How does conservation work in wormholes?

Question

My questions might seem naive as I haven't plunged into the mathematical descriptions, but wormholes are interesting even if I don't actually know them :)

First of all, is wormhole making two possible paths betwwen points (like the tunnel in most visualisations) or is it just making two points beside each other like in the game Portal? Of course, second case can be bent into a tunnel, but there would be nothing special about tunnel part and no preferred way to divide into "normal" and "tunnel" part of spacetime.

My main questions are these:

1. Is momentum conserved?
2. Is kinetic energy conserved?
3. How does potential energy work?

To elaborate... In the game Portal you can fall into floor and fly out of the wall or floor elsewhere with the same speed, but another direction (normal to the portal). So momentum is not conserved, but kinetic energy is. Is that how it would actually work? Where does the momentum go then?

And does it really allow you to increase your potential energy while retaining kinetic? It would seem fair to take away some kinetic energy if you are put in a region where you have more potential energy.

Either way is the change in energy non-continuous like in the game and discussion that's linked below? Or are there just more than one way to move through but the change in potential energy is smooth in any of those? That would mean more potential equilibrium points though...

Similar questions has been discussed here, is it correct about how the falling guys gets a lot of energy (in form of a potential one) by draining it from the mouths?

Answer 1

You need to start by noting that the wormhole solutions we know of in general relativity are very different to the wormholes so loved by science fiction fans. As it happens I've just answered a closely related question How would you connect a destination to a wormhole from your starting point to travel through it? and indeed I've just asked a related question Building a wormhole.

The key point to answering your questions is that the spacetime curvature involved in a wormhole is qualitatively no different to the spacetime curvature around, for example, the Sun. It is only in the large scale connectivity that a wormhole is special. If you were falling through a (traversible) wormhole then when you looked around your local region you would see nothing special about the spacetime curvature.

So to understand how to answer your questions for a wormhole simple ask them of trajectories near the Sun.

The gravitational field of the Sun is approximately time independant and spherically symmetric, and that means that enery and angular momentum are conserved. If you started on a trajectory that passed near the Sun your trajectory can be easily calculated using the geodesic equation. You would recede from the Sun in a different direction (how different depends on how close you got to it) but with the same energy as you started with. Your angular momentum would be unchanged but your linear momentum would have changed, and the Sun's linear momentum would have changed by an equal and opposite amount to ensure total momentum is conserved.

Now suppose we construct a spherical and (approximately) time independant Visser wormhole from exotic matter and ask the same questions. You would get exactly the same answers. After passing through the wormhole you'd emerge on the other side in some direction that depended on your exact trajectory and with the same energy. The total momentum of you and the matter making up the wormhole wouldn't have changed, but your individual mometa would have.

The point is that there is nothing special about a wormhole - it is all suprisingly boring!