on April 22, 2002
I found Wald's book to be better as an introduction than MTW. However, you'll probably want to get both books since you'll need them if you're going to really understand GR. Here are some points:
1) Wald is more concise than MTW. MTW tries to make differential geometry "intuitive" through some, in my opinion, poorly choosen concepts. So I found Wald to be much more understandable.
2) The book is much shorter than MTW so it is a little less daunting of a task. Wald still covers all the basics so you are not cheated out of any topics.
3) Do not expect to learn the differential geometry you need from Wald's Chapters 2 and 3 and appendices. A nice cheap book is Lovelock and Rund's "Tensors, Differential Forms and Variational Principles" (Dover). This book is surprisingly good and will cover the topics in a very understandable way in as few pages as possible. This allows you to get on with GR as quickly as possible. Read a chapter in Lovelock and Rund and then read the corresponding section in Wald. This allows you to understand both the concept and Wald's notation. I found the two books worked perfectly together.
There have been many books written on general relativity from both a physical and mathematical viewpoint, but this one stands out as one that is a hybrid between mathematical rigor and physical insight. It is certainly written for the physics student, but mathematicians interested in general relativity can certainly benefit from its perusal. I only read the first nine chapters of the book, so my review will be limited to these.
The first chapter is a short introduction to special relativity put in by the author for motivation. And, instead of introducing the mathematical formalism "as needed" in the book, the author chooses to outline it in detail in chapters two and three. The approach taken is a "modern" coordinate-free one, at least from the standpoint of differential geometry, but he delegates to an appendix the relevant background in topology. Since he is targeting the physicist reader, he does not hesitate to use diagrams to explain the concepts. The author introduces the idea of a dual vector using the example of a magnetic field. Tensors are then defined with great clarity from the standpoint of mathematical rigor. The physicist reader may have trouble digesting this if seeing tensors defined this way for the first time, instead of via their transformations properties, as is typically done. The abstract index notation is introduced to deal with the plethora of indices involved in manipulating tensors. In the treatment of geodesics, the author shows that it is sufficient to consider curves that are affinely parametrized, and the geodesic equation is derived in a coordinate basis. Riemannian and Gaussian normal coordinates are discussed as consequences of the unique solution of the geodesic equation. Curvature is also characterized in terms of the geodesic equation and two methods for calculating it are discussed: the coordinate component and tetrad methods, with the Newman-Penrose method briefly discussed. The existence of symbolic programming languages such as Mathematica and Maple make tensor manipulation much less laborius than the author contends in the book.
In the next chapter, the principle of general covariance is introduced as one that prohibits the existence of perferred vector fields in the laws of physics. The metric is the only quantity permitted to be related to space in the laws of physics. Thus quantities such as the Christoffel symbols, cannot appear in these laws. The author discusses in detail how general relativity views gravitation in terms of curved spacetime geometry and how Mach's principle is incorporated, the later forcing the spacetime metric to be a dynamical variable. The author discusses the difficulty in solving the Einstein equation, namely that a simultaneous solution for the spacetime metric and matter distribution is required (since the stress-energy tensor, the "source", requires knowledge of the spacetime metric for its interpretation). The linearized theory is discussed in detail along with the Newtonian limit. Gravitational waves are shown to follow from the linearized Einstein equation. The effect of energy loss on the orbital period of the Taylor-McCulloch binary star system is discussed as an experimental verification of general relativity.
Applications to cosmology are given in chapter 5, which is restricted to the case of homogeneous, isotropic cosmologies. The reader gets introduced to the famous Hubble constant, along with Robertson-Walker and Friedman solutions. A fairly lengthy overview of the evolution of the universe is given.
The next chapter is devoted entirely to the Schwarzschild solution, which is used to discuss the four experimental verifications of general relativity, namely the gravitational redshift, the precession of Mercury's orbit, bending of light by the Sun, and the time delay of radar signals. The singularities in the Schwarzschild solution are treated via the Kruskal extension.
