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Wikipedia says:

A wormhole (Einstein-Rosen bridge) is a hypothetical structure connecting disparate points in spacetime, and is based on a special solution of the Einstein field equations.

If a traversable wormhole existed, could its ends move relatively to each other in space? What would be the momentum of the energy-matter escaping from one end, in relation to the momentum of the energy-matter entering the other end?

Imagine there is a traversable wormhole, with two ends, A and B, each in a separate galaxy A* and B* where these ends do not move relative to its galaxy but the galaxies A* and B* are moving (through space) away from each other with speed v=1000 m/s. When I enter the end A with speed v=1 m/s, will I emerge from end B moving relatively to galaxy B* with speed v=1000 m/s or with speed v=1 m/s?

Can the same question be asked about white hole/black hole pair?

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General relativity with one test particle (i.e you) can be described by an action principle. The action only depends on the metric and your position, so it's translationally invariant. Hence by Noether's theorem, it has four symmetries, corresponding to translation in space and time. The conserved quantities associated with space translation can be reasonably associated with the total momentum of you and the gravitational field. So yes, for an appropriate definition of momentum, any scenario in GR such as a wormhole will conserve momentum.

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