A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity by Peter Collier (PDF)

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Teach yourself the mathematics of relativity – THE AMAZON BESTSELLERTo really understand Einstein’s theory of relativity – one of the cornerstones of modern physics – you have to get to grips with the underlying mathematics. A Most Incomprehensible Thing is aimed at the general reader who is motivated to tackle that not insignificant challenge. With a user-friendly style, clear step-by-step mathematical derivations, many fully solved problems and numerous diagrams, this self-study guide provides an accessible introduction to a fascinating but complex subject.For those with minimal mathematical background, the first chapter gives 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; the all important metric tensor gμν; tensor calculus; the Einstein field equations; the Schwarzschild solution (which gives a good approximation of the spacetime of our Solar System); black holes, relativistic cosmology and gravitational waves.Special relativity helps explain a huge range of non-gravitational physical phenomena and has some strangely counter-intuitive consequences. These include time dilation, length contraction, the relativity of simultaneity, mass-energy equivalence and an absolute speed limit.General relativity, the leading theory of gravity, is at the heart of our understanding of cosmology and black holes. Summed up in the words of eminent theoretical physicist John Archibald Wheeler: “Matter tells space how to curve. Space tells matter how to move.”“I must observe that the theory of relativity resembles a building consisting of two separate stories, the special theory and the general theory. The special theory, on which the general theory rests, applies to all physical phenomena with the exception of gravitation; the general theory provides the law of gravitation and its relations to the other forces of nature.” – Albert Einstein, 1919Understand even the basics of Einstein’s amazing theory and the world will never seem the same again.ContentsPrefaceIntroduction1 Foundation mathematics2 Newtonian mechanics3 Special relativity4 Introducing the manifold5 Scalars, vectors, one-forms and tensors6 More on curvature7 General relativity8 The Newtonian limit9 The Schwarzschild metric10 Schwarzschild black holes11 Cosmology12 Gravitational wavesAppendix: The Riemann curvature tensorBibliographyAcknowledgementsJanuary 2019 – this updated Kindle edition contains the same text as the third paperback edition (ISBN 9780957389465).

