Einstein's Theory: A Rigorous Introduction for the Mathematically Untrained

"One day," writes Oyvind Gron, "the seventy three year old philosopher Arne Naes appeared at my graduate course on general relativity. He immediately decided that a new type of introduction to the general theory of relativity is needed: an introduction designed to meet the requirements of non-science educated people wanting to get a thorough understanding of this, most remarkable, theory."Imagine you are deeply curious about concepts of Einstein' theory beyond just misleading popularized qualitative descriptions or strict quantitative textbooks, imagine you are an educated person who has forgotten some basic high school maths but likes to master the field of relativity as excellent as a graduate student, imagine you are a thinker, philosopher or a theoretician who likes to compare her/his concepts of space and time versus that of Einstein in General Relativity ... in all these cases, this book is for you if and only if you are a patient reader.The book is unique in its genre: not only it starts with low level maths to arrive at the core of one of the most sophisticated theories of the last century, but it is the fruit of corporation between an interested philosopher (Arne Naes) and a professor of physics (Oyvind Gron). It is therefore neither a vulgarized version of the theory, even when it goes into the qualitative discussions, nor a strict textbook jumping between mathematical formulas and mistaking the mathematical formalism with the reality.In other words, although the books deal with mathematics of GR (the general relativity) the authors are aware that beyond the equations are hidden wonders and curious concepts rooted in Einstein's inventive imagination.The book is in fact written for a reader whose urge for a deeper understanding of Einstein' theory is armed him/her with sufficient patience to climb, step by step, up the mathematical pyramid and decipher the glyphs in which one of the most outstanding theories of the last century has been written.Does it mean that the book is "perfect"? Not at all! Despite all the efforts by the authors, the reader might sometime feel lost, or discovers that the subject could have been explained more clearly, or the philosophical or mathematical points could be deeper or more radical but the point is that: whereas in such cases the reader may rely on many other references (including her/his own notes and ideas), hardly can s/he find any other book which is on a par with the current book.As Arne Naes puts it, according to "Sir Arthur Eddington, ... mathematical physics cannot be understood through popularisations" That is the reason beyond the structure of the book and at the end we should confirm the claim made by the authors of the book who "venture to suggest the understanding acquired by the reader [of this book] may be deeper than what is necessary for completing graduate courses ... Only patience is needed."

I agree with many reviews. If you are truly untrained at math, this is a tall order. I know calculus but am 20 years out of college. I frankly loved the upfront section and the teaching of trigonometry and basic calculus. I was impressed. The section on tensors was not sufficient though and the outer product description left me totally unclear. I used another book to help (students guide to vectors and tensors I think is the name). The gradual build up to relativity was great. I do believe there were some errors (not many) — I could not find any errata docs on the internet anywhere. Some errors were obvious, but some I’m not sure on. Most descriptions were great but I can’t say all was 100% clear (might be me).I just bought the book by Wald, thinking that perhaps I could tackle the harder stuff. Nope. It’s discussion is way way denser and assumes way more knowledge about Manifolds. I’m going to try to find something between this book and Wald’s.Oh - I read the Kindle version. Overall, fine. But in a few spots the equations were distorted. Usually easy to tell but has the potential to trip you up. Rare, but real.Still, I really am happy to have found this book. Learned a lot about general relativity and how it is applied (at least at a basic level).

This book is a rare gem. It has a solid review of the mathematics you need to be able to do special and general relativity. For that part of the book, it is a review rather than a first time presentation. Keep that in mind.The real strength of the book is when it gets to talking about the mathematics needed for general relativity. The coverage of Christoffel symbols is outstanding, and includes drawings and explanation I have not found anywhere else. I also loved the section on inflation. While basic, it shows how the inflation mechanism can come right out of the equations. Beautiful.For a physics student this is a solid companion to study relativity. For a physicist in another discipline who wants to learn relativity on their own, it is hard to beat this book. But, please realize you have to know practical advanced mathematics and have a solid physics background to use this book. It is not for the total novice.

This is an excellent book. I am about 1/2 way thru it and the presentation of the necessary math to understand relativity is the best I have ever seen. However, the title "...for the Mathematically Untrained.." is misleading. I think the "mathematically untrained" will find the book rough going; however, for those with a background in college level math, i.e. calculus and vectors in particular will find the book gives a refreshing presentation of these subjects. The careful analysis of a derivative and basis vectors is the best I have ever seen. The geometric approach for derivations such as Christhoffel symbols makes this concept much clearer than an algebraic approach (which by the way, is also shown). I have a BS in physics and a MS in EE and I always felt that school rushed one thru courses with emphasis on problem solving rather than understanding basic principles was probably not a good approach for understanding. I just wish that I had this book early in my academic career.

A difficult read for a truly mathematical untrained but, with effort, the background mathematical concepts needed to developed a revolutionary theory of spacetime and gravity reveal the interdependence of intuitive perspectives and the equations that make a theory predictive and testable. The book makes this interdependence clear and the difficulties Einstein had with the math. A continual reference to previous graphics and formula from the most basic to the more difficult, guide the reader from basic vectors through the history and development of differential calculus, to vector fields, tensors, and Christoffels. Lucid verbal explanations and comments are interspersed throughout the book giving the reader perspective.

This book provides a step by step rigorous introduction to Einstein's General Theory to general readers with some background in mathematics and calculus and willing to learn the specific mathematical framework ( Tensors, curvilinear co-ordinates etc ) needed to properly understand his field equations.

This is an outstanding book on a subject that many authors have failed to portray adequatly.I have read many books on Einstein's theories and until completing this offering was in a state of confusion having been left with many unanswered questions and a subsequent feeling of inadequacy on my part as a reader. It would seem that Eddington's quoted statement in the preface that mathematical physics cannot be understood by popularisation has proved true - intelligent readers will be left baffled by such attempts. I don't claim to be intelligent but I have certainly been baffled by countless other books on the subject - including Einsteins own attempt to explain things without recourse to maths!This book introduces all the mathematics necessary to acquire a level of understanding of the treatment of non-euclidean geometries thereby providing the foundations necessary to give the concise formulation of the Einstein field equations. This is achieved by educating the reader in vectors, differential and integral calculus, vector transformations, contravarient and covariant vectors, curvilinear coordinates systems and tensors. In order to achieve an understanding of intrinsic curvature Christoffel symbols and parallel transport are introduced which leads into the Riemann, Ricci and Einstein tensors. Along the way the authors describe philosophical difficulties that along with the deeper understanding of the maths resolve many of the questions that arise from less satisfactory treatments of the subject.I found the treatment of rotating reference frames and the related subject on gravitational time dilation to be outstanding - possibly as a result of a newly acquired understanding of the maths necessary to describe these complex situations. Similarly the concise treatment of some of the common solutions to the field equations and the predictions of the theory are a testament to the success of the authors approach. Indeed less than a third of the book, 100 odd pages, covers the theory itself, the field equations and their solutions, predictions, black holes, and cosmological implications including inflation! All of these subjects are completely amenable thanks to the remarkable mathematical tuition offered by the previous two thirds of the book.I compliment and whole heartedly thank the authors for this work. I cannot recommend it enough.

All new and difficult concepts are crystal-clear explained. I only disagree with the subtitle about ' the mathematically untrained' that will surely be unable to go throughout the entire text. All students studying the subject of Einstein's Theory on relativity should have a wonderful companion in this book.