A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity

For those who need a quick review of essential mathematics in order to understand relativity, this is the book. Look nowhere else. Thanks to Peter for making me understand relativity in an incredibly quick time. I am a doctor in the ugly side of forties who has left physics some thirty years ago and trying to assimilate complex physics topics now more leisurely trying to rediscover my passion which is physicsThis book is just for someone like me

This book is simply extraordinaryI have had the opportunity to consult many books on Einstein's relativity. I have no doubt that this book is the one that perhaps best explains this exciting branch of physics. The author, who may be a self learner in physics like me, has probably read a lot of books on the subject and one feels that he has managed to compile the best explanations from different textbooks (there are notably quite a few ideas from Schultz's excellent book which really gives the reader a geometrical intuition). The book never veers into unnecessary mathematical complexity and one feels that the author's aim is to allow the reader to understand intuitively what relativity is. To use the expression of another good physics author (Jakob Schwichtenberg), this is definitely a "student friendly" book.In my opinion, this is an excellent introduction to the subject, which can later be supplemented by more specific works.The level required to understand this book is modest and there is no need to be a genius. An understanding of fundamental analysis (derivation, integration and some basics of vector analysis) is more than sufficient. The first chapter of the book is a reminder of elementary mathematical notions useful to refresh the memory.Ce livre est tout simplement extraordinaireJ'ai eu l'occasion de consulter de nombreux ouvrages sur la relativité de Einstein. Je n'ai aucun doute pour dire que cet ouvrage est celui qui explique peut-être le mieux cette branche si passionante de la physique. L'auteur qui est peut-être un "self learner" en physique comme moi a probablement lu beaucoup d'ouvrages sur le domaine et on sent qu'il est arrivé à compiler les meilleurs explications de différents textbook (il existe notamment pas mal d'idées tirées de l'excellent livre de Schultz qui permet de vraiment donner une intuition géométrique au lecteur). L'ouvrage ne vire jamais dans la complexité mathématique inutile et on sent que le but de l'auteur est de permettre au lecteur de comprendre intuitivement ce qu'est la relativité. Pour reprendre l'expression d'un autre bon auteur de livres de physique (Jakob Schwichtenberg ), il s'agit définitivement d'un livre qu'on peut qualifié de "student friendly"A mon sens, il s'agit d'une excellente introduction au sujet qui pourra par la suite être complété par des ouvrages plus spécifiques.Le niveau requis pour comprendre cet ouvrage est modeste et il est inutile d'être un génie. Une compréhension de l'analyse fondamentale (dérivation, intégration et quelques bases de l'analyse vectorielle) sont largement suffisantes. Le premier chapitre de l'ouvrage est d'ailleurs un rappel des notions mathématiques élémentaires utiles pour se raffraichir la mémoire.

Buen libro para quien gusta de la ciencia, tiene un poco de preparación matemática y quiere ir un poco más allá que los libros de divulgación del tema...

Any book on this subject is a tradeoff between accessibility and rigorousness. This book is targeted to those who want to move beyond the “bowling ball and marbles on a rubber sheet” popularizations to find out the mathematical structure behind the theory, and are willing to make the necessary effort. It is the perfect book for people at that stage; much more accessible than the typical textbook. I particularly appreciate it when the author provides some details but says something like "you don't actually need this stuff".All the preparatory math you need, starting from high school level, is in chapter 1, but learning it on your own would be a tall order. The "but" in the review title is my opinion that without some existing knowledge of vectors and calculus, this book will be pretty hard. In that case, at least a self-study guide in those subjects would be a good idea before starting.With a BS in physics from many years ago, I'd studied Special Relativity (constant-velocity motion near the speed of light) but not GR, and was at least passable (if a bit rusty) with the math in chapter 1. That left tensors to learn – if you want to understand GR, you have to understand tensors. I cruised through the Special Relativity material, but, as expected, I found tensors slow going.This book is about as good a basic introduction as you’re likely to find, but is best supplemented by other sources, such as:1. “Student’s Guide to Vectors and Tensors” by Fleisch.2. Search Youtube for “general relativity Susskind” - listed in the bibliography.3. Search Youtube for “tensor calculus eigenchris”.4. Physics.stackexchange.com is a goldmine of questions and answers - also listed in the bibliography. You'll probably find many of your questions asked and answered there.5. Various lecture notes and papers from the web. Just search for whatever additional material you need. Some material will probably be too advanced, but you’re likely to get a useful nugget of info here and there.What you learn from this book will help you make better use of these resources.Bottom line: if you want to understand the foundation and justification of GR, get this book – but be prepared to put in a lot of work. The author is trying to show you the secrets of the universe. Why would you expect that to be easy? That said, you'll get a lot out of it even if you don't understand every bit. If it just makes the meaning of the terms in the Einstein Field Equations - the deceptively simple-looking but profound expression of GR - a bit less mysterious, it will have been worth the cost.And I finally understand tensors.

