It can be hard to know where to start with textbooks. This
bibliography includes a number of books I have some experience with,
along with my impressions of them. I've also included a few I've
seen strong recommendations for.|
- [Artin] Michael Artin, Algebra,
- "Algebra" means linear algebra -- the study of groups, rings,
modules, and vector spaces. This is an excellent text. I
took the class this was written for (long, long ago) and I've recently
read about half of the (finally!) completed book. I find Artin's
style very readable, and he covers material that's very useful in
understanding relativity. The only drawback to this book is its
price, which is rather high. If you don't have a firm grounding
in algebra, and don't have a book on hand that you like, consider this
- [Browne] John Browne, Grassmann
Algebra, Incomplete draft can be found here
Browne started to write a textbook on the algebra developed by
Grassmann, but never finished it. It's oriented toward use with a
Mathematica package (which was also never completed, I think).
book was never brought to publication quality, and the programming
sections are pretty irrelevant to most of us. So why am I talking
There's lots of good material in between the Mathematica stuff, and
this text covers the exterior algebra in an accessible,
reasonable way. If you find differential forms difficult to
understand on an intuitive level, read the first two chapters of this
book! (And you can't beat the price.)
- [Burke] William L. Burke, Applied
Differential Geometry, Cambridge University Press, 1985
- This entertaining book presents things from a different point of
view, and it can be very helpful with sharpening one's understanding of
the concepts. Burke gives many examples of applications of the
mathematics he's discussing to physics, which is nice, but he tends to
assume a familiarity with the physics which was, in my case,
unwarranted. Perhaps as a result of that, I found it hard to
actually learn the material
from this book; I found it much more useful after I'd already
encountered the material in other texts. In consequence, I'd
suggest treating this as a supplement, not a primary reference.
- [Einstein1] Albert Einstein, Relativity,
The Special and the General Theory, a Popular Exposition,
Bonanza Books, 1961
- This contains a nice introduction to special relativity, and a very
brief overview of general relativity. If you're looking for a
text to get started learning relativity you could do much worse than
this one. Math required is minimal. I find Einstein's
writing quite readable, even though this is a translation. See
- [Einstein2] Albert Einstein et al, The
of Relativity, Dover, 1952
- This contains a number of early papers on relativity by
Lorentz, Weyl, and Minkowski. It's fun to read the old papers,
and see how the ideas developed during the early years. It's also
cool to read these and realize you're seeing the text just the way it
was published the first time; this is how it was first presented to the
world. But don't try to learn relativity from
this book! It's not a textbook and was never intended as
such. Learn it from regular textbooks or from classes; read the
papers for fun.
- [Herstein] I. N. Herstein, Topics
in Algebra, Wiley
- Don't bother. Even though it's highly recommended by many
readers, and it's a nice enough text in general, it doesn't go far
enough to be of much use in learning relativity, and it's too expensive
for what you get.
- [Lang] Serge Lang, Algebra,
revised 3rd edition,
- This is an advanced text. I found it very dull, but none
the less it covers a lot of worthwhile material, and gets into some
things in much greater depth than [Artin].
Note, also, that Lang has written a number of texts on algebra, some
more elementary than this. In my opinion, his writing is dry,
but he certainly knows what he's talking about; you probably
won't go too far wrong with any of his books. They're much
cheaper than [Artin].
- [Lang2] Serge Lang, Linear Algebra,
- This is a less intimidating book than [Lang].
My first encounter with linear algebra was through an earlier version
of this text (the Second Edition, which was published many years ago by
Addison Wesley). It covers the basics, and is available used
reasonably inexpensively. Lang is a thorough and accurate
instructor, and everything you really need to know before tackling
relativity seems to be present, but I find his writing painfully dull.
- [MTW] Misner,
Thorne and Wheeler, Gravitation,
- The classic. It weighs almost 5 pounds (in
paperback!). Its coverage is broad and deep, and the authors have
tried to actually explain a lot of the concepts in an intuitive way
(with pictures), rather than just presenting the math and proving the
theorems using plain algebra. Their style, though sometimes
unusual, is extremely engaging. But watch out: This
is not an introductory text, and if you already know something
of tensor calculus and the rudiments of relativity before you start
you'll find it much easier going. The authors sometimes assume
things are obvious, when only someone who already had a strong
background in this area would find them so. They also cover such
a breadth of material that they can't help leaving loose ends, which is
occasionally somewhat maddening. In conclusion: Buy this book,
study it, enjoy it, but if the going gets too rough don't hesitate to
switch to something easier, such as [Schutz1].
- [Rindler] Wolfgang Rindler, Introduction
to Special Relativity, Oxford
- I have not used this book.
This text on special relativity comes highly recommended, and I've
looked over the table of contents, which looks good. It covers
the geometric viewpoint, which [Einstein1]
does not. It covers a lot of material which is typically just
glossed over in general relativity texts, and looks like a valuable
adjunct to the other books I've used. I may yet buy a copy, just
to fill in some of the gaps in my SR knowledge.
