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A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity
Based on the concept of four-dimensional spacetime - curved in the vicinity of mass-energy, flat in its absence - Einstein's theories of special and general relativity together form a cornerstone of modern physics. Special relativity has some strangely counter-intuitive consequences, including time dilation, length contraction, the relativity of simultaneity and mass-energy equivalence, whilst general relativity is at the heart of our understanding of black holes and the evolution of the universe.
Using straightforward, accessible language, with numerous fully solved problems and clear derivations and explanations, this book is aimed at the enthusiastic general reader who wants to move beyond maths-lite popularisations and tackle the essential mathematics of this fascinating theory. (To paraphrase Euclid, there is no royal road to relativity - you have to do the mathematics.) For those with minimal mathematical background, the first chapter provides a crash course in foundation mathematics. The reader is then taken gently by the hand and guided through a wide range of fundamental topics, including Newtonian mechanics; the Lorentz transformations; tensor calculus; the Einstein field equations; the Schwarzschild solution; the four classical tests of general relativity; simple black holes; the mysteries of dark energy and the cosmological constant; and the Friedmann equations and Friedmann-Robertson-Walker cosmological models.
Understand even the basics of Einstein's amazing theory and the world will never seem the same again.
Contents
Preface
Introduction
1 Foundation mathematics
2 Newtonian mechanics
3 Special relativity
4 Introducing the manifold
5 Scalars, vectors, one-forms and tensors
6 More on curvature
7 General relativity
8 The Newtonian limit
9 The Schwarzschild metric
10 Schwarzschild black holes
11 Cosmology
Bibliography
Appendix - Planetary motion data
Acknowledgements
This second edition consists of the original first edition text plus all those corrections and clarifications to the first edition listed in the online errata sheet.
Using straightforward, accessible language, with numerous fully solved problems and clear derivations and explanations, this book is aimed at the enthusiastic general reader who wants to move beyond maths-lite popularisations and tackle the essential mathematics of this fascinating theory. (To paraphrase Euclid, there is no royal road to relativity - you have to do the mathematics.) For those with minimal mathematical background, the first chapter provides a crash course in foundation mathematics. The reader is then taken gently by the hand and guided through a wide range of fundamental topics, including Newtonian mechanics; the Lorentz transformations; tensor calculus; the Einstein field equations; the Schwarzschild solution; the four classical tests of general relativity; simple black holes; the mysteries of dark energy and the cosmological constant; and the Friedmann equations and Friedmann-Robertson-Walker cosmological models.
Understand even the basics of Einstein's amazing theory and the world will never seem the same again.
Contents
Preface
Introduction
1 Foundation mathematics
2 Newtonian mechanics
3 Special relativity
4 Introducing the manifold
5 Scalars, vectors, one-forms and tensors
6 More on curvature
7 General relativity
8 The Newtonian limit
9 The Schwarzschild metric
10 Schwarzschild black holes
11 Cosmology
Bibliography
Appendix - Planetary motion data
Acknowledgements
This second edition consists of the original first edition text plus all those corrections and clarifications to the first edition listed in the online errata sheet.
334 pages, Paperback
First published January 1, 2012
354 people are currently reading
810 people want to read
About the author
Peter Collier
1 book11 followersLibrarian Note: There is more than one author in the Goodreads database with this name.
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Displaying 1 - 30 of 31 reviews
December 1, 2015
I always feel a bit iffy when I begin a review for a book I didn’t fully understand. There's a fine line between unfairly condemning an author for one’s own faults as a reader, and fairly criticizing a would-be teacher for their genuine shortcomings. Learning, after all, is normally a shared endeavor; and to be successful, it requires skill (though different kinds of skill) on the part of both the master and the pupil. In practice, I’m not sure it’s possible to separate out the contributions of either party; so the best we can do is attempt to be fair.
Peter Collier is, to the best of my knowledge, not a physicist; and I’m not sure if he even studied very much physics in university. He does appear, however, to have some math and science background, and generally comes across as a bright fellow. This book is thus not a tool developed by a professional, but the self-published work of an amateur. Collier taught himself general relativity; here, he attempts to teach us.
It must be said, before any evaluation of the book itself, that this book fills a vacuum in the market. There are plenty of popular science books on Einstein; and there are plenty of textbooks on relativity. But I’m not aware of another book that attempts to bridge this gap between the math-less popularizations and the mathematically advanced textbooks. As an added bonus, the book is fairly cheap—especially compared to any real textbook—so, regardless of execution, I’d say this books fills an important gap.
Collier begins with a whirlwind tour of basic math. To many people, I suspect, some of the concepts that he covers will seem ludicrously basic. For example, how many people don’t know what infinity is? Of course, it’s better to assume too little rather than too much; but frankly, if you have never learned algebra or calculus before, there are much, much better books out there to learn from. Collier’s explanations of this basic stuff are, at best, only reminders to rusty readers. (Personally, I have a very weak math and science background, but I spent some time last year refreshing myself with Khan Academy and other suchlike material.)
