The Classification of Space-time Singularities Honours Thesis -1989 (original) (raw)

The General Theory of relativity has been used to model the global geometry of the universe. One of the most important predictions concerning the more realistic models are regions where matter density and gravitational field strength become infinite, that is singular regions where the metric and indeed space-time itself become undefined. Important theorems by Penrose, Hawking and Geroch indicate that singularities are quite a general feature of solutions of Einstein's field equations, and not just a consequence of symmetry assumptions. The fact that the metric or space-time becomes undefined at a singularity causes problems in making precise a definition of singularity. Since by definition, the space-time manifold consists only of points where the manifiold and metric structure are well behaved, the singularity cannot be considered to be a regular space-time point, and consequently these points have to be ommited from the space-time manifold. This being the case, one has to deduce the existence of the singularity from the behaviour of the points left in the manifold. The existence of curvature scalars that diverge is sufficient to show that a singularity exists, which is clearly a physical singularity. The models in which this occurs exhibit a phenomonon known as causal geodesic incompleteness, -that is, causal geodesics terminate in a finite affine length at the singularity. In fact, it is causal geodesic incompleteness that is used as a criterion for establishing the singularity theorems of Penrose, Hawking and Geroch. However, more general models do exist that exhibit geodesic incompleteness but do not have an associated curvature singularity. In Section One of this paper, we will establish some basic formalism concerning causality and examine conditions affecting the propagation of causal curves in a given space-time. Further, we will formally establish geodesic incompleteness as a defining tool for singularity and examine its consequences and its downfalls. In Section Two, we will examine a definition that seems to overcome the problems associated with the geodesic incompleteness definition of space-time singularity, and which seems to be the most widely accepted definition of singularity, as such we will spend some time on this section. In Section Three, we will look at a definition that has been proposed that utilizes only the causal structure of space-time and not its differentiable properties. This definition of singularity seems to have a fundamental problem and as such we will only only examine it briefly. Section Four is a mathematical supplement that defines some of the more unusual mathematical formalism used in this paper.