Books by Penrose, Roger
- Penrose, Roger. Cycles of Time. New York: Alfred A. Knopf, 2010. ISBN 978-0-307-26590-6.
-
One of the greatest and least appreciated mysteries of
contemporary cosmology is the extraordinarily special state
of the universe immediately after the big bang. While at
first glance an extremely hot and dense mass of elementary
particles and radiation near thermal equilibrium might seem
to have near-maximum entropy, when gravitation is taken into
account, its homogeneity (the absence of all but the most
tiny fluctuations in density) actually caused it to have
a very small entropy. Only a universe which began in such a
state could have a well-defined arrow of time which permits
entropy to steadily increase over billions of years as
dark matter and gas clump together, stars and galaxies form, and black
holes appear and swallow up matter and radiation. If
the process of the big bang had excited gravitational
degrees of freedom, the overwhelmingly most probable outcome
would be a mess of black holes with a broad spectrum of
masses, which would evolve into a universe which looks
nothing like the one we inhabit. As the author has
indefatigably pointed out for many years, for some reason
the big bang produced a universe in what appears to be
an extremely improbable state. Why is this? (The
preceding sketch may be a bit telegraphic because I discussed
these issues at much greater length in my review of
Sean Carroll's
From Eternity to Here [February 2010]
and didn't want to repeat it all here. So, if you aren't
sure what I just said, you may wish to read that review
before going further.)
In this book, Penrose proposes “conformal cyclic cosmology”
as the solution to this enigma. Let's pick this apart, word by
word.
A conformal
transformation is a mathematical mapping which
preserves angles in infinitesimal figures. It is possible to
define a conformal transformation (for example, the
hyperbolic transformation
illustrated by M. C. Escher's Circle Limit III)
which maps an infinite space onto a finite one. The author's own
Penrose diagrams
map all of (dimension reduced) space-time onto a finite plot
via a conformal transformation. Penrose proposes a conformal transformation
which maps the distant future of a dead universe undergoing runaway expansion
to infinity with the big bang of a successor universe, resulting in
a cyclic history consisting of an infinite number of
“æons”, each beginning with its own big bang and
ending in expansion to infinity. The resulting cosmology is
that of a single universe evolving from cycle to cycle, with the
end of each cycle producing the seemingly improbable conditions required
at the start of the next. There is no need for an inflationary epoch
after the big bang, a multitude of unobservable universes in a
“multiverse”, or invoking the anthropic principle to
explain the apparent fine-tuning of the big bang—in Penrose's
cosmology, the physics makes those conditions inevitable.
Now, the conformal rescaling Penrose invokes only works if the universe
contains no massive particles, as only massless particles which always
travel at the speed of light are invariant under the conformal transformation.
Hence for the scheme to work, there must be only massless particles in the universe
at the end of the previous æon and immediately after the big bang—the
moment dubbed the “crossover”. Penrose argues that at the
enormous energies immediately after the big bang, all particles were
effectively massless anyway, with mass emerging only through symmetry
breaking as the universe expanded and cooled. On the other side of the
crossover, he contends that in the distant future of the previous æon
almost all mass will have been accreted by black holes which then
will evaporate through the
Hawking process
into particles
which will annihilate, yielding a universe containing only massless
photons and gravitons. He does acknowledge that some matter may
escape the black holes, but then proposes (rather dubiously in my
opinion) that all stable massive particles are ultimately
unstable on this vast time scale (a hundred orders of magnitude
longer than the time since the big bang), or that mass may just
“fade away” as the universe ages: kind of like the
Higgs particle
getting tired (but then most of the mass of stable
hadrons doesn't come from the Higgs process, but rather the
internal motion of their component quarks and gluons).
Further, Penrose believes that information is lost when it falls
to the singularity within a black hole, and is not preserved in
some correlation at the event horizon or in the particles
emitted as the black hole evaporates. (In this view he is now
in a distinct minority of theoretical physicists.) This makes
black holes into entropy destroying machines. They devour all of
the degrees of freedom of the particles that fall into them and
then, when they evaporate with a “pop”, it's all
lost and gone away. This allows Penrose to avoid what would otherwise
be a gross violation of the second law of thermodynamics. In his
scheme the big bang has very low entropy because all of the entropy
created in the prior æon has been destroyed by falling into
black holes which subsequently evaporate.
