Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

222 infinities in cosmologyspacetime. The collection of all such classes of curves (i.e., the collection of endpoints defined by them) defines the geodesic boundary (or g-boundary, for short) ofspacetime. It was Robert Geroch who elaborated all details of this construction.Let us notice the philosophy behind the g-boundary construction. We collect informationfrom inside a given spacetime (by following the behavior of geodesics in it) tolearn something about the way its structure breaks down. The apophatic character ofour knowledge is mitigated by tracing vestiges of what we do not know in the domainopen for our investigation. In this way we have learned that there are time-like, lightlike,and space-like g-boundaries, similar to “conformal infinities.” 8 However, this isonly the beginning of the story.10.4 The b-Boundary CrisisIn any spacetime, besides geodesic curves, there are plenty of other curves, and it isevident that the g-boundary method ignores them. Examples of spacetimes are knownin which all causal geodesic curves can be prolonged indefinitely (such spacetimes aresaid to be geodesically complete, or g-complete, for short) but which are “singular inother curves.” The problem is that in the case of nongeodesic curves no “prolongationprocedure” is known. A new method has to be elaborated.The job was undertaken by Bernd Schmidt (1971). His principal idea was to constructa “singular boundary” of spacetime as a collection of suitably defined end points of allcurves in this spacetime (not only of geodesic curves) that can be interpreted as historiesof physical objects. The main problem was how to define a length or a prolongationof such curves. 9 Schmidt has succeeded in doing so by using, as an auxiliary tool, aspace of all local reference frames attached to all points in spacetime. Such a space,well known in differential geometry, is a part of the structure called frame bundleover spacetime. This is why Schmidt’s boundary of spacetime has been called theb-boundary. It was soon acclaimed to be the best available definition of singularitiesin general relativity.It was fairly complicated, however. Schmidt himself was able to provide a few “toyexamples” that showed that in very simple cases his b-boundary construction workswell (in agreement with natural intuitions). However, when Bosshardt (1976) andJohnson (1977), almost simultaneously but independently, computed the b-boundaryfor the closed Friedmann world model, the disaster became manifest. It has turned outthat the b-boundary of this model consists of a single point. This means that the initialand final singularities, present in the Friedmann world, coincide. The beginning and theend of the world are the same! Moreover, from the topological point of view, all pointsof spacetime, together with the single b-boundary point, are “close” to each other. 10The reaction to this disaster was immediate. It became rather obvious that Schmidt’sdefinition was bad and that one needed to look for a better one. Several constructionswere proposed, but they either did not work or were not general enough. The so-called8 In fact, the method of conformal transformation is also helpful in investigating the structure of singularities.9 The definition had to be such that in the case of geodesic curves it should coincide with the previous one.10 More strictly, no spacetime point can be Hausdorff separated from the b-boundary.