ByRalph E. Dratmanon March 2, 2017

This is a pretty amazing book. I'm only a few chapters in, but it is filling in many gaps in my math education, mostly at an intuitive level. Better yet, it is just a sheer pleasure for me to read. Learning a lot while having fun? Not a bad way to spend my retirement.

2 people found this helpful

ByL. Jordan Bickelon September 8, 2015

I really wanted to get more out of this book and the author tried to make it work for those without the maths... but it didn't work for those without the maths. Not enough return on reading effort investment.

ByLegionOnomaMoion March 3, 2015

Writing good popular science requires an extraordinary amount of skill. The only challenge more difficult than making the material interesting without succumbing to sensationalism is finding the line between simplification and accuracy. A book that covers complex analysis in chapter 7 out of 34 and covers graduate level material only qualifies as popular science because of marketing. I purchased Sir Roger Penrose's book early on in my academic career as an undergraduate student just beginning to learn more advanced mathematics & physics. I found that often enough I had to read one or more textbooks just to really understand many chapters, and having done so no longer needed to read the material found in Penrose's voluminous book. The breadth of material covered is staggering, the author's prose delightful, and the genius mind behind the elegant presentation of mathematical physics all to apparent- providing one already knows the subject matter well-enough to make most of the book a condensed treatment of familiar topics.

For those whose knowledge of mathematical & theoretical physics is at the graduate level or beyond, this book provides the perspective of one of the world's foremost authority on innumerable topics, along with a great deal of remedial material. For those who lack such mathematical literacy, most of the book is impenetrable.

For those whose knowledge of mathematical & theoretical physics is at the graduate level or beyond, this book provides the perspective of one of the world's foremost authority on innumerable topics, along with a great deal of remedial material. For those who lack such mathematical literacy, most of the book is impenetrable.

ByWoldon December 26, 2012

I am a practicing scientist (biology) who has decent (not stellar) math skills. I took math through calculus and Physics through E&M.

Nonetheless most of this book is completely inaccessible to me.

Penrose in no Feynman. That about sums it up. I have enjoyed reading Feynman's Lectures on Physics for years and get something out of them each time. Feynman had a pedagogical point behind almost everything he said. There was math, but it was there to illustrate the physics. Most of the lectures were self-contained: if you studied them, you could understand most of what he was saying.

Penrose's book is the opposite . Penrose has no sense of the reader, no apparent desire to communicate-- and often seems to completely forget about the reader and simply babble to himself. There is disappointingly little physics-- it is swallowed up in the math. As many others have posted, you would need to be a graduate student (or PhD ) in math to understand much of this. One often gets the impression that Penrose's goal is to show how brilliant he is. However, he should have used his brilliance to think harder about how to actually explain his thinking to the reader in ways that are comprehensible.

As Feynman once said, if he couldn't explain something to a freshman physics class, then he (Feynman) probably didn't understand it himself. 90% of this book fails that basic test.

However, there are lots of "nuggets" in the book that are quite interesting and can lead one (if you use other sources) to understand various things. And, I am sure it is useful to people who are either smarter than me or who know a lot of math. Thus I give the book two stars. I'm glad the book exists and someone tried to write it. Maybe on a sabbatical someday I will be able to understand more.

Nonetheless most of this book is completely inaccessible to me.

Penrose in no Feynman. That about sums it up. I have enjoyed reading Feynman's Lectures on Physics for years and get something out of them each time. Feynman had a pedagogical point behind almost everything he said. There was math, but it was there to illustrate the physics. Most of the lectures were self-contained: if you studied them, you could understand most of what he was saying.

Penrose's book is the opposite . Penrose has no sense of the reader, no apparent desire to communicate-- and often seems to completely forget about the reader and simply babble to himself. There is disappointingly little physics-- it is swallowed up in the math. As many others have posted, you would need to be a graduate student (or PhD ) in math to understand much of this. One often gets the impression that Penrose's goal is to show how brilliant he is. However, he should have used his brilliance to think harder about how to actually explain his thinking to the reader in ways that are comprehensible.

