# A complete course in relativity.

I was wondering what book would be useful in learning special relativity and basic general relativity. I know vector calculus, multivariable calculus, and a bit of variational calculus. I'm looking for something that takes an informal approach to the underlying mathematics(but still rigorous). I'd liek the book to start from special relativity.

I obviously know basic, mechanics, basic relativity, and a bit of celestial mechanics.

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If you're planning on buying a book, I highly recommend you look through university libraries/bookstores first. You'll also probably need at least 2 sources.

I put a few free resources http://www.mathphyswiki.com/index.php?title=General_Relativity" [Broken]. I'll be adding to them as time (& interest) allows.

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Fredrik
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My suggestion is that you start with "A first course in general relativity" by Bernard Schutz. That book would have been my first recommendation even if you had said that you are only interested in SR, because the chapters about SR are what really makes that book worth reading. It will also give you an introduction to GR, but in my opinion it doesn't contain enough information about the mathematics of GR. So I suggest that you also get "General Relativity" by Robert Wald. This is a great text on GR, with lots of information about the mathematics, but it says very little about SR.

You may however find Wald's book too mathematical. Perhaps you should just start with Schutz and then decide if you want more.

My suggestion is that you start with "A first course in general relativity" by Bernard Schutz. That book would have been my first recommendation even if you had said that you are only interested in SR, because the chapters about SR are what really makes that book worth reading. It will also give you an introduction to GR, but in my opinion it doesn't contain enough information about the mathematics of GR. So I suggest that you also get "General Relativity" by Robert Wald. This is a great text on GR, with lots of information about the mathematics, but it says very little about SR.

You may however find Wald's book too mathematical. Perhaps you should just start with Schutz and then decide if you want more.
I've read some of the mathematical portions of shcutuz and siliked it for it's lack of rigour.

Chris Hillman
Beware of "simple physics" websites

I was wondering what book would be useful in learning special relativity and basic general relativity. I know vector calculus, multivariable calculus, and a bit of variational calculus. I'm looking for something that takes an informal approach to the underlying mathematics(but still rigorous). I'd liek the book to start from special relativity.

I obviously know basic, mechanics, basic relativity, and a bit of celestial mechanics.
You could try the textbook by D'Inverno. See http://www.math.ucr.edu/home/baez/RelWWW/reading.html#gtr [Broken] for some other recommendations (including the books by Schutz and by Wald which have also been suggested).

I wish to avoid debunking at PF, having already amply paid my dues in that regard elsewhere. However, I see that another poster (possibly Eugene Schubert?) mentioned a website, everythingimportant.org, which is operated by Eugene Schubert; unfortunately, this website is a typical example of "simple physics" crankery. Such sites claim that maintream textbooks "make physics look too hard". I remind serious students of Einstein's dictum: a theory should be as simple as possible--- but no simpler. I decline to discuss specifics because that is simply not worthwhile.

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Fredrik
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I've read some of the mathematical portions of shcutuz and siliked it for it's lack of rigour.
I agree. That's why I also suggested Wald for the GR stuff. It's very rigorous for a physics book. For example, the definitions of a manifold and its tangent space is done as meticulously as it would be done in a book on differential geometry. Schutz does this very poorly I think.

However, I still think Schutz's book has excellent stuff about SR. It's the best text on SR I've seen, but there may of course be others that I haven't seen.

One thing I like in particular about Schutz's book is its very careful explanation of tensors in the context of SR. That's by far the best introduction to tensors I've ever seen. When I read the definition of a tensor field in Wald's book, I was very happy that I had already learned the elementary stuff about tensors in Schutz's book.

There is not a single book I can call very good. If there is such a thing as a perfect and best book, it will be the one that you are going to write yourself. Start with a blank notebook or blank papers, start writing down notes and it will turn into a book with your own approach, taste and even originality which you can use to teach others. The sources can be from anywhere. In fact some posts in this forum can be considered quite good!
I do not agree with Chris (again!) that D'Inverno's book is good.
I think you can try to get hold of Kip Thorne's lecture notes, under his "gravitational waves" lectures and the "application of classical physics" lectures, availiable free in the internet. (Thanks Kip!) Caltech people like Kip Thorne and Richard Feynmann are reputed to be good teachers, teachers who are able to churn out physicists.
There is another problem with "good" books: they are very expensive! Try buying the Wheeler's "Phone Book". In fact I wanted to buy Schutz's book but ended up buying a Chinese book published in PRC at 10% the price. Not good? But it is cheap and at least serve some purpose.
Be careful about books that teach maths to physicist though. They are not very clear and sort of like betray the mathematicians.

