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Firstly, I understand that we have no observational evidence for 'wormholes'. They are theorised solutions to general relativity equations.

That said, if macroscopic wormholes do exist---how do they work? I've been thinking about this for the past couple of days, and I have some questions.

  • What does the mouth of a wormhole look like? I've seen this; is that more or less accurate?
  • If I place one end of a wormhole in a high magnetic field, and the other end on my desk, will all my ferrous objects fly into the wormhole?
  • If so, with what force? What does the magnetic field look like at each end of the wormhole?
  • If the distance between the mouths of the wormhole was comparable to the force that the magnets exert, would the magnet 'pull' on itself and draw itself into the wormhole?

Do any of these questions even make sense?

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A simple picture on how they work: Take a sheet of paper. Draw a point at the top and at the bottom. It seems then that the distance between them is quite large. Now fold the paper so that both points are aligned on top of each other. Now their distance is small. By deforming the space, you have lowered the distance between the points. –  Lagerbaer Mar 31 '11 at 3:11
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@Lagerbaer but would there now not be two distances between the two points? What if, at each of those points, there was an electron. The field would travel both the short distance and the long distance. –  nornagon Mar 31 '11 at 22:48
    
To be honest: I've got no idea :-) I reckon that for a proper treatment of problems of this sort you'd need to do proper electromagnetism in general relativity. –  Lagerbaer Apr 1 '11 at 0:06
    
Let's not start on how gravity would work if they were masses rather than electrons... –  nornagon Apr 2 '11 at 1:50
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2 Answers

I am not an expert by any means (just starting my studies in these topics) but i will share my understanding of the topic so to complement any further discussion of your question:

1) The example of joining two points in a sheet of paper makes good intuition but i think its physically very misleading; there is no such operation in GR as joining two space-time points, because in the case of the sheet of paper the points are joined by being close in an embedding space (euclidean 3-dimensional space), while classical GR does not have any notion of an embedding space in which space-time exists. Other theories (not sure, but probably string theory?) may have further assumptions in this regard, but as far as i can tell, such embedding spaces are not allowed to produce any direct, physical consequences

2) GR allows solutions in space-time topologies that are non-trivial (where trivial means homeomorphic to asymptotically flat minkowski space), but the dynamics of GR doesn't allow by itself any dynamical transition between space topologies.

What there exists in the literature is postulating ad-hoc the geometry in question, in this case two asymptotically flat minkowski spaces connected by the identification of two $S^3$ mouths on each space. Then one uses GR equations to derive the required $T_{uv}$ tensor that makes the geometry stable.

What there is not in the literature to my knowledge is taking flat minkowski space, and obtain a consistent evolution of trivial, flat space topology in a given time foliation parametrized as $t_{0}$ into another non-trivial space topology in another time $t_{1}$, even assuming exotic stress-energy densities, i.e: $T_{00} > T_{11} + T_{22} + T_{33}$. And the reason is that GR as a dynamical theory, is strictly a local theory (please, correct me if this is wrong, this just represents my knowledge of the subject) and as a local theory, doesn't really tell much about how the manifold behaves globally

for a much more in-depth answer, please see What is known about the topological structure of spacetime?

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Wormholes are similar to black holes, but where the even horizon is replaced by a membrane of some type of quantum field which causes geodesics to diverge. The Hawking-Penrose energy condition, in particular the weak energy condition, gives geodesics which focus inwards in a spacetime diagram. In order to get a wormhole it requires that inward focusing geodesics be “defocused” near or at the region where the event horizon of a black hole would otherwise exist. This means the geodesics are defocused into some other region of spacetime. The wormhole is then two 3-balls cut out of spacetime and where the boundaries of the two balls have points identified with each other. The special field which violates the Hawking-Penrose energy conditions defines a junction in spacetime where the curvature exhibits an abrupt change.

We might think of a wormhole as a cut in space, say the removal of a ball $B^3$ with a spherical surface $S^2~=~\partial B^3$, which is matched by another ball, where the surfaces of the two balls are identical. The two spheres $S^2$ and $S’^2$ then have points on them which are identified with each other. The two ball then in effect when sutured together define a three sphere. The junction is then a region at this surface, or just above where the curvature of spacetime jumps with a negative sign. This results in a defocusing of geodesics across the $S^2$, which forces the geodesic to emerge at $S^2$. This is a multiply connected space.

The difficulty comes if one of these openings is accelerated in a “send and return” process. This in special relativity results in the so called “twin paradox,” which for the case of a wormhole means the clock on the accelerated opening ends up in the past of the clock on the unaccelerated opening. This is a form of time machine. The multiply connected space is embedded in spacetime and by this means is transformed into a multiply connected space plus time. This is the primary difficulty with the whole idea of wormholes, for the symmetries of spacetime mean one can transform this into the funny form of a time machine. Now it might be that at the point the two openings in this send and return process have the same proper time and are connected by a null ray that something disastrous might happen. As this condition is met the vacuum will wind through the wormhole in a “pile up” or asymptotic condition approaching this null condition. This might violently destroy the wormhole before it can be transformed to a time machine.

However, the negative energy of the exotic quantum field may do that anyways. This quantum field is not bounded below, which means the ladder of states descends endlessly to $E~=~-\infty$. This means a vast amount of radiation would flow out from this strange quantum field. In fact this will always happen to the extent the negative energy field is “cancelled out.” So this negative energy or exotic quantum field seems to be self-canceling and simply can’t exist. Wormholes on a macroscopic scale most likely do not exist.

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