Well, I told you I had a definition of a straight line (in SpaceTime) JP. It doesn't expend energy. And that one is represented by uniform motion and geodesics where we live. But that is from discussing what we see and learn about SpaceTime. It does not, as I see it, discuss 'ideal parallel lines'. Although in reality we have to accept that they, if using some bendable material as all materials I know of are, will form themselves after gravity.

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Are you defining SpaceTime as a sphere for this? And then assuming some sort of geodesics on the outside, or is it Newtons spheres, in a sphere, in a sphere? If we assume that we come back from where we started and then imagine us driving that cosmic car, using the tires to define our parallel lines then? And when I think of it (gravity) assuming the inside of a sphere I get seriously lost trying to imagine gravity working as described here.