A thought experiment

[unixronin]

I refer you back to this post on the "hall of mirrors" universe.

Consider, now, this:

Imagine that you and a friend build two identical relativistic starships, and set off across the Universe on parallel courses, a short distance apart, at 0.999999C.  Assume that you have some method, be it optical, gravitational or whatever, of visually observing the other's ship, flying a few miles away from you.  In only a few years, ship time, you reach an "edge", or even more interestingly a vertex, of the universe, and you "wrap around" and transition to the opposing "facet" of the four-dimensional space.  Suppose, further, that your parallel courses happen to hit the edge - or vertex - so perfectly that you transition through one "facet", while your friend transitions through a different, adjacent "facet" touching yours only at an edge or vertex.

What happens?  Is your friend's ship still flying next to you?  Or does it, from your perspective, vanish (and you from your friend's), the two ships completing their transition billions of light years apart?  Is it possible to map the faces in such a way that, for all possible edges and vertices, points that are adjacent on one side of a "wrap transition", but on separate facets, are also adjacent on the far side of the transition, even though both faces have been "rotated" through 36 degrees?

Note:  This is almost certainly a trick question ... I think.  My intuition on this is that, from the perspective of an observer in three-dimensional space, you would always be somewhere within the body of the Poincaré dodecahedral space - or whatever shape the manifold turns out to have - and would not be able to actually approach the edges or vertices of the four-dimensional space.  (Remember, too, that we're not talking about a dodecahedral three-dimensional space, we're talking about an apparently-unbounded three-dimensional space mapped onto a four-dimensional dodecahedral space.)  But I haven't done the math - I don't even begin to know how to do the math - and so I could be wrong.  You never know, it could turn out that some related effect is responsible for cosmic megastructures such as the Great Wall or the Great Void.

Comments

[robbat2.livejournal.com]

If I recall, one of the side-effects of the 3D to 4D mapping (but really don't quote me on it, it's been years since I was involved in the maths), is that, if after transiting the vertex as you described, you then stopped, and observed in all directions, with instantaneous vision in all directions (not bounded by the speed of light), you would see your friend's ship both next to you, and on the opposite side of the universe.

[unixronin.livejournal.com]

Yes, given instantaneous vision in all directions not bounded by distance, lightspeed or absorbtion by intervening matter, you would be able to see countless multiple images of your friend's ship and your own, spaced at multiples of the diameter of the universe. Of course, you wouldn't be able to resolve the images, since you would run into the infinite-luminosity bright-sky paradox, because everything else in the universe would also be similarly imaged and re-imaged to infinity.

[radarrider.livejournal.com]

As I understand it, saying that the universe is a dodecahedron is a way of describing how objects would "wrap-around" from one side to the other. I don't think there are actual pentagonal facets such that you can go through one and your friend through an adjacent one. Since you're both essentially in the same reference frame you would experience the same effect. You'd see the universe rotated by 36 degrees but you and your friend would have the same relative position, orientation, and velocity.

[unixronin.livejournal.com]

Yup, as I said, I think it's a trick question. Or, well, not really a trick question per se, but ... the underlying symmetry geometry of the 4D spacetime manifold is not directly perceivable or navigable from 3D space, such that from the viewpoint of the observer you're always somewhere in the middle -- you cannot get to an edge. The edges, faces and vertices are always, effectively, about 20 billion LY from you, no matter where in the 3D universe you actually are.

"Which point in the 3D universe is in the center of the 4D universe?" "All of them, and none."

[radarrider.livejournal.com]

Would that then imply that the universe is relatively homogeneous? For example, we can estimate the age of the Milky Way galaxy and nearby galaxies. If I can instantly transport myself to a galaxy roughly 10 billion LY away, would it be roughly the same age and surrounded by other galaxies of similar age? Would the edge of the observable universe appear as far away?

It seems to me that the answer would be yes. Everything in the universe is the same age so it's only because we are farther away from more distant objects that they appear younger.

Here's another interesting thought. At the moment of the Big Bang, the universe was very, very small, nearly if not actually a zero-size point. If the universe had the same dodecahedral topology at the beginning, then something traveling at or near the speed of light shortly after the Big Bang could traverse the entire universe multiple times, wrapping around each time, within a relatively short period of time.

[unixronin.livejournal.com]

Both of those seem logical conclusions, yes. Of course, the universe was opaque to electromagnetic radiation for about the first 700,000 years after the big bang, but theory has it the neutrino pulse was released only 1.09 seconds after the bang (at which time the temperature was still 10¹⁰K). So you could probably have tested the multiple-traversal theory at that point, given an instrument constructed entirely of neutrinos. ;)

[jilara.livejournal.com]

I tend to favor one of the premises posited by some science fiction writers, that you can translocate through what could best be described as the origami universe, because it isn't necessarily a dodecahedron, but can be folded in many different ways and possibilities, thus creating even more possible outcomes than the model cited. And makes your brain hurt. ;-)