Local Characterization of Singularities in General Relativity

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Abstract
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.

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