As Feynman once said, if he couldn't explain something to a freshman physics class, then he (Feynman) probably didn't understand it himself. 90% of this book fails that basic test.

However, there are lots of "nuggets" in the book that are quite interesting and can lead one (if you use other sources) to understand various things. And, I am sure it is useful to people who are either smarter than me or who know a lot of math. Thus I give the book two stars. I'm glad the book exists and someone tried to write it. Maybe on a sabbatical someday I will be able to understand more.

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ByBubbaGon March 26, 2006

This work is definately not just another "golly gee whiz" account of modern science.

It is well written and concise with a presentation of some of the major philosophical problems associated with the concepts presented. The chapters on quantum decoherency and measurement are quite engrossing.

Overall, however the work suffers from a lack of continuity in presentation and this is the work's undoing. Penrose at first presents the mathematical concepts as if the reader only had a basic understanding of algebraic relationships. In places he readily assumes the reader has never heard of such concepts as a derivative or complex notation and presents the matter with a 'compute without understanding' approach. In later chapters, however, he then presents such complex formalisms as Lagrangians and Reimann metrics as if the reader was already quite familiar with the complex mathematical structures neccesary to engage in a meaningfull conversation on the subject. One reaches the point where one cannot help wonder what aduience the book is addressed too. At times Penrose will treat complex ideas with a little rigour but then revert back to the 'turn-the-crank' style.

"The Road to Reality" at times is way beyond the scope of the average layman but still lacks the formal rigour to keep the advanced student interested. Penrose tries to be too much to too many audiences and in so doing the work suffers from a lack of continuity and I suspect many will find it hard to retain an interest from cover-to-cover.

One does not need to read the book in a linear fashion, however and many would probably find it interesting and worthwile to select chapters that are of interest to them.

The last 5 chapters themselves would suffice to make an engrossing work and this is where the book really shines. Penrose presents some very insightful ideas regarding the current state of research efforts in particle physics. Penrose seems to hint that current efforts in this area have stagnated due to economic and beaurocratic red tape. University and government agencies have invested large amounts of capital into particular research methods and scientists have no choice but to be herded into these particular efforts. Massive and expensive accelerators are being built to search for elusive particles that will confirm one particular theory while other approaches are being neglected. Penrose's concluding thesis is an overhaul in research efforts is needed along with an entirely new approach to the problem.

It is well written and concise with a presentation of some of the major philosophical problems associated with the concepts presented. The chapters on quantum decoherency and measurement are quite engrossing.

Overall, however the work suffers from a lack of continuity in presentation and this is the work's undoing. Penrose at first presents the mathematical concepts as if the reader only had a basic understanding of algebraic relationships. In places he readily assumes the reader has never heard of such concepts as a derivative or complex notation and presents the matter with a 'compute without understanding' approach. In later chapters, however, he then presents such complex formalisms as Lagrangians and Reimann metrics as if the reader was already quite familiar with the complex mathematical structures neccesary to engage in a meaningfull conversation on the subject. One reaches the point where one cannot help wonder what aduience the book is addressed too. At times Penrose will treat complex ideas with a little rigour but then revert back to the 'turn-the-crank' style.

"The Road to Reality" at times is way beyond the scope of the average layman but still lacks the formal rigour to keep the advanced student interested. Penrose tries to be too much to too many audiences and in so doing the work suffers from a lack of continuity and I suspect many will find it hard to retain an interest from cover-to-cover.

One does not need to read the book in a linear fashion, however and many would probably find it interesting and worthwile to select chapters that are of interest to them.

The last 5 chapters themselves would suffice to make an engrossing work and this is where the book really shines. Penrose presents some very insightful ideas regarding the current state of research efforts in particle physics. Penrose seems to hint that current efforts in this area have stagnated due to economic and beaurocratic red tape. University and government agencies have invested large amounts of capital into particular research methods and scientists have no choice but to be herded into these particular efforts. Massive and expensive accelerators are being built to search for elusive particles that will confirm one particular theory while other approaches are being neglected. Penrose's concluding thesis is an overhaul in research efforts is needed along with an entirely new approach to the problem.

