Lorentzian Geometry, U ₄ Theories and Singularities in Cosmology

Abstract

At first we introduce the space-time manifold and we compare some aspects of Riemannian and Lorentzian geometry such as the distance function and the relations between topology and curvature. We then define spinor structures in general relativity, and the conditions for their existence are discussed. The causality conditions are studied through an analysis of strong causality, stable causality and global hyperbolicity. In looking at the asymptotic structure of space-time, we focus on conformal infinity, the asymptotic symmetry group of Bondi, Metzner and Sachs, and the b-boundary construction of Schmidt, with a mention of recent work which tries to improve Schmidt’s definition. The Hamiltonian structure of space-time is also analyzed, with emphasis on Ashtekar’s spinorial variables.

Finally, the problem of a rigorous theory of singularities in space-times with torsion is studied, describing in detail recent work by the author. We define geodesics as curves whose tangent vector moves by parallel transport. This is different from what other authors do, because their definition of geodesics only involves Christoffel symbols, though studying theories with torsion. We then prove how to extend Hawking’s singularity theorem without causality assumptions to the space-time of the ECSK theory. This is achieved by studying the generalized Raychauduri equation in the ECSK theory, the conditions for the existence of conjugate points and properties of maximal timelike geodesics. Our result can also be interpreted as a no-singularity theorem if the torsion tensor does not obey some additional conditions. Namely, it seems that the occurrence of singularities in closed cosmological models based on the ECSK theory can be less generic than in general relativity. Our work should be compared with important previous papers. There are some relevant differences, because we rely on a different definition of geodesics, we keep the field equations of the ECSK theory in their original form rather than casting them in a form similar to general relativity with a modified energy-momentum tensor, and we emphasize the role played by the full extrinsic-curvature tensor and by the variation formulae.