Soon we find ourselves in Newtonian mechanics, which for me was familiar enough territory. This is the physics you likely learned in high school (or your country’s equivalent). After this, we reach special relativity. Surprisingly, perhaps, you don’t need very difficult math to understand special relativity, which is good for me. And although special relativity requires quite different notions of space and time, I’d say it’s not ultimately that difficult to gain an intuitive understanding of it. So far, so good.
Here is where I ran into trouble. After special relativity, Collier begins introducing the student to the special mathematical tools necessary to understand general relativity. This is not easy; and indeed, from what I’ve heard, even Einstein struggled with it. How do you describe four-dimensional curved spacetime in mathematical language? It’s about as complicated as it sounds; simple things, like distance and time, what it means for a line to be straight, have to be re-examined. The mathematical apparatus used by Einstein in handling these problems is known as tensor calculus, and it’s no cakewalk.
Though no doubt I bear some responsibility for this, I think it’s a bad sign for Collier’s teaching method that, as soon as the math exceeded what I understood prior to opening the book, I felt confused; and even now, I wouldn’t say I have a firm grasp of tensor calculus. In fact, I got the impression that Collier’s own understanding of the math was a bit superficial. I suspect an expert would have been able to explain it more lucidly and succinctly. Collier’s explanations are, by contrast, sometimes a bit messy and confused. At the very least, they lack elegance.
The book improves after Collier finally gets to Einstein’s field equations, since the rest of the book (with the exception of the final chapter) just toys around with these equations. We see why Einstein’s equations approximate to Newton’s equations in weak, static gravitational fields at speeds much slower than light; and then, we get to see some of the classical tests of general relativity—red-shifted light, gravitational time-delay, the procession in the perihelion of Mercury. We move on from there to black holes, and the book ends with a chapter on cosmology. I found this last chapter rather unrelated to the rest of the book. It's mainly concerned with determining the curvature of the universe, which is indeed very interesting; but the presentation was, I found, too brief to convey any adequate understanding of the topic.
Even though there were times I enjoyed this book very much—especially since my reading coincided with the 100-year anniversary of general relativity—I’m not sure I would recommend it. For I don’t know how much I actually learned from this book. I had a decent conceptual (i.e. non-mathematical) understanding of special and general relativity before starting; and now I can say I’ve got a taste of how physicists make precise predictions with these theories. But do I understand them on a deeper level? I’m not certain.
From time to time, in fact, I found myself wondering whether learning the mathematics behind (say) general relativity or quantum mechanics is worthwhile for the layreader. Assuming that the reader wishes, as I do, to deepen her understanding of the universe, we are led to this question: are the equations of physics merely practical tools for making predictions, or are they conceptual aids to the understanding? Of course, the sensible answer is probably ‘both’, and I’m sure it depends on the person as well. For my part, even when I’m following the math easily enough, I still mentally translate the symbols into some sort of mental picture or idea. I admit, seeing the equations does help me, for example, precisely understand how one variable will affect another; but I’d say even in this case my “knowledge” of the physics in question is not ultimately mathematical.
This is likely more an indication of my lack of practice that anything else. Thinking with equations is laborious for me; words and pictures are much easier. Still, if the ability to think ably along mathematical lines is only attainable through much practice, we are lead to question books of this sort, which aim to circumvent all the practice while retaining the math. Nevertheless, many people read them, including myself, so perhaps they do help the amateur student achieve a deeper understanding. Again, I'm sure it depends much on the student. But unfortunately for any prospective buyers of this book, you can only figure out what kind of student you are by diving in and seeing what happens.
As a side note, my tenuous conclusion regarding books of this sort is that it's probably unnecessary for a layreader to learn the actual field equations that working physicists use. Much better, I think, is the approach taken by Feynman in QED, where instead of teaching the reader the equations physicists use to calculate answers efficiently, he reveals what's happening behind these equations, which is I think far more interesting, if far less efficient for any would-be experimenter. Of course, comparisons to Feynman are a bit unfair in this regard, since he was so damned good at explaining things, so I won't hold it against Collier.
Peter Collier is, to the best of my knowledge, not a physicist; and I’m not sure if he even studied very much physics in university. He does appear, however, to have some math and science background, and generally comes across as a bright fellow. This book is thus not a tool developed by a professional, but the self-published work of an amateur. Collier taught himself general relativity; here, he attempts to teach us.
It must be said, before any evaluation of the book itself, that this book fills a vacuum in the market. There are plenty of popular science books on Einstein; and there are plenty of textbooks on relativity. But I’m not aware of another book that attempts to bridge this gap between the math-less popularizations and the mathematically advanced textbooks. As an added bonus, the book is fairly cheap—especially compared to any real textbook—so, regardless of execution, I’d say this books fills an important gap.
Collier begins with a whirlwind tour of basic math. To many people, I suspect, some of the concepts that he covers will seem ludicrously basic. For example, how many people don’t know what infinity is? Of course, it’s better to assume too little rather than too much; but frankly, if you have never learned algebra or calculus before, there are much, much better books out there to learn from. Collier’s explanations of this basic stuff are, at best, only reminders to rusty readers. (Personally, I have a very weak math and science background, but I spent some time last year refreshing myself with Khan Academy and other suchlike material.)
