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A First Course in General Relativity
General relativity has become one of the central pillars of theoretical physics, with important applications in both astrophysics and high-energy particle physics, and no modern theoretical physicist's education should be regarded as complete without some study of the subject. This textbook, based on the author's own undergraduate teaching, develops general relativity and its associated mathematics from a minimum of prerequisites, leading to a physical understanding of the theory in some depth. It reinforces this understanding by making a detailed study of the theory's most important applications - neutron stars, black holes, gravitational waves, and cosmology - using the most up-to-date astronomical developments. The book is suitable for a one-year course for beginning graduate students or for undergraduates in physics who have studied special relativity, vector calculus, and electrostatics. Graduate students should be able to use the book selectively for half-year courses.
392 pages, Paperback
First published January 1, 1985
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Displaying 1 - 27 of 27 reviews
November 12, 2018
This is an informative, very readable, gentle introduction to general relativity, perfectly suitable to advanced undergraduates (it has been designed, and often used, as a reference text to one-year general relativity courses at senior undergraduate level). The book assumes the minimum of pre-requisites, without watering down the subject matter, and without compromising on mathematical depth and rigour.
The approach of the book is pedagogically excellent: after a good speedy review of the main concepts of special relativity, the author develops, in a gentle and highly intuitive manner, tensor calculus in special relativity, and in curvilinear coordinates in Euclidean and Minkowski spaces; he then deals with differential geometry in curved manifolds, and once all this necessary mathematical apparatus is properly explained, the book addresses the concepts of physics in a curved spacetime, until we finally get to the beautiful field equations of general relativity.
Remarkable conceptual lucidity, mathematical rigour and a stress on physical intuition all contribute to an excellent level of readability, supported also by a relativity low number of typos (all very easily identifiable anyway). The concept of tensor is developed with an emphasis more on geometric aspects rather than on transformation properties, an approach that I previously saw employed with some good degree of success by Roger Penrose in his masterpiece (the “Road To Reality”), but to which I personally prefer the more algebraic approach that puts the priority on transformation properties.
Non-Euclidean geometry is similarly introduced in a “constructive”, “applied” way rather than starting as abstract mathematical idealizations, which again is a question of personal tastes and objectives. On the other hand, I really liked how the author adopted the "perfect fluid" model analogy, which I found quite effective in gaining a more “physical” understanding of the stress-energy tensor which is a fundamental concept, as it describes the density and flux of energy and momentum in spacetime, thus representing the source of gravitational field in Einstein's field equations.
In more general terms, critical concepts and items such as equivalence principles, the metric tensor, parallel transport, covariant derivative, Christoffel symbols, curvature, geodesics and many others, are all explained in a concise but mathematically rigorous manner, with derivations that never leave too many missing steps to the reader, who can always complete such derivations with a very reasonable level of effort.
I particularly enjoyed the progression with which the mathematical machinery for dealing with curvature is explored, and how and its associated Riemann curvature tensor (and its contractions: Ricci Tensor, Ricci scalar, Einstein tensor) are all addressed in a very effective way.
The only issue that I have with the author's overall approach is that occasionally too much material is left as exercise to be read and completed in the problem section of each chapter – I strongly recommend that such sections be investigated as thoroughly as the main text, in order to get the maximum benefits out of this book.
Once the field equations are treated, the last few chapters of the book shift the focus to astrophysics and cosmology: gravitational waves, general solution for static spherically symmetric spacetimes, the all-important Schwarzschild metric, quantum mechanical pressure inside a star.
All these subjects are treated with concise clarity too.
Chapter 11 (which I skipped, as not of primary interest to me at the moment, because of their very specific, technical astrophysical/cosmological nature) deals with black holes, perihelion shift and gravitational lensing.
Overall , this is a very good introductory book to general relativity. A good, solid starting point, an helpful and informative stepping stone recommended to readers who then want to get into more comprehensive treatments of the subject, and an accessible and rewarding read.
4 stars - definitely recommended.
