# Conservation of potential energy for a wormhole

So you managed to build a stable traversable wormhole. Somehow you managed to acquire the exotic negative-tension materials with sufficient densities to make it all work.

Now you place opening A of the wormhole deep inside a gravitational well, and the other opening B far outside that well.

What would a traveler approaching wormhole A experience?

A traveler that is stationary in regards to point A would have to expend a considerable amount of energy to reach excape velocity, leave the gravity well and reach point B trough normal space.

Conservation of energy implies that they would need the same minimum amount when going through the wormhole.

The distance they travel however is much shorter, and I cannot see how the required escape velocity can be the same. Do they therefore experience a stronger gravitational gradient along their journey?

Are they perhaps strongly repelled from opening A and for that reason have to expend the same amount of energy they would need to move out of the gravity well in a normal manner?

In the same vein, is a traveler approaching wormhole B strongly attracted towards it, so that they can gain the same amount of kinetic energy that they would get when traveling towards the gravity well in free fall from B to A trough normal space?

How does the spacetime around the two wormhole openings in different gravitational depths look like?

The wormhole itself will of course have a considerable mass of its own, and the exotic matter used to stabilize it will have its own weird gravitational effects, but let's assume that those are negligible compared to the effects of the giant gravity well that is near one of the wormhole openings.

Related questions:

• This question deals with whether the law of conservation of energy is broken when an object that travels trough a wormhole disspears at one point an reapears at another

• This question is very similar to mine, but does not consider gravitational wells aside from the wormhole itself.

• I found an almost duplicate question. (It is just named to badly it is very hard to ind it.) Commented Nov 13, 2014 at 8:57

Disclaimer: I'm not a GR expert, but this is how this question has been explained to me by other physicists before. If I got something wrong, please correct me.

The traveler does indeed not have to exert as much work to leave the gravity well via the wormhole compared to the normal route. They are not repelled from mouth A nor attracted to mouth B by any effect having to do with the gravity of the planet.

Conservation laws are preserved, however, by interaction with the wormhole mouths themselves. When the traveler enters mouth A and leaves mouth B, no work is required to raise their mass because mouth A appears to gain equal mass to the traveler, and mouth B loses it. As far as conservation laws are concerned, it's as if the traveler crashed into and merged with an asteroid in low orbit (mouth A), and then an identical copy of the traveler got assembled out of the mass of another asteroid (mouth B) and ejected in high orbit.

So, if you try to generate infinite energy by throwing something through the wormhole and then running a generator off it as it falls back down, your plans will be foiled by mouth A becoming steadily more massive while mouth B becomes steadily less massive, until mouth A collapses into a black hole.

The answer to this question depends on your assumption about the continuity of the "metric", the descriptor of the gravitational field.

Case 1: Assume that the metric can be discontinuous.

In this case the gravitational potential increases abruptly as one crosses the wormhole's throat. It seems that an object entering the lower mouth (A) and immediately emerging from the higher mouth (B) has magically gained potential energy. However, this energy is stolen from the wormhole, whose mass is reduced by an equal amount.

Case 2: Assume that the metric cannot be discontinuous.

This is the standard assumption. [It's motivated by the fact that a discontinuous change in the metric across a boundary would mean that the geometry of the boundary itself would not be well defined, which would be unphysical.] In this case the gravitational potential is conservative, as usual. This means that outbound travel through the wormhole would be "uphill", the gravitational potential gradient within the wormhole would be much higher. As in Case 1, the traversing particle's gain in potential energy would equal the resultant loss in the wormhole's mass.

"How does the spacetime around the two wormhole openings in different gravitational depths look like?"

In Case 1 the spacetime would be locally flat and curvature-free in the vicinity of each mouth, right up to the discontinuity at the throat. In Case 2, the spacetime would still be locally flat (all spacetimes are) but there would be curvature. In short, there would be nothing unusual about the spacetime in the vicinity of the mouths in either case.

• Doesn't the discontinuity in the metric tensor automatically mean presence of matter in the place of discontinuity? (Since the derivatives of the metric are connected to non-zero stress-energy tensor.) In such case, the wormhole would probably be filled with very thin and very dense strip of matter that couldn't be avoided.) Also, wouldn't the discontinuity in metric cause very sudden and strong forces on passing object? Commented Nov 9, 2014 at 17:32
• This question is very similar to mine, but does not consider gravitational wells aside from the wormhole itself.

Actually I think that question does consider a gravitational well that exists in the surrounding space.

What seems to me to be different about your question is that you're asking about the forces exerted on objects as they move around in this spacetime. There is a problem here because in GR, gravity isn't considered a force. Your space traveler will follow an inertial path through spacetime and will be weightless the whole time. Their proper acceleration will be zero. So the answer to your question is no, there is no such force, for the trivial reason that gravity isn't a force.

• I probably have not expressed myself clearly then, but this is not the gist of my question. Does a traveler have to exert less work to leave the gravity well via the wormhole than via a normal route? If the work is not the same, then that would be a violation of conservation of energy, and one could gain infinite energy by repeatedly traveling through the wormhole from A to B and then back through normal space time. But if the work is the same, how does that work manifest when traveling trough the wormhole? (I'll try to edit my question to clarify) Commented Nov 5, 2014 at 9:21