Presumable to be a coordinate system it would have to exist at more than one point! Does he mean to define a Riemann normal coordinate system as a chart such that [itex]g_{\mu\nu}[/itex] takes its canonical form and the first derivatives [itex]\partial_\sigma g_{\mu\nu}[/itex] vanish wrt the coordinate bases at every point where this chart is defined, and to say that each point p, q, r etc. in M is covered by at least one such a chart (although not necessarily the same chart for p, q and r)?

Or is he only saying that each point p is covered by at least one chart in whose coordinate basis at that point [itex]g_{\mu\nu}[/itex] takes its canonical form and the first derivatives [itex]\partial_\sigma g_{\mu\nu}[/itex] vanish, although at another point q covered by the same RNC chart, [itex]g_{\mu\nu}[/itex] doesn't necessarily take its canonical form, wrt the coordinate basis at q, and the first derivatives [itex]\partial_\sigma g_{\mu\nu}[/itex] don't necessarily vanish?

Carroll is only saying that every point on the manifold has a set of unique coordinates that are Minkowskian. It's a set of coordinates because a rotation or boost of a Minkowski coordinate system gives you another Minkowski coordinate system.

For the arguments that follow, we don't really care what the coordinate system looks like at other points. What is relevant is the metric and it's derivatives. The Minkowski metric space at a given point is a tangent space at that point so the first derivative is the same as the underlying manifold, which is good to know. From the second derivative develops the notion of curvature.

After what is clearly understood what the manifold is like at a point, we can start connecting points on the manifold with parametric curves.

"Such coordinates are known as Riemann normal coordinates"

More precisely it should be said "Riemann normal coordinates with respect to the point p". They are sometimes useful when you need to calculate and compare some geometric quantities at this particular point.

And the properties in Proposition 5.11 on page 78 give an intuitive feel for Riemann normal coordinates. I hope to say more about this "intuitive feel" later, but right now I have to do real work.

Thanks, everyone, for clearing that up for me. So, to paraphrase, for each point p, there exists a RNC chart with p as its origin, x(p) = 0 = (0,0,0,0). And a RNC chart is one for which the components of g take their canonical form in the coordinate basis, [itex]dx^\mu \otimes dx^\nu[/itex], at this point:

There are other kinds of "normal" coordinate systems, e.g. Fermi normal coordinates. Aren't those two conditions just what defines "normal"? (Maybe that's just the first condition?) I haven't seen all the definitions carefully explained, but I expect that "normal" is defined by conditions like this, and "Riemannian", "Fermi" etc., are defined by specifying what geodesics to use to construct the coordinate system.

Incidentally, I get the impression that the word local is used in two different senses in the context of differential geometry and GR:

(1) non-global: "In each local chart a Riemannian metric is given by smoothly assigning a 2 x 2 positive definite matrix to each point"; (2) at one point only: "Local Lorentz covariance, which follows from general relativity, refers to Lorentz covariance applying only locally in an infinitesimal region of spacetime at every point."

Sometimes even open to either interpretation in the same expression, as in this thread: "A local Lorentz frame only exists in an infinitesimally small region of spacetime" (JesseM); "The local Lorentz frame is usually well-defined in a large region, but it only agrees with a grid of rulers and clocks in an infinitesimal region" (Fredrik). Is the name meant to indicate a non-global chart that's locally Lorentzian in the sense of being Lorentzian at one point (one location)? Or is it a chart that's local in the sense of being non-global, or not necessarily global, and also happens to be Lorentzian at one point?

When Carroll writes "Such coordinates are known as Riemann normal coordinates, and the associated basis vectors constitute a local Lorentz frame" (2:20), the message I'm hearing is that he's referring to a frame which is "Lorentz" (orthonormal) at one point, and only one point (although, of course, that's a very special case of non-global).

I also occurs to me that it's often hard to avoid an ambiguity in English between specific and nonspecific indefinite pronouns, e.g. "they asked if I had a book". (Did they have a particular book in mind, or were they just asking if there was any book that I had?) Almost always the context makes it clear in everyday life. But when the topic is something as abstract as "a point" (and when the reader is someone as ignorant as me!), that ambiguity can become significant.

Oh dear, I'll have to have a more careful look at Lee then. Perhaps Carroll was only intending this as a passing mention of RNCs, rather than a definition. One more for the list of things that "normal" can mean (unit length, orthogonal, possibly both...).

For the record, I'm still confused about "local Lorentz frames", so don't take anything I may have said about them too seriously. I think that "local Lorentz frame", "local inertial frame", and "normal coordinates" are the same thing. I also think that in GR, there are several different kinds, Riemannian, Fermi, etc., while in SR, the definitions of those special cases are equivalent.

Local Lorentz frame (at a point) means connection coefficients ("forces") vanish, and this happens when the derivatives of the metric tensor vanish - thus in normal coordinates centered at this point. But there are many normal coordinates centered at a given point. So, we can try to choose among them those with other "nice" properties.