Are Wormholes Physically Possible in Our Universe?

[stevendaryl]

Space-time having a non-trivial fundamental group does not imply the existence of a wormhole. It is a sufficient but not necessary condition.

So what is the definition of a "wormhole"? The descriptions I've seen (in Misner, Thorne and Wheeler, for example), demonstrate by reducing space to two-dimensions, a flat sheet. A wormhole is depicted as a "tunnel" connecting two distant points in space. For example:
http://www-tc.pbs.org/wnet/hawking/strange/assets/images/ss.wormholes.jpg

Are you saying that having that topology (or the 3D analog, rather) is not sufficient to have a wormhole?

 

[WannabeNewton]

The definition of a wormhole is actually more restrictive than that of simply having a non-trivial fundamental group. Wikipedia has a good description of the restriction: http://en.wikipedia.org/wiki/Wormhole

For a more detailed and more mathematical treatment of wormholes, see section 10.7.2. of "Introduction to Black Hole Physics"-Frolov and Zelnikov

 

[stevendaryl]

The definition of a wormhole is actually more restrictive than that of simply having a non-trivial fundamental group. Wikipedia has a good description of the restriction: http://en.wikipedia.org/wiki/Wormhole

For a more detailed and more mathematical treatment of wormholes, see section 10.7.2. of "Introduction to Black Hole Physics"-Frolov and Zelnikov

Hmm. The definition in Wikipedia is this:

If a Minkowski spacetime contains a compact region Ω, and if the topology of Ω is of the form Ω ~ R x Σ, where Σ is a three-manifold of the nontrivial topology, whose boundary has topology of the form ∂Σ ~ S2, and if, furthermore, the hypersurfaces Σ are all spacelike, then the region Ω contains a quasipermanent intra-universe wormhole.

That's a little hard for me to parse, but taking it one step at a time:

• "the topology of Ω is of the form Ω ~ R x Σ". I think that's just saying that the region can be split into space + time, with time being represented by the real numbers, as usual.
• "Σ is a three-manifold of the nontrivial topology". The spatial part of the wormhole is an ordinary chunk of 3-space.
• "whose boundary has topology of the form ∂Σ ~ S2". Here's where my understanding of higher-dimensional geometry gets a little fuzzy. S2 is just a sphere. So I think this is just saying that the boundary of the wormhole is a sphere. The analogy in 2-dimensions is that the boundary is a circle. It seems to me that in the 2-D case, the wormhole is a cylinder, so the boundary would be a PAIR of circles, one on each end of the wormhole. So I would think that in the 3-D case, the boundary would be a pair of spheres, one on each end.
• "the hypersurfaces Σ are all spacelike". That's just saying that the spatial part of the wormhole really is spatial.

So I'm not sure exactly what this definition is saying, nor am I sure of how it differs from the intuitive idea of the 3-D analog of the 2-D case of two distant circles joined by a cylinder.

 

[HarryRool]

Hmm. The definition in Wikipedia is this:That's a little hard for me to parse, but taking it one step at a time:

• "the topology of Ω is of the form Ω ~ R x Σ". I think that's just saying that the region can be split into space + time, with time being represented by the real numbers, as usual.
• "Σ is a three-manifold of the nontrivial topology". The spatial part of the wormhole is an ordinary chunk of 3-space.
• "whose boundary has topology of the form ∂Σ ~ S2". Here's where my understanding of higher-dimensional geometry gets a little fuzzy. S2 is just a sphere. So I think this is just saying that the boundary of the wormhole is a sphere. The analogy in 2-dimensions is that the boundary is a circle. It seems to me that in the 2-D case, the wormhole is a cylinder, so the boundary would be a PAIR of circles, one on each end of the wormhole. So I would think that in the 3-D case, the boundary would be a pair of spheres, one on each end.
• "the hypersurfaces Σ are all spacelike". That's just saying that the spatial part of the wormhole really is spatial.

So I'm not sure exactly what this definition is saying, nor am I sure of how it differs from the intuitive idea of the 3-D analog of the 2-D case of two distant circles joined by a cylinder.

"The spatial part of the wormhole is an ordinary chunk of 3-space."
No. "Nontrivial topology" essentially means that the space contains a hole (whose boundary is ∂Σ).

"So I think this is just saying that the boundary of the wormhole is a sphere."
Not quite. It's saying the boundary has the topology of a sphere (in the same sense that oranges and bananas both have the topology of a sphere in that their surfaces can be incrementally deformed without surgery into a sphere).

"It seems to me that in the 2-D case, the wormhole is a cylinder, so the boundary would be a PAIR of circles, one on each end of the wormhole. So I would think that in the 3-D case, the boundary would be a pair of spheres, one on each end."
No. A wormhole is defined here as a spacetime in which the spatial part has a hole in it. To obtain what we normally consider to be a wormhole (with its two mouths) we must mathematically identify that hole with another hole elsewhere in the space.

The problem with this (Visser's) definition, as mentioned in the Wikipedia article, is that "nontrivial topology" could be interpreted to mean more than merely possessing a hole. It could be interpreted to mean that the space possesses a "handle" in the Euler sense. In this case, a connection between two universes, which we would normally consider to be a wormhole, would not satisfy the definition.

The proper fix, as implied in the article, is to dispense with the term "nontrivial topology" and describe the space as possessing a hole. This is equivalent to describing the space as having the property of preventing the incremental shrinking of some enclosed surfaces to a point (an enclosed surfaces of arbitrarily small surface area).

 

[stevendaryl]

...The proper fix, as implied in the article, is to dispense with the term "nontrivial topology" and describe the space as possessing a hole. This is equivalent to describing the space as having the property of preventing the incremental shrinking of some enclosed surfaces to a point (an enclosed surfaces of arbitrarily small surface area).

But a torus (or the 3-D equivalent) has that property, as well. That's sort of what I was getting at--with the intuitive idea of what a wormhole is, I would think that a wormhole connecting distant parts of a 3-sphere would produce a 3-torus.

 

[HarryRool]

But a torus (or the 3-D equivalent) has that property, as well. That's sort of what I was getting at--with the intuitive idea of what a wormhole is, I would think that a wormhole connecting distant parts of a 3-sphere would produce a 3-torus.

You're right. A wormhole connection between two location in a 3-sphere universe has the topology of a torus. We do not, however, want to define a wormhole as a space with the topology of a torus. Were we to do that, a connection between to two universes (two distinct 3-spheres) -- that is locally identical to what we would normally call a wormhole -- would violate our definition. The topology of two 3-spheres joined in this way (to produce a dumbbell) is that of a sphere, not a torus.