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ByJyotirmoy Bhattacharyaon August 11, 2008

It is not possible to express the ideas of modern physics without using mathematics very different from what one studies in high school. But a popular physics book can hardly assume more than a high school level of math. Therefore popular physics books are impossible.

Penrose's 'The Road to Reality' is a demonstration of this proposition. Penrose must be congratulated for facing the problem head on, not shying away from the formulae and trying to teach his readers all the mathematics needed. Penrose is more capable than most for such an undertaking, and often he comes up with clever, intuitive ways of explaining difficult concepts. But ultimately the beautifully-crafted intuition collapses due to the lack of a supporting structure of necessary technical details and hard proofs and the reader is left holding fuzzy ideas which he cannot independently apply.

The book would be a great way for a graduate student in physics or mathematics to see the big picture. Others would do well to stick either with less ambitious popularizations or to go straight to the textbooks. For the former, my recommendation would be Penrose's own The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics (Popular Science) while for the latter there is no better place to begin than Singer and Thorpe's Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) and Needham's Visual Complex Analysis.

Penrose's 'The Road to Reality' is a demonstration of this proposition. Penrose must be congratulated for facing the problem head on, not shying away from the formulae and trying to teach his readers all the mathematics needed. Penrose is more capable than most for such an undertaking, and often he comes up with clever, intuitive ways of explaining difficult concepts. But ultimately the beautifully-crafted intuition collapses due to the lack of a supporting structure of necessary technical details and hard proofs and the reader is left holding fuzzy ideas which he cannot independently apply.

The book would be a great way for a graduate student in physics or mathematics to see the big picture. Others would do well to stick either with less ambitious popularizations or to go straight to the textbooks. For the former, my recommendation would be Penrose's own The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics (Popular Science) while for the latter there is no better place to begin than Singer and Thorpe's Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) and Needham's Visual Complex Analysis.

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ByRonald W. Satzon September 28, 2013

Obviously Penrose is a super mathematician. The book is comprehensive and well organized, and the illustrations are very helpful. The first 400 pages concern mathematics, the remaining 650 pages concern physics, both of the microcosmos and the macrocosmos. The equations are very abstruse, with very few actual computations provided, and very little physical data are provided either. As a theoretical physicist and systems/mechanical engineer, I felt obligated to read this work, which took about 10 days to complete. It was worth the effort, because Penrose does critique the existing theories and, in particular, explains the conflict between General Relativity and Quantum Mechanics. He clearly knows that a better theory would be desirable, but doesn't provide one; his own theory, Twistors, has no testable predictions. No one seems to be able to come up with a viable theory of "quantum gravity." What's irritating to me about this work is that Penrose goes through practically every idea in modern theoretical physics--except for the one I've been working with since 1965: the Reciprocal System of Dewey B. Larson. There is no excuse any more for the physics community to keep ignoring this unified, general theory--which is precisely the kind of theory which Penrose says he wants. The Reciprocal System does feature discrete space-time, which Penrose himself advocates. But Penrose will not budge from General Relativity and curved space-time, though all the recent space observations show that the universe is Euclidean. Larson's work and my papers are freely available on the Web; after you read Penrose's book, you might want to look at the Reciprocal System. I do recommend Penrose's book to mathematical physicists as I do think that his critique of the current situation in modern physics is right on. I just think he's ignoring the solution!

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ByMark Weitzmanon April 5, 2005

As an admirer of his previous books ( e.g. shadows of the mind ) I was very disappointed by this massive effort on Penrose's part to describe the physical universe. As a dropout from the Phd. program in theoretical physics at Caltech I still try and keep up with the latest developments in theoretical physics particularly in regards to quantum field theory and string theory. The problem with this book is if you aren't already familiar with the mathematics underlying much of modern physics the book is useless and if you are the book should have been only 200 pages long instead of 1050. Basically I bought the book to get Penrose's views on string theory and the measurement problem in quantum mechanics but here other than some useful criticism he added little to his previous works. Though the book is inexpensive I still feel a condensed version written for those who can really understand this book would have been more useful. For those who are not familiar with the mathematics and the physics presented here dont even think of learning it from this book - it will be impossible.

