Introduction

Although many of our considerations will be purely geometrical, treating space-time as a pseudo-Riemannian manifold and asking whether or not this geometrical structure is breaking down, it must always be remembered that we are really working with a physical theory, governed by particular physical equations for fields and particles, and that it is the breakdown of the physics that is primarily of interest. The breakdown of the geometry is simply one possible manifestation of the breakdown of the physics.

Unfortunately there is a conflict between the mathematical contexts appropriate to, on the one hand, geometry and, on the other hand, physically significant differential equations. In differential geometry one deals with geodesies, domains of dependence and so on. For this to be valid one requires that the connection should satisfy a Lipshitz condition, which ensures the existence of unique geodesies and normal coordinate neighbourhoods. Providing this holds, the differentiability of the metric has little geometrical significance and it is customary to require it to be C∞ for convenience. By contrast, in the study of hyperbolic differential equations (a type to which Einstein's equations belong) questions of differentiability are crucial. The differentiability chosen reflects the character of the solutions allowed: by choosing a low differentiability one admits solutions like shock-waves or impulse-waves which may be very significant; conversely, by choosing too high a level of differentiability one will brand as “singular” shock-wave solutions that from the point of view of fluid dynamics may be entirely legitimate.