The approach of the book is pedagogically excellent: after a good speedy review of the main concepts of special relativity, the author develops, in a gentle and highly intuitive manner, tensor calculus in special relativity, and in curvilinear coordinates in Euclidean and Minkowski spaces; he then deals with differential geometry in curved manifolds, and once all this necessary mathematical apparatus is properly explained, the book addresses the concepts of physics in a curved spacetime, until we finally get to the beautiful field equations of general relativity.
Remarkable conceptual lucidity, mathematical rigour and a stress on physical intuition all contribute to an excellent level of readability, supported also by a relativity low number of typos (all very easily identifiable anyway). The concept of tensor is developed with an emphasis more on geometric aspects rather than on transformation properties, an approach that I previously saw employed with some good degree of success by Roger Penrose in his masterpiece (the “Road To Reality”), but to which I personally prefer the more algebraic approach that puts the priority on transformation properties.
Non-Euclidean geometry is similarly introduced in a “constructive”, “applied” way rather than starting as abstract mathematical idealizations, which again is a question of personal tastes and objectives. On the other hand, I really liked how the author adopted the "perfect fluid" model analogy, which I found quite effective in gaining a more “physical” understanding of the stress-energy tensor which is a fundamental concept, as it describes the density and flux of energy and momentum in spacetime, thus representing the source of gravitational field in Einstein's field equations.
In more general terms, critical concepts and items such as equivalence principles, the metric tensor, parallel transport, covariant derivative, Christoffel symbols, curvature, geodesics and many others, are all explained in a concise but mathematically rigorous manner, with derivations that never leave too many missing steps to the reader, who can always complete such derivations with a very reasonable level of effort.
I particularly enjoyed the progression with which the mathematical machinery for dealing with curvature is explored, and how and its associated Riemann curvature tensor (and its contractions: Ricci Tensor, Ricci scalar, Einstein tensor) are all addressed in a very effective way.
The only issue that I have with the author's overall approach is that occasionally too much material is left as exercise to be read and completed in the problem section of each chapter – I strongly recommend that such sections be investigated as thoroughly as the main text, in order to get the maximum benefits out of this book.
Once the field equations are treated, the last few chapters of the book shift the focus to astrophysics and cosmology: gravitational waves, general solution for static spherically symmetric spacetimes, the all-important Schwarzschild metric, quantum mechanical pressure inside a star.
All these subjects are treated with concise clarity too.
Chapter 11 (which I skipped, as not of primary interest to me at the moment, because of their very specific, technical astrophysical/cosmological nature) deals with black holes, perihelion shift and gravitational lensing.
Overall , this is a very good introductory book to general relativity. A good, solid starting point, an helpful and informative stepping stone recommended to readers who then want to get into more comprehensive treatments of the subject, and an accessible and rewarding read.
4 stars - definitely recommended.
November 18, 2025
There is a surfeit of wonderful books about general relativity - Carroll, Wald, Hartle. But I found this marvellous book in the library, years before I discovered those, and spent many, many hours with it, gradually understanding more and more about this most extraordinary of theories!
September 24, 2017
usually, physics/ math textbooks give me giant yawns, but this one is definitely exceptional, I enjoyed reading it a lot. thank god Bernard Schutz skips all that torturous index gymnastics of differential geometry and jumps straight into special and general relativity (where half of the quantities in diff geom happily die and leave us in peace). his introduction to tensor calculus is very helpful. lots of clear explanations, for example why mutual length contraction or time dilation is not a contradiction at all (this one has ALWAYS got me), or how Einstein followed his particular line of reasoning find analogy of Newton's gravity- no preferred coordinate system- local conservation of energy- special relativity in the locally inertial frame to develop the field equation. gets a bit more technical when it comes to black holes and neutron stars, but bearable. fascinating book!
March 3, 2016
Very good book to start general relativity. The first four chapters carefully work out the necessary mathematics, in more geometric forms than just emphasizing transformation rules. The fluid analogy was especially helpful for understanding stress-energy tensor which is the source of gravitational field.
The next four chapters brought the readers to the real essence of general relativity - the physics and mathematics of curvature itself. Somehow he managed to bring out the flavour really clearly. While the chapters on cosmology and gravitational waves are not as useful unless explicit astronomical calculations related to interferometry and telescopes are needed, the book itself has presented good overview and detail of the theory. The closure using Schwarzschild black hole was a good one, good enough to launch readers to the next subsequent books with more details.
