# Wormholes and energy conservation

#### Azael

A friend of mine came up with this idea that I would like to ask here.

Just assume wormholes do exist and can be stable.

Now put 2 wormholes over the surface of the earth. One 100meters above the ground and the other 1 kilometer straight above the first one. Link them so the first wormhole ends in the second.

Now drop a object into the first wormhole. It will arrive at the second wormhole opening, fall straight down into the first and appear at the second wormhole again and so on. Constantly accelerating(neglect any air friction).

Now rig up a device that takes advantage of the kinetic energi from the fall of the object. Wouldnt this be a unlimited energy supply?

This must surely break energy conservation laws right? Since the object gets moved to a position with higher potential energy without spending any energy to get there.

Related Special and General Relativity News on Phys.org

#### franznietzsche

Azael said:
A friend of mine came up with this idea that I would like to ask here.
Just assume wormholes do exist and can be stable.
Now put 2 wormholes over the surface of the earth. One 100meters above the ground and the other 1 kilometer straight above the first one. Link them so the first wormhole ends in the second.
Now drop a object into the first wormhole. It will arrive at the second wormhole opening, fall straight down into the first and appear at the second wormhole again and so on. Constantly accelerating(neglect any air friction).
Now rig up a device that takes advantage of the kinetic energi from the fall of the object. Wouldnt this be a unlimited energy supply?
This must surely break energy conservation laws right? Since the object gets moved to a position with higher potential energy without spending any energy to get there.

Thats not quite how it works. The problem is you're mixing paradigms here. The first, Newtonian physics where objects move in 3 dimensions as a function of some parameter time. The other General Relativity where objects move in four dimension. Both have conservation of energy, but it works differently. In General Relativity there is no gravitational force, per se. There is simply a the warping of space time by mass. That is to say, an object falling in a gravitational field is not accelerating in 4 dimensions, its moving in a line with constant speed. In 3 dimensions its accelerating yes, but not in 4. An object passing through a wormhole, would be moving along a similar straight 4-dimensional line. Since no work is done on the object as it passes through the wormhole (otherwise it would not be moving in a straight 4-dimensional line) it does not gain any energy, in 4 dimension, even though it appears in 3 dimesions to gain potential energy.

#### JesseM

franznietzsche said:
Thats not quite how it works. The problem is you're mixing paradigms here. The first, Newtonian physics where objects move in 3 dimensions as a function of some parameter time. The other General Relativity where objects move in four dimension. Both have conservation of energy, but it works differently. In General Relativity there is no gravitational force, per se. There is simply a the warping of space time by mass. That is to say, an object falling in a gravitational field is not accelerating in 4 dimensions, its moving in a line with constant speed. In 3 dimensions its accelerating yes, but not in 4. An object passing through a wormhole, would be moving along a similar straight 4-dimensional line. Since no work is done on the object as it passes through the wormhole (otherwise it would not be moving in a straight 4-dimensional line) it does not gain any energy, in 4 dimension, even though it appears in 3 dimesions to gain potential energy.
Just because something is moving along a geodesic (what you call a 'straight 4-dimensional line'), that doesn't mean it does not gain or lose energy, does it? For example, I thought that one way of thinking about why light is redshifted as it climbs out of a gravity well is that it loses energy on the climb out.

Anyway, energy conservation is complicated in GR--this page points out that energy conservation doesn't even hold globally in general, although it does still apply locally, and in certain spacetimes like the Schwarzschild metric.

#### derz

JesseM said:
Just because something is moving along a geodesic (what you call a 'straight 4-dimensional line'), that doesn't mean it does not gain or lose energy, does it? For example, I thought that one way of thinking about why light is redshifted as it climbs out of a gravity well is that it loses energy on the climb out.
Kinetic energy is a relative concept (and remember, photons possess only kinetic energy).

#### JesseM

derz said:
Kinetic energy is a relative concept (and remember, photons possess only kinetic energy).
Sure...but what are the details of how this works in a particular choice of coordinate system, in a spacetime where energy is conserved? The Baez page said that global conservation of energy would still make sense in the Schwarzschild metric, for example. If we use Schwarzschild coordinates, is the photon losing kinetic energy as it climbs out of the gravity well? Do you have to define some notion of "gravitational potential" in this metric to make conservation of energy work out?

