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

224 infinities in cosmologythe scientific image of the universe. Even if this is true, the classical singularity theoremstell us something important about the structure of the universe. Every quantumgravity theory must, in the limit of low energies, recover classical geometric theory ofspacetime (general relativity). This means that it must contain enough powerful mechanismsthat would be able to smooth out singularities with all associated infinities, butthe mechanisms must not be so powerful as to erase them completely. They have to bereproduced in the limiting case. In other words, the looked-for quantum gravity theorymust be “singular in the limit.” In this sense, “infinities” should potentially be presenteven in “finite theories” of our universe (see Heller 1993).Let us try to collect our results. First, in the standard cosmology the problemof infinity appears in two forms: in the form of an “infinitely distant,” and in theform of an “infinitely divergent” (singularities). For both of these cases, mathematicalmethods have been elaborated to deal with infinities. Although we are unable to workdirectly with infinities, we apply to them various versions of the “strategy of limits.”The standard “going to the limit” in a convergent series is the simplest and bestknownmethod of this kind. Conformal transformations and attaching different kindsof boundaries to spacetime are more sophisticated ones. We can say that althoughinfinities are beyond our direct reach, they can be mathematically “tamed,” and onlywhen they have been suitably tamed can they be incorporated into a sound physicaltheory or model.This is a warning against considerations, not quite rare in popular books, on “cosmologicalinfinities” that are based on a naive concept of infinity. Infinity should notbe regarded as something “very, very, very . . . big,” but rather as something irreducibleto all “big things” we know. Without the aforementioned mathematical “taming” ofinfinities, it is better to refrain from speculations than to pronounce uncontrollableutterances.Second, from the way we have tamed infinities in classical cosmology (i.e., withouttaking into account quantum gravity effects), we learn an important lesson: infinitiesare not “local effects.” They are not strictly localizable at “infinitely distant” regionsof spacetime or within the singularities in the beginning and end of the universe.Their presence can be “felt” everywhere in spacetime. For instance, the structure ofthe conformal boundary or of the b-boundary determines various causal properties ofspacetime, which in turn have direct influence on local physics. 14The “nonlocality” of singularities can best be seen in a “pathology” they produce inthe Friedmann model (when they are understood as b-boundary points): as soon as wetry to incorporate singularities into the geometric description of the world, everythingcollapses to a single point.10.6 Infinity of UniversesOne of the reasons the multiverse idea gained popularity was purely ideological. Itsaim was to neutralize teleological interpretations seemingly implied by the fine-tuning14 For instance, when the past conformal boundary is space-like, there are particle horizons in the model; whenthe past conformal boundary is null, there are no particle horizons in the model (see Hawking and Ellis 1973,p. 128).