Soon we find ourselves in Newtonian mechanics, which for me was familiar enough territory. This is the physics you likely learned in high school (or your country’s equivalent). After this, we reach special relativity. Surprisingly, perhaps, you don’t need very difficult math to understand special relativity, which is good for me. And although special relativity requires quite different notions of space and time, I’d say it’s not ultimately that difficult to gain an intuitive understanding of it. So far, so good.
Here is where I ran into trouble. After special relativity, Collier begins introducing the student to the special mathematical tools necessary to understand general relativity. This is not easy; and indeed, from what I’ve heard, even Einstein struggled with it. How do you describe four-dimensional curved spacetime in mathematical language? It’s about as complicated as it sounds; simple things, like distance and time, what it means for a line to be straight, have to be re-examined. The mathematical apparatus used by Einstein in handling these problems is known as tensor calculus, and it’s no cakewalk.
Though no doubt I bear some responsibility for this, I think it’s a bad sign for Collier’s teaching method that, as soon as the math exceeded what I understood prior to opening the book, I felt confused; and even now, I wouldn’t say I have a firm grasp of tensor calculus. In fact, I got the impression that Collier’s own understanding of the math was a bit superficial. I suspect an expert would have been able to explain it more lucidly and succinctly. Collier’s explanations are, by contrast, sometimes a bit messy and confused. At the very least, they lack elegance.
The book improves after Collier finally gets to Einstein’s field equations, since the rest of the book (with the exception of the final chapter) just toys around with these equations. We see why Einstein’s equations approximate to Newton’s equations in weak, static gravitational fields at speeds much slower than light; and then, we get to see some of the classical tests of general relativity—red-shifted light, gravitational time-delay, the procession in the perihelion of Mercury. We move on from there to black holes, and the book ends with a chapter on cosmology. I found this last chapter rather unrelated to the rest of the book. It's mainly concerned with determining the curvature of the universe, which is indeed very interesting; but the presentation was, I found, too brief to convey any adequate understanding of the topic.
Even though there were times I enjoyed this book very much—especially since my reading coincided with the 100-year anniversary of general relativity—I’m not sure I would recommend it. For I don’t know how much I actually learned from this book. I had a decent conceptual (i.e. non-mathematical) understanding of special and general relativity before starting; and now I can say I’ve got a taste of how physicists make precise predictions with these theories. But do I understand them on a deeper level? I’m not certain.
From time to time, in fact, I found myself wondering whether learning the mathematics behind (say) general relativity or quantum mechanics is worthwhile for the layreader. Assuming that the reader wishes, as I do, to deepen her understanding of the universe, we are led to this question: are the equations of physics merely practical tools for making predictions, or are they conceptual aids to the understanding? Of course, the sensible answer is probably ‘both’, and I’m sure it depends on the person as well. For my part, even when I’m following the math easily enough, I still mentally translate the symbols into some sort of mental picture or idea. I admit, seeing the equations does help me, for example, precisely understand how one variable will affect another; but I’d say even in this case my “knowledge” of the physics in question is not ultimately mathematical.
This is likely more an indication of my lack of practice that anything else. Thinking with equations is laborious for me; words and pictures are much easier. Still, if the ability to think ably along mathematical lines is only attainable through much practice, we are lead to question books of this sort, which aim to circumvent all the practice while retaining the math. Nevertheless, many people read them, including myself, so perhaps they do help the amateur student achieve a deeper understanding. Again, I'm sure it depends much on the student. But unfortunately for any prospective buyers of this book, you can only figure out what kind of student you are by diving in and seeing what happens.
As a side note, my tenuous conclusion regarding books of this sort is that it's probably unnecessary for a layreader to learn the actual field equations that working physicists use. Much better, I think, is the approach taken by Feynman in QED, where instead of teaching the reader the equations physicists use to calculate answers efficiently, he reveals what's happening behind these equations, which is I think far more interesting, if far less efficient for any would-be experimenter. Of course, comparisons to Feynman are a bit unfair in this regard, since he was so damned good at explaining things, so I won't hold it against Collier.
January 5, 2016
This book is astonishing, really. A non-physicist decided to teach himself relativity theory, using some text books and the full range of publicly available online resources. That's no mean feat. But then he did something really brave - arguably crazy as all truly brave acts are - and wrote a book for other people who are attempting to do the same thing.
If you've followed my status updates you might be thinking it's just a catalogue of errors but that would not be fair; most of the mistakes I found are simple slip-ups, probably typos in several cases. That type of thing occurs alarmingly often even in texts by professionals that have been checked by professionals. The trouble here is that they may cause more trouble for the intended audience than they would for people with considerable mathematical/scientific training.
More worrying are the small number of errors that show lack of understanding of the physics and/or maths, as these are likely to confuse and mislead the intended audience. I suggest that readers keep an eye on the online errata page, which the author has successively updated in the light of reader comments.
