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

conformal infinity 219In Sections 10.2 through 10.5, we consider the problem of infinities in relativisticcosmology and focus on mathematical methods to work with them. Essentially,one meets two kinds of infinities in cosmology: regions “infinitely distant” in spacetime,and “regions,” called singularities, at which the standard structure of spacetimebreaks down. When one approaches such regions, some physical magnitudes tend toinfinity. The strategy one adopts in cosmology is to learn something about these “forbiddenregions” by collecting information from regular domains of spacetime as oneapproaches the “forbidden domains.” In this way, one constructs various spacetimeboundaries, some elements of which represent singularities, and some other “infinitelydistant” regions.In Section 10.6 we deal with various aspects of the infinity problem as it arises withinthe multiverse ideology. We present some attempts to subdue this concept to a moredemanding analysis and indicate limitations in which all such attempts are involved.In Section 10.7 we try to draw a philosophical and theological lesson from theprevious considerations.10.2 Conformal InfinityIf we want to discuss the problems of space and time infinities in relativistic cosmology,we should keep in mind that these two kinds of infinities are but two components ofthe one spacetime infinity problem. Quite often, for both technical and philosophicalreasons, we prefer to discuss this problem in its “two-component” form (space and timeseparately) rather than in its spacetime invariant form, but then we are immediatelyconfronted with additional computational and conceptual questions that have to besolved. In the standard cosmological models, owing to their high geometric symmetries,these questions are solved quite naturally, but in more general cases they often requireskill and ingenuity.As mentioned in the introduction, in relativistic cosmology we meet two kinds ofspacetime infinities: points at spacetime infinity (i.e., points infinitely distant from theobserver) and singularities (such as the one in the Big Bang). Both of these kinds ofinfinities are a challenge to our horror of boundaries: we hate everything that limitsthe field of our potential knowledge.Points at infinity are inaccessible, by definition. In general relativity there are variouscategories of “points at infinity.” For instance, an infinite time could be needed to reachsuch a point, or a signal of infinite velocity would be required to send a message tosuch a point. Can we say something meaningful about such infinitely distant regions?In some cases, and to a certain extent, the answer is positive. The so-called conformaltransformations, well known in modern geometry, rescale distances in spacetime in sucha way that points at infinity are brought to finite distances; consequently, we are ableto investigate these “infinities” by studying various patterns that they form after beingconformally rescaled. However, the price we must pay for this is that any conformaltransformation distorts various geometric properties of a given spacetime, with oneexception: the structure of light propagation remains unaffected. In general relativity,light beams (photons) propagate along curves called null or light-like geodesics, andthe geometry of these curves is the same before and after any conformal transformation