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The change in energy of an object as it moves from point A in space to point B in space is the integral of the net force it experiences over it's path of travel. If an object returns to it's starting location, and the force field it is within is constant over time, then it is provable through calculus that the change in energy of the object will be zero for force induced by electric fields, magnetic fields, and for gravity exerted by non-rotating bodies. The theorem which proves this, however, makes assumptions about the topology of space: space must be continuous (without holes), and that there's no method to jump from one point in space to another point in space.

In the case of a rotating mass, energy is still conserved as whatever energy an object gains from the rotation will be lost by the rotating object. This is why the moon is further from earth than it once was, and why the earth spins slower than it once did. In the case of a wormhole, however, the topology of space is radically altered. Assuming that one could create the negative energy density necessary to create the wormhole, is it possible to use a wormhole to violate conservation of energy?

Edit: To clarify, I'm talking about the sort of situation in which the wormholes in the presence of fields (electric, magnetic, or gravitational). For example: if one end of the wormhole is on the floor and the other is positioned directly above on the ceiling, and something is dropped into the wormhole on the floor, won't it continue falling indefinitely?

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Suppose that you traverse a wormhole from Galaxy A to Galaxy B. There are 2 cases to consider:

1) The length of the wormhole’s neck is zero.
That is, you can straddle the wormhole’s throat and be in Galaxies A & B simultaneously. In this case, energy conservation is maintained by the fact that observers in the Galaxy B will see the wormhole lose mass (equal to yours) as you emerge from the wormhole.

2) The length of the wormhole’s neck is greater than zero. In this case, energy conservation is maintained by the fact that observers in Galaxy B will see the wormhole lose mass equal to yours corrected by the potential energy that you gained or lost due the gravitational potential difference between Galaxy A and Galaxy B.

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  • $\begingroup$ Wormhole "mass loss" - you're assuming two asymptotically flat regions (inter-universe wormhole) and calculating and interpreting the two ADM masses separately when only the total is conserved and meaningful, and if there's only one flat region (intra-universe wormhole) there is only one ADM component and that will not change as something passes through the wormhole. I don't know how a global measure like ADM mass could be interpreted locally as "wormhole mass". $\endgroup$ – Julian Moore Dec 4 '16 at 9:45

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