I feel that anyone without a physics degree but wanting to learn relativity theory in all its mathematical glory should read Relativity: The Special and the General Theory before attempting this book and perhaps following up with a professional introductory text on the General Theory if sufficient interest remains.
If you've followed my status updates you might be thinking it's just a catalogue of errors but that would not be fair; most of the mistakes I found are simple slip-ups, probably typos in several cases. That type of thing occurs alarmingly often even in texts by professionals that have been checked by professionals. The trouble here is that they may cause more trouble for the intended audience than they would for people with considerable mathematical/scientific training.
More worrying are the small number of errors that show lack of understanding of the physics and/or maths, as these are likely to confuse and mislead the intended audience. I suggest that readers keep an eye on the online errata page, which the author has successively updated in the light of reader comments.
I feel that anyone without a physics degree but wanting to learn relativity theory in all its mathematical glory should read Relativity: The Special and the General Theory before attempting this book and perhaps following up with a professional introductory text on the General Theory if sufficient interest remains.
January 1, 2019
The book lives up to its name. It offers what it promises and is , indeed, quite a gentle introduction, as gentle as the subject matter allows it to be. Peter Collier is a guy like you and me. He is not a physicist. But he is someone who is passionate about learning physics. So he decided to teach himself special and general relativity. This book is precisely for those of us like him who want to become self-taught. On one hand, most popular science books lack substance. On the other hand, most academic textbooks are too heavy to read for the likes of us. And this book is the golden mean or the Goldilock Zone between these two for the beginners.
April 9, 2015
This book mantains what it promises. Almost one half of the book gives too much room to introductory stuff, I mean very basic math, and I found it useless, as I think, one willing to face GR has to have the very basics in his background. So, I had the impression the author wanted to emulate Leonard Susskind's style without having the same capability and effectiveness. On the other hand, the central part of the book up to the second-last chapter is very good, it makes tricky things really easy to grasp. That is why I recommend this book. I didn't like the last chapter, as the material was presented without giving the rationale for it. I think most of the effort spent in teaching trivial math could have been better spent in the clarification of the part dedicated to Cosmology.
November 26, 2013
Well, it lives up to its name! Much of this proved incomprehensible to me. I do not blame the author at all, however, as this is an excellent run through the complexity of calculus, tensors, and matrices that are needed to appreciate modern cosmology, special, and general relativity. The math is very dense. That said, beginning as I did with a rudimentary understanding of all the topics covered, the cantor through the maths provided an appreciation for what was involved. I confess to skimming some of the more dense Tensor calculus. Excellent book for the committed reader. I would supplement it with the online Susskind lectures called the theoretical minimum.
October 11, 2014
The vector calculus is pretty intense, but it's nice to go beyond the coverage of the popular physics books.
December 30, 2025
The title comes from Einstein's famous quote: "The most incomprehensible thing about the world is that it is at all comprehensible."
The author, Peter Collier, uses this phrase to capture several meanings: (1) The mysterious, mind-bending nature of relativity theory itself - a theory so counterintuitive that even our everyday intuitions about space and time completely break down. (2) How despite this incredible complexity, mathematics somehow manages to describe the universe with stunning accuracy. (3) A humble admission about how difficult the subject matter is to grasp, and a reflection on the author's own learning journey when he realized that popular science books had only scratched the surface of understanding relativity.
A Most Incomprehensible Thing" by Peter Collier (revised edition 2019) is not a light read for casual curiosity. Rather, this is a serious, mathematically rigorous introduction to Einstein's theory of relativity intended for the dedicated self-learner. What makes it extraordinary is that it somehow manages to be both mathematically honest and genuinely accessible. Collier bridges the gap between over-simplified popular science books on one side, and impenetrably dense university textbooks on the other. The result is something rare: a book that treats you like an intelligent person who is willing to work hard, rather than talking down to you.
The author spent twelve months obsessively learning relativity on his own (between 2010-2011) after watching Stanford University physics lectures by Leonard Susskind. Frustrated by not finding a "Goldilocks" textbook pitched at just the right level, he decided to write one. The book reads like a letter from a fellow traveler who has just completed a difficult journey and wants to help others walk the same path.
What Makes This Book Exceptional
★ The foundation chapter is extraordinary. Rather than assuming readers know mathematics, it teaches every concept needed, from basic functions to calculus to tensors, with full worked examples.
★ Unlike many textbooks that say "it can be shown that..." and skip the work, Collier shows every step. This means you actually understand how equations are derived, not just that they're true.
★ The author acknowledges struggling with these concepts himself. He reminds readers that Einstein took ten years to develop general relativity. Permission to struggle is psychologically important.
★ Nearly every section includes fully solved problems. Seeing how mathematics is applied in practice is the best way to understand it.
★ Despite technical depth, the writing is clear and often witty. The author avoids jargon when possible and explains jargon when necessary.
★ The book clearly shows how general relativity reduces to special relativity, which reduces to Newtonian mechanics in appropriate limits. You understand the hierarchy of physical theories.