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Bygoldsbaron November 8, 2007

As an occasional reader of some of the more general books on the universe, I was looking for something with a little more detail, something that didn't run away from math. A quick read of the intro and I thought this would be the perfect book. Penrose claimed that he could help people with limited ability to work with fractions. Fractions?!? No problem I thought.

A few pages later and I was drowning. This isn't simple math. Problem is, he doesn't start from the beginning and build up (which granted would take more like 10,000 pages). Instead, this book jumps hardcore into some serious math. I'm sure this is great for some people and many will deservedly give this book 5 stars. My issue is that it is being sold as a "don't worry, I'll teach you what you need to know" type of book. In reality, the reader better have a serious math background. For the record, my math background was generally statistics and business related math via an undergrad and an MBA program.

A few pages later and I was drowning. This isn't simple math. Problem is, he doesn't start from the beginning and build up (which granted would take more like 10,000 pages). Instead, this book jumps hardcore into some serious math. I'm sure this is great for some people and many will deservedly give this book 5 stars. My issue is that it is being sold as a "don't worry, I'll teach you what you need to know" type of book. In reality, the reader better have a serious math background. For the record, my math background was generally statistics and business related math via an undergrad and an MBA program.

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ByKen Braithwaiteon November 24, 2008

Alas I must give thumbs down to this book. If you know the material then it can be a useful roadmap with some interesting highlights, but when he gets into an area you don't already know the book has little value. He doesn't explain the math, so it come across as a list of facts. Does a list of facts about tensors really sound useful? Compare this to the brilliant job Kolmogorov and others did in "Mathematics".

Mathematics: Its Content, Methods and Meaning Buy that instead.

Mathematics: Its Content, Methods and Meaning Buy that instead.

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ByMr. R. Brissendenon February 15, 2006

We urgently need to place the scientific enterprise in a more sober perspective where its limitations are laid bare to the layperson. Unfortunately that great proselytizing mathematician Penrose has produced a new but disappointing blockbuster - The Road To Reality.

On the cover it is described as a complete guide to the physical universe. In other words the TOE. Coming from the arch-skeptic Penrose this really is a bit of a shock. He seems to have been got at.

In the book Penrose covers page after page with pure mathematics covering most aspects of modern physics in astonishing detail. For the layman it is a hopeless challenge. For a mathematician it is a joyous read that takes one back to one's student days and beyond, as one recalls Laurent series, line integrals, rings, groups, and the truly magical behaviour of complex numbers. His description of how this stuff becomes physics is a tour de force that few other authors could emulate. But in what way is it a complete description of the physical universe?

In section 28.5, after many hundreds of pages, we find out: the tone starts to change. Here Penrose demolishes comprehensively and convincingly, even for laypersons, the Inflationary Theory - a major corner stone of modern Cosmology. In his demolition job he seems to come very near indeed to saying the inflationary physicists are quite stupid: Humanity starts cheering him on. What sort of guide to the TOE can this be? Well, in the following pages, he proceeds to demolish the physics pretensions of most of the mathematical groundwork painfully laid out in the first 800 pages. Unfortunately his demolition of strings, that dodgy life-raft to which most theoretical physicists now cling, is so lengthy and esoteric it can only really be relished by a specialist like himself. But I have no doubt at all that he has hit his intended target.

Penrose continues with a dismal picture of the state of modern experimental measurement. Penrose describes the National debt-sized expenses, the half a lifetime timescales and the consequential scarcity of experiments. He turns next to the research difficulties; the young researchers locked into the latest fad theory as part of some big team effort competing for limited funds; the lack of acid tests; and indeed the putting aside of these vital correctives; he describes the new "theories" that are so flexible they could fit round a pig's ear anyway. The book ends essentially by confessing the universe is a great mystery requiring yet more new insights.

Why on earth then did he label his book a TOE? Thank goodness his skeptical instincts eventually did kick in to provide a long-over-due corrective. But unhappily, far too late for the average reader who would long since have dropped the book onto the coffee table or into a drawer.

