Introduction to General Relativity: A Math and Physics Perspective

• Mathematica
• andytoh
In summary: Assuming the student is reasonably well-prepared to begin with (i.e., they have a strong mathematical background and are comfortable working independently), I would recommend Ohanian, Carroll, and Ohno as excellent starting points. Ohanian's Linearized GR starts from the ground up,covering everything from gravitational waves to the bending of light. Carroll's book is a bit more modern, mixing in topics like tensors and differential manifolds without getting bogged down in too much topology. Ohno's book is a bit more traditional, focusing exclusively on general relativity. After reading these three texts, I think it is important for the student to get his/her hands
andytoh
I think most students develop their mathematical background up to classical tensors before digging into the physics of general relativity.

I think this approach is good, but once this "Introduction to General Relativity" is finished, then the student should refine his mathematical background further and study tensors (with the modern coordinate-free approach), differential manifolds, point-set topology, differential topology, group theory, fiber bundles, etc... and then study general relativity again with a more mathematical general relavity book. This way, he gets a strong taste of general relativity from the both the physicist's and mathematician's point of view, with which he can then decide upon how to specialize. What do you think?

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I think that the three tier format of MTW Gravitation more or less implements your idea. In any case, what books would you recommend for the three stages (prliminary tensorss, physical introduction, mature restudy) that you suggest.

I like the approach in Ohanian. He introduces linearized GR from the beginning, which allows him to cover the bending and retardation of light, gravitational lenses and gravitational waves, before getting into tensor analysis. If I were teaching a course, I think I'd use this approach so that the class isn't so frontloaded with mathematics.

For the math itself, I like the approach in Carroll, which mixes modern ideas with the "tensors are objects that transform as..." approach. I do think differentiable manifolds should be introduced at the beginning (as Carroll does); it's just too useful an organizing principle. But I don't think it's useful to inject a lot of topology into a first course, even a graduate course. Topology, Lie groups, fiber bundles, etc. would be topics for math methods course (e.g. from Choquet-Bruhat et al.)

One question to ask yourself is
"At what level and for what purpose are you studying General Relativity?"
(Certainly a similar question can be asked of any subject.)

andytoh said:
I think most students develop their mathematical background up to classical tensors before digging into the physics of general relativity.

I think this approach is good, but once this "Introduction to General Relativity" is finished, then the student should refine his mathematical background further and study tensors (with the modern coordinate-free approach), differential manifolds, point-set topology, differential topology, group theory, fiber bundles, etc... and then study general relativity again with a more mathematical general relavity book. This way, he gets a strong taste of general relativity from the both the physicist's and mathematician's point of view, with which he can then decide upon how to specialize. What do you think?
I think it is better to start with mathematics and not with physics and then accidentally move into the field of general relativity. Then, if you are really good, and work very hard, you could probably understand most of GR

I really like parts of Sachs & Wu because they get straight to the point, connecting the mathematics with the physics it is trying to model.

Some general advice for would-be autodidacts of gtr

Hi, Andy,

I dare say that every teacher discovers with experience that how a given student learns best varies greatly (and often, unpredictably!) between individuals. This is one of the major reasons why classroom teaching, especially of technical topics, is so challenging. And why any advice offered here might be good for some students but very bad for others, so caution is advised. Ultimately, students should probably weigh any advice they may be offered according to how well they guess it corresponds to their individual learning style and how well they guess the advisor knows the subject. (Incidently, are you a teacher mulling how to advise a student, or a graduate student seeking advice for yourself?--- I am guessing the former.)

Students who are enrolled in university have the benefit of an established curriculum, which has generally been carefully thought out to ensure that all dependencies are satisfied and that the intellectual demands made upon the students in each course are not unreasonable. Even better, graduate or undergraduate students generally can seek guidance from advisors who have some notion of their individual background and abilities, and who can help the student tailor coursework to best meet individual needs. Therefore, it seems reasonable to assume that in this thread we are tacitly discussing SELF-education in general relativity, rather than university coursework. (Of course, at the second year graduate level or above, the distinction rapidly becomes blurred.)