Skipped gravitational radiation chapter for the time being, while reading cosmology as introduction. Overall, this book really is good especially if you bother to do the exercises - very instructive among them being computing by hand Einstein tensor, Riemann tensor, Ricci scalar, etc. which will take very long time but doing once helps. For example, it becomes clear how Christoffel symbol takes part in computation of these quantities and why GR is a computationally intensive subject. It was also clear by doing manual computation why Bianchi identities implies only 20 out of 256 components of curvature tensor are independent.
The next four chapters brought the readers to the real essence of general relativity - the physics and mathematics of curvature itself. Somehow he managed to bring out the flavour really clearly. While the chapters on cosmology and gravitational waves are not as useful unless explicit astronomical calculations related to interferometry and telescopes are needed, the book itself has presented good overview and detail of the theory. The closure using Schwarzschild black hole was a good one, good enough to launch readers to the next subsequent books with more details.
Skipped gravitational radiation chapter for the time being, while reading cosmology as introduction. Overall, this book really is good especially if you bother to do the exercises - very instructive among them being computing by hand Einstein tensor, Riemann tensor, Ricci scalar, etc. which will take very long time but doing once helps. For example, it becomes clear how Christoffel symbol takes part in computation of these quantities and why GR is a computationally intensive subject. It was also clear by doing manual computation why Bianchi identities implies only 20 out of 256 components of curvature tensor are independent.
October 20, 2015
Probably not a good textbook (as it is advertised) but a good reference and guide for physical clarity and intuition. Builds from SR to GR step by step.
not-ready-for-it-yet
October 27, 2025Despite its name this book is not sufficiently deep into advanced geometry or mathematical generality concepts, which need to be covered more thoroughly in more advanced textbooks before properly appreciating its contents.
To be revisited at some later date
To be revisited at some later date
June 20, 2015
A really good book! I recommend this book for everyone who want to know how genius Einstein was. By the structure point of view, this book is spot on. It starts with mathematical tools to work with a curved space like vectors, one-forms, dual space, and tensor algebra. Then it proceeds to the redshift experiment and the importance of curved space by introducing Riemann tensor with Ricci and Einstein tensor afterwards.
Fun things start to happen when one start to work with physics in curved space. Gravitational waves come into play here and suddenly one can work with fascinating stuff like Schwarzchild metric and black holes. At this points, one can overlook to recent infamous paper like holographic principle, Hawking radiation, or Alcubierre drive.
The last chapter of the book tells us about cosmology and upcoming issues with general relativity like Robertson-Walker metric, accelerating universe, etc.
This book is one of the best books that I can recommend to all physics enthusiasts (or nerds).
Glad midsommar!
Fun things start to happen when one start to work with physics in curved space. Gravitational waves come into play here and suddenly one can work with fascinating stuff like Schwarzchild metric and black holes. At this points, one can overlook to recent infamous paper like holographic principle, Hawking radiation, or Alcubierre drive.
The last chapter of the book tells us about cosmology and upcoming issues with general relativity like Robertson-Walker metric, accelerating universe, etc.
This book is one of the best books that I can recommend to all physics enthusiasts (or nerds).
Glad midsommar!
February 11, 2020
excellent book
June 19, 2016
I said a while back that Shankar's "Principles of Quantum Mechanics" was the best textbook I ever used, I want to amend that and put this on top. Very clear and extremely interesting (But manageable!) exercises.
Although he assumes nothing beyond vector calculus and linear algebra, I can't help but come to the conclusion that I would have had a lot harder time had I no differential geometry background.
Highly recommended!
Edit 2016: I really liked this book as an introduction to SR/GR and definitely still recommend it. Although, I've found its not something that I refer to that much anymore. There are much more in-depth treatments of the topics covered (i.e. MTW, Carroll, Wald etc.)
Although he assumes nothing beyond vector calculus and linear algebra, I can't help but come to the conclusion that I would have had a lot harder time had I no differential geometry background.
