Local Characterization of Singularities in General Relativity

We formulate a new approach to singularities: their local description. Given any incomplete space-time M, we define a topological space, the ``g boundary,'' whose points consist of equivalence classes of incomplete geodesics of M. The points of the g boundary may be thought of as the ``singular points'' of M. Local properties of the singularity may now be described in a well-defined way in terms of local properties of the g boundary. For example, the notions: ``dimensionality of a singularity,'' ``past and future of a singular point,'' ``neighborhood of a singular point,'' ``spacelike or timelike character of a singularity,'' and ``metric structure of a singularity'' may all be expressed as properties of the g boundary. Two applications of the g boundary outside of the realm of singularities are discussed: (1) In the case in which the space-time M is extendable (for example, Taub space), the g boundary is shown to be that regular 3-surface across which M may be extended in this case, the Misner boundary between Taub and Newman-Unti-Tamburino (NUT) space. (2) With a slight modification of the definitions, the g boundary of an asymptotically simple space-time is shown to be Penrose's surface at ``conformal infinity.'' The application of the g boundary technique to singularities is illustrated with a number of examples. The g-boundary structure of one particular example leads to our consideration of non-Hausdorff space-times.