Teaching oneself gtr from a standing start poses a nontrivial challenge, no doubt about it! Fortunately, autodidacts are blessed by an extraordinary number of truly superb textbooks on general relativity--- this is certainly NOT true for many other subjects of equally compelling beauty and interest. My own recommendations as of 2005 are at http://www.math.ucr.edu/home/baez/RelWWW/reading.html and they do conform to the "helical method" you espouse, in which the student revisits material once he has acquired greater sophistication.

From the posts I've just looked over in this forum (unfortunately I have noted quite a bit of misinformation offered by well-intentioned posters, as well as some good contributions from a few posters who obviously DO know enough about this beautiful but subtle subject to offer guidance to newbies), it seems safe to say that many readers here will greatly enjoy and benefit from the two popular books by Robert Geroch and Robert Wald listed at the beginning of my reading list (see link above). Popular books on physics in general and gtr in particular are in my experience generally more misleading than helpful, but these two are truly exceptional--- no doubt because the two authors happen to be leading experts on gtr who work at the University of Chicago (one of the great centers for gtr research). Wald is of course also the author of a standard graduate level gtr textbook.

Regarding textbooks, two of the texts mentioned so far, by MTW and by Ohanian and Ruffini, are on my list and I think very valuable. Sachs and Wu is not, because I happen not to consider this as valuable as the books I do list. (Certain techniques pioneered by Sachs in re Bondi radiation theory and optical scalars are essential topics for serious students to master, but I feel that other books, such as the textbooks by D'Inverno and Carroll, or the book by Poisson, do a better job of introducing these topics--- at least, for most students.) In general, I urge serious students to pay top dollar, as it were, for a recent book such as Carroll or Poisson, rather than paying one tenth as much for a book which was published twenty or thirty years ago. In my view, with a few exceptions such as the wonderful textbooks by MTW and Weinberg, the problem book by Lightman et al., books first published before 1975 or so are generally too outdated in notation and topics covered to efficiently bring the student to the point where he (or she) can easily read recent papers or arXiv preprints. Unfortunately, Dover books are often old books, and in some cases were not very useful even when new. There are exceptions, such as the wonderful book by Flanders on differential forms.

Regarding specific topics which are invaluable to serious students of gtr, there are some fairly obvious prerequisites which should not be omitted even by autodidacts, including:

1. abstract algebra (linear operators, quadratic forms, trace and ideally a bit of invariant theory, matrices as representations of linear operators, vector space bases etc. are all essential),

2. differential equations (the geodesic equations are ordinary differential equations; the field equations are partial differential equations; up to and including some local versus global existence; "local versus global" is a constantly recurring theme in gtr, and failure to recognize this critical distinction is one of the most frequent causes of student confusion),

3. mathematical methods generally (e.g. basic notions of real and complex analysis including integral theorems, the standard theory of the basic equations of mathematical physics such as the heat equation and the wave equation, e.g. solution by separation of variables or by power series, special functions including Bessel functions, Legendre polynomials, spherical harmonics, and hypergeometric functions, Sturm-Liouville theory, potential theory for the Laplace equation, multinomial Taylor series, asymptotic expansions, perturbation theory),

4. differential manifolds (including vector fields as first order linear partial differential operators, tensor fields, exterior forms, Lie algebras and Lie groups) and ideally some prior exposure to surface theory or even abstract Riemannian geometry.

But in addition to these, some essential prerequisites often overlooked (I think) by autodidacts include:

1. mathematical models (especially the notion of a "theory" in mathematics viz. physics, plus the interplay of experiment and theory in physics and the other "hard" sciences),

2. electromagnetism at the level of Landau and Lifschitz (because some essential topics in gtr are best motivated, at least initially, via presumably familiar topics in EM, and also because some of the most important applications of gtr involve EM on curved spacetimes in some way),

3. undergraduate physics generally (e.g. some interesting topics in gtr, which is a classical theory, turn out to be closely connected mathematically to the beautiful theory of the Schroedinger equation!)

My own particular interests in this area seem to center around exact solutions. Here, a graduate student wishing to learn the theory of the Ernst equation or colliding plane waves will want to be familiar with the lovely theory of point symmetries and variational symmetries (at least) of (systems of) ordinary and partial differential equations, to have encountered the notion of Baecklund automorphisms, and to know something about theta functions.