Penrose is a pure mathematician and his very welcome iconoclasms can be clearly put down to his non-physics training, which must make him rather irritating to physicists because he also happens to be a very great mathematician. It is extremely unfortunate that he never quite sticks to his guns. There is little doubt that if he were so to choose, he could define clearly the unbridgeable limitations of science.

On the cover it is described as a complete guide to the physical universe. In other words the TOE. Coming from the arch-skeptic Penrose this really is a bit of a shock. He seems to have been got at.

In the book Penrose covers page after page with pure mathematics covering most aspects of modern physics in astonishing detail. For the layman it is a hopeless challenge. For a mathematician it is a joyous read that takes one back to one's student days and beyond, as one recalls Laurent series, line integrals, rings, groups, and the truly magical behaviour of complex numbers. His description of how this stuff becomes physics is a tour de force that few other authors could emulate. But in what way is it a complete description of the physical universe?

In section 28.5, after many hundreds of pages, we find out: the tone starts to change. Here Penrose demolishes comprehensively and convincingly, even for laypersons, the Inflationary Theory - a major corner stone of modern Cosmology. In his demolition job he seems to come very near indeed to saying the inflationary physicists are quite stupid: Humanity starts cheering him on. What sort of guide to the TOE can this be? Well, in the following pages, he proceeds to demolish the physics pretensions of most of the mathematical groundwork painfully laid out in the first 800 pages. Unfortunately his demolition of strings, that dodgy life-raft to which most theoretical physicists now cling, is so lengthy and esoteric it can only really be relished by a specialist like himself. But I have no doubt at all that he has hit his intended target.

Penrose continues with a dismal picture of the state of modern experimental measurement. Penrose describes the National debt-sized expenses, the half a lifetime timescales and the consequential scarcity of experiments. He turns next to the research difficulties; the young researchers locked into the latest fad theory as part of some big team effort competing for limited funds; the lack of acid tests; and indeed the putting aside of these vital correctives; he describes the new "theories" that are so flexible they could fit round a pig's ear anyway. The book ends essentially by confessing the universe is a great mystery requiring yet more new insights.

Why on earth then did he label his book a TOE? Thank goodness his skeptical instincts eventually did kick in to provide a long-over-due corrective. But unhappily, far too late for the average reader who would long since have dropped the book onto the coffee table or into a drawer.

Penrose is a pure mathematician and his very welcome iconoclasms can be clearly put down to his non-physics training, which must make him rather irritating to physicists because he also happens to be a very great mathematician. It is extremely unfortunate that he never quite sticks to his guns. There is little doubt that if he were so to choose, he could define clearly the unbridgeable limitations of science.

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ByRamon March 19, 2007

Like many other physicists, I consider Penrose's achievements in proving singularity theorems, and his work with Hawking on black holes as a crowing achievement of classical gravity. It may be safely said that Penrose brought tools into General Relativity that we needed desperately in the 60s and 70s.

