Carroll's book
takes a rather standard approach, even somewhat old-fashioned approach,
albeit from a quite modern perspective. He first spends quite a lot of time trying to make sure that the
reader understands that part of the language of mathematics relevant to an understanding of
the physics of gravitation and its application, i.e., Einstein's theory of general relativity:
this actually amounts to about 2/3 of the text, although some little of that is also interspersed
with immediate applications. It does quite a splendid job of
- first, in Chapter 1, introducing the basic machinery
of tensors via a "review" of special relativity, where the
reader is assumed/presumed to be more familiar with such things (I recall here Wheeler's description of a tensor: a
machine, something like a winepress, into which vectors are inserted, and other objects, perhaps including
more vectors are created);
- then, already still in Chapter 1,
introducing the vector spaces usually referred to as "tangent spaces" and making
the most natural
choices of basis vectors for them seem perfectly natural and reasonable. These are
the tangent vectors along
any choice of coordinate curves, xi, i = 1, 2, 3, 4, these vectors being
denoted in terms of the partial-derivative
operators which evaluate changes of functions along those curves, i.e., ð/ðxi.
Using the chain rule from calculus, it is easy to see that
these operators transform
"covariantly" as one changes
from one coordinate system to another;
- and explaining, now in Chapter 2, with good, although not many, explicit
examples of manifolds---curved spaces---noting the
correct definitions while emphasizing the important parts (only), and
- striving, rather successfully, to make natural the very important physical
notions of affine connection, "maps" from vector spaces to those infinitesimally close,
and curvature, tidal gravitational forces, now in Chapters 3 (for the
math) and 4 (for the physical interpretation),
- all the while adding in some important additional mathematical notions,
- such as Lie
derivatives---essential for understanding symmetries, for example,
- and Penrose diagrams, a very important, modern device for more careful descriptions of the
several kinds of "infinities" and "singularities" that occur in spacetimes,
- and differential forms---objects
which are also vectors, sometimes called co-vectors, but which have different transformation
properties, and for which the natural choice of basis quantities is dxi, which transform
"contravariantly" as one changes from one coordinate system than another.
We can also note that Carroll comes from a heritage and training as a particle physicist,
instead of having begun as a
researcher in (mathematical) general relativity. This makes his descriptions
somewhat easier to
understand for the beginning student, but makes his presentation somewhat
"old-fashioned,"
since most published research work involves somewhat different approaches and notation.
Three points in particular will be discussed in more detail during our studies:
- more modern approaches tend to more often use coordinate-independent and basis-independent presentations,
instead of one which is so dependent on components: we will try to use his "easy-walking" approach, but
use a slightly different "middle ground";
- differential forms are actually much more important, and more useful, than he suggests: we will try
to be more careful about all that;
- non-holonomic basis forms, which give the correct physical intuition for vector and tensor
components: we will spend a fair amount of time emphasizing the difference, and why.
- I will also take a different approach from him relative to action principles and derivations from
Lagrangian densities. I will take those parts very, very lightly relative to his approach; this will
save a bit of time for the "additions" above.
Applications of the mathematics to real physical problems are discussed well, and in some detail,
although not always with complete derivations now; however, because of the time spent on
mathematics, above, the number of applications is somewhat limited. There are detailed discussions of
- single stellar systems, our sun or a "black hole," and the experiments one can do there,
- "weak" gravitational waves, and some relation to "distant" sources, and
- cosmology a la Friedman-Robertson-Walker, which is presented in a much more modern
way than the alternative, earlier, "best-version" which was in Weinberg's book.
Schutz's book
takes a much slower, mathematically-more-careful approach to our subject matter, spending
the first 4 chapters in the special-relativistic mode. It is therefore a very good place
to go when you have temporarily become lost; it should give you more detail to read, and
to therefore formulate questions that should help you inquire for help about finding your
way back!
- Its re-introduction to special relativity is really rather thorough, in Chs. 1 and 2.
- A good, serious introduction
to tensors (in flat spacetimes) is given in Chapter 3, including both tangent vectors and
differential forms. [One notes that the material in Carroll on differential forms is
rather terse, and we will be using them considerably more than he does.]
- The material on fluids, in Ch. 4, is in much more detail and care than I want, but,
therefore, the material on the stress-energy tensor (Ch. 4.4) could be quite useful.
- The three chapters, Chs. 5, 6, and 7, are, again, a slowly-moving, physically-motivated,
guide to the simplest way to understand the curvature tensor and Einstein's equations; we
will take a somewhat more modern approach, so this could be a useful "fallback."
- The chapter on gravitational radiation spends far too much time on how to measure these
things---no such measurements having yet been made, of course---and no time at all on
exact solutions that describe gravitational waves. Nonetheless surely 8.3 and 9.4 are useful.
- The details about static and stationary solutions for stars and black holes are very
good, with lots of graphs, details, etc.
- His (last) chapter is on cosmology, and is basic and useful, but the models are
only briefly discussed.
Last updated/modified: 16 January, 2007
Comments to Daniel Finley---
finley@tagore.phys.unm.edu
