Zee, in his book "Einstein Gravity in a Nutshell", is particularly sarcastic about sophisticated mathematics in general relativity. Here are some quotes from the book:
It might seem that the first approach is much more direct. One writes down (2) and that
is that. The second approach appears more roundabout and involves some “fancy math.”
It might even provoke an adherent of the first “more macho” approach to wisecrack, “Why,
with a bit of higher education, sine and cosine are not good enough for you any more? You
have to go around doing fancy math!” The point is that the second approach generalizes
to higher dimensional spaces (and to other situations) much more readily than the first
approach does, as we will see presently. Dear reader, in going through life, you would be
well advised to always separate fancy but useful math from fancy but useless math.
The set of D-by-D matrices R that satisfy these two conditions forms the simple orthogonal group SO(D),
which is just a fancy way of saying the rotation group in D-dimensional space.
Fancy people call the upper index contravariant and the lower index covariant—I can never
remember which is which. If you like big words, go for it.
I feel that it would be good for those readers seeing Riemannian geometry for the first time
to work through some “classical” differential geometry dealing with curves and surfaces,
“real” stuff that you could actually see and “hold in your hands.” Throughout this chapter,
we will be living in good old 3-dimensional Euclidean space. I am going to tell you how the
greats like Frenet and Gauss thought about curves and surfaces. None of the fancy tangent
bundle talk for us; we will just do it. Action, not talk!
Some sophisticated types favor a fancy-schmancy index-free notation.
This is analogous to the vector notation v that you are fluent with, instead of the index
notation v_i . But it takes considerable effort to learn the index-free notation, and when push comes to shove, in an
actual calculation, even a sophisticate might have to descend to indices. Besides, you have to learn to walk before
you can fly, and I think that for a first introduction to Einstein gravity, grappling with indices is an essential and
ennobling experience.
Unaccountably, some students are twisted out of shape by this trivial act of notational
sloth. “What?” they say, “There are two kinds of vectors?” Yes, fancy people speak of
contravariant vectors (p^μ for example) and covariant vectors (p_μ for example), but let
me assure the beginners that there is nothing terribly profound going on here. Just a
convenient notation.
Meanwhile, a mathematician friend of yours—could have been James Joseph Sylvester,
a rather astute fellow, since he demanded that his salary from Johns Hopkins University
be paid in gold before accepting its invitation to move from England to a scientifically
impoverished but economically upstart country called the United States—told you about
some fancy-pants math called matrix theory.
Because of general coordinate invariance, we have considerable freedom in choosing
the coordinates. This corresponds to picking a gauge in electromagnetism. At this point,
the rich man with his or her wealth of fancy terms starts talking about Killing vectors, and
possibly even foliation. We will get to all that later. But for the moment, it is pedagogically
more transparent to follow the poor man’s way,∗ using explicit nuts and bolts, wearing no
fancy pants.
You get the idea before I run out of Greek letters! As before, fancy people who like big
words call the upper indices contravariant and the lower indices covariant.
At this point, our friend the rich man could start spouting fancy talk about the covariant
derivative, presumably without writing down a single index and disdaining such “quaint
old-fashioned notions” as transformation, and thus cause our other friend the Jargon Guy
to become flush with joy. Instead, let’s be more modest and, together with our friend the
poor man, try to understand what the covariant derivative really means by working out a
simple example. Again, a tale best told through a fable.
... we see that speaking of Killing vectors is just an extra fancy way of saying
that the static isotropic metric in chapter V.4 does not depend on the coordinates t and φ.
But at the level of this book, it is
a small price to pay to avoid going into fiber bundles and other fancy mathematical topics.
No need to take a fancy course in partial differential equations!
Beginning students often snicker at this sort of getting an answer by “winging it,” compared to solving a
partial differential equation in all its glory, complete with factors of 2π and what not. But in fact, in cutting edge
research, the ability to do the former is often much more prized than the ability to do the latter. On the cutting
edge, the analog of the partial differential equation is typically not known, but the truly great theorists are often
able to grope for what they want in the dark “by the seat of their pants.”