In addition to these, one should mention that acquiring considerable familiarity with standard software which is extremely useful at all levels for mathematical physics generally, such as Maple and Mathematica, should not be neglected. In particular, I highly recommend GRTensorII, which is ideally suited for student explorations of gtr at a variety of levels. Note that the book by Poisson is best studied with a working installation of GRTensorII at hand. GRTensorII does not always conveniently support passing information to some other valuable Maple packages, but it works very well with powerful commands like "casesplit" (e.g. for solving the vacuum field equations or the Killing equations.)

Speaking of Bondi, I have important news both good and bad. The good news is that GRTensorII is free; see http://grtensor.phy.queensu.ca/. The bad news is that it runs under Maple, which is certainly not free. To be sure, Maple is invaluable for all kinds of computations in mathematical physics (far more than can be covered in a single course, in fact, although many universities do now offer courses in using symbolic computational engines like Maple or Mathematica.) Some might suspect that anyone who is unwilling to purchase software costing a few thousand dollars in order to study mathematical physics probably isn't sufficiently determined to succeed in their studies anyway, but I do regret that this poses a serious obstacle for some enthusiasts. (Registered university students can obtain Maple at something like one twentieth the list price, incidently, and I urge them not to pass up the opportunity!)

GRTensorII is ideally suited to making the kind of computations with specific spacetimes which students will find most valuable for gaining physical intuition into the theory, but is not suitable for symbolic computation in the style of "tensor gymnastics". (I should note that GRTensorII is also useful for working with approximations such as in the above-mentioned Bondi radiation theory, or in perturbations of black holes, not to mention working with large families of solutions such as the Ernst vacuums, so I don't wish to give the impression that it is only useful for working with specific exact solutions.) However, other packages are available which can handle that kind of thing to greater or lesser degree. As always, a wise student will beware of possible bugs.

This is off the top of my head, so no doubt I have forgotten some essential topics and will feel very silly once I realize what I omitted!

One last thought: I tend to feel that any student who assumes (not that I really think you were doing this!) that he must specialize in either mathematical or physical approaches to gtr will have limited success. In such a mature subject, any contributor worthly of note will be, I think, reasonably familiar with all relevant mathematical techniques and theorems (e.g. existence and uniqueness results), as well as well as being constantly mindful of the difficult and all too often neglected issue of physical interpretation of all our impressive mathematical techniques. I have recently been studying the collected papers of Chandrasekhar (you can picture me winding my way up my own helix here!) and enthusiastically echo those who would urge serious students to try to take him as something of a role model. Anyone who assumes from casual perusal of his monographs that Chandra exhibited much mathematical insight but comparatively little physical insight has greatly underestimated his genius.

Chris Hillman

(The same "Chris Hillman" cited in wolram's sticky above; this is my very first PF post, so please note that I hope to avoid discussing psychoceramics here; the "debunking" page cited by wolram was only a tiny part of a larger website offering guidance to on-line resources for serious students of gtr.)

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AlephClo
Chris Hillman said:
Hi, Andy,
Some might suspect that anyone who is unwilling to purchase software costing a few thousand dollars in order to study mathematical physics probably isn't sufficiently determined to succeed in their studies anyway

Someone who was really serious would write their own package.

As a follow up to Chris Hillman's post, these resources [directed to teachers of relativity] are worth mentioning:

http://arxiv.org/abs/gr-qc/0511073 Wald, "Teaching General Relativity"

http://arxiv.org/abs/gr-qc/0506075 Hartle, "General Relativity in the Undergraduate Physics Curriculum"

http://www.aapt-doorway.org/TGRU/ "AAPT Topical Workshop: Teaching General Relativity to Undergraduates" (see the articles, slides-from-talks, and the posters)

By the way, I would add Thomas Moore's "A Traveler's Guide to Spacetime:
An Introduction to the Special Theory of Relativity"
http://www.physics.pomona.edu/faculty/prof/tmoore/travguide.html
to the list of resources. (Moore made some insightful comments at the workshop above.)

