James Vickers (Southampton): 

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Singular solutions of Einstein's equations 

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ABSTRACT: 

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Abstract: According to the theory of General Relativity, spacetime is
described by a manifold together with a Lorentz metric which is assumed
to be sufficiently differentiable that Einstein's equations are defined
(for example $C^2$). One then detects the presence of singularities by
showing that the (maximal extension) of the spacetime is incomplete in
some sense. In the standard approach to singularities (see e.g. Hawking
and Ellis) a singularity is regarded as an obstruction to extending a
geodesic. However this does not correspond very closely to ones physical
intuition of a singularity. To remedy this one needs to enlarge the
class of solutions that one allows to include all the physically
reasonable solutions. Unfortunately due to the non-linear nature of
Einstein's equations this is not straightforward. In this talk I will
show how one can use generalised functions in the sense of Colombeau to
widen the class appropriately and furthermore show how one may introduce
a new way of looking at singularities in which they are regarded as
obstructions to the propagation of test fields on the background
spacetime rather than obstructions to extending geodesics.