Methods for obtaining physically realistic solutions are discussed in chapter 7, most of these being obtained by exploiting stationarity and symmetry properties. Perturbation theory is discussed very briefly with no explicit examples given.
Topics of a more mathematical nature appear in chapter 8, wherein the causal structure of spacetime is discussed. The discussion is qualitative and not based on Einsteins equation, and so is applicable to general spacetimes. One wonders when reading it if the obtained framework can be based on an analytical (or possibly numerical) treatment of the Einstein equation, instead of pure differential geometry. It is shown that null geodesics are The discussion here sets the tone for the next chapter on singularities, wherein the author derives criteria for determining when a timelike geodesic is not a local maximum in proper time between two points, and for when a null geodesic fails to remain on the boundary of the future of a point or two-dimensional surface. By using the local positivity of the stress-energy tensor (this is the only place the Einstein equation gets used) to get an inequality on the Ricci tensor, the author shows that timelike geodesics cannot be maximal length curves and null geodesics cannot remain on past or future boundaries. However, using compactness properties of the space of causal curves allows one to prove the existence of timelike and nullike curves of maximal length in globally hyperbolic spacetimes. The singularity theorems are shown to follow from this contraction, giving the result that spacetime is timelike or nulllike incomplete. A very detailed discussion of the definition of a singularity in physics is given. In all of the author's discussion, it is very interesting to note that the Einstein equation is only used once in obtaining the bound on the Ricci tensor. One naturally wonders if this framework is more general than what is available via general relativity, namely a question to ask is whether the Einstein notion of gravity can be derived from a consideration of singularities. Enforcing the presence (or absence) of singularities may allow the derivation of gravitational theories that are not the same as Einsteins, and yet have the same experimental success.
on August 25, 2001
This book was a little scary to read the first time I opened it. Abstract Indices all over. OMG, What does this upside down triangle mean? Where did this strange L come from? These are the sort of questions you will be asking yourself if you try to read this book without adequate preparation in Differential Geomtery. Sure Wald has 2 chapter devoted to this, but thats like asking you to learn all the vocabulary that you have in english from 5 little summary sheets. However once you do know soemthing about Riemannian Geomtery(an excellent elementary source is the book by Bishop and Goldberg "Tensor Analysis by Manifolds"), this book is a joy to read. Every explanation is crystal clear, and makes for a very enlightening experience overall. There's no need to read between the lines that some books expect you to, and Wald dosent insult his reader's intelligence either. This books is written for serious students of relativity, be it applied mathematicians or physicists. For the people willing to patiently read the book, and learn the details he presents, this book is probably the best preparation to general relativity. One complaint however is the noticeable shortage in exercises. And the ones supplied arent particularly difficult either. But all in all, an amazing read.
on February 27, 2007
Wald's book was the standard text for two graduate courses in GR that I took during my PhD (one was an introductory grad course on GR and the other was an advance special topics course on black holes). The first six chapters lay the groundwork for classical GR, starting with a quick recap of the tensor notation (Wald's Index free notation is very useful), a little bit of differential geometry and the Einstien's equations. The Initial value problem of GR is treated in an elegant chapter that concludes the introduction. Advanced topics like black holes, area theorems, singularity theorems etc are treated in latter chapters, along with a nice chapter on QFT in curved space-time and the Hawking effect. I found Wald's book most useful for understanding the singularity theorems, which have been discussed very lucidly without sacrificing much rigor (some of the more technical details are best left to Hawking and Ellis).
There is a priceless discussion on Penrose diagrams, asymptotic infinity, ADM energy and the BMS group which to my knowledge have never appeared in another book (one has to go back to papers of Ashtekar and Penrose to find this information).
