Hopefully this is the correct forum for this. I felt that Physics Overflow may not be the correct place. I had a student approach me ask me what kinds of mathematics goes into the study of wormholes. She specifically asked whether there is any amount of topology involved, but I'd appreciate it if I could give her a more full answer. So, topology answers are the best but other areas of math would be nice. I am a topology Ph.D. student but know almost zero physics, just in case my background helps with your answer. I didn't know which tag(s) to add. Thanks.

  • $\begingroup$ If you're talking about physicsoverflow.com, that site is defunct now - we took over their business, so to speak. So yes, this is the right place. $\endgroup$
    – David Z
    Apr 19 '11 at 0:17

The answer to your initial question is "mostly differential geometry, with a little bit of topology"

Consider the Kruskal diagram for the Schwarzschild spacetime:


This represents the largest posible manifold covered by the coordinate system $(t,r,\theta,\phi)$ in which the Schwarzschild spacetime's metric is usually rendered (these coordinates only cover a patch of this spacetime). The hyperbola labeled $r=0$ is a region in which the spacetime curvature is infinite.

Diagonal directions on the diagram represent the paths traveled by light rays, while directions more vertical than this represents the possible paths of observers. The diagonal region labeled II is therefore constrained to intersect the surface $r=0$ at some point--this is the interior of the black hole. (note that all histories in region III have the r=0 surface in their past, and they all must exit the diagonal rays--this is the interior of the white hole).

This is not the case for regions I and IV, however. Observers in these surfaces need never intersect the $r=0$ surface, either in their past or their future (see the line labeled $r=const.--this could represent a circular orbit happily orbiting in a circle forever).

In fact, regions I and IV are similar enough that one could imagine at some point far to the right, a surface in region I is identified (in the sense of topology) with a surface in region IV. Observers in this region would say that they are "nearby," but travelling in this way would take a long time, as you would have to go far to the right, and then to the surface, and back around. A faster route may be to travel through the point where region I and region IV intersect. It should be clear that this is not a possible path for this diagram, however--you would have to travel at a steeper angle than $45^{\circ}$ in order to take this trip. This wormhole is not transversible.

It turns out, however, that for perturbations of the Schwarzschild geomerty, however, you do get transversible wormholes, which you can picture by just having the two lines coming from the black hole intersect somewhere higher than the (T,X) origin, and the two lines coming from the white hole doing the symmetric thing. Then, you are allowed to travel in either direction to get to the region where the two spheres are identified.

  • $\begingroup$ This is the basic outline. I haven't looked very closely at wormhole models, so just consider this as a quick introduction as to how one could get wormholes in General Relativity. The physics behind models like this creates a bunch of problems, and these things aren't actually believed to exist. But in concept, here it is. $\endgroup$ Apr 19 '11 at 1:33

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