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

220 infinities in cosmologyof spacetime. In this sense, we can say that light propagation is a skeleton of the entirespacetime geometry.In some cases, after the conformal rescaling and an additional procedure, calledcompactification, one can drawn a nice picture of a given spacetime. The compactificationattaches to the picture clear boundaries that represent “points at infinity.” Suchpictures are called Penrose diagrams and are often used to study the “infinity structure”of different relativistic spacetimes. The boundaries of Penrose diagrams perspicuouslydisplay various properties of “conformal infinities.” Some parts of them are formedby the end points of curves that are histories of freely moving massive particles (theso-called time-like geodesics); they are said to represent “time-like infinities” of agiven spacetime. Some parts of them are formed by the end points of curves that arehistories of photons (null geodesics); they represent “null” or “light-like infinities.”Other parts of them are formed by the end points of curves that represent no physicalobjects (particles moving along such curves, called space-like geodesics, would have tomove with velocities greater than that of light); they form “space-like infinities.” Takinginto account the fact that causal influences can be propagated along time-like andlight-like curves, Penrose diagrams are said to represent the causal structure of variousspacetimes.From this short reminder of well-known facts about spacetime architecture 2 wecan learn an interesting philosophical lesson. In general relativity “infinitely distant”regions of spacetime are not mere shadowy horizons. They make their presence inthe regular domains of spacetime through their causal properties. Geometry of light isinvariant with respect to conformal transformations, and it creates a skeleton supportingthe causal structure of spacetime, which, in turn, serves as a groundwork for the entirephysics. The structure of conformal infinity is an essential ingredient of global physics.10.3 Infinitely DivergentOther barriers for our knowledge are situated at finite distances from us in spacetime.They are formed by singularities of various kinds. There are mild singularities thatcan be removed by simple prolongations of a given spacetime (the so-called regularsingularities); there are singularities similar to stringlike inhomogeneities in spacetime(quasi-regular singularities); and there are malicious singularities, called strongcurvature singularities, having the property that on approaching them the spacetimecurvature and some other physical magnitudes blow up to infinity. To the latter kindbelong the initial singularity in the Big Bang cosmological models and the Big Crunchsingularity in the closed standard cosmological model. The initial singularity is situatedin our finite past, and the final singularity in our finite future. These expressionsare meaningful only with respect to cosmological models in which, owing to theirhigh symmetries, spacetime naturally decomposes into one-dimensional time and thefamily of three-dimensional instantaneous spaces. In such cosmological models wecan meaningfully speak about the unique world’s history, and possibly about its abrupt2 The fundamental reference is Hawking and Ellis (1973). The reader could find a good introduction in Gerochand Horowitz (1979) and, on the more popular level, Geroch (1978).