Also worthy of reference:

http://arxiv.org/abs/gr-qc/0506065
"Classical General Relativity" by Malament
(who was also at UChicago; discussion of deeper foundations)

"General Relativity" by Ludvigsen

"Relativity: The Special Theory" and "Relativity: The General Theory" by J.L. Synge [for being one of the first "geometrized" (emphasizing invariance) treatments of relativistic physics]

Finally, here's my old list of references:
http://www.phy.syr.edu/courses/modules/LIGHTCONE/references.html

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On a personal note, I first studied General Relatvity at the undergraduate level with only the minimum required mathematical background. Getting an A+, I thought I was the guru of GR. Then I started perusing Hawking's "Large Scale Structure of Spactime" and was overwhelmed by the level of math and realized that I didn't know so much GR after all.

To overcome my new lack of confidence, I then studied differential geometry (Wasserman's "Tensors and Manifolds" being my favourite) and topology, only to realize that I found topology to be my favourite subject and studied that in great depth (too many topology books to mention here). But I did not forget that I initially had my interest in GR and so currently, my specific interest is to do earn a PhD by doing research solely in the topological aspects of GR, notably proofs, just as did Zeeman, King, McCarthy, Penrose, and Hawking himself have done. I am open to other mathematical aspects of GR as my mathematical interests continue to evolve but point-set topology and differential topology are by far my greatest interest.

I don't think I am walking down a mathematical dead-end street because I do have a good physical background on GR and so am prepared to give physical interpretations of my topological findings. Thanks for the advice guys, especially Dr. Hillman.

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Andy, that's a great personal history! More power to you!

andytoh,
Are you studying the arguably more-physical topologies (e.g., the Zeeman, Alexandrov, and fine topologies)? Have you looked at the Fullwood-McCarthy topology?

Hmm, I got a copy of Wasserman (1st ed; I didn't realize there was a 2nd edition) for \$10, but wasn't too impressed with it, finding it too mathematical for my taste. Guess I'll have to give it another try.

robphy said:
andytoh,
Are you studying the arguably more-physical topologies (e.g., the Zeeman, Alexandrov, and fine topologies)? Have you looked at the Fullwood-McCarthy topology?

At one time I wanted to specialize only in point-set topology and nothing else. But my interest in switching to researching topology in GR began when I studied Zeeman's "fine" topology as his alternative to the Euclidean topology for Minkowski Spacetime. I realized that I was hooked when I then studied Hawking, King, and McCarthy's "path topology" for Minkowski spacetime which had the same homeomorphism group as Zeeman's fine topology, and which amazingly required in its proof nothing more than my beloved point-set topology. I have not looked at the Fullwood-McCarthy topology in great detail yet, as I am still only a graduate student (and thus still trying to discover how I would like to approach GR/topology/differential geometry), but I am not biased to either physical or theoretical topology in GR.

In my opinion, if history's greatest topologists did some research in general relativity, we would certainly understand general relativity better today.

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How I got to be such a knowitall

Hi, Andy,

andytoh said:
On a personal note, I first studied General Relatvity at the undergraduate level with only the minimum required mathematical background. Getting an A+, I thought I was the guru of GR. Then I started perusing Hawking's "Large Scale Structure of Spactime" and was overwhelmed by the level of math and realized that I didn't know so much GR after all.

To overcome my new lack of confidence, I then studied differential geometry (Wasserman's "Tensors and Manifolds" being my favourite) and topology, only to realize that I found topology to be my favourite subject and studied that in great depth (too many topology books to mention here). But I did not forget that I initially had my interest in GR and so currently, my specific interest is to do earn a PhD by doing research solely in the topological aspects of GR, notably proofs, just as did Zeeman, King, McCarthy, Penrose, and Hawking himself have done. I am open to other mathematical aspects of GR as my mathematical interests continue to evolve but point-set topology and differential topology are by far my greatest interest.

I don't think I am walking down a mathematical dead-end street because I do have a good physical background on GR and so am prepared to give physical interpretations of my topological findings. Thanks for the advice guys, especially Dr. Hillman.

