Mathematical Structures of Space-Time

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 the asymptotic symmetry group of Bondi, Metzner and Sachs, and the b-boundary construction of Schmidt. The Hamiltonian structure of space-time is also analyzed, with emphasis on Ashtekar's spinorial variables. Finally, the question of a rigorous theory of singularities in space-times with torsion is addressed, 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 the 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 studying the generalized Raychaudhuri 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. Thus, it seems that the occurrence of singularities in closed cosmological models based on the ECSK theory is 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

63 pages, plain-tex, published in Fortschritte der Physik, volume 40, pages 1-30, year 1992

References in corpus (2)

• The Singularity Problem for Space-Times with Torsion
• Hamiltonian Structure of a Friedmann-Robertson-Walker Universe with Torsion

Cited by in corpus (10)

• f(T) teleparallel gravity and cosmology
• The Raychaudhuri equations: a brief review
• On the structure and applications of the Bondi-Metzner-Sachs group
• Raychaudhuri equation in spacetimes with torsion
• New Action Principle for Classical Particle Trajectories in Spaces with Torsion
• From Spinor Geometry to Complex General Relativity
• Singularities and n-dimensional black holes in torsion theories
• Toward Singularity Theorems with Torsion
• Complex Geometry of Nature and General Relativity
• Singularity Theory in Classical Cosmology