Is a wormhole detour in spacetime really shorter in distance or time than a straight line? 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?
 A: 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.
A: 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.