I've told this story before, but you (or others) might not have seen those long-ago posts (in sci.physics.*):

I encountered MTW before I knew any math other than high school trig and algebra, or any electromagnetism or even a solid grounding in Newtonian physics, with one key addition: by a happy accident, I had recently studied (as an extracurricular activity) Taylor & Wheeler, so I was familiar with hyperbolic trigonometry and Minkowski geometry. When I saw MTW, I was immediately entranced by the pictures of hyperslices in the Schwarzschild geometry and so on. I thought: "hey, if the point is to be able to draw a picture, I can do this stuff!" So I started studying MTW and bye and bye was able to come up with some interesting results which were not mentioned in the book.

For example, I remember embedding various hyperslices other than those they discuss. I also embedded the FRW dust and radiation fluid and discovered they look like an American football and a lemon, respectively. Since I didn't know any calculus at that time, I simply used a table! :-/ I later found the FRW dust picture in the book The Shape of Space, by Jefferey Weeks Later still I was able to ask him if he had carried out the computation or just guessed, and he said he had just guessed--- rather impressive because the exact shape very closely resembles what he drew. I had a similar conversation with Teukolsky in which he was able to draw this shape correctly without doing any computations. This might be of interest here because in my sample of three individuals who -can- make computations, it is obvious that all three have enough intuitive understanding to know what the answer is without actually computing anything.

I also remember that, inspired by the discussion of the Eddington chart in MTW, I had the idea of trying to transform the Schwarzschild chart to "pull down" hyperslices orthogonal to the world lines of the LeMaitre observers (who fall in freely and radially from rest "at infinity") so that they appeared as coordinate planes. Again, I used a table of integrals (I did know enough differential calculus to verify by differentiation that I was using the correct indefinite integral),
and noticed that the slices are locally isometric to $$E^3$$, a fact which many might find surprising (and which amplies the discussion in MTW about the Coulomb field showing up in gtr with suprisingly little change from Newtonian gravitation). Much much later I learned that this apparently first done by Painleve in 1921! But his chart has only recently become popular; see http://arxiv.org/find/gr-qc/1/ti:+Painleve/0/1/0/all/0/1

Although I didn't know about the semipopular books by Geroch and Wald at the time, when I did come across them I saw that they would have been perfect for me at this stage, incidently, which is one reason why I recommend them so enthusiastically to anyone curious about gtr who has little mathematical background.

While I wouldn't recommend anyone to try to duplicate my own path (to state the obvious, MTW is a graduate level textbook intended for students with a strong background in math and physics), my experience does show that what counts in learning gtr is a strong visual imagination. My experience probably also shows that a certain amount of ability and a pragmatic approach (e.g. using a table of integrals under the assumption that I would soon pick up integral calculus) can be helpful. It also shows that it is quite possible to learn gtr before learning Newtonian gravitation (although I wouldn't recommend this route).

Continuing my story: I soon discovered Feynman's Lectures on Physics and started filling in my almost complete lack of background in undergraduate physics. I remember being very impressed with his introduction to complex numbers and his remark about the same differential equations playing the central role in the theoretical analysis of quite different physics phenomena. (A kind of force multiplying effect, as they would say in the Pentagon.) I soon decided I was more interested in differential geometry, differential equations, group theory, and information theory than physics (probably not quite the effect which Feynman and MTW had intended!), so I suddenly (and quite "illegally") declared myself a math major and insisted on enrolling in the abstract algebra and real analysis courses. I think my undergraduate advisor, John Hubbard of Hubbard-Mandlebrot set fame, expected a train wreck, but fortunately I did very well in those courses and eventually went on write a Ph.D. thesis on generalized Penrose tilings. Thus, ironically enough, I might be the only entity other than Roger Penrose which is familiar with both Penrose tilings and Newman-Penrose tetrads :-/ I once pointed out to Penrose that you can endow a rhomboidal Penrose tiling (with the vertices deleted) with a Lorentzian metric, so that both "thin" and "fat" rhombs correspond to
$$ds^2 = -dt^2 + dx^2, \; 0 < t, \, x < 1$$
Then the "vertices" correspond to deficits and excesses as in the Regge calculus, and the null geodesics resemble "curves" which had been introduced by J. H. Conway. This proved amusing!

