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

infinite problems 183approaching the speed of light so that the wall forms a “cone” in spacetime. Insidethe cone are surfaces of φ = const., progressing from φ = φ M (which we can take todefine the bubble wall) to φ = φ T .What is very interesting is that if we define φ = const. slices to be equal-time slicesin Fig. 8.2), then at each time the spatial sections are infinite,negatively curved, homogenous, and isotropic, that is, the interior of the bubble isdescribed by the FLRW metric with k =−1 Coleman and Luccia (1980). Therefore,even if the exterior space can be described as always finite (although exponentiallyexpanding), the interior – spawned by a local nucleation event – is at all times spatiallyinfinite. The infinite time (and asymptotically infinite volume) of the exterior space hasbeen converted into infinite space at all times inside. 4Now, the bubbles will often run into each other, but as it turns out, this does notchange the infinite nature of the bubble interiors. As discussed in Aguirre et al. (2007a),Garriga et al. (2006a), in considering bubble collisions there is a center of the bubble.Far from this center, the probability to avoid being hit by another bubble approacheszero, but the physical volume at that distance diverges more quickly, so that the totalphysical volume is still infinite even if regions hit by bubbles are considered completelydestroyed (which is itself unlikely; see Aguirre and Johnson (2008), Aguirre et al.(2007a), Bousso et al. (2006a)).Although all of these aspects are clearly defined in the double-well case of Figure 8.2,they also appear to hold in the “single-well” case in which everlasting inflation is drivenby quantum fluctuations. In this case, a surface in spacetime can be defined where theinflaton falls just below the critical value to sustain everlasting inflation. This regionshould then just classically roll down the potential. Once again we can define constanttimeslices to be slices of constant φ, and analysis shows that these slices are spatiallyinfinite Linde (1986).In sum, then, we have an infinite number of times. At each of these, a large orinfinite number of bubbles can form, so that an infinite number of bubbles are formedthroughout the universe’s evolution. Each of these bubbles is spatially infinite inside.This is a rather amazing level of physically realized infinity to contemplate.(like the slices t1 ′′,t 2 ′′8.4 Infinite Problems8.4.1 The Inflationary MultiverseThis hierarchy of infinities not only is mind-boggling, but also causes enormousheadaches in cosmology. The reason is that while the different bubble universes maybe essentially identical, their properties may instead differ. In the past decade, theoristsstudying string/M theory have realized that the theory yields many different possibleversions of low-energy physics Douglas (2003), Susskind (2003), some may have differentvalues for the particle physics coupling constants. Others could have different4 It can be seen that this sort of model bridges the distinction between “actual” and “potential” infinity in theAristotelian sense of an infinite set that is completed all at one time versus one that is given by a sequence ofinclusions into the set. Here, these simply correspond to two different ways of decomposing spacetime, whichin GR are equally valid.