Could one model a ‘universe’ as a solid sphere (i.e. ball) rolling on a landscape manifold, perhaps with a different topology (i.e. different continuum)? Thus the always rolling solid sphere and infinite in extent landscape manifold are always apposed (geometrically contiguous); hence always mixed continua? Could one then also include non-uniform endless motion for such solid sphere, by adding lumpy-bumpiness to landscape? Or perhaps a shallow or deep bowl effect?
Would the latter then be similar to all motion (i.e. orbital and hence non-uniform), inclusive of comets? That is, is all motion in the manifold (‘universe’) non-uniform and orbital? However is some of such motion (explosiveness) also exponential in drop off, such as for supernova, Big Bang, and fireworks? Then likewise exponential drop off for associated Big Expansion (i.e. Hubble expansion) for 3-volume of manifold? Would fireworks then serve as a demonstration of the universality of exponential drop off for all explosive phenomena? Is this analogous, in universality sense, to ‘frequency not scaling with mass impact, for all wave phenomena’ ?
There are many exotic curvilinear mathematical representations, some varying exponentially and otherwise; but not utilized much (at all?) for physical descriptions. If utilized for physical manifolds, then would such exotic trajectories of manifolds need be restricted to planes, in order to prevent intersections; hence consistent with theme of preserving invariance of manifolds?
Is alleged exponential drop off in explosiveness for Big Bang, not only consistent in it’s universality for all explosiveness, but also consistent with maximizing of entropy generation (a modified time construct MTC), in comparison to alternative scenarios; such as for Archimedes spiral? Likewise for all exponentially changing MTCs (including tangent vectors for mod global trajectory), abstracted and generalized as modified 1-manifold? Thus is just exponential fall off both necessary and sufficient, for non-uniform change, without resort to any more exotic non-uniform mathematical constructs? TMM
Lawrence J. Dennis, 1972, A Catalog of Special Plane Curves, Dover Publications, N.Y.