Though I haven’t reached a point in my tensor-analysis self-study where I can describe them mathematically, unusual spacetime geometries and topologies have always fascinated me, and it’s always kind of annoyed me that they don’t get more mathematical press. No matter how improbable they might be, and now matter how insoluble the equations are, I’d really like to read a study of these odd spaces.
For example, what about a toroidal spacetime manifold? This is the only nonstandard spacetime that ever gets discussed. From what I’ve read, an observer in such a universe would see an “extra” image of whatever they could see at a distance (call it M), and another extra image at a distance of aM (where a is the ratio of the “tube” radius to the “toroidal” radius). The set of closed geodesics would be infinite in such a universe.
Or, what about some sort of fractal spactime? For example, begin with a “spherical” spacetime, and join up six more spherical spacetimes (smoothing the junctions, to avoid discontinuities), and then add four such spacetimes to each exterior sphere in the same way, ad infinitum. There would be infinite geodesics in the spacetime, even if the geodesic does not have both ends at infinity.
Or, consider an infinitely long “cylindrical” spacetime. There would be an infinite set of closed geodesics, but also an infinite number of open ones.
And finally, the mother of them all: non-orientable spacetimes! What about a Klein spacetime? Geometrically similar to the toroidal spacetime, but with some reversals along the way. Or a Möbius spacetime? Or one based on the projective plane? The possibilities are enormous!