Needle in a haystack

I’m well on my way reading Penrose’s Road to Reality, and (as you may expect) having a great time. The first part, devoted to maths, was just delicious and, to me, an excellent refresher that has prompted me to dust from the shelves some books on rediscovered topics like complex analysis (“Visual Complex Analysis” (Tristan Needham)) or fiber bundles &co. (“Gauge Theory and Variational Principles” (David Bleecker)).

As for the physics chapters afterwards, i find their quality more uneven. In general, my impression is that one needs a pretty good basis on the topics discussed by Penrose to really appreciate and fully understand many a discussion in the book. I’d even say that pretty good means a university background in a technical area. Albeit, admittedly, it’s hard to judge from my standpoint, i don’t buy the claim (made at the preface) that even someone don’t getting rational numbers can understand (parts of) the book. For instance, i’ve found that the chapters on Relativity provide an insightful overview for people in the know, but, in my opinion, will fail to convey the deep principles of the theory to newbies. This impression was reinforced after reading the chapters on Quantum Field Theory: i’ve already forgotten most of what i learned about QFT at the university, and, besides, i never really understood it in depth. As a result (i think), those chapters of the book devoted to QFT have been frequently hard to understand to me, and i’ve finished them with a strong feeling of being missing many important points. Oh well, maybe it’s just me.

GodDespite the above knit-picks, the book is absolutely worth reading. In particular, i’ve enjoyed immensely the chapters dedicated to Cosmology and the problem of the low entropy of the Big Bang. If we are to believe in the second law of thermodynamics, the entropy in the universe has been increasing since its origin some 13 thousand million years ago. In other words, its path through the phase space has been traversing regions of greater and greater volume, where a region stands for a set of microscopic configurations giving rise to the same macroscopic behaviour (for instance, there are virtually an infinite number of possible positions and velocities for the atoms in the air of my room which gives raise to macroscopic values for pressure, temperature or smell that are indistinguishable by means of macroscopic measurements). Taking into account that the most entropic objects in the current universe are black holes (using the famous Bekenstein-Hawking relationship between a black hole’s entropy and the area of its horizon; see here and here for details) and an estimation of the number of black holes in the current universe, Penrose concludes that the volume of the phase space compatible with our Big Bang is about one part in 10^{10^{123}} of the total available. As shown in the figure, god must have had a hard time finding the exact point to start all this! If you don’t have Penrose’s book (or want to read a nice summary), you can read about these intriguing issues in his excellent survey (Space-time and Cosmology (PDF), part of the freely available Tanner Lectures on Human Values from the University of Utah [1], or hear and see his three lectures at Princeton University. There you’ll find why inflation and the anthropic principle, fashionable cures to everything as they may be, are not a solution to this conundrum (at least according to Penrose; you may find amusing to think about it a bit before reading the answer ;-)). Enlightening.

[1] As an aside, you’ll find lots of other interesting Tanner Lectures in their site, including this one by Richard Dawkins on Science and Religion.

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One Response to “Needle in a haystack”

  1. allan Says:

    I had a similar experience with “Road to Reality,” in that it seems to me to be a good book for people already “in the know.” Except that for me the roles of QFT and relativity were reversed. I really don’t see anyone with, well, a less than university level of physics understanding most of the “deep and important points.” But perhaps I am just more cynical about the average person’s brain than Penrose.

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