Perfect For:
People genuinely curious about how gravity and the universe work
Physics students wanting deeper understanding before university courses
Self-educated people pursuing knowledge for its own sake
Anyone willing to invest genuine effort in understanding
Not Suitable For:
Casual readers wanting entertainment
People expecting to learn relativity in an afternoon
Those uncomfortable with mathematics
People wanting intuition without calculations
My rating:5/5 stars
The author, Peter Collier, uses this phrase to capture several meanings: (1) The mysterious, mind-bending nature of relativity theory itself - a theory so counterintuitive that even our everyday intuitions about space and time completely break down. (2) How despite this incredible complexity, mathematics somehow manages to describe the universe with stunning accuracy. (3) A humble admission about how difficult the subject matter is to grasp, and a reflection on the author's own learning journey when he realized that popular science books had only scratched the surface of understanding relativity.
A Most Incomprehensible Thing" by Peter Collier (revised edition 2019) is not a light read for casual curiosity. Rather, this is a serious, mathematically rigorous introduction to Einstein's theory of relativity intended for the dedicated self-learner. What makes it extraordinary is that it somehow manages to be both mathematically honest and genuinely accessible. Collier bridges the gap between over-simplified popular science books on one side, and impenetrably dense university textbooks on the other. The result is something rare: a book that treats you like an intelligent person who is willing to work hard, rather than talking down to you.
The author spent twelve months obsessively learning relativity on his own (between 2010-2011) after watching Stanford University physics lectures by Leonard Susskind. Frustrated by not finding a "Goldilocks" textbook pitched at just the right level, he decided to write one. The book reads like a letter from a fellow traveler who has just completed a difficult journey and wants to help others walk the same path.
What Makes This Book Exceptional
★ The foundation chapter is extraordinary. Rather than assuming readers know mathematics, it teaches every concept needed, from basic functions to calculus to tensors, with full worked examples.
★ Unlike many textbooks that say "it can be shown that..." and skip the work, Collier shows every step. This means you actually understand how equations are derived, not just that they're true.
★ The author acknowledges struggling with these concepts himself. He reminds readers that Einstein took ten years to develop general relativity. Permission to struggle is psychologically important.
★ Nearly every section includes fully solved problems. Seeing how mathematics is applied in practice is the best way to understand it.
★ Despite technical depth, the writing is clear and often witty. The author avoids jargon when possible and explains jargon when necessary.
★ The book clearly shows how general relativity reduces to special relativity, which reduces to Newtonian mechanics in appropriate limits. You understand the hierarchy of physical theories.
Perfect For:
People genuinely curious about how gravity and the universe work
Physics students wanting deeper understanding before university courses
Self-educated people pursuing knowledge for its own sake
Anyone willing to invest genuine effort in understanding
Not Suitable For:
Casual readers wanting entertainment
People expecting to learn relativity in an afternoon
Those uncomfortable with mathematics
People wanting intuition without calculations
My rating:5/5 stars
December 15, 2020
There is a difference between knowing the steps required to ride a bike – start the bike moving a bit, hop on, start pedaling – and knowing how to actually ride a bike. I was reminded of this distinction reading A Most Incomprehensible Thing by Peter Collier. Collier’s book is ambitious. His goal is to take a reader who knows no more than basic arithmetic to an understanding of Einstein’s field equations in less than 200 pages.
This is an impossible task. It means that the first 82 pages of the book are a review of basic primary to high school math and physics, beginning with the concept of a function and trigonometry identities, moving up to multivariate calculus. This treatment takes years of development when it taught in schools. If someone did not understand these basics, the book would move far too fast. On the other hand, for a reader who has understood these basics, these sections are superfluous.
Because so much of the book was taken up by the math primer, once Collier gets to tensor calculus, which was new for me, I felt he moved far too fast. Collier seems to believe that the knowledge is knowing the rules for manipulating algebraic objects. Several times he writes in capital letters that calculations in the text are not necessary to actually manipulate tensors, seemingly giving the reader permission to not understand the deeper meaning behind calculations.
After studying the mere thirty-two pages which go from introducing tensors to geodesics, parallel transport, and the Riemann curvature tensor, I felt I couldn’t continue reading without turning to an outside source. I could understand the rules for manipulating tensors. I could read the Riemann curvature tensor, but Collier does not explain where it comes from (there is an appendix where he derives the Riemann tensor, but to my taste the derivation there did not contain any additional intuition). Understanding this tensor, and in particular its contraction, the Ricci tensor, is fundamental to understanding the Einstein field equations, as it makes up their left-hand sides.
General relatively is fundamentally about the geometry of space-time. While physicists describe the geometry using algebraic symbols, there is strong geometric intuition behind the objects they study. I personally consider this intuition as important for understanding as knowing the rules for manipulation of the algebraic objects involved. In Euclidean space, for example, one-forms can be interpreted as a series of hyperplanes. The dot product of a one-form and a vector tells us the number of hyperplanes pierced by the vector. The reason a contravariant vector is contravariant is that when we lengthen the basis vectors, the vector components shrink. And so forth.