- [Rudin] Walter Rudin, Principles
Mathematical Analysis, Second Edition, McGraw Hill
- Mathematical analysis is "calculus done over, done
If you learned elementary calculus from Apostol you may not need
anything more. But if you're like most of us, you learned
it in just 3 dimensions with some differential stuff in higher
dimensions, and that's all.
Rudin covers the basics of analysis, including compact sets, limits,
integration, implicit functions, and all the rest of what should be in
an elementary analysis course. It's all done in n
dimensions, which is the major difference between this and elementary
calculus. If you don't know what an "open cover" is, or why
"closed and bounded" is the same as "compact", you should seriously
consider learning some analysis before trying too hard to understand
Note that whether you use this text, or some other, you can safely skip
the sections on Lebesgue integration and the higher-dimensional analog
of Stokes' theorem. Lebesgue theory is largely inapplicable in
real world, and Stokes' theorem is probably more easily learned from a
more advanced differential geometry text.
Bernard F. Schutz, A first course in general relativity,
- Excellent book! Buy it!
Schutz follows the same approach that [MTW] use but
his coverage is much, much narrower than theirs. His target
audience is undergraduates, and that makes a big difference to how he
presents the material. He deals with special relativity, general
relativity, and that's all (no electrodynamics, no miscellaneous
differential geometry thrown in for fun, no extras). His writing
is pleasant, his proofs are accessible, his explanations are mostly
comprehensible. He does a nice job of explaining special
relativity, as well (but you will still find the going a lot easier if
you've already learned special relativity from a simpler book, such as [Einstein1]). And the book is small and
light enough to carry along on the bus, to the beach, or anyplace else,
which may be significant when you consider that any general relativity
text, even this one, is likely to take you at least a year to read!
- [Spivak] Michael Spivak, Calculus
Manifolds, Benjamin, 1965
- This book is a classic. It's a monograph on
and integration theory on manifolds. Most of the text is devoted
to developing the theory of differential forms, and in the last couple
chapters Spivak presents Stokes' theorem and the classical theorems
which are its corollaries.
I have a couple of problems with this text. Spivak, for all that
his writing is very clear and engaging, makes little attempt to explain
what a differential form means, nor does he attempt to show any
intuitive meaning for the exterior derivative of a differential
form. Stokes' theorem, when properly presented, is intuitively
obvious -- even in n dimensions. When I first read this
book, however, I was left struggling to figure out what it was
about. The whole statement of the theorem fits on one line and
uses only 9 symbols, but their significance was largely lost on me.
My other objection is that Spivak tried too hard to keep it
elementary. He treats only manifolds which are explicitly
embedded in Rn. That should simplify things,
and in this area, any simplification is to be applauded. However,
the consequence is that he seems to have a lot of useless mechanism
lying around doing nothing, because nearly all of what he's talking
about is just Euclidean space. I also found it substantially
harder to keep track of which maps were going in which direction,
because so much of the mapping that's going on seems superfluous, and
because every map is from Rn to Rm. If the
theory is done on abstract manifolds, instead, none of the maps are
superfluous, and it's easier to see the significance of what's being
done ... or, anyway, that's how it seemed to me.
If you want to understand differential forms, you should take a peek at
[Browne] before you dive into calculus on
Keith R. Symon, Mechanics, 3rd edition,
- This is largely unrelated to relativity. It's a standard
"intermediate-level" mechanics text. If you would like to learn
something about the Lagrangian formulation of mechanics, which isn't
typically taught until the junior year in college, this may be of
use. The first half-dozen chapters cover the Newtonian
formulation (f=ma) and then he gets into the more advanced stuff.
The drawback: I found this text painfully dull. Goldstein
is more advanced and consequently harder to understand, but Goldstein's
writing is less soporific, in my opinion.
Robert M. Wald, General Relativity, University of
Chicago Press, 1984
- This is a really fine text. The treatment is very
the coverage is very deep. His viewpoint is very abstract, which
means he doesn't specialize needlessly and hence gloss over aspects of
the subject matter. He includes a review of differential geometry
in the appendices which is also worthwhile. But beware: This
book is very dense. Learn differential geometry and
general relativity first, then tackle this book.
- [Warner] Frank W. Warner, Foundations
of Differentiable Manifolds and Lie Groups, Scott, Foresman,
- This is one of the references cited by [Wald].
It contains a
very nice introduction to the theory of differentiable manifolds.
Drawback: That nice introduction is in the first chapter; after
that he hits the accelerator and it's bye, bye. The rest of the
text is very dense. (But it's a skinny book...)
However, even if all you ever read is the first two chapters, this
could still be a worthwhile book, because it presents the "modern"
theory of differentiable manifolds in a thorough, more or less
- [WarnerS] Seth Warner, Modern
- I have not used this book.
I've seen this recommended as a good place to start, and I've looked
over the table of contents. It certainly covers all the topics
that are really needed. Reviews I've seen have been mixed; some
like it and find it useful, some say it's too thick and leaves too much
in the exercises. But one thing's certain: It's cheaper
than dirt. If you want an inexpensive intro to algebra, you
can pick up a used copy for about ten bucks. If you don't like
it, you haven't lost much!
Page created in 2004, and last updated -- just a reformat! -- on 11/16/06