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

infinitely divergent 221beginning and end. However, this very particular situation cannot serve as a model fora general definition, or even as an adequate description, of singularities. 3In the theory of relativity, decomposition of spacetime into space and time separatelydepends on the choice of the local reference frame; consequently, the concept of theworld history, as measured by time, has no invariant meaning. On the other hand, thehistory of a single observer, or of a massive particle, or of a photon has a well-definedmeaning because it is represented by a suitable curve in spacetime: by a time-likecurve in the case of an observer or of a massive particle, and by a null geodesic curvein the case of a photon. The concept of the prolongation of a curve in the theory ofrelativity is a tricky one (given that length depends on the reference frame); however,in the case of geodesic curves there is a sense in which one can meaningfully speakabout its prolongation. 4 Suppose that all time-like and light-like geodesic curves ina given spacetime can be indefinitely prolonged (in the above sense). 5 This wouldmean that histories of all physical objects never meet any obstacles – they can happenindefinitely. In such a case, spacetime is said to be singularity-free. In contrast, if atleast one such geodesic curve cannot be indefinitely prolonged, this would mean that ithas encountered a singularity. 6 In the standard cosmological models, all time-like andlight-like geodesic curves break down at the initial or final singularities.With the help of this construction it was possible to prove, in the 1960s and 1970s,several theorems on the existence of classical singularities (i.e., without taking intoaccount possible quantum gravity effects). It has turned out that singularities are notby-products of some simplifying assumptions, as was so far commonly believed, butrather that they are deeply rooted in the structure of the present theories of gravity.Let us notice that in this approach singularities are not “points” or some “pathologicalregions” in spacetime but are just a name announcing the fact that the structure ofspacetime somewhere breaks down. Indeed, if we know only that a curve cannot beprolonged, we know that something is going awry, but we do not know the natureof the obstacle. To borrow the analogy from theology, we could say that singularitiesare defined apophatically, 7 that is, purely negatively, by denying to them the structureof spacetime. What can we do to overcome this docta ignorantia situation? We tryto organize the end points of geodesics that break down at singularities into somemeaningful totality that would inform us, at least to some extent, about the structure ofthe spacetime edge representing something called singularity. To this end we investigatethe behavior of time-like and light-like geodesics, as they are more and more prolonged,in order to decide which of them have the same end points. The class of curves havingthe same end point defines this end point. Instead of investigating the end points, weinvestigate the corresponding classes of curves to which we have access from within3 References cited in footnote 2 are also recommended for this section.4 We say that a geodesic curve can be indefinitely prolonged (technically, it is said to be complete) if the affineparameter along it can assume arbitrarily large values.5 We do not take into consideration space-like geodesics because they are not histories of physical objects.6 It goes without saying that this is a very rough, intuitive description. To change it into the working definitionwould require a host of technical details. The reader interested in them should consult the classical monographby Hawking and Ellis (1973). Earman (1995), a more philosophically oriented book (but also highly technical),is also worth reading.7 For more about apophatic theology see Section 10.7.