#### pervect

Staff Emeritus
Azael said:
A friend of mine came up with this idea that I would like to ask here.
Just assume wormholes do exist and can be stable.
Now put 2 wormholes over the surface of the earth. One 100meters above the ground and the other 1 kilometer straight above the first one. Link them so the first wormhole ends in the second.
Now drop a object into the first wormhole. It will arrive at the second wormhole opening, fall straight down into the first and appear at the second wormhole again and so on. Constantly accelerating(neglect any air friction).
Now rig up a device that takes advantage of the kinetic energi from the fall of the object. Wouldnt this be a unlimited energy supply?
This must surely break energy conservation laws right? Since the object gets moved to a position with higher potential energy without spending any energy to get there.
Energy will be conserved in this case via the gain and loss of mass of the mouths of the wormhole.

This, however, will not stop the "perpetual motion" machine from operating :-(. It is possible that some unanalyzed mechanism might eventually cause the wormhole itself to fail, other than that this is a definite issue.

The wormhole mouth at the bottom will get "heavier" because of all the matter entering it. The worm hole mouth at the top will get lighter, eventually acquiring a negative mass!

Here's a source for why this has to happen:

http://www.npl.washington.edu/AV/altvw69.html
As this scenario emerged from our discussion, the focus of the Workshop turned to the question of how, if such natural wormholes exist, we might search for and find them. And we invented a way.

If a positive electric charge Q passes through a wormhole mouth, the electric lines of force radiating away from the charge must thread through the aperture of the wormhole. The net result is that the entrance wormhole mouth has lines of force radiating away from it, and the exit wormhole mouth has lines of force radiating toward it. In effect, the entrance mouth has now been given a positive electric charge Q, and the exit mouth acquires a corresponding negative charge -Q. Similarly, if a mass M passes through a wormhole mouth, the entrance mouth has its mass increased by M, and the exit mouth has its mass reduced by an amount -M.

In the early universe these mass changes might create a dynamically unstable situation. If one natural wormhole mouth begins to increase in mass, its twin will correspondingly be reduced in mass until it acquires a net negative mass. The mouth with positive mass will attract more mass to it, while its negative-mass twin will gravitationally repel any nearby mass. Thus, this mass imbalance should grow until it eventually it is damped by the growing distance scales from the expansion of the universe.
For clarity, it is the ADM mass of the mouths that will change.

another source:

http://golem.ph.utexas.edu/string/archives/000550.html

An interesting fact about wormholes is that they change in mass as an object passes through them. To see this, imagine a wormhole connecting two distinct asymptotically flat spacetimes. In each spacetime, the ADM mass is conserved. Thus, if we pass an object of mass m through the wormhole, from A to B, the ADM mass on either side cannot change. This means that the mass of the mouth A increases by m and the mass of mouth B decreases by m. In other words, the wormhole has measured the mass of the object. Note that this argument applies in any dimension in which the ADM mass makes sense.
There's also a paper by Suskind and a rebuttal to said paper also by Suskind that talks about this in passing, though Suskind is mainly interested in the quantum mechanical aspects of the situation.

http://www.arxiv.org/abs/gr-qc/0503097
http://www.arxiv.org/abs/gr-qc/0504039

The rebuttal paper abstract is rather amusing, BTW.

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#### Azael

Thanks for the replies. Looks like I have some reading to do :)

#### franznietzsche

JesseM said:
Just because something is moving along a geodesic (what you call a 'straight 4-dimensional line'), that doesn't mean it does not gain or lose energy, does it? For example, I thought that one way of thinking about why light is redshifted as it climbs out of a gravity well is that it loses energy on the climb out.
Anyway, energy conservation is complicated in GR--this page points out that energy conservation doesn't even hold globally in general, although it does still apply locally, and in certain spacetimes like the Schwarzschild metric.

My familiarity with relativity stops with the Schwarzschild metric, so i was not aware that energy would not be conserverd globally.

However, as I understand the theory, travelling on a geodesic means there is no change in energy. But i could be wrong. However, light is gravitationally redshifted in a Schwarzschild spacetime even though energy is conserved.

#### pervect

Staff Emeritus
It all hinges on what one defines as energy. In the Schwarzschild metric, $E_0$, commonly called the "energy-at-infinty" is a conserved quantity for light following a lightlike geodesic, however, the usual defintion of energy as defined from the energy-momentum 4-vector $\sqrt{|E_0 E^0|}$ is not, nor is the "local" energy, which is the energy defined in a local basis of locally orthonormal vectors, using local clocks to measure time and local rods to measure distances.

#### Entropy

It is really hard to speculate about wormhole physics because he know so little about them. Wormholes are really at the level of science-fiction. I would imagine if wormholes do exist that the momentum of an object would have to be conserved somehow. Meaning the direction you enter a wormhole has to be the same when you exit one. Just like when a roller coaster does a loop, the normal force of the rail changes the direction of motion of the car, the wormhole would have to apply a force, and therefore supply energy, to the object passing through it.

