Is it possible for there to be a spacetime which shares one of our dimensions? Imagine two 2D spacetimes, one with $x,y$ axes and the other with $x,z$ axes. They share a dimension in common with the x axis. Is it possible that a spacetime which shares at least one dimension with our four $(x,y,z,t)$ dimensions and may have other dimensions not in common with our 4 dimensions exists? In other words,can other hypothetical spacetimes have dimensions which are represented by axes such as $(x,z), (x,y,w), (y,z,t,w)$? [The $w$ denotes the 4th spatial dimension] I know the axes representing a space is arbitrary, but what I mean here is that taking our spacetime as $(x,y,z,t)$ can there be a spacetime which shares some of our dimensions and have others that we don't have?
(This is my first question asked here,so I'm not sure if this type of question is appropriate here)
 A: Time is a bit complicated, so let's ignore that for a moment.
If two, 2 dimensional flat surfaces crossed (imagine two pieces of paper) then the intersection is a line.  A slice of an object in on "space" would briefly appear (assuming we could see objects in the other space, which is another issue) in another when that object passed across that line.  If two spheres intersected, then the intersection region would be a circle.
In 3 dimensions, the intersection would be a 2 dimensional surface.  If the two spaces were flat, then the intersection would be a plane.  If the two spaces where actually hyperspheres, then the intersection would be a sphere.
If the times were separate, which is tricky, it seems like the objects at the intersection would be frozen in place from the perspective of another space.
Practically, there would be some difficulties.  The biggest one is probably the interaction of things in one space-time with another.  The interactions of particles takes time and are pretty exclusive between a charge and a charge carrier.  Electric charges interact with photons, but would photons of one space-time interact with electric charges from another?  If not, then the objects in one space would be invisible to objects in another, even at the intersection region.
