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[This text is taken from the Introduction of RS' Ph.D. thesis]

General relativity--the fascinating theory of space, time and gravitation as formulated by A. Einstein 85 years ago--dramatically differs from all other field 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 field 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 differential equations of second order for the coefficients of the metric tensor; hence one has to assume the metric to satisfy sufficient differentiability conditions. Usually one supposes the metric to be smooth; C2 suffices to do classical differential geometry, whereas C3 is needed to ensure energy conservation. C2- (i.e., the first 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 first 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 difficulty in defining a singularity in general relativity is deeply connected with another of the theory's main principles: diffeomorphism 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 find a different 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 definition a spacetime is called singular if there exist incomplete geodesics, i.e., geodesics of finite affine 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 defined 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. [4], 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 definition.

Consequently, the recent development of the study of spacetime singularities has focussed more upon a study of the field 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 conflict between the mathematical contexts appropriate to, on the one hand, partial differential equations and, on the other hand, geometry. In the differential 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 differenatibility of the metric is of little geometrical significance. By contrast, in the study of hyperbolic PDEs the question of differentiability is crucial; the differentiability chosen determines the character of the solutions allowed. By choosing low differentiability one admits solutions like shock waves or impulsive waves, which, on the other hand, are ruled out as ``singular'' when insisting on high differentiability.

Accepting the field equations to play the primary role, they should determine the choice of differentiability. A singularity should be an obstruction to the existence of a solution to the field equations. Clearly, this cannot be determined in general. On the other hand, there are existence theorems that tell when it is possible to construct a solution, i.e., we know what is not a singularity. However, in general it is difficult to link back a certain differentiability condition necessary to prove (local) existence of solutions to Einstein's equations (Sobolev conditions, usually imposed on the Riemann tensor on specific hypersurfaces in a particular class of coordinate systems, well suited for the formulation of the Einstein equations as an initial value problem) to geometry, that is the differentiability of the metric on the entire spacetime manifold.

Summing up the entire discussion so far, it would be very desirable to have a description of singularities as internal points of the spacetime manifold where the field equations are satisfied in a weak (probably distributional) sense (see also [2]). Hence one would wish to significantly lower the ``geometric differentiability bound'' (i.e., C2-) on the metric. Indeed, a recent monograph on the subject of spacetime singularities [1] reaches the conclusion that the answer to many of the questions raised above ``involve detailed considerations of distributional solutions to Einstein's equations, leading into an area that is only starting to be explored [...]'' In particular, one wants to be able to describe spacetimes containing matter whose density function is e.g. unbounded but integrable or confined to a submanifold in spacetime (both scenarios amounting to a finite mass per unit volume). Important examples in this class include thin cosmic strings and impulsive gravitational waves.

Another strong motivation for a study of distributional spacetimes arises from the fact that by an argument of Isham [5] the latter will substantially contribute to a path integral description of (a yet to be formulated theory of) quantum gravity.

The big challenge in setting up a framework that might be called a ``distributional geometry'' adapted to the needs of general relativity of course resides in the immanent nonlinearity of the latter theory; calculating the Riemann tensor from the metric is an essentially nonlinear operation. On the other hand, classical distribution theory as founded by L. Schwartz in his famous book [6] is a linear theory. This deep conceptual problem obviously is the main reason why applications of distribution theory to general relativity have been rare and either limited to special situations or lacking the necessary mathematical rigor. Moreover, Geroch and Traschen ([3]) have shown that a physically sensible and mathematically sound framework based on linear distribution theory cannot handle such interesting sources of the gravitational field as strings and point particles.

At this point the theory of algebras of generalized functions enters the field. It provides a rigorous mathematical framework for simultaneously treating singular (i.e., distributional) objects, nonlinear operations and differentiation. Moreover recently the question of diffeomorphism invariance of the construction--which is vital in the contex of General Relativity-- has been finally setteled. Hence algebras of generalized functions provide a suitable setting for describing general relativistic spacetimes of low differentiability; indeed there has been a strong line of research following this approach.

For a more detailed discussion of ''singularities as internal points'' and distributional curvature in this context see this page by Jonathan Wilson. For a recent ovierview of applications of generalized functions in General Relativity see the review article by James Vickers [7].


C. J. S. Clarke, ``The Analysis of Space-Time Singularities,'' Cambridge Lecture Notes in Physics, (Cambridge, 1993).

C. J. S. Clarke, ``Singularities: boundaries or internal points,'' in Singularities, Black Holes and Cosmic Censorship edited by P. S. Joshi, 24-32 (IUCCA, Bombay, 1996).

R. Geroch, ``Limits of spacetimes,'' Comm. Math. Phys. 13, 180-193, (1969).

S. W. Hawking, G. F. R. Ellis, ``The Large Scale Structure of Space-Time,'' (Cambridge University Press, Cambridge, 1973).

C. J. Isham, ``Some Quantum field theory aspects of the superspace quantization of general relativity,'' Proc. R. Soc. A 351, 209-232 (1976).

L. Schwartz, ``Théorie des Distributions,'' (Herman, Paris, 1966).

J.A. Vickers, ``Nonlinear generalised functions in general relativity,'' in Nonlinear Theory of Generalized Functions, Chapman & Hall/CRC Research Notes in Mathematics 401, 275-290, eds. M. Grosser, G. Hörmann, M. Kunzinger, M. Oberguggenberger (Chapman & Hall/CRC, Boca Raton 1999).
Last changed January 3, 2007 by M.K. and R.S.