This book offers a relatively gentle, yet semi-rigorous mathematical introduction to Einstein's theories of Special and General Relativity. I think that one way to approach learning these theories is to repeatedly expose yourself to the material. This book succeeds admirably at helping with that aim. It provides enough maths yet done so in a clear and easy-to-follow way that the reader can feel a little more confident in tackling more formal and advanced texts on the subjects after completing this book.Some thoughts on the title. First, whilst it may be "gentle" compared to a standard textbook on the subject(s), it is by no means gentle. It still uses the same mathematical "machinery", yet the author does proceed step by step and assumes very little prior knowledge. Secondly, it really isn't a book about the mathematics of relativity, but instead, is more just relativity using the required mathematics. At the start of the book the essential maths needed is covered, but to be honest, it is just a quick summary, and I suggest that for a better understanding and to actually be able to work many problems in General Relativity will require a more thorough understanding of the maths. In particular, being proficient in multi-variable calculus, as well as a good understanding of Tensor Analysis would make things a lot easier. That said, the book is self-contained, in that to understand what is presented only requires understanding the maths at the level given at the start of the book.After the introductory maths section, he covers Newtonian mechanics in Chapter 2, which is pretty straight forward. Chapter 3 is Special Relativity. My guess is that this book is more focused on General, rather than Special relativity (SR), given that only one chapter is devoted to SR. To be honest, if you've never studied Special Relativity before, then I suggest you pick up a book or take a course devoted specifically to Special Relativity, rather than this book. Why? I feel that this book only touches upon Special Relativity, most of the main ideas are here, but it is too brief and seems to lack a bit of coherence. I've studied Special Relativity extensively, and there is nothing like starting with just the two postulates, and then systematically and methodically deriving or showing everything, bit by bit, from time dilation and length contraction, to simultaneity, and then the invariant interval, along with the many paradoxes. This book's single chapter might leave you wondering - if it is your first exposure to Special Relativity - what is Special Relativity really all about anyway? But if you are familiar with Special Relativity, then this book has some useful and good insights. Particularly I liked the way proper time was discussed, and the notion of parameterized proper time. I think this chapter's main strength is its tie-in with General Relativity, in particular, the invariant interval, which leads to the line element, then the metric and metric tensor, and also the various 4-vectors. It ties in well with General Relativity, with Special Relativity and its metric being a limiting case of General Relativity when the spacetime (and associated metric) are for flat spacetime.Chapter 4 introduces the Manifold, and this was a useful but less rigorous discussion of manifolds.Chapter 5 discusses scalars, vectors and tensors. This is where things start to get a bit heavier. The distinction is made between contravariant vectors ("vectors") and covariant vectors ("one forms"). I was actually struggling a bit with this chapter, as the distinction between contravariant vectors and covariant vectors was stated to be that the former lies in the tangent space, whilst the latter lies in the cotangent space. Huh? This was not enlightening for me, but after some internet searching, and discovering that contravariant vectors, if we (say) double the basis vector length, we then halve the corresponding vector component, whereas with covariant vectors, if we double the basis vector we also double the corresponding component, then this chapter started feeling a lot easier to me.Chapter 6 covers more curvature ideas and concepts, including the "monster", the Riemann curvature tensor. Also the author covers connection coefficients (Christoffel symbols), and the Ricci tensor and Ricci scalar, the latter two which are derived from the Riemann tensor. The two Ricci components actually appear in Einstein's field equations. I feel that whilst this chapter was pretty good if not a little challenging, I would have benefitted from some diagrams. It would have been nice to be able to visualize the connection coefficients.Chapter 7 is the showpiece, where Einstein's field equations are derived. The field equations really are the heart of General Relativity. The author goes step by step through a progression on how Einstein arrived at the field equations, except for the last few steps, where we leap from an "almost finished" set of field equations to the final form. I found the explanations of the energy-momentum tensor (aka the stress-energy tensor) to be very good, even if it was challenging.Chapter 8 covers the Newtonian limit, that is, what does General Relativity become in a weak and static (unchanging over time) gravitational field environment.Chapter 9 and 10 cover the Schwarzschild metric and Schwarzschild black holes, and these chapters were excellent and very well written. I think the only criticism of this part of the book is that the notions of proper vs. coordinate time and proper vs. coordinate length at times were not completely clear to me. In Special Relativity, proper time is clear and easy to define, and for a given pair of events or interval, is the time elapsed on a clock present at both events. However, in this book, the notion of proper time for a distant observer (assuming distant from both the gravity field and the events) is used, which, whilst this may be the correct way, it threw me as I was used to noting proper time is only for an observer/clock present at both events.Chapter's 11 and 12 cover Cosmology and Gravitational Waves, respectively. Again, a worthwhile read, although sometimes I found that steps could have been expanded upon or explained more fully.This review may sound like I have a lot of criticisms of the book, but really I found this book to be an excellent and well written "first exposure" on these topics. I now feel more confident in trying to tackle harder texts, so I feel this book has succeeded admirably in offering a gentle set of notes on a mathematical treatment of both Special and General Relativity.