Let me ask some basic questions that I'm getting out of Misner/Thorne/Wheeler; MTW.
If you watch an ant walking around an apple
, do you really believe that ant thinks its walking in a straight line? Do you think the apple is flat and geometry is somehow curved, or do you think the apple is actually meaningfully round
in some global reference frame?
If you see light bending around the sun
in a solar eclipse, do you think that the light traveled in a straight line, but space was somehow bent, or do you think that the light actually meaningfully changed directions
in a global refence frame?
If you see two satellites travel along geodesic paths
and somehow meet at the same point twice, do you think they both traveled straight lines in some warped space-time, or do you think that the satellites actually meaningfully changed directions
in a global reference frame?
From the writing, I have this sense that the authors MTW really truly believe there is no global reference frame, and that in each instance, the ant, the light, and the satellites really are
moving in straight lines. They manage to convince themselves of this by saying that the light and the satellite feel no forces; thus they must be traveling in straight lines.
These are the arguments they are using to entice me down the rabbit hole, as Einstein said "Why were another seven years required for the construction of the general theory of relativity? The main reason lies in the fact that it is not so easy to free oneself from the idea that coordinates must have an immediate metrical meaning."
I have no such desire to free myself from this idea. To me, any legitimate coordinate system should have immediate metrical meaning. And all coordinate transformations simply change the metrical meanings.
When I think of the distance between two points, I think of the shortest
distance between two points. Integrating along the path to find the distance between two points represents the distance traveled
along the path between the two points. I don't see how this is a change in space-time. It is a change in what you mean by distance.