I ended up watching an excellent series of Youtube videos by the user eigenchris on tensors (specifically his “Tensors for beginners” series and his “Tensor calculus” series). Just like the section of A Most Incomprehensible Thing on tensors, the videos in this series start by introducing tensors and end with the Riemann curvature tensor. This long series of videos was exactly what I was looking for. After watching these videos I now feel like I understand where the curvature tensor comes from and why it is important for understanding the intrinsic curvature of space-time.
I tried to go back to Collier after watching the video series, but I found his subsequent development of the momentum tensor – the right-hand sides of the Einstein field equations – equally opaque. So I did not finish the book. Instead I am waiting for eigenchris to finish his new series on relativity. As I write this review, he is finishing up with special relativity, and he releases a new video around once a month.
I salute Collier for his attempt, which according to him grew out of his own curiosity about general relativity, and frustration with the available textbooks, which were at too high a level for a layperson to understand. The hurdle, unfortunately, proved to be too high. Instead, I recommend that someone with an understanding of high school math and physics instead watch eigenchris, and share my anticipation as he progresses toward general relativity.
This is an impossible task. It means that the first 82 pages of the book are a review of basic primary to high school math and physics, beginning with the concept of a function and trigonometry identities, moving up to multivariate calculus. This treatment takes years of development when it taught in schools. If someone did not understand these basics, the book would move far too fast. On the other hand, for a reader who has understood these basics, these sections are superfluous.
Because so much of the book was taken up by the math primer, once Collier gets to tensor calculus, which was new for me, I felt he moved far too fast. Collier seems to believe that the knowledge is knowing the rules for manipulating algebraic objects. Several times he writes in capital letters that calculations in the text are not necessary to actually manipulate tensors, seemingly giving the reader permission to not understand the deeper meaning behind calculations.
After studying the mere thirty-two pages which go from introducing tensors to geodesics, parallel transport, and the Riemann curvature tensor, I felt I couldn’t continue reading without turning to an outside source. I could understand the rules for manipulating tensors. I could read the Riemann curvature tensor, but Collier does not explain where it comes from (there is an appendix where he derives the Riemann tensor, but to my taste the derivation there did not contain any additional intuition). Understanding this tensor, and in particular its contraction, the Ricci tensor, is fundamental to understanding the Einstein field equations, as it makes up their left-hand sides.
General relatively is fundamentally about the geometry of space-time. While physicists describe the geometry using algebraic symbols, there is strong geometric intuition behind the objects they study. I personally consider this intuition as important for understanding as knowing the rules for manipulation of the algebraic objects involved. In Euclidean space, for example, one-forms can be interpreted as a series of hyperplanes. The dot product of a one-form and a vector tells us the number of hyperplanes pierced by the vector. The reason a contravariant vector is contravariant is that when we lengthen the basis vectors, the vector components shrink. And so forth.
I ended up watching an excellent series of Youtube videos by the user eigenchris on tensors (specifically his “Tensors for beginners” series and his “Tensor calculus” series). Just like the section of A Most Incomprehensible Thing on tensors, the videos in this series start by introducing tensors and end with the Riemann curvature tensor. This long series of videos was exactly what I was looking for. After watching these videos I now feel like I understand where the curvature tensor comes from and why it is important for understanding the intrinsic curvature of space-time.
I tried to go back to Collier after watching the video series, but I found his subsequent development of the momentum tensor – the right-hand sides of the Einstein field equations – equally opaque. So I did not finish the book. Instead I am waiting for eigenchris to finish his new series on relativity. As I write this review, he is finishing up with special relativity, and he releases a new video around once a month.
I salute Collier for his attempt, which according to him grew out of his own curiosity about general relativity, and frustration with the available textbooks, which were at too high a level for a layperson to understand. The hurdle, unfortunately, proved to be too high. Instead, I recommend that someone with an understanding of high school math and physics instead watch eigenchris, and share my anticipation as he progresses toward general relativity.
January 9, 2016
I had been looking for a book that would give me a mathematical introduction to special and general relativity. I wanted a book that would be rigorous but that would not completely overwhelm my math knowledge, which stops at second year college calculus. For the most part this book delivered. The explanations of special relativity were all completely comprehensible. I got a good understanding of Lorentz transformations and the derivation of E=mc2 was very clear. It got a lot muddier for me when the book got into general relativity. I was not able to fully absorb the Riemann curvature tensor despite rereading the relevant sections several times and supplementing the book with some side reading on the Internet, but I don't really blame the book for this. This is difficult stuff, so it takes more than a bit of casual reading over the Christmas holiday to fully understand. But I did get a good general sense of how the tensor math works and its physical significance for general relativity, which is really all that I could have reasonably expected. A deeper understanding is going to take some further study. This book won't work for a lot of people -- too hard for some, too easy for others, and parts of it were too basic or too advanced for me, but it was mostly just right for my level of math and physics knowledge, so I was very happy to have found it.
