Two textbooks which should answer all your curvature questions
Hi again, Michel,
You will probably want to read in MTW, Gravitation, about Schild's ladder (section 10.3), geodesic deviation (section 11.2), and about curvature generally in Riemannian/Lorentzian geometry and its applications in gtr. Don't be frightened off by the "weighty" appearance of this beautiful book! (Assuming of course, that you are not already using it.) It is not nearly as forbidding as some like to claim.
Another textbook, which is much shorter and which may present a more friendly appearance, is Gravitational Curvature by Theodore Frankel. (Unfortunately, out of print). This book features an unusually detailed explanation of several interesting and important ways to understand curvature and its central role in gtr.
RandallB:
1. Wherever you said "dimensions", I think you meant to say "directions", or even better, "coordinate basis vector fields".
2. When you said "simply standing still and allowing time to pass requires that we must be moving in all of 4 of those GR dimensions", you probably shouldn't have. I don't think it would be easy to turn this into a sensible statement which corresponds to how gtr actually models gravitational phenomena.
3. Somewhere in there, I think you are (possibly deliberately, trying to simplify things for the original poster) conflating two distinct concepts which can be associated with a "timelike congruence" (the world lines of an idealized family of observers whose world lines smoothly fill up a region of spacetime without any mutual intersections; such a congruence is defined by giving a timelike vector field, not neccessarily a timelike geodesic vector field but without loss of generally a timelike UNIT vector field). To wit: first, frame fields (aka "orthonormal basis of unit vector fields") and, second, a slicing of some region of spacetime into spatial hyperslices.
The first allows us to describe any tensor, say the Riemann tensor, with respect to a frame associated with some family of observers. (Some authors speak of "physical components" of tensors, in contrast to the components with respect to a coordinate basis, which in general have no operational significance in physics.)
In particular, we can employ the Bel decomposition to break the Riemann tensor (in a four dimensional Lorentzian manifold) into three pieces, the electrogravitic tensor (which corresponds in gtr to the tidal tensor), the magnetogravitic tensor (which in gtr describes such effects as spin-spin forces, if any), and the topogravitic tensor (which corresponds fairly closely to what you would probably consider "spatial curvature"). This decomposition is independent of physics, but plays a key role in gtr. Note that a "nonspinning geodesic frame" is the closest we can get in a curved spacetime, according to gtr, to Lorentz frames. (This means: a frame field in which the timelike unit vector field yields an geodesic congruence, and the three spatial unit vector fields have vanishing Fermi derivatives with respect to the timelike vector field; see MTW.)
The second only works for "irrotational" timelike vector fields (aka "hypersurface orthogonal" vector fields or congruences). If so, we can define the three-dimensional Riemann curvature tensor of each hyperslice (its content is fully captured by the three-dimesional Ricci tensor). This can be computed from the spacetime curvature together with the irrotational congruence; MTW has a very detailed discussion of this, but the same material is well covered in more recent books, such as the one by Eric Poisson, A Relativist's Toolkit, Cambridge University Press, 2004.
One often sees statements the effect that gtr models gravitation as a curvature effect, and indeed in some sense IDENTIFIES "the gravitational field" with the Riemann curvature tensor. Such statements are correct, as far as they go. However, maybe you were only saying that it would be misleading to state that "this is all there is to general relativity". I don't think anyone said that here, but I would probably agree that claims of the form "that is all there is too gtr" invariably appear misleading when you take a closer look.
You did make one point with which I certainly agree (indeed, MTW and other textbooks stress this point): strictly speaking, the notion of covariant differentiation belongs to the domain of Riemannian (Lorentzian) geometry, and is particularly closely associated with the differential geometry of curves in (not neccessarily curved!) manifolds. As such, it is almost unavoidable when discussing scenarios involving accelerating observers in special relativity (as does the notion of frame fields!), but it does not neccessarily have anything to do with gravitation, or even to physics.
Cesium frog: I know what you mean, and your remark is essentially correct, I think, but be careful! You might have overlooked an important point: it is not easy to define directions GLOBALLY in curved spacetimes (that is, to define a specific unit vector field, in this case a spacelike unit vector field, on a given manifold, using only geometric properties of the metric tensor or derived tensor such as the curvature tensor). Sometimes you can appeal to geometric features which particular Lorentzian spacetimes may possess, such as Killing vector fields or distinguished null directions, to well-define some "distinguished direction" at each event. In this case, you are appealing to the coordinate vector field @/@r (clearly I need to learn how to obtain some mathematical markup around here, if this is possible!), but this is not a Killing vector field in the Schwarzschild vacuum solution.
But about the nifty things which congruences can do only in curved manifolds: if you don't already know about the beautiful Clifford congruence on S^3, you will love this! Even better, a good place to find a nice picture of this congruence is in one of the best short introductions to gtr, a lovely expository article called "The Geometry of the Universe", by none other than Roger Penrose. I made some attempt to get this on-line, but AFAIK it is currently only available in a (very nice!) out of print book, or rather two editions by different publishers, both out of print. The cheaper one was Mathematics Today, edited by Lynn Arthur Steen, Vintage, 1980. Be careful: there is a third book with the same title which is completely different!
So what is this Clifford congruence? Well, a picture is worth a thousand words, but it consists of a family of great circles on S^3 which twist around each other at just the right rate to balance the geodesic convergence with the divergence due to twisting--- think of two skew lines in euclidean space--- so that they maintain constant distance in a suitable sense. This congruence has many lovely interpretations in mathematics, e.g. the "Hopf fibration" and the geometry of the algebra of quaternions. For our purposes, it is a beautiful illustration, in Riemannian geometry, of the notion of the vorticity of a vector field. (I can't resist adding that these great circles, aka "Hopf circles", also define a family of nested flat two dimensional tori, the "Hopf tori", which turn up in gtr, for example in the geometry of certain "pp-waves".)
Los Bobos: I don't think we want to get into a discussion of either the history or the philosophy of gtr, but it rather appalling how much ink has been spilled over the passage you quoted from Einstein's 1916 exposition of his then new theory of gravitation! In particular, this passage later assumed new significance in the seemingly endless controversy (more imaginary than real) over the so-called "Ehrenfest paradox" (be wary of recent papers and arXiv eprints on suchissues, which vary widely in quality).
Chris Hillman
But then, didn't Einstein himself consider their precise form something of an interim solution?
