How Do Multiply Connected Spaces Work with Three Different Spaces?

[Jonny_trigonometry]

I'm not sure if this is the right forum, so move if desired. I read Michio Kaku's Hyperspace a couple years ago, and he breifly spoke of multiply connected spaces. Think of a sheet, bend it in half, and make a cut through eash side, and connect the cuts together to make a "wormhole". Extend this to 3D space etc... From what I know, I've only heard of two different spaces being multiply connected. How would three spaces work? Suppose you want to travel through the wormhole, which one would you exit if there are three connected together? You have two choices. Could you say that you exit both at the same time, kind of like a superposition thing, and you can't know which one until you go through the wormhole and find out? Much like the half dead and alive cat in Schroudinger's cat paradox, or the electron passing through both slits at the same time in the double slit experiment.

 

[FSC729]

The notion of a space being "connected" is a topological one. Here is a good explanation of what it means for a set to be "connected".

http://mathworld.wolfram.com/ConnectedSet.html"

Now a multiply connected set is one that is connected by not simply connected, once again here is an explanation by mathworld:

http://mathworld.wolfram.com/MultiplyConnected.html"

Now what it means for a space to be multiply connected is explained by mathworld:

http://mathworld.wolfram.com/ConnectedSpace.html"

If you want a good introduction to the notion of a connected space please refer to the book http://www.bestwebbuys.com/books/compare/isbn/0131816292" by Munkres.

You might want to ask this question in the "Tensor Analysis and Differential Geometry" Forum.

John G.

 

[Jonny_trigonometry]

ok cool, I should check out those things. I was also wondering how people interpret all these concepts in a physical sense. ie. what happens when a particle travels through a domain where three different parts of a space are connected?

 

[selfAdjoint]

Take a torus - surface of a donut - it's "doubly connected" because you can find two independent curves that you can't shrink to a point within the surface. Namely a circle going around the hole and a circle going through the hole. And you can write any closed curve on the torus surface as the sum of these two, up to continuous deformation (homotopy). We say the torus is the "Cartesian product of two circles" and denote it as $$\mathbb {T}^2$$.

Now consider the analogous thing (can't visualize it) with three circles: the three torus $$\mathbb{T}^3$$. It will be three-connected. And now look at the theoretical setup physicists call "periodic boundary conditions". This consists of imagining the physics happening in a rectangular box, with the conditions on opposite faces of the box forced to be identical. Think about it and see that that setup is just a three torus! Hence it's three connected and that should affect what the physicists calculate.

 

[selfAdjoint]

I want to mention in addition two physics phenomena which are due to multiply connected configuration spaces: the Aharonov-Bohm effect and the geometric phase. I recommend googling on both of them, or you could go to http://web.mit.edu/redingtn/www/netadv/" and click on 'A' for A-B and 'G' for GP.

 

[Jonny_trigonometry]

Take a torus - surface of a donut - it's "doubly connected" because you can find two independent curves that you can't shrink to a point within the surface. Namely a circle going around the hole and a circle going through the hole. And you can write any closed curve on the torus surface as the sum of these two, up to continuous deformation (homotopy). We say the torus is the "Cartesian product of two circles" and denote it as $$\mathbb {T}^2$$.

Now consider the analogous thing (can't visualize it) with three circles: the three torus $$\mathbb{T}^3$$. It will be three-connected. And now look at the theoretical setup physicists call "periodic boundary conditions". This consists of imagining the physics happening in a rectangular box, with the conditions on opposite faces of the box forced to be identical. Think about it and see that that setup is just a three torus! Hence it's three connected and that should affect what the physicists calculate.

yes, this is the kind of stuff I'm talking about. The types of configurations of space where it connectes with itself. Imagine a room, and picture a sphere in the middle, not touching any walls or the floor or ceiling. The sphere doesn't have to have a definate surface, but when you look at it, you actually are looking through a 3d "window" to a different location. You can crawl through it into the other space, from any direction. This is the particular situation I'm thinking of. Where you exit is just another location in space, like say, the beach. When on the beach you see this sphere floating in thin air, and when you walk up to it, you peer into a room, and then you can enter that room by moving through it. In this example the two spaces are multiply connected, and it's easy to visualize what it would be like to throw a baseball through. you can throw it through the sphere-window in any direction and it just travels through without any disturbance into the other location.

This is the idea I've been picturing. Is this a well known type of space configuration?

 

[Jonny_trigonometry]

I was wondering if the scenerio I described above is an accepted type thing. I know that a 2d "window" is accepted, but I don't know about a 3d one. (by accepted, I mean that it works out mathematically)

 

[selfAdjoint]

I think the idea you presented, when expressed in 4-d specetime, is topologically equivalent to the "Einstein-Rosen Bridge". Einstein and Rosen in the 1930's set out to express the Schwartzschild metric for a nonrotating uncharged gravitating masspoint in light cone coordinates. This could be described as "the viewpoint of a photon" except that a photon, which experiences no proper time, actually has no viewpoint.

The geometry they then derived has two "sheets", each a copy of spacetime, with a bridge between them. The mouth of the bridge would appear to an observer near it in 3-space as the sphere you described.

 

[Jonny_trigonometry]

huh, that's interesting. So this is an accepted configuration... Ok, now on to the next part. What if, not just two "sheets" but three are connected in this way? suppose you tried to shoot an electron through it, which place would it exit? Could it exit out of both at once, (kind of like the double slit experiment, where the electron travels through both slits at the same time), but then its wavefunction would collapse when we measure it in one of the two "sheets"?