Curvature within the Earth
Let us see how spacetime is curved within our planet, the Earth. The original material is in Misner, Thorne, and Wheeler, p. 39; but the same material is arranged as the "Boomerang Project" by Wheeler and used in his book (1999). Idealizations are assumed: the Earth has a uniform density, and does not rotate; test particles do not attract each other and do not collide. I presume you know "spacetime diagram" and "embedding diagram". We already know how a free motion looks, viewed from the fixed frame and from the frame freely falling with the body in motion, in question (see Free Fall). The following example is a sort of free fall, but since it has a "rising" part as well, "free float" may be a better name.
As in the figure (the coordinates are set at the top of the Earth, and may be regarded as Minkowskian), each particle oscillates, and the two trajectories corss, despite they are separated initially, at a regular period (84 minutes, for one cycle). According to general relativity, this shows the curvature of space (the vertical plane crossing the center of the earth). Particle A is free-floating, and hence its trajectory is a geodesic (in spacetime), seen from particle A; since B's trajectory cross A's geodesic, it must be regarded as curved, seen from A. Actually, such curvature is intrinsic; measurements on the surface where A and B move can reveal this curvature.
And whether seen from A or B, or from whatever system (e.g., the first Minkowskian frame), such curvature can be expressed in an invariant term (the same in each system); thus spacetime must be regarded as curved. As an illustration, let me add another view from A, but this time A chooses a "curved" coordinate system which is obtained by a continuous deformation (diffeomorphism, in technical terms) from the previous one. Still, by general covariance, this new view expresses exactly the same relationship between A and B. Notice that A's curved trajectory is itself the time axis for A.
It may be instructive to notice that Einstein explained his insight as regards general covariance to Paul Ehrenfest in similar terms; see The Genesis of General Relativity: (6) How did Einstein overcome the "Hole Argument"?. What is important is that gravity can be expressed by curvature of spacetime, as an invariant feature (independent of a choice of coordinates). And we have a convenient way to visualize such curvature (but remember that it's not space curvature, but spacetime curvature, that fully explains gravitation and its action; the following diagram shows only spatial curvature).
References
Misner, Thorne, and Wheeler (1973) Gravitation, 607-617.
Wheeler, J. A. (1999) A Journey into Gravity and Spacetime, Scientific American Library, 1999.
Last modified July 23, 2002.(c) Soshichi Uchii suchii@bun.kyoto-u.ac.jp