April 18, 2017
Ok, you probably won't understand it. That's why you read "a most incomprehensible thing" in the title. Because it is. After reading a lot of Michio Kaku, Stephen Hawking (for dummies), Brian Greene, Lee Smolin, and stuff like that I became more interested into the physics details, i.e.the freaking mathematics. Since I'm not a physicist but a programmer, you can imagine how difficult is to try to grasp this book. So why bother, you may ask? Well, you can't truly admire Mozart without understand music. You can't truly admire Einstein without knowing what he really did. And this means the whole mathematics around his theory.
This book tries to explain in a step by leap-of-faith-step the real thing. Sometimes it is raw, brutal, non romantic way of doing that. This is the "show me the money" book. This is the bad medicine (oh oh oh, shake it up) book. And often you see the author going from A to Z in the formula transformation in a rush, assuming that you did your home work previously explained. This is hard to grasp, but all in all I learned a trick or two.
This book tries to explain in a step by leap-of-faith-step the real thing. Sometimes it is raw, brutal, non romantic way of doing that. This is the "show me the money" book. This is the bad medicine (oh oh oh, shake it up) book. And often you see the author going from A to Z in the formula transformation in a rush, assuming that you did your home work previously explained. This is hard to grasp, but all in all I learned a trick or two.
May 24, 2017
I am literally proud of myself for finishing this book.
It is exactly what I was looking for when I found it. This book smashes through all the gory mathematical details of cosmology, and i do mean gory because there are definitely times in this book when you will feel like you're taking a mathematical bloodbath! Unless (I imagine), you've previously endured advanced physics and two or three levels of calculus. I am among those of us who have not! My highest math in college was the first calculus course offered with trigonometry as the prerequisite. Enough of my personal history, this book, besides being exactly what I was looking for, is exactly what the author says it is. A detailed description of the fundamental concepts of special and general relativity, heavy on the math. I understood a lot and I feel I have gained even more knowledge about cosmology, astronomy, and physics than expected. Fantastic. l'll certainly read it again, soon even.
It is exactly what I was looking for when I found it. This book smashes through all the gory mathematical details of cosmology, and i do mean gory because there are definitely times in this book when you will feel like you're taking a mathematical bloodbath! Unless (I imagine), you've previously endured advanced physics and two or three levels of calculus. I am among those of us who have not! My highest math in college was the first calculus course offered with trigonometry as the prerequisite. Enough of my personal history, this book, besides being exactly what I was looking for, is exactly what the author says it is. A detailed description of the fundamental concepts of special and general relativity, heavy on the math. I understood a lot and I feel I have gained even more knowledge about cosmology, astronomy, and physics than expected. Fantastic. l'll certainly read it again, soon even.
September 27, 2015
The author has succeeded in accomplishing what he set out to do - give a readable, fascinating, and entertaining introduction to the mathematics of general relativity - which is in itself no small feat. Highly recommended for the (mathematically competent) interested reader. For the serious student of relativity, this book may sit comfortbly alongside the more typical textbooks, yielding conceptual as well as mathematical intuition where others yield rigorous proof.
The pricetag is worth mentioning. The author states in the introduction that he indeed intended for the book to be "inexpensive verging on the downright cheap". Given the ridiculous cost of the typical textbook, you're getting a lot of value for your money here.
I thoroughly enjoyed this book.
The pricetag is worth mentioning. The author states in the introduction that he indeed intended for the book to be "inexpensive verging on the downright cheap". Given the ridiculous cost of the typical textbook, you're getting a lot of value for your money here.
I thoroughly enjoyed this book.
May 1, 2019
It really is incomprehensible .
My math gave out on me when the author got to tensor calculus and geometric interpretations of space time. I felt like my head would explode and I just couldn’t follow the discussion. So I was ok through a little more than half the book. Need to build up my math chops if I ever want to really understand general relativity. I took a quick look at one of Sean Carroll’s books in the library.it might help. This stuff is really only for serious students.
My math gave out on me when the author got to tensor calculus and geometric interpretations of space time. I felt like my head would explode and I just couldn’t follow the discussion. So I was ok through a little more than half the book. Need to build up my math chops if I ever want to really understand general relativity. I took a quick look at one of Sean Carroll’s books in the library.it might help. This stuff is really only for serious students.
September 19, 2015
If you have not taken courses through at least integral calculus it would probably be hard to finish this book. However, it brought me back to college differential equations and calculus, and gave me a much better insight into the mathematical concepts behind the theories. Congratulations to the author for a great attempt at making the mathematics of relativity relatively accessible (pun intended).
February 1, 2016
This book is brilliantly and succinctly written, but it's absolutely true that you'll need plenty of time to absorb the difficult equations. Alas, I know my memory will fail me when trying to remember the math, especially without regular use in daily life. But it was nice to know what the Einstein field equations are, for however transient that experience might be.
September 3, 2015
The simplest true mathematical description of special and general relativity that I have ever found. Beautiful style. The book starts by explaining derivatives, and ends juggling with tensors. It is also very complete yet brief, easy to read, giving an excellent synthesis.
