PENROSE'S NEW CLOTHES
Bearing in mind the classic fairy tale by Hans Christian Andersen "The Emperor's New Clothes", I wouldn't be surprised if someone says, loud enough for the others to hear: "Look at the Penrose's new book. It's beautiful!"
At first glance, the book does look stunning, and once we open it, we cannot miss the following statements printed on the inside flap of the paper sleeve: "The Road to Reality is the most important and ambitious work of science for a generation. It provides nothing less than a comprehensive account of the physical universe and the essentials of its underlying mathematical theory. (…) In a single work of colossal scope one of the world’s greatest scientists has given us a complete and unrivalled guide to the glories of the universe which we all inhabit."
It is not known who is the author of these statements, however. This is the first shocking omission.
It seems to me that Roger Penrose’s "The Road to Reality: A Complete Guide to the Laws of the Universe" is a very personal book, written by Penrose, and for Penrose. The first sixteen chapters are devoted to a galloping outline of the "underlying mathematical theory", but the author has failed to mention whether such an outlook has been helpful to any undergraduate or graduate student. I suspect that Penrose has not tested his mathematical outlook on anyone but himself.
In my opinion, there are even more serious and alarming omissions. Penrose uses a well-known three-world diagram to present "the three forms of existence" (mental, physical, and Platonic mathematical worlds; cf. Fig. 1.3, pp. 17-18), but does not mention the proposition due to Leibniz (Monadology, 78) about ONE entity which emanates two complementary forms of existence, material and mental. Speaking about Leibniz, he discusses the notion of ‘dx’ as denoting an ‘infinitesimal quantity’ (p. 230), but fails to mention the paradoxes which we encounter by modeling ‘dx’ with a mathematical "point" (recall Zeno’s paradoxes and Thompson's lamp paradox of 1954, after James Thompson). Which brings us to the most important part of Penrose’s book, dealing with his insights and speculations in quantum physics and Einstein’s theory of gravity.
Let’s recall some well-known facts. There is a profound mystery in our universe: 96 per cent of it is of some unknown nature, called ‘dark matter’ and ‘dark energy’. To quote from T. Reichhardt, Nature 421, 777 (20 February 2003): "The new data imply an age for the Universe of 13.7 billion years, and a distribution of mass and energy in which 4% of the Universe is normal matter (atoms), 23% is dark matter, and 73% is dark energy". The very first question regarding the so-called dark energy, which supposedly drives the expansion of the spacetime along the cosmological time arrow with constant acceleration, is about its physical substrate: energy of what? Let’s call it ‘energy of X’, and formulate the next, and perfectly obvious, questions: I look at my wristwatch and record an instant, t1 . At some later time t2 , say, t2 - t1 = 8 min, the spacetime will be expanded by the dark energy of X. Where is the pool of X located at the first instant, t1 , and what kind of time would you attribute to it, relative to t1 ?
It shouldn’t be surprising that we encounter immense difficulties. Penrose does acknowledge that "gravitational energy is a genuinely non-local quantity" (p. 467; see also Ch. 19.5), but provides only an example of mass-energy conservation law defined at ‘null infinity’, under the stipulations that (i) the spacetime is "asymptotically flat", and (ii) some hypothetical "gravitational waves" were emitted from some physically unrealistic "isolated system", "in the limit when the system becomes completely spatially isolated from everything else" (Fig. 19.10, p. 468). (This "isolated system" sounds like astrology to me, but maybe I’m wrong.) Also, Penrose does not make any effort to compare the two kinds of non-localities, from Einstein’s gravitational energy and the holistic nature of quantum entanglement (p. 407 and Ch. 23; see also pp. 667-668). He only explains the reason why he has been unable to provide the actual dynamics of the "reduction of state" (p. 855): there is a fundamental conflict with Einstein’s principle of general covariance (p. 850). Fair enough. However, in the discussion of Einstein’s cosmological constant [lambda], which is the first issue we address to clarify the puzzle of ‘dark energy of X’, Penrose writes the following (p. 777): "Thus, any non-constancy in [lambda] would have to be accompanied by a compensating non-conservation of the mass-energy of the matter. It is certainly much more theoretically comfortable to have [lambda] constant – as indeed consistent with observation".
It is certainly "theoretically comfortable" to bypass the so-called coincidence problem (cf. N. Straumann, gr-qc/0208027), but only in a book reflecting the personal inclinations of the author. Had Roger Penrose chosen a different title for his book (say, "The Road to Reality: A Personal, Biased, and Incomplete Guide to the Laws of the Universe"), I wouldn’t have any objections to it.
To sum up, I would recommend Penrose’s book mostly to graduate students in mathematical physics (a good exercise is suggested on p. 319), provided they have previously studied H. Weyl’s "Space-Time-Matter" (Fourth Edition, Dover Publications, New York, 1951, p. 270), as well as two fundamental papers, by T. Levi-Civita (Rendiconti della Reale Accademia dei Lincei, vol. 26, 381 (1917); English translation in A. Loinger, physics/9906004), and again by H. Weyl (Amer. J. Math. 66 (1944) 591; see Appendix in A. Loinger, physics/0407134). Then perhaps they will enjoy Claus Kiefer's "Quantum Gravity" (Oxford University Press, 2004, 320 pages; ISBN: 0-19-850687-2). It’s being written for graduate students, with great clarity and professional scrutiny.
Perhaps the road to quantum gravity should include the physics of the human brain, particularly its amazing self-acting faculty: we think about the brain, with the brain. Perhaps this could be a hint to the nature of gravitational field: it is not only nonlinear in its own coupling, but also makes all matter fields self-interacting. Our mind is not material, and yet it can interact with its brain, just as the geometry of matter is not material but does interact with all physical fields: "Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve" (John Wheeler). However, the intrinsic "timing" of this bi-directional talk is missing, and I anticipate a "fundamental conflict with Einstein’s principle of general covariance" (p. 850), too. We can probably find comfort in the last lecture by Albert Einstein (Princeton University, April 14, 1954, according to notes taken by J. A. Wheeler): "The representation of matter by a tensor was only a fill-in to make it possible to do something temporarily, a wooden nose in a snowman."
As Roger Penrose put it (p. 1045),
"Perhaps what we mainly need is some subtle change in perspective – something
that we all have missed…" Here I agree
with Sir Roger. Only the change in perspective
may not be subtle.