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"Before" the Big Bang? Understanding entropy.
on 2 January 2011
After being delighted with Penrose's "Road to Reality" (2004) I couldn't wait to see what he would say about cosmology. Penrose's whole argument revolves around the consideration of the constraints put on cosmological theories by the Second Law of Thermodynamics, constraints he hinted at in the "Road to Reality".
These constraints he elaborates in a deep discussion of the nature of entropy, and what is so very special about the Big Bang. The book has three parts, "The Second Law and its Underlying Mystery", "The Oddly Special Nature of the Big Bang"; and the speculative proposal he concludes with : "Conformal Cyclic Cosmology".
Penrose takes no hostages : this is a deeply mathematical book, as is "The Road to Reality". He is a Platonist, he believes there is something there to tell us about! The first two sections of the book are "standard physics", But, as Seth Lloyd said in his Physics World review of the previous book, "When he represents the well established, nailed-down parts of mathematics and physics, Penrose is a joy to read. ... Penrose's treatment is ... deep; he is witty; he provides elegant insights." So his first section, which covers Bolzmann's definition of entropy, Liouville's Theorem, and similar matters, manages to explain the gigantic nature of phase space, the remarkable fact that although the equations of motion are symmetrical with time the path taken though phase space is definitely time-asymmetrical, and the robustness of the definition of entropy despite its apparent subjectivity in the details of counting states in phase space; all in only 45 rather small pages.
The second section now takes this "elementary" treatment and systematically applies it at a cosmological scale. There is a very strange peculiarity here which becomes very obvious in this Part. I think that Penrose thinks that his explanations could be followed by Everyman with a little application, since he carefully explains the difference between natural logarithms and logarithms with base 10. But he then launches into an intricate exposition of conformal geometry as it applies to the metric tensor of General Relativity! His purpose here, never mind who can understand it, is to use the constraints implied by the Second Law on a cosmological scale to constrain the geometry of space-time at the Big Bang.
And it appears that the constraints are very real. Because the entropy at the Big Bang is of necessity extraordinarily low, it must be (it seems) that gravitational degrees of freedom cannot have been excited. More explicitly, he draws a mathematical analogy between the electromagnetic field tensor F and the charge-current vector J in the Maxwell equations on the one hand, and (respectively, from General Relativity) the conformal Weyl tensor C and the Einstein or Ricci tensor E on the other; where E provides the source of the gravitational field (involving the mass-energy density tensor) and C characterises the curvature of space-time. Penrose asserts the "Weyl curvature hypothesis" C=0 at the Big Bang to represent its special low-entropy state.
But, and now here is the trick, a smooth C at the Big Bang invites a mathematical expression of this assertion that implies a smooth C prior to the Big Bang. Prior? Fear not! This is only a mathematical fiction. Or is it? Penrose then opens his third section where he piles speculation upon speculation to show that it is not irrational to consider the possibility of continuity "before" the Big Bang and "after" the what I shall call the Big Crunch for brevity. The background of this is the old belief of physicists, nearly universally held since Newton, that the Universe (the totality of everything physical that is) is really infinite in time. Today, it is conventional to say that spacetime itself originated at the Big Bang, and to speak of events prior to the Big Bang is to speak literal nonsense. But, Penrose suggests, this may not be necessarily true. And, he goes further to suggest, it is the detailed structure of the irregularities in the cosmic microwave background that may enable us to look behind the Big Bang without invoking inflation theories.
I must confess to being way out of my depth in this section. What is clear though is that Penrose believes that quantum theory, despite its magnificent observational successes, is still only a provisional theory; a position for which he claims the support of no less than Dirac himself. Everyone knows that a quantum theory of gravity is yet to be achieved, so that it is clear, even without the embarrassing anomalies of the mysterious dark matter and dark energy, that our ignorance is still profound. Central to Penrose's case in this third section is his account of information loss in black holes, and the consequent necessary non-unitary nature of Nature, a consequence that he has no hesitation in linking with the quantum mechanical problem of the collapse of the wave function during observations (the problem of Schrödinger's cat).
I am personally disposed to believe that the Universe is finite in time, at least towards the past. Perhaps this is something beyond observational proof, but in any case I think that Penrose's discussion, whether you believe him or not, is elegant and profound, and I sincerely hope that the new generation of mathematical physicists will take him very seriously. I think he is pointing to the next revolution in physics, with the development of quantum gravity, a consequent revolution in cosmology, and progress at last in some glimmer of appreciation of what consciousness could possibly be. The Universe is intelligible, and the systematic demand for intelligibility has always stimulated revolutions in our understanding.