I would like to make an argument regarding John Davis' flat earth model. In his Relativity & Flat Earth page, he says this:

Consider a theoretical object in a perfectly stable orbit around a theoretical planet in a traditional round earth manner. Remember from Newtons laws of motion: an object in motion tends to stay in motion and in the direction it is in motion. We can certainly say that the object in orbit that it feels no experimentally verifiable difference in force or pseudo-force - which is equivalent to saying it is experimentally not accelerating (and thus not changing direction or speed.) Remember, Einstein disillusioned our naive view of space based on the equivalence principle.

Our sight would lead us to believe this might be foolish, but if space is curved (and Relativity relies on the assumption that it is) it would be silly to not question our visual representation of space since by all accounts it appears as if our observational (and theoretical) language is ill equipped to deal with description of it.

(original page)

http://theflatearthsociety.net/relativity.htmlMy understanding of this is that John Davis is saying that orbits are flat/straight relative to space time, and it is the curve of space time according to Einstein's theory of General Relativity which renders orbits as curved, and celestial surfaces as convex. But in round earth terms, we don't truly consider a line drawn along the surface of a sphere to be perfectly straight, but as circles. Inexperienced round earther's are often pedantically reminded of this fact. It may be straight relative to the surface, but is really a geodetic curve. Likewise, if a line in space-time is geodetic, it may be straight relative to space-time, but ultimately must be considered curved.

Now to point out a complication with the above argument. We can observe the true curve in a geodetic line drawn on a sphere or torus, because we observe in 3 dimensions, and our 3 dimensions are not constrained to the 2 dimensions of the surface of the sphere. If our observation were constrained to the sphere's surface, the lines would curve differently because photons would be incapable of travelling parallel to one another. The appearance of that curve, however would change based on our position. The line would always appear concave, as if we were inside of it. If you try to cross the line, the line will appear to change its bend to appear to bend around you. If the surface was toroidal, on the other hand, geodetic lines would all appear straight, because photons could move parallel to one another in toroidal space.

If spacetime was indeed curved in such a way that the orbital paths and surface of the earth were geodetic to spacetime, for spherical space, they would appear to change magnitude relative to the 3 dimensional reference frame of our craft exploring the solar system, and for toroidal space, every orbit and planet would appear straight or flat.

The appearance of flatness of a round earth is explained easily. The radius of a sphere can be so large compared to the distance of us, the observers, that it subtends a nearly 180 degree angle for us. An infinite plane, coincidentally would subtend exactly 180 degrees for us. This logically demonstrates the fallacy of assuming something is flat because it appears to be flat.