I had no prior exposure to differential geometry when I started reading the book (indeed my background at this point was an undergrad degree in Electronics, so my knowledge of physics when I took this course was rudimentary to say the least). I however found no difficulty in following this book, and indeed this book was the most exciting grad level book that I read until Polchinski's two tomes on String Theory. I would recommend Wald's book for anyone who likes to understand General Relativity and especially Black Hole Thermodynamics...and last but not least, the exercises in the book are all interesting and in some cases are pretty nontrivial. I learnt a lot of GR working out these exercises and highly recommend them to anybody studying this book...it is definitely worth spending time on these exercises.
on February 9, 2000
This book is really very useful for anybody who is interested in modern mathematical presentation of the General Relativity. It is a pity that the role of Hilbert's variational principle is underestimated -- look into a small book by Utiyama and you find a very elegant consize derivation of the Schwarzschild solution. R.Wald is wrong when he says that Friedmann has given the solution only for the closed Universe. In his second paper in 1924 (Z. Phys., 12, 326) A.Friedman has found the open, hyperboloid Universe, so ALL the solutions in Table 5.1 are rightfully referred to as Friedmann ones.
on January 2, 2016
The first book I use for GR was Schutz, that is a wonderful book on its own but is only as it is named "A first course". Then I went through Carroll, to be honest GR has never been an easy one for me. Having said that I believe to had acquire a better mathematical and physical maturation for when I decided to pick Wald. I like the tensorial notation used, also it has the best derivation of Einstein's equations I have come across so far, I would say much, much better than Carroll's one, it is also self contained and I like the mathematical level used throughout. It consists of two parts, part one called FUNDAMENTALS has 6 chapters, part two titled ADVANCED TOPICS continues from chapter 7 till 14, I have studied chapters 1-13: Introduction, Manifolds and Tensor Fields, Curvature, Einstein's Equations, Homogeneous Isotropic Cosmology, The Schwarszchild Solution, Methods for Solving Einstein's Equations, Casual Structure, Singularities, The Initial Value Formulation, Asymptotic Flatness, Black Holes, Spinors. The only chapter I have not read yet because of lack of time is the last Chapter 14 "Quantum Effects in Strong Gravitational Fields". What I most like of this book is the crystal clear explanatory level that R. Wald shows through out the entire material making evident that he is a brilliant expositor and teacher, for example, is the first time that I really understood the singularity theorems of Penrose and Hawking (mathematically), also for the Initial Value Formulation chapter, he provides and explains what is needed in the theory of partial differential equations which motivated me even further to look at another book to go more in depth into partial differential equations and their relations with GR; and just for the last example the "Spinors" chapter is a tour de force which after going through it made me (I believe) understood everything about the representations of the Poincare group and things like why the Klein Gordon equation describes a unitary representation of the Poincare group acting on physical states of spin zero, Dirac's equation acts on particles of spin 1/2, Maxwell's acts on particles of spin 1 and (linearized) gravity on particles of spin 2. As if this were not enough it also brings 6 Appendices which complement the whole book! All in all the best book on GR I have come across so far, I also prefer it much more rather than Weinberg.
on January 25, 2011
This book should in my opinion only be taken up after the reader has mastered a more introductory level text on general relativity.
I originally bought this book and tried to read it without any prior preparation in GR and I failed miserably. It is simply too dense and too abstract for an introduction to the subject.
However, I turned to Schutz's First Course in GR and really spent the time mastering it. Schutz's book, if the reader takes time and does as many exercises as possible (and masters the coordinate-version tensor calculus), really brings one to a deep understanding of GR both from a physical and mathematical perspective. However, after having mastered Schutz, the reader is left with a mild sense of distaste in that the coordinate-version of tensor calculus fails to capture the essential idea of GR that the laws of physics are coordinate-independent and 'exist' in spacetime independent of how and whether coordinates are introduced. Schutz gives the reader some idea of this but the coordinate-free math is not thoroughly and consistently developed.