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vanesch
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You could try the textbook by D'Inverno.
I second that. It is a great read! Of course it is only a stepping stone, but at least it gets you going. Don't look as of now for what you might read after that: you'll be able to make a better choice once you are there.

As to those objecting to a certain lack of rigor, that's not the point. First of all you have to get yourself used to the material on an intuitive level, and that's what D'Inverno does very well. As I said, it is just a stepping stone.

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Schutz is quite good, but he may leave you a little unsatisfied in some areas (hard not too on a first pass). He does a good job on SR, but you still may want to fill in details with, say, Rindler's SR book.

Wald is quite a difficult book IMO. I think he has a tendency to overformalize. I would try Carroll before Wald, and his book is fun to read.

So I've narrowed it down to sean caroll's and d'inverno's. How much do each cover?

Our Chris Hillman has a http://math.ucr.edu/home/baez/RelWWW/reading.html#gtr [Broken]. Carroll assumes some knowledge of SR.

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So I've narrowed it down to sean caroll's and d'inverno's. How much do each cover?
You're studying a subject, not a book.

Fredrik
Staff Emeritus
Gold Member
I'm answering a private message here, so I can check and edit the LaTeX stuff.

Ragnar said:
I know wald's book takes a rigorous approach, but is it formal as well? I hate mathematical formality and struggle with notation.
I'm not sure what you're asking really. Isn't it the notational stuff more of a struggle when the book isn't very formal? Anyway, a big part of GR is differential geometry, and no matter what book you read about that subject, you will find that it involves a lot of struggle with the notation.

A good thing about Wald is that it uses the abstract index notation for tensors. That makes a lot of mathematical expressions much easier to write, compared to the index free notation (which is the most common in pure math books, I think).

Here's an example. Given a tensor T, of rank (1,2), i.e. a map

$$T:V^*\times V\times V\rightarrow\mathbb{R}$$

and a vector v, we define a tensor S of rank (1,1) by

$$(\omega,u)\mapsto T(\omega,v,u)$$

for all

$$(\omega,u)\in V^*\times V$$.

Now that's the index free notation. Here's the same thing (all of it, including what I described verbally) in the abstract index notation:

$${S^a}_c={T^{a}}_{bc}v^b$$

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Ragnar,

I would advise "Field theory" by Landau-Lifchitz.
Reading "Mechanics" first by the same authors would help.
It looks a bit hard at first sight, but it goes always to the point with the most useful approach.

Reading some more elementary book before could be useful.
However, personally, I have found the long literary discussions quite useless and counter producting.
You know ... these discussion with light rays and signals ...

The central point in special relativity is the invariance of the 4-distance ds².
This is a direct consequence of the Maxwell's equations, or the constancy of the speed of light.
Replacing the Galilean transformation by the Lorentz transformation is a consequence, while the changes in mechanics are quite logical and natural.

You could also read the lessons on special relativity by Einstein.

Michel

"The central point in special relativity is the invariance of the 4-distance ds"

I agree!!!

Come on, I first learnt my relativity at the age of ten with this book : The large scale structure of space-time, Hawking.
:)

The central point in special relativity is the invariance of the 4-distance ds².
I disagree, invariance of 4-distance is not something special. What is special is the way that distance is measured.

robphy
Homework Helper
Gold Member
Reading some more elementary book before could be useful.
However, personally, I have found the long literary discussions quite useless and counter producting.
You know ... these discussion with light rays and signals ...
In my experience, I learned the formalism first from portions of numerous sources (course textbooks: Skinner, Lawden, Landau, Wald; various other textbooks: Taylor-Wheeler, MTW, Weinberg) ... but that's all it was.. formalism.