I also hugely enjoyed reading his popular book "Emperor's New Mind" so I decided to read up his new book "Road to Reality". Most of the book deals with basic mathematics which may be useful for laymen, but I cannot comment on those parts since I skipped them entirely. My comments pertain to aspects of Physics that Penrose discusses, specifically when he expounds on EPR paradox, state vector reduction, gravity's possible role in state reduction, and in later chapters when Penrose expresses his misgivings against string theory. First I must say that a majority of the discussion on state vector reduction in this book is very simplistic and Penrose's misgivings in this regard are his alone, and not shared by most of the Physics community. For a very good reason too, as there is absolutely no reason for any correlation between gravity and state vector reduction. More serious errors abound in his discussion of string theory. I am actually surprised that Penrose did not bother to run this section by string theory experts who would have been able to disabuse him of some very basic misunderstandings. For example, a very basic claim that Penrose makes is that extra dimensions are unstable...The simple version of the argument Penrose gives for this statement is essentially this: consider a space-time that is a product of four flat dimensions R^4 and six curled up dimensions in the form of a compact manifold X(a Calabi-yau manifold if you want supersymmetry, but this is inessential for Penrose's arguement). This is a solution to vacuum Einstien equations if X is Ricci-flat. Now consider time dependent perturbations of X. Then X evolves as a function of time like a closed universe, and from a theorem of Hawking and Penrose such a closed universe always evolves to a singularity if the perturbation is what is termed as generic. The upshot is simply that a typical perturbation of the internal space leads to singularities in finite "time". So far Penrose is right. What however is the correct conclusion? Penrose concludes that extra dimensions are unstable. Unfortunately this is a wrong conclusion. First of all, Poincare invariant perturbations are not of this class, so in taking an extra dimensional theory as a background and perturbing around this background we will never see the instability Penrose is worried about. However, in a cosmological setting one can justifiably ask what happens if we perturb the internal space the way Penrose wishes. In this case something interesting happens: even though the space-time evolves to a singularity in the classical approximation, this does not happen in the quantum theory. The moduli space approximation is valid at all times, and if we properly account for degres of freedom that become light near the singular regions, we can continue the evolution in moduli space. This can be shown in string theory, and there are analogs of this phenomenon in gauge theory. The upshot therefore is that Penrose's misgivings regarding the "instability" of extra dimensions is a misunderstanding of the tractable quantum effects that resolve such singularities.

A second objection of Penrose is his viewpoint on computation of black hole entropy in string theory. Determining the microstates of black holes has been one of the holy grails of Quantum Gravity, and String theory made a huge breakthrough in this problem circa 1995 when Strominger and Vafa (building on work by Sen and Susskind among others) were able to account for the entropy of extremal black holes by counting microstates of a collection of D-branes. Later the work was extended by other to near extremal black holes. In every case computed so far the agreement between Bekenstein-Hawking expression for entropy and the string theory calculation is remarkable. What is more, of late it has been possible to compute higher order corrections to black hole entropy (in some cases to all orders in perturbation theory) and once more, there is precise agreement. Ok, what does Penrose think about this? Penrose believes that maybe the math works for some unknown reason, but the physics may be incorrect...this would be laughable if it did not originate from a brilliant mathematical physicist like Penrose.

A separate misgiving Penrose expresses is the fact that all calculations of black hole entropy so far have been performed for extremal or near extremal black holes, which are qualitatively different from Schwarzschild black holes (for example specific heat is positive for near extremal black holes). First of all, extremal black holes are the simplest objects to compute in string theory simply because a certain amount of supersymmetry is preserved by these black holes and supersymmetry typically lets us analytically continue in coupling without encountering phase transitions, allowing us to typically match up microstates of D-branes and the corresponding black holes. Near extremal black holes break supersymmetry but still entropy computations work, leading us to believe that there is no phase transition between the D-brane phases and the black hole phase even in the non supersymmetric case. Schwarzschild black holes of course are much harder to model in string theory. String theorists believe this to be a technical difficulty which we are trying to solve. At this stage there is no reason to believe there is a fundamental difference between the near extremal and far from extremal black holes. Certainly Penrose does not provide any input to this question. I can go on discussing the flaws in this book but for reasons of time and space I have to terminate my review here. I think this is a nice book to learn some of the math behind modern physics but this book pushes the author's misguided ideas a whole lot more than the space such ideas deserve.

The readers will benifit from this book as long as they keep in mind that certain parts of the book are outdated and some parts are wrong entirely. If you are trying to understand the modern crisis in particle theory and (or ) want to understand in layman's terms the progress already achieved in quantum gravity and field theory this is not the book for you.

