According to the Cosmic Censorship hypothesis realistic singularities should be hidden by an event horizon. However there are many examples of physically realistic space–times which are geodesically incomplete, and hence possess singularities according to the usual definition, which are not inside an event horizon. Many of these counterexamples to the cosmic censorship conjecture have a curvature tensor which is reasonably behaved (for example bounded or integrable) as one approaches the singularity. We give a class of weak singularities which may be described as having distributional curvature1. Because of the non–linear nature of Einstein's equations such distributional geometries are described using a diffeomorphism invariant theory of non–linear generalised functions2. We also investigate the propagation of test fields on space–times with weak singularities. We give a class of singularities3,4 which do not disrupt the Cauchy development of test fields and result in space–times which satisfy Clarke's criterion of 'generalised hyperbolicity'. We consider that points which are well behaved in this way, and where Einstein's equations make sense distributionally, should be regarded as interior points of the space–time rather than counterexamples to cosmic censorship.
On the Existence of a Maximal Cauchy Development for the Einstein Equations: a Dezornification
