Graduate Alternative topology of our 3D space

[kochanskij]

TL;DR

In a topology where three different 3D spaces meet at a plane, what determines which way a moving object will go?

Mathematically, a flat 2D surface folded down over the edge of a desk has no curvature, so a Flatlander could not detect the fold. Same for us in space if our 3D space was folded down in a 4 dimensional sense. Suppose 3D space was folded down in two different perpendicular directions, both perpendicular to all three of our space dimensions. If we picture it embedded in 5D hyperspace, the 3D space containing the X,Y,Z axes and the space containing X,Y,W axes and the space containing X,Y,V axes all meet at the common X,Y plane.

If this were our real universe, an object following Newton's laws of motion in X,Y,Z space goes in a straight trajectory thru this X,Y plane. It wouldn't see the fold of space or detect anything unusual. It could end up in the X,Y,W space or the X,Y,V space. What determines which 3D space it enters?

There is no gravity or any other forces on the object. Does it randomly go into one or the other space? How could it be random? Newtonian mechanics is deterministic. This is not a quantum object. It is a large rock or planet, not an atom. Would each molecule independently "choose" a space, breaking the object apart. Or would binding forces hold the object together? What if the object is a swarm of dust particles? Would their tiny gravity hold the swarm together?

Perhaps such a topology is impossible in the real universe? If so, what law of physics does it violate? It is consistent with general relativity, isn't it?
Is there some unknown law of physics that determines which 3D space the object enters? Is there some unknown law that prevents this topology from occurring?

 

[Ibix]

I don't think such a topology meets the definition of "manifold" used by GR. A point away from your junction is "near" points in two directions so is homeomorphic to $\mathbb R^2$ locally, but a point on the junction is "near" points in three directions, so is not homeomorphic to $\mathbb R^2$ locally. There's a more technical discussion of how a cross fails to be a 1d manifold in this current thread in Topology.

So, in short, such topologies are excluded from GR by hypothesis. There might be unknown physical laws covering such non-manifolds, or they may just be impossible. Either way, GR can't give you an answer to what happens.