I also hugely enjoyed reading his popular book "Emperor's New Mind" so I decided to read up his new book "Road to Reality". Most of the book deals with basic mathematics which may be useful for laymen, but I cannot comment on those parts since I skipped them entirely. My comments pertain to aspects of Physics that Penrose discusses, specifically when he expounds on EPR paradox, state vector reduction, gravity's possible role in state reduction, and in later chapters when Penrose expresses his misgivings against string theory. First I must say that a majority of the discussion on state vector reduction in this book is very simplistic and Penrose's misgivings in this regard are his alone, and not shared by most of the Physics community. For a very good reason too, as there is absolutely no reason for any correlation between gravity and state vector reduction. More serious errors abound in his discussion of string theory. I am actually surprised that Penrose did not bother to run this section by string theory experts who would have been able to disabuse him of some very basic misunderstandings. For example, a very basic claim that Penrose makes is that extra dimensions are unstable...The simple version of the argument Penrose gives for this statement is essentially this: consider a space-time that is a product of four flat dimensions R^4 and six curled up dimensions in the form of a compact manifold X(a Calabi-yau manifold if you want supersymmetry, but this is inessential for Penrose's arguement). This is a solution to vacuum Einstien equations if X is Ricci-flat. Now consider time dependent perturbations of X. Then X evolves as a function of time like a closed universe, and from a theorem of Hawking and Penrose such a closed universe always evolves to a singularity if the perturbation is what is termed as generic. The upshot is simply that a typical perturbation of the internal space leads to singularities in finite "time". So far Penrose is right. What however is the correct conclusion? Penrose concludes that extra dimensions are unstable. Unfortunately this is a wrong conclusion. First of all, Poincare invariant perturbations are not of this class, so in taking an extra dimensional theory as a background and perturbing around this background we will never see the instability Penrose is worried about. However, in a cosmological setting one can justifiably ask what happens if we perturb the internal space the way Penrose wishes. In this case something interesting happens: even though the space-time evolves to a singularity in the classical approximation, this does not happen in the quantum theory. The moduli space approximation is valid at all times, and if we properly account for degres of freedom that become light near the singular regions, we can continue the evolution in moduli space. This can be shown in string theory, and there are analogs of this phenomenon in gauge theory. The upshot therefore is that Penrose's misgivings regarding the "instability" of extra dimensions is a misunderstanding of the tractable quantum effects that resolve such singularities.

A second objection of Penrose is his viewpoint on computation of black hole entropy in string theory. Determining the microstates of black holes has been one of the holy grails of Quantum Gravity, and String theory made a huge breakthrough in this problem circa 1995 when Strominger and Vafa (building on work by Sen and Susskind among others) were able to account for the entropy of extremal black holes by counting microstates of a collection of D-branes. Later the work was extended by other to near extremal black holes. In every case computed so far the agreement between Bekenstein-Hawking expression for entropy and the string theory calculation is remarkable. What is more, of late it has been possible to compute higher order corrections to black hole entropy (in some cases to all orders in perturbation theory) and once more, there is precise agreement. Ok, what does Penrose think about this? Penrose believes that maybe the math works for some unknown reason, but the physics may be incorrect...this would be laughable if it did not originate from a brilliant mathematical physicist like Penrose.

A separate misgiving Penrose expresses is the fact that all calculations of black hole entropy so far have been performed for extremal or near extremal black holes, which are qualitatively different from Schwarzschild black holes (for example specific heat is positive for near extremal black holes). First of all, extremal black holes are the simplest objects to compute in string theory simply because a certain amount of supersymmetry is preserved by these black holes and supersymmetry typically lets us analytically continue in coupling without encountering phase transitions, allowing us to typically match up microstates of D-branes and the corresponding black holes. Near extremal black holes break supersymmetry but still entropy computations work, leading us to believe that there is no phase transition between the D-brane phases and the black hole phase even in the non supersymmetric case. Schwarzschild black holes of course are much harder to model in string theory. String theorists believe this to be a technical difficulty which we are trying to solve. At this stage there is no reason to believe there is a fundamental difference between the near extremal and far from extremal black holes. Certainly Penrose does not provide any input to this question. I can go on discussing the flaws in this book but for reasons of time and space I have to terminate my review here. I think this is a nice book to learn some of the math behind modern physics but this book pushes the author's misguided ideas a whole lot more than the space such ideas deserve.

The readers will benifit from this book as long as they keep in mind that certain parts of the book are outdated and some parts are wrong entirely. If you are trying to understand the modern crisis in particle theory and (or ) want to understand in layman's terms the progress already achieved in quantum gravity and field theory this is not the book for you.

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