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐(This review is for the second edition.)When I tried to study general relativity on my own and look for an appropriate book, I realized that I had already three general relativity books.1. Gravitation by Misner, Thorne, and Wheeler2. Introducing Einstein’s Relativity by D’inverno3. Introduction to the Theory of Relativity by BergmannAlthough I majored physics in university and have a doctoral degree in mathematics, the above three books were difficult to me. More precisely, in my opinion, any good book should be read thinking “so… so… why? … so… so… ok. I see. So…” But the above books seemed to force me to accept many things without clear understanding. Relativity is a well-established part of physics, so in principle, it is possible to give reasonable explanations to any statement in relativity books. But for people who learn such a discipline with critical or logical spirit, easy and detailed explanations are essential.Accessibility is the best merit of the book, A Most Incomprehensible Thing by Peter Collier. In Preface, the author says,This book is written for the general reader who wishes to gain a basic but working understanding of the mathematics of Einstein’s theory of relativity, one of the cornerstones of modern physics… I never found my ideal volume but instead had to make use of many different sources… The book I had in mind would assume little prior mathematical knowledge (even less than my own patchy sixth-form/high school maths if it were to be suitable for the general reader)…Some days ago, I finished the book. I was satisfied with the book as a whole, and I felt that I learned somethings, and I am more prepared with more advanced books in general relativity like Relativity, Gravitation, and Cosmology by Chen and Hartle’s famous book on introduction to relativity.But I have to tell you bad news. This book is not easy also. If an instructor specializing in general relativity teaches with this book, then even a freshman could understand it. But as a self-study book, I think this book is difficult. If you can successfully choose topics that you need in the book, focusing important parts, passing unimportant parts, and coming back to necessary parts again and again, finding some typos or errors, not expecting to understand everything in the book, then you may be able to finish the book on your own and be satisfied with the book. But if you want to thoroughly understand the book, you must know a lot of things before reading the book. To do that, you must have taken one year course of freshmen level calculus and physics including somewhat detailed chapters on special relativity. If you are not already familiar with special relativity at least the level of the popular science books, you may have difficulty in understanding the topics related to special relativity in the book. In particular, it would be good to know about time dilation and length contraction in advance.Besides that, you may know that relativity regards space-time as a differentiable manifold and relativity is formulated with the language of tensors. In the book, there are introductory chapters on differentiable manifold and tensors. But they are insufficient. If you want to comprehensively understand differentiable manifolds and tensors, you must study independently with some sources like the followings.Regarding tangent vectors,1. Related sections in Differential Geometry of Curves and Surfaces by do Carmo2. Related sections in Elementary Differential Geometry by O’NeillTo understand differentiable manifold and tensor, you must know the basics of differential geometry and to do that, you must be familiar with the concept of tangent vector. Differential geometry books deal with tangent vectors in detail. If you have taken one year course of calculus and one-semester course of linear algebra, then you would have no difficulty in reading differential geometry books.Regarding tensors,1. Chapter 5- Multiple Integrals, Chapter 6- Vector Analysis, and Chapter 10- Coordinate Transformations in Mathematical Methods in the Physical Sciences by Boas. Try to understand what a line element is and topics related to coordinate transformations.2. Quick Introduction to Tensor Analysis by Sharipov. You can get this on the internet. If you are familiar only with calculus, linear algebra, then you can read the book. Within only 46 pages, it contains the essentials you must know to understand the tensor formulations in relativity. Exercises are really helpful.Regarding covariant derivatives (Connection),1. Differential Geometry of Curves and Surfaces by do CarmoGeneral relativity says that energy makes the space-time curved. What is the meaning of ‘a differentiable manifold is curved’? If a covariant derivative is given to the space, then we can say about how much the space is curved. In my opinion, do Carmo’s book mentioned above is the best to learn covariant derivative among differential geometry books that I’ve ever seen.2. Chapter 2 – Riemannian Geometry in Morse Theory by MilnorDo Carmo’s book deals with covariant derivative of a 2-dimensional manifold embedded in R^3. But to understand general relativity, you must go further. We should know how to define covariant derivative on an abstract differentiable manifold, not only on a 2-dimensional manifold embedded in R^3. The mentioned chapter of Milnor’s book is about only 20 pages, but contains all the basic essentials of covariant derivative in the generalized sense, also with a concise introduction to Riemann curvature tensor that is also needed to understand general relativity.3. Riemannian Geometry by do CarmoIf you can be satisfied with the two books mentioned, it would be good. But if you want to understand what a differentiable manifold is and the concepts like metrics, geodesics, more on covariant derivatives, Riemann curvature tensor, the relationship between the Riemann curvature tensor and the Gaussian curvature (this would be the curvature you might think when you hear that a surface is curved), this book is recommended.4. An Introduction to Differentiable Manifolds and Riemannian Geometry by BoothbyIf you can be satisfied with the three books mentioned, it would be good. But if you think that do Carmo’s Riemannian Geometry is not logically so reasonable in some parts, then you can consult the classic Boothby’s book. For example, covariant derivative is a global concept. But to prove properties of covariant derivative, the author uses local concepts. It’s possible, but there is no enough explanation of doing that. Boothby explains that in detail. Besides that, if you want to more comprehensively understand the important concepts stated above, that is, tangent vectors, metrics, geodesics, covariant derivatives, Riemann curvature tensor, the relationship between the Riemann curvature tensor and the Gaussian curvature, then Boothby’s book would be perfectly helpful.I’d like to reemphasize the best merit of the book. There were huge level-gaps between the level of special relativity in the books for freshmen and of general relativity in the existing books like D’inverno’s or Ohanian’s for undergraduate students. The author exactly pointed out the gap and succeeded at large.Here are detailed points and my questions.1. If a real beginner would read the book, then he wouldn’t understand the equation at page 96 regarding the divergence of a gravitational field. The appearance of the mass density was abrupt. He would also had difficulty in understanding Section 3.3.3 dealing with introducing the second observer’s space-time diagram in the first observer’s space-time diagram not because the content is hard to understand, but because he had no motivation to follow the not-so much interesting long arguments. And he would fail to understand the Ricci tensor (and so the Ricci scalar) because of typos at several places. If he already knows Ricci tensor, these typos would be trivial things. But if he is really a novice in relativity and differential geometry, this kind of typos can be a big obstacle to read the book.2. If a real beginner would read the book, then he would fail to understand the meaning that Schwarzschild metric is a solution of the Einstein field equation. It would be better if the author had explained more explicitly that a solution of the Einstein field equation is a metric and the Schwarzschild metric is a solution under some conditions. As for the latter, the book is good, but as for the former, not so good. The derivation of Einstein field equation was good. But the derivation of the Lorentz transformation was not good. In my opinion, it would be better if the author postulated them as axioms. The explanation of the Einstein field equation was appropriate, but as for the Lorentz transformation, it was too lengthy.3. About the Friedmann equations, I feel the lack of a detailed explanation of how the Friedmann equations are derived from the Einstein field equations. I don’t want to see and follow every mathematical detail, but want to see the logic.4. In page 151, it says that the scalar product of two four vectors is invariant under Lorentz transformations. It’s true, but I think that detailed explanations are needed.5. In page 152, it is said that relativistic momentum is conserved in all inertial frames. What’s the meaning of this?6. In page 207, it says that small regions of spacetime are locally flat. Does locally flat mean that the Riemann curvature tensor is 0? So it is not curved? But Einstein field equation says that the energy makes spacetime curved. It’s absurd.7. In page 221, it says “We mentioned earlier that, using dummy index, the covariant derivative of a tensor field is equivalent to the divergence of that field.” But I cannot find any relevant remarks about it in earlier pages.8. In page 249, it says “We’ve actually slipped an assumption here: that the coordinate time difference between two events in Schwarzschild spacetime is the same as that recorded by a distant observer.” But I cannot understand where the assumption is used.9. In page 301, the density of radiation is proportional to the inverse of the fourth power of the scale factor. But the argument seems insufficient.As my first general relativity book, I am satisfied with the book as a whole, and I thank the author. Now I am reading my (possibly) second general relativity book.

