What he and I said is that acceleration doesn't curve spacetime. If you define "space" as a hypersurface (in Minkowski space) of constant time coordinate, then the curvature of space will of course depend on what coordinate system you're using.

The coordinate system I would associate with the motion of the point at the center of a rotating disc has flat hypersurfaces of constant time. To get a curved "space" you would have to do something like pick a point on the edge and define a local coordinate system by taking a segment of the world line of that point to be the time axis, and then use the standard synchronization convention to define the rest of it.

I don't like Greene's explanation of these things.

I could have equally well said I thought acceleration could be thought to curve spacetime....as well as space....

I doubt I really understand yet....I thought Einstein's equivalence said that acceleration and gravitational potential had the "same" effects....

Greene also uses a three dimensional disgram of an object moving thru spacetime (FABRIC OF THE COSMOS, page 61) ...say x,y are space and z is time, t....so a particle with constant velocity will trace out a straight line along, say, the t axis; rotational motion motion appears as a corkscrew along the t direction, and uniform accelerated motion as a curved trajectory...I believe I also saw this in another author's text but can't find it...

I'm not an expert, but I believe the equivalence principle states that they have the same effect locally. However tidal effects occur due to the second derivative in the Riemann tensor, which cause neighbouring geodesics to diverge.

I like them. Epstein uses a similar diagram to show an alternative way to visualize curvature. In the case of rotational motion (orbiting a bigger mass) the corkscrew is caused by a "density gradient" of the 2+1 spacetime, that bends the worldline around a column of higher density, because it advances slower in the denser region. This slower advance also models gravitational time dilation. Analogy: light rays are bent in a media with varying optical density / variable propagation speed.

What it says is that an experiment that measures some of those effects won't be able to distinguish between gravity and acceleration if the region of spacetime in which the experiment is performed is small enough.

I don't dislike spacetime diagrams at all. In fact, I think they are by far the best way to explain almost anything in special relativity.