It didn't start to all "click" until I was introduced to the "more physical" operational definitions using light-rays and signals and interpreting spacetime diagrams and the geometry of Minkowski and more general spacetimes. I would recommend (in addition to some of the above suggestions) the deceptively simple book "General Relativity from A to B" and the enlightening chapter on Minkowski space in "Mathematical Physics" (both by Bob Geroch), "Flat and Curved Space-Times" (by Ellis and Williams), and possibly "Relativity: The Special Theory" (by J.L. Synge). Now, the formalism makes more sense to me.

Furthermore, it leads some down a road which asks WHY does Minkowski spacetime (or a more general Lorentzian-signature spacetime) have the various structures we now take for granted [manifold, metric (and its signature), connection, curvature, topology, and causal structure].

robphy,

I understand your point of view, but my personnal conclusion is really very much the opposite.

For me, questions like "why the Minkowski's geometry" are meaningless.
Analysing signal experiments and assuming c is a universal constant is nothing more than a special case of the Maxwel's equations. And these thought experiments are not better at getting the basic ideas.

It is impossible to explain "why the Minkowski's geometry".
It is only possible to make plausible the idea that geometry is part of physics.

However, I also feel that theoretical physics can be a black hole if it is not supported by a real link with experience. That's why I feel important to learn about this aspect. The classical examples are the clock slowing down, the GPS, the MMX, the gravitational redshift and its observation (like mossbauer), ...

Strong technical training in this respect is more useful than endless discussions of the first steps.
The first step must fast and determinate.
Going back to the basics is my motto, but the basics are mathematical.

Michel

PS
==
When I was very young I read a tale about a bug living in a bowl ...
The idea that geometry can be explored was obvious from this tale.
The space-time aspect doesn't change anything.
And it is useful even for thinking about special relativity.

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robphy
Homework Helper
Gold Member
For me, questions like "why the Minkowski's geometry" are meaningless.
Analysing signal experiments and assuming c is a universal constant is nothing more than a special case of the Maxwel's equations. And these thought experiments are not better at getting the basic ideas.

It is impossible to explain "why the Minkowski's geometry".
It is only possible to make plausible the idea that geometry is part of physics.
For end-user classical applications (where a classical spacetime is merely the arena for physics), I would probably agree with you. However, for many approaches to quantum gravity, the question is very meaningful.

Going back to the basics is my motto, but the basics are mathematical.
Agreed... but realize that various approaches to quantum gravity challenge what classical gravitation regards as basic and may take for granted. One way to approach the problem is to ask Which (if any) structures are most fundamental, which leads to the rest of structures in the classical limit?

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MeJennifer,

I disagree, invariance of 4-distance is not something special. What is special is the way that distance is measured.
I never said the invariance of ds² is something special. I don't know what "special" could mean.
But this invariance is the most important message we received from the Maxwell's equations.
The invariance of ME is equivalent to the ds² invariance.

My point is that the Maxwell's equations tell us that geometry is part of physics.
Because there is no way of fitting the Maxwell's equation in a galilean spacetime.
And similarly we can be led to the idea of riemanian spacetime.

And "measuring" can only be done according to the laws of physics.
And how else can we coordinate spacetime?

Michel

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robphy:
Yeah, quantum gravity, singularities, cosmological models and stuff like that question our basic asssumptions.
Recently someone published a paper on cyclic cosmology using the dark energy recently observed and eliminated the singularity and inflation in one fell swoop. Makes me wonder what happen to those beautiful mathematics which Hawking and Penrose wrote so much about.
http://arxiv.org/abs/hep-th/0610213

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I myself read the first few chapters of a first course on general relatvity. I thought chapters one through four were amazing. The chapters after that were terrible from a mathematical point of view. Is wald's book good?

I myself read the first few chapters of a first course on general relatvity. I thought chapters one through four were amazing. The chapters after that were terrible from a mathematical point of view. Is wald's book good?
I've had dreams about Wald's text. I'm walking down a dark alley. I turn my head around slightly. And there it is. A book! The book seems to be moving in my direction. The faster I walk the faster the book moves. I come into the light as does the book. I can now see the title and the authors name on the front cover. It reads "General Relativity" by Robert Wald. I then scream and start running. The book runs after me flapping its pages. I keep screaming and running, screaming and running and running and running and running ... Its at this point I wake up in a cold sweat!

So no. I don't like Wald's book. :rofl:

Pete