Abstract

The main topics in space-time edge geometry are introduced in an essential way, and their applications to the study of gravitational singularities are discussed. A simple but modern introduction to the concepts of Differential Geometry, which are most relevant to these topics, is also given.

Introduction to Part I: Fundamental concepts in Differential Geometry

In recent years, the importance of differential geometry in theoretical physics has greatly increased, and field theories are now formulated largely in terms of topics such as the theory of connections, principal bundles, lagrangian and symplectic geometry, and so on. However, there are still differences between the approach of physicists and that of mathematicians to this crucial area of common interest. In fact, usually physicists prefer to deal with coordinate calculations, thinking that a more abstract and rigorous formalism is difficult to handle and unessential for physical understanding, while mathematicians often consider the coordinate notation as untidy and obscuring. In my opinion, the present state of the art enables us to get over these differences. In fact, many ideas in differential geometry can be introduced in a very clear way by adopting an abstract and general framework, and this clearness turns out also to be undoubtedly very useful in practice; in particular, we have now a powerful, neat and simple theory of connections on fibred manifolds, which includes all the particular cases (such as principal bundles and tensor bundles), and allows clearer understanding and easier handling. On the other hand, the coordinate formalism can be set in an absolutely rigorous way, and this increases both its power and its reliability. This part I is intended to present to the physicist some of the main ideas in this field, in an essential way and without boring details. Its prerequisites are only standard undergraduate courses in analysis and linear algebra, but of course it is not sufficiently expanded to give a deep insight into the subject; however, it could be useful reading before tackling a book on differential geometry.

Introduction to Part II: what is a space-time singularity?

It is well-known that the study of Einstein's equation of general relativity leads to predict the existence of space-time "singularities"; in other terms, it produces models of the physical world in which some observer arrives, in a finite interval of proper time, to the "edge of the universe". Moreover, this edge cannot be included in an extension of the space-time, for example because the observer experiences unbounded curvature; the best known examples are the Schwartzschild and Friedmann solutions. It had been proposed (Lifshitz et al.) that singularities could arise from an excess of imposed simmetry, and that they would not be present in physically more realistic solutions. But, now, this seems not to be true; in particular, Hawking and Penrose have proved theorems which say that singularities are unavoidable when some general (and physically plausible) conditions are satisfied. Then, relativistic singularities seem to have a kind of "stable" character, and to be physically very important. Hence the problem of the definition of singular space-time: we would like to have a general criterion for deciding whether a given space-time has singularities and, possibly, a classification of them. The natural thing to do is to try to imitate the riemannian framework, where problems of this kind has a natural and straightforward setting (§4.12). Essentially, we must find a suitable criterion of completeness analogous to that of metric completeness (in other terms: how can we decide whether a given space-time has "cuts" or "holes"?) and then examine the problem of extendibility. As for completeness, one might think of using the concept of geodesic completeness, which is in fact sufficient in many space-times; however, this is not a satisfying criterion from a general point of view. In fact, there are examples of space-times which are geodesically complete, but admit world lines of finite proper-time length and bounded acceleration which are inextendible, i.e. have no endpoint in the space-time. In physical terms, this means that a spaceship with an engine of finite power and a finite amount of fuel could arrive, after a finite interval of proper time, at the "end of the universe", and then would cease existing. It is clear that, in a physically satisfying definition of completeness, such a space-time must turn out to be incomplete. Thus, although this difficulty does not arise in the commonly studied space-times, geodesic completeness is not sufficient for the study of singularities. The construction of the b-completion (i.e. bundle-completion ) seems to solve these problems in a satisfying manner. A b-boundary is attached to any manifold with a connection; in particular, it provides endpoints for all curves in a space-time which we would consider incomplete from a physical point of view; hence, we may consider the singularities of a space-time as points in its b-boundary. Thus we have the concept of b-completeness (i.e. the emptiness of the b-boundary), which turns out to be a generalization of that of geodesic completeness (b-completeness implies g-completeness) and also of that of metric completeness (for riemannian manifolds, the metric completion and the b-completion coincide). Thus, in principle, the study of space-time singularities proceeds in two steps. First, we should determine the b-boundary. Second, for each point of the b-boundary, we should study extendibility; this is so because, analogously to the riemannian case, the b-boundary does not distinguish between true singularities and singularities which can be eliminated by extending the space-time; in other words, a space-time obtained by removing points from a b-complete one is b-incomplete, and its b-boundary is constituted exactly by the topological boundary of the set of removed points. Of course, this procedure may present difficult mathematical problems but, in principle, is currently considered as the right setting for the study of space-time singularities. In this part II we shall discuss these ideas and some of their main developments.