Potential Energy in Wormholes: Charged Particles & Electric Fields

[Haorong Wu]

TL;DR

If a particle travels through a wormhole, what will its potential energy become?

Suppose a charged particle is in an electric field and feels an electric potential. Then the particle travels through a wormhole to another electric field and the particle feels a different electric potential. The potential energy of the particle will change. So what will that part of potential energy become? Will it transfer into kinetic energy or into heat? What if the potential energy increase? How can the particle gain enough energy to increase its potential energy?

 

[Ibix]

A wormhole, if one could be created, wouldn't be magic. The electromagnetic field would have a value defined everywhere, including inside the wormhole region. So the situation you describe is just a particle moving in an electromagnetic field, but in a complicated spacetime. The calculations might be more complex (and you may have to be careful about how you define energy in a complicated spacetime), but on a general level the answer to your question is the same as a particle in a varying potential in any other spacetime. It may or may not be able to reach certain places depending on its initial state of motion.

 

[Haorong Wu]

Thanks, @Ibix. I forgot that the electric field can go through the wormhole as well.

 

[pervect]

Summary: If a particle travels through a wormhole, what will its potential energy become?

Suppose a charged particle is in an electric field and feels an electric potential. Then the particle travels through a wormhole to another electric field and the particle feels a different electric potential. The potential energy of the particle will change. So what will that part of potential energy become? Will it transfer into kinetic energy or into heat? What if the potential energy increase? How can the particle gain enough energy to increase its potential energy?

I think Visser's "Lorentzian Wormholes" talks about this. There's also the paper by Cramer, et al, in Phys reiveiw D, "Natural Wormholes as gravitational lenses", see the abstract at https://journals.aps.org/prd/abstract/10.1103/PhysRevD.51.3117. Cramer also wrote a few popularizations in the science fact column of the science fiction magazine, "Analog", which are online. One of them is at https://www.npl.washington.edu/av/altvw33.html. Of cours, these are popularizatons and not peer reviewed papers, the peer reviewed paper is the one I referenced above.

Basically, when a particle goes through a wormhole, the mass , momentum and charge of the particle is added to the wormhole entrance. There will in general be electric and or gravitational fields through the throat of the wormhole. Similarly the mass, momentum, and charge of the exiting particle is deducted from the exit of the wormhole.

The abstract mentions the mass change, and the possibility that the exit wormhole has a negative mass, though it doesn't go into all of the details. The full paper does better, if you can get it. Here's the abstract.

Once quantum mechanical effects are included, the hypotheses underlying the positive mass theorem of classical general relativity fail. As an example of the peculiarities attendant upon this observation, a wormhole mouth embedded in a region of high mass density might accrete mass, giving the other mouth a net negative mass of unusual gravitational properties.

The electric field is an easier case to analyze than the mass. Talking about energy in general relativity gets rather involved, electric fields are much simpler. The electric field lines of the charged particle do not and cannot "break" when the particle goes through the wormhole. The formal interpretation of this is the continuity equations, with the field lines being represented by the Maxwell tensor or it's dual, the Faraday tensor. (I'm a bit hazy nowadaus on which one of the two represents the electric field lines and which one the magnetic without looking it up, but they are duals of each other). A wormhole pair initially without field lines will change when a particle passes through it. It will wind up with field lines exiting from the source, field lines threading the wormhole, and field lines entering the wormhole exit from the particle that passed through it.

As far as energy goes, my recollection is that it's not single valued. For the electric field case, the field would still be the divergence of a potential function, but in a multiply connected topology, the potential function isn't necessarily single valued.

Some old posts of mine were written when my memory was fresher, see for instance https://www.physicsforums.com/threa...he-mouth-of-the-wormhole.956098/#post-6062061, or search for my name with the keyword wormhole.

 

[pervect]

I wasn't quite satisfied with my first post, so I'm going to add another that I hope will be more clear.

I'm going to stick with electric fields rather than gravitational ones, as it's much easier to deal with.

Let's set up a situation. We have a negatively charged ball, and a positive charge near the ball without enough energy to escape to infinity without a wormhole.

Then we add in the wormhole. As part of specifying the wormhole, we specify the electric field in it's throat, and we'll assume that that field is initially zero.

Then the positive charge that can't escape via normal means can still escape through the wormhole. After it escapes, the wormhole throat will no longer have no field lines, because the field lines from the charge can't break, and are "dragged through" the throat of the wormhole when the charge traverses it.

Where did the energy come from that allowed the charge to escape to infinity? The charge didn't have enough energy to escape on it's own. Basically, the answer is that the energy was "borrowed" from the source wormhole. As Cramer notes, it's possible via this mechanism to wind up with the source wormhole having a negative net energy.

What happened to the charge? We use Gauss law to find the charge of the entrance and exit wormholes, which we get by integrating the field lines. Initially, there we assume there's no field lines anywhere, and hence the source and exit each have zero charge. But after the charge passes through the wormhole, integrating the field lines shows that the entrance to the wormhole appears to have a positive charge, and the exit of the wormhole has a negative charge. So charge is conserved.

And, as noted, we needed to borrow energy from the source wormhole for the charge to pass through.

As far as potential goes, after the charge threads the wormhole with field lines, we have a situatio where we conclude that the integral of the electric field along a closed loop is zero if and only if the loop does not go through the wormhole. This is perfectly fine with the differential form of Maxwell's equations, but some of the integral forms of said equations that are usually taught need to be modified a bit. Mathematically, the difficulty with the integral form can be traced back to the muliply connected topoology.