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This is more like a conceptual question. Wormholes are tunnels connecting two different parts of the same universe or connecting two parts in different universes. Taking the former one as an example; It makes sense to think about them if the two mouths are spacelike separated since this way it makes the interstellar travel shorter.

According to Wikipedia (even though it's not a perfectly reliable source)

A wormhole could connect extremely long distances such as a billion light years or more, short distances such as a few meters, different universes, or different points in time.[2]

What happens if the mouths are so close to each other like a few $mm$ even $nm$ ? If they are too close, I can't think of a configuration where it would take shorter period of time to travel a distance through a wormhole than travelling between the points. (Unfortunately, we have covered Einstein-Rosen bridges superficially so I don't have the right machinery to carry out the math)

This way of thinking leads to a conclusion that there might be a lower limit to the distance between two mouths of an Einstein-Rosen bridge, I wonder what this limit is and how to find it, to be more precise is there even a lower limit?

Possible Way To Resolve This Dilemma

  • Einstein-Rosen bridges don't necessarily make the travel time shorter between two points hence the distances shorter so it's reasonable to have as short distances as $mm$, $nm$ between the two mouths.

Yet I'm not sure of whether this assumption would be correct or not.

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    $\begingroup$ If we're talking about classical GR, and you're making no specification about masses/energies involved, surely there's no associated length scale either? $\endgroup$
    – Eletie
    Mar 25 at 23:31
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For the Einstein-Rosen bridge in the maximally extended Schwarzschild spacetime, there is only one distance scale, which is the Schwarzschild radius. But this is not a traversable wormhole.

If you want a traversable wormhole, then it can't be a vacuum spacetime, and it's going to violate an energy condition. You can make the metric be anything you want, and therefore you can make your distance be anything you want. Then the Einstein field equations will dictate the stress-energy tensor.

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