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When we talk about Physics (Einstein's) laws we use notions of distance and velocity. Take for example objects A and B. In A's base at specific moment of time there's definite distance to B and definite B's velocity.

But if there's a short wormhole staring near A and ending near B there's no more definite distance. There're actually two distances from A to B - e.g. long and short.

If the B's end of wormhole moves at B's speed (in A's base) then there're also two B's velocities - one normal and one equal to zero measured via wormhole.

Doesn't that mean that notions of distance and velocity are incorrect in the space with wormholes, i.e. multiconnected space?

== UPDATE ===========================================

It seems that I should reformulate my question in a more strict way.

There're two relativity point objects A and B. There's also a wormhole W.

First wormhole mouth Wa is near A and is connected to A. Second mouth Wb is near B and it is moving together with B at B's speed in A's base. But the wormhole length (the distance between Wa and Wb measured through wormhole) is constant.

Now in A's base we have some relativity formulas for B's energy, time shift, etc. In these formulas we put Vb - B's velocity in A's base.

The question is: how do we measure Vb to use in the formulas?

There're two ways:

  1. We measure Vb in ordinary way out of wormhole. In this case Vb have some large value.

  2. We measure Vb through the wormhole. In this case Vb is equal to 0.

Which value should we use in the formulas?

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    $\begingroup$ A wormhole is a shortcut through space. let me give an analogy. If you're standing right where you are and I'm standing on the exact opposite side of Earth, what's the distance between us? Most would say it's about 20000 km, half the circumference of Earth. And that's a perfectly valid measurement. But now what if somehow there was a way to travel directly through the core of Earth? Then you could say the distance is also around 12742 km (the diameter of Earth). Does that mean notions of distance are meaningless now? $\endgroup$ – Jim Jan 10 '15 at 14:05
  • $\begingroup$ @Jim The notion of distance between points A and B in Euclid or Minkovsky space is definite. So is the notion of velocity. $\endgroup$ – Andrey Minogin Jan 10 '15 at 17:43
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    $\begingroup$ Yes, my point was that notions of distance and velocity similarly do not become invalid in multiconnected spaces, they simply require you to specify how they were measured. Aircraft measure airspeed velocity as well as ground velocity because both are not the same always. We can list a physical distance as, example, "10 km driving but only 4km as the crow flies". Having multiple valid ways of measuring something doesn't make the notions incorrect, it makes it necessary for specification of how the measurement was made to accompany the value as well. $\endgroup$ – Jim Jan 10 '15 at 17:53
  • $\begingroup$ Please see my update. $\endgroup$ – Andrey Minogin Jan 11 '15 at 10:02
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You don't need a wormhole for there to be different distances between two points, because gravitational lensing does that already.

In its most symmetric form the light flow round a gravitational lens looks something like:

Lens

The light from the object can follow one of two paths $a$ or $b$ shown by the solid lines to reach the observer, and as a result the observer sees two images. Actually the observer sees a ring, because the setup is axially symmetric about the line between the observer and object. This is the famous Einstein ring. Note that the paths $a$ and $b$ look curved to us because we're looking at a curved spacetime. As far as the light rays are concerned the paths are straight. So we have two different straight lines from the object to the observer.

In this most symmetric setup the length of all the paths $a$, $b$ and the ones out of the plane of the diagram are the same. However break the symmetry slightly and now the paths $a$ and $b$ have different lengths. Let me emphasise this point. Light travels along straight lines - null geodesics - and yet there are multiple different paths to get from the object to the observer, and they can all have different lengths.

So we have exactly the same situation as you describe for your wormhole, though obviously the difference is far larger with a wormhole. My point is that this is routine in general relativity and not regarded as anything special. The notion that there is a unique distance between two points simply doesn't apply in GR - it is part of the baggage that you have to abandon.

Response to comment:

You assume that two objects taking the long way round and going through the wormhole would end up with different velocities. The trouble is that an object passing through the wormhole would experience acceleration because the spacetime in a wormhole is not flat so the transit through the wormhole would change it's velocity. This shouldn't be surprising as the wormhole has to be held open by enormous masses of exotic matter.

But to calculate how an object moves in the wormhole you would need to write down a metric, calculate its geodesics and likewise for the long way round. However we don't have any analytic metrics describing traversible wormholes so we could only do this calculation numerically.

However, provided the geometry is time independant we can be sure that energy is conserved - Noether's theorem tells us this. That means it's impossible to go round the loop and end up with more energy than you started with, and therefore that whichever way round you went you would end up with the same kinetic energy and therefore velocity.

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    $\begingroup$ Do you mean two solid line paths? $\endgroup$ – Kyle Kanos Jan 10 '15 at 19:57
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    $\begingroup$ @KyleKanos: I meant what I typed, though I concede it wasn't very clear. Hopefully I've clarified it a bit now. $\endgroup$ – John Rennie Jan 10 '15 at 20:06
  • $\begingroup$ Much clearer now (and thinking in this way, it is more clear what the original statement was intending). Thanks $\endgroup$ – Kyle Kanos Jan 10 '15 at 20:09
  • $\begingroup$ You mean that in Einstein's physics there's no definite distance between two points (in one base)? OK. What about velocity? $\endgroup$ – Andrey Minogin Jan 10 '15 at 20:53
  • $\begingroup$ @AndreyMinogin: I mean that in Einstein's physics there's no unique straight line between two points, and those straight lines may have differnt lengths. I've updated my answer to discuss the velocity. $\endgroup$ – John Rennie Jan 11 '15 at 6:23

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