Cycles of Time: An Extraordinary New View of the Universe

by Roger Penrose

there is a conundrum to be faced. It is a conundrum that appears to confront us irrespective of CCC. The issue has to do with the evident fact that the entropy of our universe—or the current aeon, if we are considering CCC—seems to be vastly increasing, despite the fact that the very early universe and the very remote future appear to be uncomfortably similar to one another. Of course they are not really similar in the sense of being nearly identical, but they are alarmingly ‘similar’ according to the usage of that word commonly applied in Euclidean geometry, namely that the distinction between the two seems to be basically just an enormous scale change. Moreover, any overall change of scale is essentially irrelevant to measures of entropy—where that quantity is defined by Boltzmann’s marvellous formula (given in §1.3)—because of the important fact, noted at the end of §3.1, that phase-space volumes are unaltered by conformal scale changes.[3.40] Yet, the entropy does seem to increase, vastly, in our universe, through the effects of gravitational clumping. Our conundrum is to understand how these apparent facts are to square with one another.