The EigenChris channel covers “General Relativity Basics” in a 36-minute video, that, for all the math and equations, mentions some concepts actually important to my sci-fi writing and screenplay. So, for today, we’ll do a math video.
A manifold is a structure of space that behaves linearly in small spaces but it a closed structure without apparent boundaries on a large scale. The Earth, as a sphere, is a manifold. A flat map of the world always enlarges the distances toward the poles. On the ground, there seem to be no boundaries, and you are always at the “center” of your space reference, and calculations are linear. Special relativity applies. On the large scale, gravity appears and general relativity applies.
So the universe is a 4-d manifold, with 3 dimensions of space and one of time. At any point in space, everything seems to look like it has three dimensions.
Gravity would in a sense result from the way the manifold is closed as to not have edges.
There are also ideas like equivalence principle, and differentiation by “parallel transport”.
The inside of an Oneill Cylinder would seem to be a manifold if large enough. It would be more interesting to fold the cylinder into a torus (making it a true manifold), but for artificial gravity the bites of the doughnut would have to be artificially segmented with separate rotations. But mathematically it would seem to be a manifold and if the structure were large enough, the residents would not notice it.
Picture embed from Wikipedia: 4 simple manifolds, click for attribution