No, not really. Well, a little then

But my point, ahem, is that the the straight line in this 3D idealization (+ time) actually is a closed circle and that even thought they 'come back' to the same space coordinates in a different time, the same can be said for any type of circle made, as long as times arrow is involved. And doing it without involving time(s arrow) seems impossible to me?

So? a closed circle representing a straight line in SpaceTime, but it all hanging upon if my understanding is correct, that 'all geodesics represent the shortest path in a curved SpaceTime'. As for 'flat SpaceTime' I understand that to be a generalization, for cases where one don't need to take into consideration curved SpaceTime, but i may be wrong there?

You wrote "For example, a circle in space is made up of all points equidistant from a common center. If you try to construct that in space-time, you get a hyperbola." Can you give me an example of why that is so? And are you talking about lifting a two-dimensional figure up in a 3D space? No you're not are you, you're talking about using time or not using time. But that becomes even weirder, I'm sure that you can do it mathematically, but considering doing anything inside our SpaceTime without involving time becomes impossible, unless we are talking about 'changing' time in adapting it to different frames of reference? And having no-time at all is then only given to photons/light? And that frame isn't allowed to use, is it? But even so, the point is still that whenever you create a 'closed circle' it should stay closed, no matter how it will transform itself geometrically in a different space/time. I doubt that you can 'open' it by simply placing it 'elsewhere'. On the other hand, considering that it should be the equivalence to a straight line it have to be open when transformed into ??

Ah well, I like it but I will need to think about it. And if you see some way to clear my confusion JP, you're most welcome to it

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In some way it actually fits in how I think about space sometimes, like some 'surface' presenting a 3D image for us in times arrow. I've thought about that before, wondering if space just is 'surface(s)' as all those paths do exist simultaneously. No matter if light 'travels' or not, space will still express itself the same if we would to measure a photons path. For example, sending one photon at a time splitting it in equal distances from source to sink, constantly backing the sink, will present us one 'surface'. Move the laser a little and do it again, and you will have another description of how space bends, with another 'surface' depicted. Like invisible layers but not arranged upon each other at all, more like very weirdly folded in all directions, and then wondering how you could fold that pattern out of a one-dimensional surface? And now I'm perfectly certified, am I not

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One might argue that for to call it a 'real circle' all paths in it have to be taken simultaneously, Possibly? Thinking of drawing a circle on paper and when then looking on it its path will be covered simultaneously in all its parts? If you see how I think. A circle can only be a circle when there are no openings in it under times arrow, and to cover that situation one would have to build a ring circling the perfect sphere instead? But how would one argue then, considering that this perfect rings path also should represent a straight line in a curved SpaceTime..

And if you instead consider space as 'visible points', each one deforming under gravity, each point should look differently, depending on from where you looked upon it too, with all points uniquely deformed. But I also think that you will find a 'likeness' to it if you scaled the points up or down, that is, the points deformations would still 'hang together' having a relation. Well, that's rather obvious of course, but still

Could one see that as a fractal behavior? Naah. It wouldn't be, would it?

oh yes, headache warning indeed