That is where Wald's textbook comes in. Once you understand Schutz, this book is really, as one reviewer stated, a tour de force. It really helps to have some advanced math course that involves proof writing (e.g. real analysis, abstract algebra, or linear algebra) to be able to appreciate and understand the rigor of the arguments. With this background, however, the reader quickly realizes that Wald's mathematical development of the subject is PERFECT and impeccable. Furthermore, it really prepares the reader for modern advancements in GR including causal structure, singularity theorems, and quantum gravity, for example.
Relativity is really a beautiful subjects. However, it is in my opninon a very difficult subject to learn both because of the amount of new mathematics that the student has to learn and also because of the very deep thought required to fully understand the physical implications of the theory. However, a couple of helpful points. The math is perfectly analogous to studying the (locally) Euclidean geometry of curved surfaces and non-cartesian coordinates in two or three dimensions; it is unfortunate that this math is not taught more commonly at the high school or college level, but aside from this, it is really not too hard in principle to learn, just tedious. The physical understanding of the theory is sometimes lost in textbooks (including in my opinion in Schutz and even in Wald) as the mathematical machinery is grinded out, but I encourage the student of GR to constantly try to break the arguments down to 'real-world' situations; for example, the reader should always try to envision creating some constructable coordinate system in any physical situation rather than say that some coordinate system exists in which the metric is nearly flat, etc. Incidentally, Wheeler's Gravitation text does a very nice job of bringing the physical picture to the fore, although I do not recommend it as a primary learning tool for GR (really because its size and constant digressions make the subject seem to require endless amounts of time to learn).
I totally encourage the reader (including any self-students such as myself) to take the time to learn this beautiful subject. This series of books worked well for me:
AP French Special Relativity (Read and understand thoroughly)
Wheeler's spacetime physics (after reading French, can read this relatively quickly, focusing on the 'geometric' nature and formulation of SR)
Bernard Schutz First Course in GR
If you want, the reader can really hammer home his understanding of GR by doing Lightman's problem book in GR after Schutz. Some of the material is not covered in Schutz, but with the (brief) introductions in Lightman plus maybe a little external reading, most of the problems in Lightman are doable.
Then, if you are ready to take the leap and try to understand the fully modern view of GR as well as its current-day theoretical research, I recommend Wald. Again before reading this, please try to study at least one advanced math book that involves introductory proof writing (e.g. linear algebra, abstract algebra, and/or real analysis).
If this seems to long to the new student, don't necessarily worry: each step along the way (French's SR, Wheeler, Schutz) is in and of itself deep, important, exciting, and philosophically satisfying.
on June 24, 2004
I used this text for a course after taking an undergraduate GR course based on Shutz. I found Shutz to be a much clearer and pedagogical text, and don't think I would have learned GR as easily if I had started with Wald. I think one requires greater mathematical preparation than I possess to fully appreciate the discussions involving topology in the second chapter and appendix. Oddly, however, this text becomes clearer as the reader advances through it: later chapters were more straightforward and still concise.
on October 16, 1998
Offers the clearest introduction available (using the best notation) to the mathematical background (e.g. the connection). Concise, careful, and clear. Particularly strong on singularity theorems, causality, and black hole thermodynamics. Narrower coverage than Stephani or d'Inverno, but provides the best introduction to these topics. Includes problems. Should appeal particularly to mathematically minded readers. This book might look daunting at first glance but I think it is actually very "reader-friendly"-- I find I appreciate it more each time I return to it.
on December 11, 2006
Wald's book stands out as the clearest presentation of general relativity yet produced. The downside is that the conciseness often makes it inaccessible to the beginner. If you try to learn from this book, you *need to do exercises* (from this book or another). It is too hard to follow if you don't have the experience of computations under your belt. But once you do get to the point where you follow Wald, you will follow him easily and pleasurably, as he writes with effortless clarity.
A common myth is that this book is overly mathematical. On the contrary--some of the highlights are where Wald discusses the role of Mach's principle in Einstein's formulation of the theory, and the role of our "philosophical projudices" in our choice of cosmology. Wald's talent is the ability to state the interesting physical or philosophical stuff without having to ramble on like other authors.