So I've been revising my notes on general relativity, and I've found several things worth mentioning.
1. Equivalence Principle. The equivalence principle gives us geometry. This is often poorly described (I too committed this error in my drafts).
The equivalence principle tells us neither the composition of a body nor its mass determines its trajectory in a gravitational field. So gravity determines paths, and this gives us geometry.
Moreover, there are different equivalence principles which should be mentioned. I yielded to this, and became incoherent (alas!). The trick is to stick this into a box, for the interested reader to find out more about it, but not obstruct the writing.
2. Coordinates for Black Hole. Different coordinates for the Schwarzschild solution are described beautifully in Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, and Eduard Herlt's Exact Solutions of Einstein’s Field Equations (Cambridge University Press, 2d edition, 2009).
3. Manifolds, Mathematics. I think I ought to examine Christopher Isham's Modern Differential Geometry for Physicists for a Physicist's differential geometry.
I should like to discuss the exponential map, which relates paths to geometry (as alluded in the equivalence principle discussion).
Most readers probably will agree that "Part II" of my notes (which specifically discuss differential geometry) are the toughest part of the notes.
Probably, I should mention a few examples of manifolds and explicitly study their coordinates in lecture 5.
3.1. Functions. I never discussed what it means for a function on a manifold (a) to exist, (b) to be smooth.
Really, this let us discuss curves too. Why? A curve is just a function γ:I→M where I is just a closed interval, and M is the manifold.
3.2. Diffeomorphisms. This word is thrown around a lot, but never defined rigorously (or at all!). So I should re-investigate this a bit.