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In this question I am coming from a mathematical perspective. Apologies if this question has already been answered in layman's terms. But here I am trying to understand the issues in mathematical terms.

Consider travelling between two points which are very far appart, say the distance of 1000 times the distance to our nearest neighbouring galaxy, Canis Major. Say you travel directly toward that location, then since our universe is flat, you will pretty much travel in an straight line. To a good approximation, anyway.

Our spacetime will be embedded in an higher dimensional flat space, like the way a 2D sheet of paper exists in 3D. If the spacetime metric was a true metric (distance must be positive) then by the axiom of the triangle inequality, any detour out of the plane of flat space would be a greater distance of travel; as it would be a curved path between the same two points.

QUESTION. Can a detour through a wormhole really be shorter in distance or time than the straight line path?

As the spacetime metric is a pseudo-metric (distance can be negative), I realise that triangle inequality will probably no longer apply. Is that the reason why wormholes can allow a faster journey between the two points?

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  • $\begingroup$ When you travel to the other side of the universe by means of a wormhole it seems to me that you have traveled a shorter distance than taking a long way home. $\endgroup$
    – Deschele Schilder
    May 31, 2021 at 14:52

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While any $n$-dimensional manifold can be embedded in $R^m$ for some $m>n$ that doesn't mean the metric in the embedding space corresponds to the metric on the manifold. Of course, usually we do embeddings so the induced metric corresponds to the intended metric for the manifold but this is not always feasible.

One wormhole manifold that is allowed by GR (by assuming the existence of some exotic matter to make the field equations match up) is flat space with two spherical volumes excised and their surfaces identified. The hand-wavy exotic matter is assumed to be on these surfaces (any argument that you cannot get this matter requires some argument from outside GR, like the energy conditions). This is a valid solution, and a path through the wormhole can clearly be shorter than a classical path. Note that this manifold will require a fairly high-dimensional space for embedding, and will not inherit its metric.

Whether physically possible wormholes are (1) possible, and (2) shortcuts depends on the actual matter fields available (and what rules there are for topology-change). The classical wormhole papers assumed exotic matter that enabled shortcuts, but it is not clear such exotic matter exists (or not). Some recent papers with no shortcut wormholes seem to be based on more realistic (?) matter fields. But it is not clear what the actual answer is yet.

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  • $\begingroup$ Thanks. Yes, I asked previously, if by the Nash Embedding theorems if higher dimensional flat space of a compatible metric must exist. The answer was yes. My intuition would be that such a flat space, with a compatible metric, would be the type we should consider, but I am a relative novice. I know a lot of the maths, but not necessarily all the physics. $\endgroup$
    – Rory Cornish
    Jun 1, 2021 at 13:41
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There has been research on this lately:

https://arxiv.org/abs/1608.05687

https://arxiv.org/abs/2008.06618

I'm no expert on this, but I think that any time a physically realizeable traversible wormhole is found, it somehow always takes longer than taking the scenic route (in the frame of the outside observer). That's a good thing. We don't want to violate causality.

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    $\begingroup$ In Classical GR, multiply connected spacetime would not violate causality. a change in spacetime topology would create closed timelike curves (famous result due to Geroch), but even CTC's are not inconsistent with causality, so I don't understand the concern. "Traversable Wormholes via a Double Trace Deformation" (link 1) is highly technical, and I don't get the technicalities because it's non-classical. In general, re OP's question, whether or not the wormhole is shorter depends on how long it is :) "wormhole" is colloquial for shortcut, but it doesn't have to be one. $\endgroup$
    – Julian Moore
    Jun 1, 2021 at 8:19
  • $\begingroup$ I you can go through the wormhole faster than outside the wormhole, then an observer in a different frame could see you come out the wormhole before you went in. (Maybe that would mean that a "you"-anti-"you" pair was created and the anti-"you" annihilated with you in the wormhole? What about no-cloning?) If you're comfortable with that, then I agree there is no concern. $\endgroup$
    – Eric David Kramer
    Jun 1, 2021 at 11:20
  • $\begingroup$ An observer could see you come out before you were seen going in, and that has nothing to do with speed; and we already have observations of the same thing via different paths because of gravitational lensing. $\endgroup$
    – Julian Moore
    Jun 1, 2021 at 11:34
  • $\begingroup$ I'm not talking about seeing. I'm talking about actual time. The observer would calculate the actual time (on his/her clock) that you went in and the time you came out and would conclude that you came out before you went in. It's the standard causality argument that I'm sure you are familiar with. $\endgroup$
    – Eric David Kramer
    Jun 1, 2021 at 11:42
  • $\begingroup$ There is no "actual" time; there is coordinate time and there is proper time measured by an observer. The observer who "concludes" (which is logical inference and not physical reality) that you came out before you went in is proceeding from the false premise that spacetime is simply connected (but other faulty assumptions are possible). In the absence of a closed timelike curve, coordinate time, t_entry < t_exit. $\endgroup$
    – Julian Moore
    Jun 1, 2021 at 12:51

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