All of this is very original, clever, and the mathematics is quite
beautiful, but it's nothing more than philosophical speculation
unless it makes predictions which can be tested by observation
or experiment. Penrose believes that gravitational radiation emitted
from the violent merger of galactic-mass black holes in the previous
æon may come through the crossover and imprint itself
as concentric circles of low temperature variation in the cosmic background
radiation we observe today. Further, with a colleague, he argues that
precisely such structures
have been observed in two separate surveys of the background
radiation. Other researchers
dispute
this
claim,
and the
debate continues.
For the life of me, I cannot figure out to which audience this book
is addressed. It starts out discussing the second law of thermodynamics
and entropy in language you'd expect in a popularisation aimed at
the general public, but before long we're into territory like:
Ahhhh…now I understand! Seriously, much of this book is tough going, as technical in some sections as scholarly publications in the field of general relativity, and readers expecting a popular account of Penrose's proposal may not make it to the payoff at the end. For those who thirst for even more rigour there are two breathtakingly forbidding appendices. The Kindle edition is excellent, with the table of contents, notes, cross-references, and index linked just as they should be.We now ask for the analogues of F and J in the case of the gravitational field, as described by Einstein's general theory of relativity. In this theory there is a curvature to space-time (which can be calculated once knows how the metric g varies throughout the space-time), described by a [ 04]-tensor R, called the Riemann(-Christoffel) tensor, with somewhat complicated symmetries resulting in R having 20 independent components per point. These components can be separated into two parts, constituting a [ 04]-tensor C, with 10 independent components, called the Weyl conformal tensor, and a symmetric [ 02]-tensor E, also with 10 independent components, called the Einstein tensor (this being equivalent to a slightly different [ 02]-tensor referred to as the Ricci tensor[2.57]). According to Einstein's field equations, it is E that provides the source to the gravitational field. (p. 129)
- Penrose, Roger. Fashion, Faith, and Fantasy. Princeton: Princeton University Press, 2016. ISBN 978-0-691-11979-3.
- Sir Roger Penrose is one of the most distinguished theoretical physicists and mathematicians working today. He is known for his work on general relativity, including the Penrose-Hawking Singularity Theorems, which were a central part of the renaissance of general relativity and the acceptance of the physical reality of black holes in the 1960s and 1970s. Penrose has contributed to cosmology, argued that consciousness is not a computational process, speculated that quantum mechanical processes are involved in consciousness, proposed experimental tests to determine whether gravitation is involved in the apparent mysteries of quantum mechanics, explored the extraordinarily special conditions which appear to have obtained at the time of the Big Bang and suggested a model which might explain them, and, in mathematics, discovered Penrose tiling, a non-periodic tessellation of the plane which exhibits five-fold symmetry, which was used (without his permission) in the design of toilet paper. “Fashion, Faith, and Fantasy” seems an odd title for a book about the fundamental physics of the universe by one of the most eminent researchers in the field. But, as the author describes in mathematical detail (which some readers may find forbidding), these all-too-human characteristics play a part in what researchers may present to the public as a dispassionate, entirely rational, search for truth, unsullied by such enthusiasms. While researchers in fundamental physics are rarely blinded to experimental evidence by fashion, faith, and fantasy, their choice of areas to explore, willingness to pursue intellectual topics far from any mooring in experiment, tendency to indulge in flights of theoretical fancy (for which there is no direct evidence whatsoever and which may not be possible to test, even in principle) do, the author contends, affect the direction of research, to its detriment. To illustrate the power of fashion, Penrose discusses string theory, which has occupied the attentions of theorists for four decades and been described by some of its practitioners as “the only game in town”. (This is a piñata which has been whacked by others, including Peter Woit in Not Even Wrong [June 2006] and Lee Smolin in The Trouble with Physics [September 2006].) Unlike other critiques, which concentrate mostly on the failure of string theory to produce a single testable prediction, and the failure of experimentalists to find any evidence supporting its claims (for example, the existence of supersymmetric particles), Penrose concentrates on what he argues is a mathematical flaw in the foundations of string theory, which those pursuing it sweep under the rug, assuming that when a final theory is formulated (when?), its solution will be evident. Central to Penrose's argument is that string theories are formulated in a space with more dimensions than the three we perceive ourselves to inhabit. Depending upon the version of string theory, it may invoke 10, 11, or 26 dimensions. Why don't we observe these extra dimensions? Well, the string theorists argue that they're all rolled up into a size so tiny that none of our experiments can detect any of their effects. To which Penrose responds, “not so fast”: these extra dimensions, however many, will vastly increase the functional freedom of the theory and lead to a mathematical instability which will cause the theory to blow up much like the ultraviolet catastrophe which was a key motivation for the creation of the original version of quantum theory. String theorists put forward arguments why quantum effects may similarly avoid the catastrophe Penrose describes, but he dismisses them