General relativity|the theory of space, time and gravitation as formulated by A. Einstein 85 years ago|
dramatically di ers from other eld theories. Spacetime is no longer given beforehand but rather it is described
by a 4-dimensional manifold together with a Lorentzian metric, which itself is subject to eld equations.
More precisely, the curvature of the metric is related to the energy momentum content of the spacetime
via the so-called Einstein equations, which|written in coordinates|form a complicated hyperbolic system
of 10 nonlinear partial di erential equations of second order for the coecients of the metric tensor; hence
one has to assume the metric to satisfy sucient di erentiability conditions. Usually one supposes the metric
to be smooth; C2 suces to do classical di erential geometry, whereas C3 is needed to ensure energy conservation.
C2�� (i.e., the rst derivatives Lipschitz continuous) guarantees at least unique solvability of the
geodesic equations and local boundedness of the Riemann curvature tensor.
A singularity in a general relativistic spacetime intuitively is a \place" where curvature blows up or some
other \pathological behavior" of the metric occurs. There are two main obstacles, however, to make this
notion precise. The rst originates from the fact that we only may speak of an event at all (i.e., of a point in
spacetime) if|according to the above|the metric is, say, C2 \there." Hence a singularity has to be viewed
as a \singular boundary point" rather than a point in spacetime. Since a Lorentzian metric|contrary to a
Riemannian one|does not give rise to a distance function, the construction of a topological boundary is a
non-trivial matter. In fact, no fully satisfactory general notion of a singular boundary of spacetime exists.
The second diculty in de ning a singularity in general relativity is deeply connected with another of the
theory's main principles: di eomorphism invariance. The fact that the components of, say, the Riemann
tensor blow up along a curve may simply be due to a bad choice of coordinates in the following sense. It
might be possible to nd a di erent coordinate system which allows to extend the spacetime beyond the
\critical point" with the Riemann tensor perfectly well-behaved. Moreover, there is a big variety of examples
showing that the blowup of some curvature quantity is not an adequate tool to detect singularities.
From the above it should become clear why it is tempting to characterize singularities in general relativity
by their geometrical properties rather than by their analytical ones. In fact, by the standard de nition a
spacetime is called singular if there exist incomplete geodesics, i.e., geodesics of nite ane parameter length
which may not be extended. Obviously designed to capture the intuitive notion of a \hole in spacetime,"
there are, however, also some problems associated with this \geometric" approach. First, it does not provide
an ultimate answer to the question what a singularity actually is; note that we have only de ned the notion
of a singular spacetime. Instead there is a quite lengthy catalogue of possible ways in which a spacetime could
\break down" (non-smoothness, unboundedness or local non-integrability of the Riemann tensor, spacetimes
\created" with a primordial singularity and the like). Moreover, by the singularity theorems of Penrose and
Hawking (see e.g. [80], chap. 8) many physically reasonable spacetimes (in particular, all realistic models of
an expanding universe and of gravitational collapse) are singular with respect to this de nition.
Consequently, the recent development of the study of spacetime singularities has focussed more upon a study
of the eld equations. General relativity as a physical theory is governed by particular physical equations;
what is of primarily interest is the breakdown of physics which may, or may not, result in a breakdown of
geometry. Unfortunately, there is somehow a con
ict between the mathematical contexts appropriate to, on
the one hand, partial di erential equations and, on the other hand, geometry. In the di erential geometric
study of singularities one deals with geodesic equations which are uniquely solvable provided the metric
is C2�� (as already remarked above); beyond this, the di erenatibility of the metric is of little geometrical
signi cance. By contrast, in the study of hyperbolic PDEs the question of di erentiability is crucial; the
di erentiability chosen determines the character of the solutions allowed. By choosing low di erentiability
one admits solutions like shock waves or impulsive waves, which, on the other hand, are ruled out as \singular"
when insisting on high di erentiability.
http://www.mat.univie.ac.at/~stein/research/PhD/PhD.pdf
Roland Steinbauer
http://www.mat.univie.ac.at/~stein/rese ... ations.php