⭐This is the Must-Have Book on Both Special & General Relativity. I would give it 10 Stars if I could. Read this when the first time you studied Special Relavity (SR) was a haze. Review SR the second time with this book it becomes crystal clear. This is my first time and book on General Relativity (GR). General Relativity (GR) was to me the Most Incomprehensible Mountain! After reading through this book, GR is now A Comprehensible Mountain for me (and you too). The author takes you by the necessary/essential equations (i.e. your core guide line/rail) gently climbing the equations up from the bottom to the top summit without skipping an equation (i.e. step). Don’t worry! He has strung and weaved together elegantly all the needed equations so that if you get the former equation/step you will understand the next equation/step and so on up to GR and a bit beyond. I vouch for it.The core guide line/rail smakes from SR all the way high up to GR and also extends beyond to your final sweet treat – Applying GR. What a better way to grasp GR that actually applying it to understanding the common Schwarzschild Black Holes and Relativistic Cosmology. The First Equations for the Cosmos/Universe. The Big Bang, Big Chill, Big Crunch, or Big Bounce. Yes, I did finished this book down to the last page and last equation. Whew! I must admit I snail’ed slowly through the latter half of the book. I am the type who likes to ant scrutinize closely every equation in a book. The concept clarity and math rigor is satisfyingly enough. It took me a whole summer 3 months as a side read when I can. This is your foundational GR book to branch off and then pursue advanced GR topics. I will see what I like to learn next.The author introduces the foundation mathematics in Chapter 1. You can skim through it if you have taken 1 yr Calculus and 1/2 yr Differential Equations. I would skim through Chapter 2 though for a quick review on Newtonian Mechanics even if you have taken 1 yr Physics. Do start reading SR in Chapter 3. The author does present 4-dimensional curvature math in spacetime really well as a prelude/interlude to GR. He presents all the required advanced math as he goes and as you will need but no more than necessary, such as Tensors, to keep the book in good length before easing you comfortably into the GR equations. No need for another side advanced math book, but this book does provide the advanced math topics for you to delve deeper in more pure subject-matter math textbooks if you want.Have you ever wander how Action-at-Astronomical Distances can be Instantaneous?! This ‘appears’ to be happening. How can the attraction between the planets and the Sun make the planets orbit the Sun at such vast distances? This Action-at-a-Distance attraction seems instantaneous, but light has finite speed! Nothing can connect the planet to the Sun in zero no time, not even light itself! Yet, something is connecting the planet to the Sun instantaneously for sure. Newton’s Law of Universal Gravitation just describes and quantifies the attraction that’s all, but doesn’t give the underlying why or reason that gives rise to this attraction. General Relativity gives the thesis for this Action-at-Astronomical Distance attraction. To understand the why/reason literally requires us to go up into another higher dimension. 4-dimensional spacetime. That’s right. 4-dimensions for us 3-dim creatures.General Relativity provides the why/reason that gives rise to this attraction as the Sun’s heavy mass causes curvature in the surrounding 4-dimensional spacetime. The 4-dim spacetime curvature extends all the way out to the planets and beyond. Think of the sun as the heavy bowling ball resting on at the center of a taut vast sheet fabric. That vast sheet fabric would represent our hypersurface called 4-dim spacetime. A marble like the earth would continually fall in its orbit around the heavy bowling ball our Sun. To understand Action-at-Astronomical Distance that seems Instantaneous requires us to think beyond 3-dimensions and into 4-dimensions. But, we are only 3-dimensional creatures and can’t see beyond that! Fortunately, we have developed 4-dimensional math as our main eyes to think in 4-dim and see in 4-dim with the help of 3-dim analogies. Ref. the Riemann Curvature Tensor.Throughout the book, the author not only provides the concepts and equations, but the historical contexts and characters that contributed to our current understanding. Galileo, Newton, Minkowski, Riemann, Einstein, Schwarzschild, Kerr, Robertson/Walker, Friedmann, Hubble, and others I missed mentioning from the book.Space and Time are not separate physical entities, but fused together as one single 4-dimensional Spacetime Fabric. 4-dim spacetime is not a human fabrication, but an experimental physical reality we should start to think in terms as one. “Matter tells Spacetime how to curve. Spacetime tells Matter how to move”. This is one written ‘Art Piece’ of Knowledge!