as no more than arm waving. If you want to understand the functional freedom argument in detail, you're just going to have to read the book. Explaining it here would require a ten kiloword review, so I shall not attempt it. As an example of faith, Penrose cites quantum mechanics (and its extension, compatible with special relativity, quantum field theory), and in particular the notion that the theory applies to all interactions in the universe (excepting gravitation), regardless of scale. Quantum mechanics is a towering achievement of twentieth century physics, and no theory has been tested in so many ways over so many years, without the discovery of the slightest discrepancy between its predictions and experimental results. But all of these tests have been in the world of the very small: from subatomic particles to molecules of modest size. Quantum theory, however, prescribes no limit on the scale of systems to which it is applicable. Taking it to its logical limit, we arrive at apparent absurdities such as Schrödinger's cat, which is both alive and dead until somebody opens the box and looks inside. This then leads to further speculations such as the many-worlds interpretation, where the universe splits every time a quantum event happens, with every possible outcome occurring in a multitude of parallel universes. Penrose suggests we take a deep breath, step back, and look at what's going on in quantum mechanics at the mathematical level. We have two very different processes: one, which he calls U, is the linear evolution of the wave function “when nobody's looking”. The other is R, the reduction of the wave function into one of a number of discrete states when a measurement is made. What's a measurement? Well, there's another ten thousand papers to read. The author suggests that extrapolating a theory of the very small (only tested on tiny objects under very special conditions) to cats, human observers, planets, and the universe, is an unwarranted leap of faith. Sure, quantum mechanics makes exquisitely precise predictions confirmed by experiment, but why should we assume it is correct when applied to domains which are dozens of orders of magnitude larger and more complicated? It's not physics, but faith. Finally we come to cosmology: the origin of the universe we inhabit, and in particular the theory of the big bang and cosmic inflation, which Penrose considers an example of fantasy. Again, he turns to the mathematical underpinnings of the theory. Why is there an arrow of time? Why, if all of the laws of microscopic physics are reversible in time, can we so easily detect when a film of some real-world process (for example, scrambling an egg) is run backward? He argues (with mathematical rigour I shall gloss over here) that this is due to the extraordinarily improbable state in which our universe began at the time of the big bang. While the cosmic background radiation appears to be thermalised and thus in a state of very high entropy, the smoothness of the radiation (uniformity of temperature, which corresponds to a uniform distribution of mass-energy) is, when gravity is taken into account, a state of very low entropy which is the starting point that explains the arrow of time we observe. When the first precision measurements of the background radiation were made, several deep mysteries became immediately apparent. How could regions which, given their observed separation on the sky and the finite speed of light, have arrived at such a uniform temperature? Why was the global curvature of the universe so close to flat? (If you run time backward, this appeared to require a fine-tuning of mind-boggling precision in the early universe.) And finally, why weren't there primordial magnetic monopoles everywhere? The most commonly accepted view is that these problems are resolved by cosmic inflation: a process which occurred just after the moment of creation and before what we usually call the big bang, which expanded the universe by a breathtaking factor and, by that expansion, smoothed out any irregularities in the initial state of the universe and yielded the uniformity we observe wherever we look. Again: “not so fast.” As Penrose describes, inflation (which he finds dubious due to the lack of a plausible theory of what caused it and resulted in the state we observe today) explains what we observe in the cosmic background radiation, but it does nothing to solve the mystery of why the distribution of mass-energy in the universe was so uniform or, in other words, why the gravitational degrees of freedom in the universe were not excited. He then goes on to examine what he argues are even more fantastic theories including an infinite number of parallel universes, forever beyond our ability to observe. In a final chapter, Penrose presents his own speculations on how fashion, faith, and fantasy might be replaced by physics: theories which, although they may be completely wrong, can at least be tested in the foreseeable future and discarded if they disagree with experiment or investigated further if not excluded by the results. He suggests that a small effort investigating twistor theory might be a prudent hedge against the fashionable pursuit of string theory, that experimental tests of objective reduction of the wave function due to gravitational effects be investigated as an alternative to the faith that quantum mechanics applies at all scales, and that his conformal cyclic cosmology might provide clues to the special conditions at the big bang which the fantasy of inflation theory cannot. (Penrose's cosmological theory is discussed in detail in Cycles of Time [October 2011]). Eleven mathematical appendices provide an introduction to concepts used in the main text which may be unfamiliar to some readers. A special treat is the author's hand-drawn illustrations. In addition to being a mathematician, physicist, and master of scientific explanation and the English language, he is an inspired artist. The Kindle edition is excellent, with the table of contents, notes, cross-references, and index linked just as they should be.