#### pervect

Staff Emeritus
Entropy said:
It is really hard to speculate about wormhole physics because he know so little about them. Wormholes are really at the level of science-fiction. I would imagine if wormholes do exist that the momentum of an object would have to be conserved somehow. Meaning the direction you enter a wormhole has to be the same when you exit one. Just like when a roller coaster does a loop, the normal force of the rail changes the direction of motion of the car, the wormhole would have to apply a force, and therefore supply energy, to the object passing through it.
Wormholes appear a lot in science fiction, but there are serious physics journal articles about them too.

If you read the above thread you'll see some links to some serious papers, a smattering of some of the more famous serious papers about the topic are

W. G. Morris and K. S. Thorne, American Journal of Physics 56, 395-412 (1988);

W. G. Morris, K. S. Thorne, and U. Yurtsever, Phys. Rev. Letters 61, 1446-9 (1988).

Matt Visser, Physical Review D 39, 3182-4 (1989).
S. V. Krasnikov, "Toward a Transversable Wormhole", LANL preprint

gr-qc/0003093, Proc. of the STAIF-2000 Conference, Albuquerque, NM, February 1-4, 2000.

S. V. Krasnikov, "A Transversable Wormhole", LANL preprint gr-qc/9909016 (September 26, 1999)

Speculation is not needed to describe what happens to the energy and momentum of an object entering a wormhole, at least when the wormhole is assumed to be embedded in an asymptotically flat space-time so that the energy and momentum can be unambiguously determined.

Both the energy and momentum of the entering object _must_ be transfered to the wormhole mouth. The remarks I made about this happening for the mass of the wormhole mouth also apply to the momentum and energy of the mouth.

This is because the energy, momentum, and mass of the wormhole mouth can all be determined from the metric of space-time a long ways away from the mouth. Another way of saying this is that it is the properties of the metric "at infinity" that determine the energy, mass, and charge.

These properties "at infinty" are separated by a space-like interval from the object entering the wormhole mouth. Hence, they simply cannot change when an object enters the mouth - they are "fixed" in the imprint of the metric.

The simplest place to start to understand this is to re-read the quote I posted earlier from John Cramer - who is a physicist as well as being a science fiction author. One can determine the charge in an enclosed surface by Gauss's law. What happens to this intergal when a charge passes through the wormhole mouth? The answer is that it does not change. You can think of the electric field lines as remaining connected, and being "dragged through" the wormhole when the charge passes through it. This gives the entrance mouth an effective charge of +Q, where Q is the charge passing through the wormhole. (The exit mouth gains an effective charge of -Q).

The gravitational situation is a bit more complicated, because gravitational field lines are not well defined in GR. But the same basic logic applies - you have an intergal "at infinity" which is a conserved quantity, and it simply cannot change when a mass passes through a wormhole mouth. This means that the object transfers its energy and momentum to the mouth - in a sense, you can think of an object that travels through the wormhole as "leaving behind" its mass and momentum.

Because this is an interesting point, I will take the liberty of repeating myself by reposting the quote in question.

Remeber that the author is BOTH a physicist AND a science-fiction author.

http://www.npl.washington.edu/AV/altvw69.html

As this scenario emerged from our discussion, the focus of the Workshop turned to the question of how, if such natural wormholes exist, we might search for and find them. And we invented a way.

If a positive electric charge Q passes through a wormhole mouth, the electric lines of force radiating away from the charge must thread through the aperture of the wormhole. The net result is that the entrance wormhole mouth has lines of force radiating away from it, and the exit wormhole mouth has lines of force radiating toward it. In effect, the entrance mouth has now been given a positive electric charge Q, and the exit mouth acquires a corresponding negative charge -Q. Similarly, if a mass M passes through a wormhole mouth, the entrance mouth has its mass increased by M, and the exit mouth has its mass reduced by an amount -M.

In the early universe these mass changes might create a dynamically unstable situation. If one natural wormhole mouth begins to increase in mass, its twin will correspondingly be reduced in mass until it acquires a net negative mass. The mouth with positive mass will attract more mass to it, while its negative-mass twin will gravitationally repel any nearby mass. Thus, this mass imbalance should grow until it eventually it is damped by the growing distance scales from the expansion of the universe.

#### pervect

Staff Emeritus
Something I should add - the issue of the "fields" in the throat of the wormhole is real, but is difficult to address quantitatively. This is something that really needs to be considered when determining the exit velocity of something passing through a wormhole, and it could even cause an object to fail to "pass through".

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