Chris Hillman

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Chris Hillman said:
I might be the only entity other than Roger Penrose which is familiar with both Penrose tilings and Newman-Penrose tetrads

This reflects my general researching motif. If I study in depth the level of topology that topologists would understand but general relativists would not, and study in depth the level of general relativity that general relativists would understand but topologists would not, then something new is just bound to be found.

Thanks for your anecdote and time in writing it.

andytoh said:
... then study general relativity again with a more mathematical general relavity book. This way, he gets a strong taste of general relativity from the both the physicist's and mathematician's point of view, with which he can then decide upon how to specialize. What do you think?

I would add philosopher's and historian's point of view, if you have a philosopher of physics or historian of physics who teaches relativity in your institution.

I have been lucky to have taken and sat-in on numerous undergraduate and graduate relativity courses, ranging from line-element-based classical differential-geometric to more-abstract tensor-geometrical to philosophical-foundations-of-physics. Each course taught me different aspects of the subject (which are often not mentioned in the other courses). Taking these courses from various professors, of course, gives you a wide range of viewpoints and skills. (Unfortunately, I've never taken a relativity course taught by an experimental relativist, a cosmologist, or a historian of physics.)

When it becomes your turn to teach the subject [including courses for nonscience majors], you can often learn a lot from some of the good introductory books, old textbooks, lecture notes, and pedagogical articles that you may not have encountered [or overlooked] in the classes you took.

(Some of these recollections reminds me of Synge's "My Relativistic Milestones". I'm curious to read of others.)

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i thought a lot about it but it is very hard i have no idea sorry.

I've looked at Dr. Hillman's updated list of recommended books in http://www.math.ucr.edu/home/baez/RelWWW/reading.html and looked for the book that approaches GR in the most mathematically elegant way, while still providing fully the physical aspects of GR.

I first considered the most mathematical book "General Relativity for Mathematicians" but felt that it would not go deeply enough into the physics of GR. I thus have decided to read from front to back the same classic GR book that stumped me two years ago, which is "The large scale structure of Spacetime" by Hawking and Ellis. Though the book still does not go as deeply with differential topology as I have already studied, the mathematics in this book still looks quite elegant and the topological approaches in the singularity theorems seem suitable for my taste. My only reserve is that the book was written 33 years ago and so I feel that too much new mathematical approaches in GR may be missing from the book and/or many significant new findings in GR itself are missing. I wish Hawking had written a more updated book since then. Nevertheless, the notations don't seem out-of-date, and I think it is the best book availiable for my combined interest in differential topology research in general relativity.

By the way, another problem with the book is that it does not have any exercises. In order to sort out my own understanding, I will write out my own exercises (with help of other GR books) and type out my solutions with the most mathematically elegant approach that I am capable of, using differential topology beyond the scope of the book wherever I feel suitable. I will post links to these exercises with solutions in this thread so that can get some opinions. Thanks.

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andytoh said:
By the way, another problem with the book is that it does not have any exercises. In order to sort out my own understanding, I will write out my own exercises (with help of other GR books) and type out my solutions with the most mathematically elegant approach that I am capable of, using differential topology beyond the scope of the book wherever I feel suitable. I will post links to these exercises with solutions in this thread so that can get some opinions. Thanks

Of course Hawking and Ellis is not a textbook but their exposition of their original research, and the aftertones of that original clang are still echoing, so I don't think you need to be worried about being out of date. You become and expert on that stuff and you'll be currently in a small minority of well-regarded mavens (but remember I said expert).

The traditional way to do "excercises" on this kind of material is to fill in the gaps in the exposition. Nathaniel Bowditch, who did this in translating Laplace's Mecanique Celeste into English sighed that "Every time I see aise a voir in the text I know I have two days of work ahead of me."

The traditional way to do "excercises" on this kind of material is to fill in the gaps in the exposition. Nathaniel Bowditch, who did this in translating Laplace's Mecanique Celeste into English sighed that "Every time I see aise a voir in the text I know I have two days of work ahead of me."