August 18, 2020
Compelling
Even though I am a pure mathematician specializing in c ommutative algebra and and haven't had much applied math, the math I didn't know was explained clearly enough that I was able to follow the discussion.
Even though I am a pure mathematician specializing in c ommutative algebra and and haven't had much applied math, the math I didn't know was explained clearly enough that I was able to follow the discussion.
April 11, 2016
A great book to understand Basic concepts especially in tensor calculus
January 18, 2017
Gentle does not necessarily mean easy. After decades since Practicing this type of math, I was able, with the author's guidance, to appreciate again relatively in its own language.
October 24, 2019
This is a “go to” text for understanding the the ideas behind transformational mathematics and Einstein’s theory of relativity. Very easy to read with helpful, solved equations. I find myself picking it up throughout the day and reading (or re-reading) a few pages and reviewing the math as well as the place the math is relevant in Einstein’s theory. I recommend this to anyone who would like to “peek” into one of the most important thought experiments in history.
May 16, 2017
Excellent!
This is the clearest explanation of General Relativity I have ever seen. You can go through this quickly if you are looking for a refresh course.
This is the clearest explanation of General Relativity I have ever seen. You can go through this quickly if you are looking for a refresh course.
November 30, 2023
The book, A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity by Peter Collier is about, as the title suggests, the mathematics behind the theory of relativity. This book provides enough background information on the mathematics used in the theory of relativity for someone who isn’t even through high school to understand. This book is very well written. Not only this, but it covers a truly fascinating subject in a way anyone can understand, as long as you’re good at comprehension and application. I do have a little bias on this because I love math more than any other subject, so I love how this book is structured with frequent examples of this math. On top of that, I love how the background and explanation text are there, but only in short chunks, allowing for more room for the actual math. This is all coming from someone who hates English, so if you like to read for the sake of reading, don’t listen to me and know that I could never imagine living a life like yours.
December 29, 2015
I gave up after reading the first 227 pages, right after the derivation of the Einstein tensor. It's just too hard for a lay reader. Nevertheless, the author has done a very good job. For instance, the reader actually can appreciate what tensors and connection coefficients signify. The technique of describing a Riemannian manifold is described, and that provides the background to understand the mathematical formulation of the general relativity. It helps if prior to reading this, one has already familiarized oneself with the concept of special relativity (e.g. by reading David Mermin's introductory text). Despite its difficulty and at times incomprehensibility, I'll still give it five stars.
May 6, 2016
Terrific book. Somewhere in it, the author quotes Stephen Hawking relating his publisher's comment that for every extra equation his book sales would drop by one half. If that were the case with Mr Collier's book there would be no readers; yet, there are and for good reason. If you're at all attracted to mathematics and have some background up to calculus it is possible to follow the equations all the way up to the "Holy Grail" of the Einstein field equations. I can't say it will be easy, but it is possible and the struggle can be pleasurable -- if you have that certain turn of mind.
December 22, 2014
Probably the finest introduction - *real* introduction - to special and general relativity available. It helps if you've got some familiarity with calculus beforehand, but my own mathematics are largely self-taught and I had next to no experience with vector calculus beforehand.
Nonetheless, after studying this text I felt more than prepared to progress to more serious works on GR and cosmology.
Nonetheless, after studying this text I felt more than prepared to progress to more serious works on GR and cosmology.
February 15, 2016
I was not fooled by the title of the book when I bought it, but I thought I would give it a try anyway. I could read up to 50% of the book, and then I had to admit it was getting REALLY complicated. The beginning was very nice, like a review of all concepts of mathematics and calculus. Then came Newton and then special relativity. At the end of that, when relativity really started, with manifolds, scalars, vectors, one-forms and tensors I had to give up!
October 22, 2016
This is a great book for a basic understanding of the mathematics of relativity. In particular, it is the best comprehensive guide to mathematics in short form that I have seen, from algebra through calculus to tensors. Does this mean I'm now a mathematician? Not likely, but it did make sense to me, an amateur dilettante, and an excellent companion to 'Quantum Computing since Democritus' by Scott Aaronson (2013).
December 8, 2014
Not for the faint hearted, but an excellent step by step into the maths of relativity. Will need a reread with a pen and paper to hand in the future.
I think you'd need at least A level maths or 1st year undergraduate - it goes a bit beyond standard calculus/matrices.
I think you'd need at least A level maths or 1st year undergraduate - it goes a bit beyond standard calculus/matrices.
April 14, 2015
A great book about a fascinating subject!!
I enjoyed the book immensely and although I couldn't follow all the mathematics, I could follow enough to get the idea that the author was trying to deliver!! Such a mind boggling subject!!
I enjoyed the book immensely and although I couldn't follow all the mathematics, I could follow enough to get the idea that the author was trying to deliver!! Such a mind boggling subject!!
April 1, 2016
Beautifully written
Clearly written by an enthusiast. The book delivers on its promise, the math behind relativity. An amazing subject to learn.
Clearly written by an enthusiast. The book delivers on its promise, the math behind relativity. An amazing subject to learn.
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