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

structure of singularities 223causal boundary of spacetime, proposed by Kronheimer and Penrose, 11 gained somepopularity, but without some hybrid combination with the g-boundary construction itwas unable to distinguish between singularities and points at infinity. However, it hasturned out that there is an aspect in which Schmidt’s construction is useful. It showswhat can happen to geometry when some physically meaningful magnitudes (likecurvature) go to infinity. Let us look at the singularity problem from this point of view.10.5 Structure of SingularitiesLet us try to understand what is going wrong with the Friedmann singularities whenan attempt is made to describe them with the help of Schmidt’s construction. The standardmethod of working with differential manifolds (every spacetime is a differentialmanifold) is to use coordinates (like in elementary geometry). There is, however, anequivalent method: instead of coordinates, one can consider all smooth functions on agiven manifold and define all relevant geometric magnitudes on this manifold in termsof these functions. This method can be generalized in the following way: If we have aspace (which is no longer manifold) with edges, cusps, boundaries, or some other kindof singular regions, we can find a suitable family of functions that could play the roleof “smooth” functions on this space 12 and, in terms of it, develop the geometry of thisspace analogously to the manifold case. This method has been elaborated by, amongothers, Roman Sikorski (1972). It is interesting to see what happens if we apply thisapproach to the Friedmann spacetime with its two initial and final singularities as theyare described by Schmidt’s b-boundary. If, for the time being, we disregard singularities,we have the Friedmann spacetime manifold with the collection of smooth functionson it, and everything behaves in the standard way. However, if we try to extend thisfamily of functions to the singularities, only constant functions survive this attempt. Inother words, on the Friedmann spacetime with singularities, only constant functionscan be defined. However, constant functions assume the same value everywhere, thatis, they do not distinguish points. Everything collapses to a single “neighborhood.” 13The aforementioned geometric properties of singularities are strict mathematicalresults, and as such they tell us something about the behavior of infinities quite independentlyof whether they apply to the real universe or not. However, there are strongindications that they do refer to the real world. As mentioned in Section 10.3, in the1960s and 1970s several theorems were proved demonstrating that singularities arenot artifacts of simplifying assumptions made in the process of model constructions(as was at that time commonly believed), but rather, inherent structural properties of abroad class of gravity theories, including general relativity (Hawking and Ellis 1973).The important point is that these theorems refer to classical singularities, that is, they donot take into account quantum gravity effects. This is a serious limitation because thereare strong reasons to believe that the very early stages of the cosmic evolution weregoverned by so-far-unknown quantum gravity laws. According to the prevailing view,the quantum gravity theory, when finally discovered, will eliminate singularities from11 This construction is explained in Kronheimer and Penrose (1967); see also Penrose (1979).12 Functions belonging to this family need not be smooth in the usual sense.13 This phenomenon was studied in detail in Heller and Sasin (1995).