Ok. So I wrote out a couple of simple exercises to "fill in the gaps" in Hawking's Large-Scale Structure of Spacetime. Here is the link:

http://www.sendspace.com/file/5d5r9a

The download link is in the bottom of the page. No pure general relativity exercises yet, since I just started and the exercises are only based on the first definition Hawking gives in his Differential Geometry chapter (page 11). With this thoroughness in gap-filling, I could come up with over 1000 exercises for the book!

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andytoh said:
Ok. So I wrote out a couple of simple exercises to "fill in the gaps" in Hawking's Large-Scale Structure of Spacetime. Here is the link:

http://www.sendspace.com/file/5d5r9a

The download link is in the bottom of the page. No pure general relativity exercises yet, since I just started and the exercises are only based on the first definition Hawking gives in his Differential Geometry chapter (page 11). With this thoroughness in gap-filling, I could come up with over 1000 exercises for the book!

Way to go, man! And the great advantage of this is that in addition to confirming your progress, this fixes the definitions in your memory, like "Locally Lipsh*tz", in this case. Use it and you won't (as easily) lose it.

andytoh said:
I've looked at Dr. Hillman's updated list of recommended books in http://www.math.ucr.edu/home/baez/RelWWW/reading.html and looked for the book that approaches GR in the most mathematically elegant way, while still providing fully the physical aspects of GR.

I first considered the most mathematical book "General Relativity for Mathematicians" but felt that it would not go deeply enough into the physics of GR. I thus have decided to read from front to back the same classic GR book that stumped me two years ago, which is "The large scale structure of Spacetime" by Hawking and Ellis.

Judging from the table of contents only, Relativity on Curved Manifolds might be another choice. The chapter on physical measurement looks particular interesting (well, I just ordered a copy, because that subject interests me).

A lovely little book is Frankel's Gravitational Curvature, but he doesn't go into much depth.

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Gaps in the literature

Hi, Andy,

andytoh said:
I've looked at Dr. Hillman's updated list of recommended books in http://www.math.ucr.edu/home/baez/RelWWW/reading.html and looked for the book that approaches GR in the most mathematically elegant way, while still providing fully the physical aspects of GR.

It will surprise few who have encountered by other attempts at exposition that I have been compiling notes for my own gtr book for years. Curiously enough, given my background, I feel that for most students the most pressing need is for an introduction which emphasizes geometric and physical intuition rather than mathematical technique. For more advanced students, there are some new monographs on some modern solution generating techniques, but it seems to me that there is little point in finding a solution (except as a possibly dazzling illustration of mathematical technique) if you can't give a good physical interpretation!

I go by "Chris" except in the most formal academic situations, BTW--- "Dr. Hillman" just might lead to requests for medical advice!

andytoh said:
I thus have decided to read from front to back the same classic GR book that stumped me two years ago, which is "The large scale structure of Spacetime" by Hawking and Ellis. Though the book still does not go as deeply with differential topology as I have already studied, the mathematics in this book still looks quite elegant and the topological approaches in the singularity theorems seem suitable for my taste. My only reserve is that the book was written 33 years ago and so I feel that too much new mathematical approaches in GR may be missing from the book and/or many significant new findings in GR itself are missing. I wish Hawking had written a more updated book since then.

Yes, there has been much progress and I agree that a completely new monograph is sorely needed. I agree that even after such a book appears, Hawking and Ellis will likely remain a valuable source, however.

Daverz said:
Judging from the table of contents only, Relativity on Curved Manifolds might be another choice. The chapter on physical measurement looks particular interesting (well, I just ordered a copy, because that subject interests me).

A lovely little book is Frankel's Gravitational Curvature, but he doesn't go into much depth.

I DID include those books in my list (not that anyone claimed otherwise). After much thought, however, I did not include them in my list of a dozen or some books each of which can, I feel, serve as an excellent primary textbook. (I didn't list Hartle's textbook because I am not yet familiar with its contents--- this is probably a serious oversight since this book appears to be widely used at present, no doubt for good reason.)

Chris Hillman

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WOAH! This thread is incredible, thanks thanks thanks to everyone.
I am very interested in SR and GR. I am still in my first semester at university, so I can't take modules in higher maths or physics (damm prerequisites). But I am thinking of slinking at the back of lectures next sem (auditing?) so that I get a general picture of some maths and SR at least. I am going to read those books recommended too. This thread has been a great help!