⭐Most of the top rated reviews for this book seem to come from those who are fairly well versed in University-level maths. I’m not surprised, because, for them, this book must be a very easy way to get to understand the mathematics of General Relativity. I am not at that level (and – to be honest – don’t intend to be), and I did not attempt to follow in detail the more complex mathematical techniques and derivations in the second half of the book. But – you know what? It didn’t matter that much – I still had a much better understanding of Relativity and Cosmology after reading it than before. It is very well structured and I can see how someone who is already into maths but not physics (well, at least not Relativity) could lap this up. I now at least know about Tensors and have an appreciation of their place in this field of endeavour: Which is absolutely central. Anyone who has done maths at A-level should be able to get something out of this book. Give it a try!

⭐If you’re interested in quantum physics and have read a few books, you will realise (like me) that you can’t really escape the maths. It’s been years since I did any calculus (etc) so this was tough for me. But it’s worth it as there is an absolute limit to what the written word and creative analogies can do to describe or explain the subject – its natural language is maths.Here the writer uses prose very well indeed (so it’s not just solid maths), and leads you through the necessary maths step-by-step, right up to Schrodinger’s equations and Einstein’s field equations (and beyond, not got past that yet).Won’t say it’s easy and I reckon you need A level maths no matter how long ago (!). But it is so worth the effort, mainly because along the way, you begin to see the maths explain what words struggle with.A brave book to write and publish (given Stephen Hawking’s quote: ‘ for every equation you put in the book you’ll halve the sales’)… well now!!And even if you do get stuck on the maths, the writer’s written explanations of many topics are very good indeed.A strong coffee, pencil & paper nearby – and don’t expect to plough through it in an evening. But very worthwhile.

⭐I was really tempted to give this very good book five stars but refrained from doing so because I think there are improvements that can be made. These ‘improvements’ are only small however and the overall idea, structure and content are all excellent (so far). No I haven’t finished yet, but I’m thoroughly enjoying studying (you have to try – quite a bit sometimes) this intriguing subject. The author (I think of him as my clever friend Peter now) introduces all the required mathematics at the beginning in good simple (as possible) terms with plenty of worked examples. There’s plenty more mathematics in the rest of the book which I found superb; it’s great to know what a Lorenz transformation is and how to work it out as well as other esoteric terms like geodesic, tensor, manifold and metric etc. If you’ve studied mathematics to age 18 (i.e. ‘A’ level here in the UK) you’ll enjoy this book. Of course anyone can read it as Peter assumes little prior knowledge but you’ll have to work that bit harder. Overall a wonderful book that does what it says and plugs the gap between popular descriptions of relativity and undergraduate texts. At the price amazing value for money.

⭐I’m still working my way through this, but given the type of book it is I feel I’m justified putting in a review already.This is a very well thought out textbook, detailing everything from first principles (including basic arithmetic!), so it doesn’t matter how much maths you know, you can follow along. This may take a bit of effort in places, where new concepts are introduced. Collier does a very good job of explaining them in a simple fashion, but it’s entirely possible you will want to read around the edges. If you do, then there is enough information here to give you a good jumping off point so that you know exactly what you are looking for.This is the book I’ve been looking for for years, without actually knowing I was looking for it.

⭐This is a well written accessible book for what is a complex subject. It may not be perfect but it helped give a different perspective on ideas of other books I have on the subject. If you want a more ‘gentle’ introduction to General Relativity then this book might be for you – it’s still a ‘full fat’ treatment and has some difficult equations, ideas and whole sections but it leads up to it at a gentler pace than some and explains a lot of things clearly (not taking them as read which some books do). Highly recommended and a bargain at the price I paid.

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