Do wormholes require higher dimensions?

[friend]

Every picture I've seen to illustrate wormholes is always a shortcut from one point on a 2D surface to another. And it's easy to see that the distance is shorter through the wormhole since we are view it from a 3D perspective. This makes me wonder if higher dimensions are required to construct wormholes. It seems the wormhole must travel through the higher dimension in order to create a shorter distance through the regular space. Do wormholes need higher dimensions to exist?

 

[Nabeshin]

Every picture I've seen to illustrate wormholes is always a shortcut from one point on a 2D surface to another. And it's easy to see that the distance is shorter through the wormhole since we are view it from a 3D perspective. This makes me wonder if higher dimensions are required to construct wormholes. It seems the wormhole must travel through the higher dimension in order to create a shorter distance through the regular space. Do wormholes need higher dimensions to exist?

No, there is no higher dimension required. It's merely a product of trying to embed the wormhole geometry so that our human brains can visualize it (as you mention, embedding R^2 with a wormhole in R^3). To see this, you can just write down the wormhole geometry referring to only 4 coordinates, the normal (t,x,y,z). No reference to any fourth spatial dimension is required in the mathematics. (If this is uncomfortable to you, note that you can write down the surface of the sphere using only 2 coordinates, or the surface of a 3-sphere using only 3 coordinates. These both exist independently of the higher dimensions we typically try to embed them in.)

 

[friend]

No, there is no higher dimension required. It's merely a product of trying to embed the wormhole geometry so that our human brains can visualize it (as you mention, embedding R^2 with a wormhole in R^3).

I imagine, however, that a wormhole would change the topology of the universe, from a sphere to a toroid, right?

 

[Nabeshin]

I imagine, however, that a wormhole would change the topology of the universe, from a sphere to a toroid, right?

This actually depends on the nature of the wormhole. You can have what we might classically think of as a 'wormhole' (the example I'm seeing here is that of a narrow throat connecting to an umbilical 'baby universe', like this: http://images.quickblogcast.com/4988-4889/onedrop3.gif ), but nevertheless the global topology is still trivial.

I do think it's true that if you have a genuine wormhole to either a) a second, asymptotically flat universe, or b) another region of the same universe, that you create a torus-like topology (in terms of genus I'm not too sure how it actually changes). Note however that there is no local way to distinguish between the two situations I've just described, so attempting to discern anything about the topology from a wormhole is not advisory.

 

[clamtrox]

No, there is no higher dimension required. It's merely a product of trying to embed the wormhole geometry so that our human brains can visualize it (as you mention, embedding R^2 with a wormhole in R^3). To see this, you can just write down the wormhole geometry referring to only 4 coordinates, the normal (t,x,y,z). No reference to any fourth spatial dimension is required in the mathematics. (If this is uncomfortable to you, note that you can write down the surface of the sphere using only 2 coordinates, or the surface of a 3-sphere using only 3 coordinates. These both exist independently of the higher dimensions we typically try to embed them in.)

Is that true? I always thought there has to be a jump in the extrinsic curvature in wormhole geometries. Having a jump in extrinsic curvature basically means that your manifold is embedded to a higher dimensional manifold in a funny way.

 

[Bill_K]

It's true. Neither a wormhole nor any other curved spacetime needs to be embedded in a higher dimensional space. We like to draw pictures of an embedding just for the purposes of visualization.

 

[George Jones]

Is that true? I always thought there has to be a jump in the extrinsic curvature in wormhole geometries.

There does not have to be a jump in extrinsic curvature, but there can be, i.e., it is possible to "make" a wormhole by gluing together two 4-dimensional spacetimes along a 3-dimensional hypersurface such that the extrinsic curvature of the hpersurface is different on each side. Then, there is a delta function source for T on the hypersurface, just as there is a delta function charge density for charge sprayed onto the surface of a sphere.

Standard Morris-Thorne wormholes are not like this, but Visser gave examples of wormholes that are like this.

Having a jump in extrinsic curvature basically means that your manifold is embedded to a higher dimensional manifold in a funny way.

[edit]Bill_K posted while I was writing and thinking.[/edit]

No. Extrinsic curvature in standard general relativity usually refers to the the extrinsic curvature of a lower-dimensional surface embedded in 4-dimensional spacetime.