- Penrose, Roger. The Road to Reality. New York: Alfred A. Knopf, 2005. ISBN 0-679-45443-8.
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This is simply a monumental piece of work. I can't think of any comparable
book published in the last century, or any work with such an ambitious
goal which pulls it off so well. In this book, Roger Penrose presents the
essentials of fundamental physics as understood at the turn of the
century to the intelligent layman in the way working
theoretical physicists comprehend them. Starting with the
Pythagorean theorem, the reader climbs the ladder of mathematical
abstraction to master complex numbers, logarithms, real and complex
number calculus, Fourier decomposition, hyperfunctions, quaternions
and octionions, manifolds and calculus on manifolds, symmetry groups,
fibre bundles and connections, transfinite numbers, spacetime, Hamiltonians
and Lagrangians, Clifford and Grassman algebras, tensor calculus, and
the rest of the mathematical armamentarium of the theoretical physicist.
And that's before we get to the physics, where classical mechanics
and electrodynamics, special and general relativity, quantum mechanics,
and the standard models of particle physics and cosmology are presented
in the elegant and economical notation into which the reader has been
initiated in the earlier chapters.
Authors of popular science books are cautioned that each equation they include (except, perhaps E=mc˛) will halve the sales of their book. Penrose laughs in the face of such fears. In this “big damned fat square book” of 1050 pages of main text, there's an average of one equation per page, which, according to conventional wisdom should reduce readership by a factor of 2−1050 or 8.3×10−317, so the single copy printed would have to be shared by among the 1080 elementary particles in the universe over an extremely long time. But, according to the Amazon sales ranking as of today, this book is number 71 in sales—go figure.
Don't deceive yourself; in committing to read this book you are making a substantial investment of time and brain power to master the underlying mathematical concepts and their application to physical theories. If you've noticed my reading being lighter than usual recently, both in terms of number of books and their intellectual level, it's because I've been chewing through this tome for last two and a half months and it's occupied my cerebral capacity to the exclusion of other works. But I do not regret for a second the time I've spent reading this work and working the exercises, and I will probably make a second pass through it in a couple of years to reinforce the mathematical toolset into my aging neurons. As an engineer whose formal instruction in mathematics ended with differential equations, I found chapters 12–15 to be the “hump”—after making it through them (assuming you've mastered their content), the rest of the book is much more physical and accessible. There's kind of a phase transition between the first part of the book and chapters 28–34. In the latter part of the book, Penrose gives free rein to his own view of fundamental physics, introducing his objective reduction of the quantum state function (OR) by gravity, twistor theory, and a deconstruction of string theory which may induce apoplexy in researchers engaged in that programme. But when discussing speculative theories, he takes pains to identify his own view when it differs from the consensus, and to caution the reader where his own scepticism is at variance with a widely accepted theory (such as cosmological inflation). If you really want to understand contemporary physics at the level of professional practitioners, I cannot recommend this book too highly. After you've mastered this material, you should be able to read research reports in the General Relativity and Quantum Cosmology preprint archives like the folks who write and read them. Imagine if, instead of two or three hundred taxpayer funded specialists, four or five thousand self-educated people impassioned with figuring out how nature does it contributed every day to our unscrewing of the inscrutable. Why, they'll say it's a movement. And that's exactly what it will be.