The previously mentioned Taylor & Wheeler is a good book for SR, though I'm not familiar with the new edition. I was sad to read that they dropped the use of rapidity ( $\theta = \tanh^{-1} \frac{v}{c}$ ), and that the book no longer includes solutions to all the problems. But you can probably find the older edition in the library.

Another SR book I'd recommend is N. David Mermin, It's About Time (this is an updated edition of his Space and Time in Special Relativity, which you may also be able to find in the library.

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Woe is we: a dirge on the demise of rapidity

They dropped rapidity?! That is terribly unfortunate, particularly in the context of gtr, where it is desirable to rapidly recognize the linear first order partial differential operator $$z \, \partial_t + t \, \partial_z$$ as the vector field generating a one parameter family of boosts, parameterized by the rapidity, for example in discussing Killing vector fields.

For those who don't know, this is just the hyperbolic analog of angle. The analogy between hyperbolic and elliptic trigonometry was the most valuable thing I got out of Spacetime Physics; become aware of this was literally invaluable when I later encountered MTW and I doubt I could have taught myself this subject had I not known about it.

Chris Hillman

Chris Hillman said:
They dropped rapidity?!

Edwin Taylor told me that folks were apparently not using it, and so was dropped from the newer edition. Although the new edition does have some nice features, the classic maroon one with the worked solutions is much better. That's why I have both editions.

You can get surprisingly far with rapidity and a little fine-tuning of your Euclidean geometric and trigonometric intuition. I've been making the rounds at the AAPT meetings trying to get the word out.

ComplexVar89
I just finished creating 33 exercise questions (with my full solutions, taking up 22 pages of typing, some easy, some hard) for the first section of the first chapter of Hawking's "Large Scale Structure of Spacetime." The section is Differential Manifolds, from his Differential Geometry Chapter (which itself has nine sections, preceding his next chapter that finally begins general relativity). Let me polish up my work (I think I might need some more submanifold questions) before I give out the link for those who want to do some differential manifolds exercises for general relativity studies.

These exercises only cover 4 and a half pages in the book, so at this rate of thoroughness, I could come up with about 3300 exercises (and about 2200 pages of work) for the book!

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Suggestion: It might be more digitally-accessible [now and in the future] if you use $$\LaTeX$$ rather than Microsoft Word. There are editors available if you need to use that pictorial menu system. Here is a supplementary program made by the makers of the Equation Editor: http://www.dessci.com/en/products/texaide/

I've been using MathType 5, but I have problems drawing shapes for my differential manifold questions. For drawing shapes and mappings, I've been using only MS Paint. Here is an example from my work:

http://www.sendspace.com/file/eav3p2

What program should I use to make the drawings I want, including for later on when I draw Minkowski timegraphs and Penrose diagrams?

Whatever you choose, you should pick a format that you can easily edit later. I'd suggest something http://en.wikipedia.org/wiki/Vector_graphics" )

For something that does $$\LaTeX$$, which can be adapted to be used in this forum, is http://jpicedt.sourceforge.net/ . Another option is xfig http://www.xfig.org/ (for windows, there's winfig [but the site seems down now] ).

Unless you're ready to publish, I'd suggest using pencil-and-paper-and-scanner (or a https://www.physicsforums.com/blog/2006/05/20/tabletpcs-for-science-and-science-teaching/" ) and drawing something "good-enough" for now... and focus on the subject content rather than the visual appearance. [I'd make a similar suggestion regarding the "typesetting"... focus on the content first.]

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There's a great big wide world out there, operatingsystemwise

And you should avoid assuming that everyone is as Microsoft-centric as you are!

LaTeX is the universal standard for papers in physics and math, as well as many other fields; LaTex (and other Tex-based document formating systems) is so superior to any other document formating system that it makes little sense to use anything else, in fact using anything else tends to brand you as a bit of a bumpkin.

(Don't worry, I'm not entirely serious. But I'm not entirely joking either.)

Chris Hillman

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