The Cauchy Data in Spacetimes With Singularities
Résumé
This article shows a method to formulate well-posed Cauchy problems for spacetimes with singularities. The formulation is identical to the prescriptions of General Relativity at the points where the metric is non-singular. The standard formulation of General Relativity doesn't prescribe the topology at the points where the metric is singular, regarding them as not belonging to the spacetime. This allows us to extend the non-singular part of the spacetime to a manifold foliated as a product between a three-dimensional manifold corresponding to the space, and a time coordinate. On the spacelike hypersurfaces from such a foliation, Einstein's field equation can be replaced by an equation with which it is equivalent at non-singular points, but which extends smoothly, by continuity, to the singular points. We provide examples for the neutral and charged, rotating and non-rotating, primordial and non-primordial, evaporating and eternal black holes. In these examples we show that the conformal structure thus find is globally hyperbolic, hence the Cauchy data is well-defined and preserved during the time evolution, and the singularities are not harmful to the time evolution.
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