Highly recommended!
Edit 2016: I really liked this book as an introduction to SR/GR and definitely still recommend it. Although, I've found its not something that I refer to that much anymore. There are much more in-depth treatments of the topics covered (i.e. MTW, Carroll, Wald etc.)
January 1, 2021
I have a PhD in electrical engineering from UCLA. 20 years ago, just out of my desire to better understand cosmology, I spent a year studying this book and working the problems on my own. I learned a lot about General Relativity. Having a solution set with at least the answers to each chapter’s problems would help a lot.
July 9, 2024
First contact I had with GR that I could understand a bit, I will always be grateful. Recomend the first topics to get some feeling (many of the exercises are very useful), never saw the advanced ones. The derivative notation could be better.
Read
November 10, 2021The way in which special relativity is taught at an elementary undergraduate level – the
level at which the reader is assumed competent – is usually close in spirit to the way it was
first understood by physicists. This is an algebraic approach, based on the Lorentz transformation (§ 1.7 below). At this basic level, we learn how to use the Lorentz transformation to
convert between one observer’s measurements and another’s, to verify and understand such
remarkable phenomena as time dilation and Lorentz contraction, and to make elementary
calculations of the conversion of mass into energy.
level at which the reader is assumed competent – is usually close in spirit to the way it was
first understood by physicists. This is an algebraic approach, based on the Lorentz transformation (§ 1.7 below). At this basic level, we learn how to use the Lorentz transformation to
convert between one observer’s measurements and another’s, to verify and understand such
remarkable phenomena as time dilation and Lorentz contraction, and to make elementary
calculations of the conversion of mass into energy.
June 7, 2012
As the title indicates, a handy introduction to GR. A grasp of high school mechanics, college level calculus and linear algebra is all that you need to start digging into this book. The books starts with a refresher chapter on SR with importance laid on the geometric construct of SR rather than the analytic way that is read in high school. Few chapters familiarizes the reader with 4-vector concepts before delving into the equivalence principle that forms the core of GR. Non-euclidean geometry is introduced later in more of an applied way rather than starting as abstract mathematical construct. I haven't gone through the last two chapters that are dedicated to gravitational waves and application to cosmology.
July 22, 2012
very good introduction to general relativity, i think. the introduction to tensors is especially great. i didn't read very much on gravitational radiation, and some of the later chapters were not as good as the earlier ones. but overall, very good. and though i don't have the most qualified opinion, i think it prepares you pretty well for MTW (if you're into that).
January 13, 2015
The problems weren't that great. They were mostly just asking you to explore derivations in the text more fully rather than asking interesting questions about the material. I didn't find his explanations very clear either, although in class we used a different notation so that's probably the source of that issue.
March 6, 2018
This was our textbook in a course on General Relativity and Cosmology taught by Professor Chien at the University of Minnesota in the early 2000s. Love it. My first exposure to GR in "modern" notation. I first borrowed a copy from the old British Council Library in New Manila when I was in highschool. Good times.
December 12, 2024
This is by far the most easily comprehensible textbook on GR. In fact it may be the most pedagogical textbook I've ever worked with. Anyone with an understanding of calculus can get this and yet it still has depth. It forsakes rigorous proofs in some cases for the sake of clarity but there's always MTW for that :p
May 19, 2017
Too dense.
Read
July 30, 2011Dude, I know everyone makes typos, but in complex physical theories the difference between a 1 and a 0 or a + and a - can be kind of significant.
Read
August 3, 2011The best introductory text on the subject that exists. One of the best texts I've ever read.
September 26, 2011
Easy-to-read undergrad-level introduction to general relativity (with a review of special relativity).
July 5, 2012
Well written from a mathematics viewpoint. Not 100% application but still a good text. Chapter 6 on Curved Manifolds was my favorite.
August 19, 2017
The explanations were often unclear with vague examples. I learned 80% from class lectures.
Want to read
April 11, 2020General relativity — for when you get serious
May 27, 2025
now start doing exercises
September 10, 2007
This is where I started learning modern mathematics (as well as GR).
Displaying 1 - 27 of 27 reviews