are applicable to a widespread literature on domain walls, branes and braneworlds, gravitational layers, impulsive gravitational waves, and the like. Moreover, they are of a geometric nature, and thus they can be used in any theory based on a Lorentzian manifold.Comment: 32 pages, no figures. New paragraph and new footnote, plus some added references. Version to be publishe

The universal 'energy' operator

The "positive square" of any tensor is presented in a universal and unified manner, valid in Lorentzian manifolds of arbitrary dimension, and independently of any (anti)-symmetry properties of the tensor. For rank-m tensors, the positive square has rank 2m. Positive here means future, that is to say, satisfying the dominant property. The standard energy-momentum and super-energy tensors are recovered as appropriate parts of the general square. A richer structure of principal null directions arises.Comment: 13 pages, no figures; minor improvements and corrections. This is a larger, expanded version, containing proofs and explanations not available in the short Note to be published in CQ