Gravitational effects of portals at different heights

Question

Summary: I am looking for a portal transportation mechanism that is consistent with the laws of thermodynamics.

Portals (or wormholes or gateways, something that connects two different locations in space-time), are usually portrayed with interesting side-effects:

• When one of the portals is placed directly above the other, you get a perpetual motion machine - things that fall into the lower portal instantly appear from the higher portal, accelerate, then fall into the lower portal again.

• When one portal is placed at the bottom of an ocean, water comes gushing out at high pressure from the other one.

• when one portal is placed in space, air is sucked out from the other location

• by logical extension, portals at different heights and atmospheric pressures would generate a constant and strong gust of air trough the portals, although that one is usually not portrayed much.

These effects are usually either actively exploited (Valves Portal game), or prevented by some special security mechanism (Stargates in the tv show disassemble and later reassemble matter)

However I am looking for a 'realistic' portal mechanism that does not have those sort of effects. The standard portals strike me as unrealistic for the following reasons:

For one thing, such a portal clearly violates conservation of energy, or it would need some sort of special mechanism and power reserve to account for the difference in potential energy between the two portal locations. Any custom mechanism that converts the energy difference into something else risks violating the second law of thermodynamics.

For another, should a portal not transport electromagnetic and gravitational forces just as well as matter and light? Electromagnetic and nuclear forces at least have to work through the portals. Since these forces hold matter together, a solid object would fall apart when it is transported through the portal.

I would therefore expect that when a electrically positive charge is placed next to one portal, the other portal would attract negatively charged particles.

Likewise, if a planet is placed next to one portal, the other portal should attract matter. Therefore the air from the planet would not escape into space through the portal, since all air that passes trough is strongly attracted back towards the portal.

I have however some trouble envisioning all logical consequences of such a portal mechanism.

• If two portals are placed a different heights, someone approaching the higher portal should perceive a gravitational pull towards the portal, right?

• Would someone approaching the lower portal also perceive a 'push' away from the portal, essentially some sort of anti-gravity force? After all, when passing through the portal they would gain potential energy, which cannot be gained for free.

• If one portal is placed in the ocean, would a bubble of air form around that portal, or a bubble of water around the other portal?

I realize that these sort of questions could be answered kind of arbitrarily, given that no such portal system exists right now. I am looking for the most consistent and natural mechanism, in line with the laws of thermodynamics; in particular the law of conservation of energy.

Is there a way to get such a consistent gate mechanism, without hand-waving all difficulties away as 'magic'?

I am mainly interested not in the physical mechanism that would make such portals possible, but in the observable consequences such portals would have, assuming they are possible.

(As far as i understand, the current best physical concept for wormholes, a traversable Einstein–Rosen bridge, involves a large mass within the wormhole itself, as well as surrounding negative-density stabilization structures. So the wormhole itself would have a couple weird gravitational effects. For now, I would like to ignore these additional effects and thread them as negligible)

Answer 1

Conservation of Energy

One possible answer is: Both ends of the portal must have same potential with respect to gravitation and with respect to electromagnetic fields. This is very natural, since both portal ends are basically the same point of the space and there should not be anything special on the border.

How does this manifest? As the gravity field of Earth leaks through the portals, the gates acquire false gravitational mass. (Not to be confused with inertial mass - it exerts forces, but does not make the gates harder to accelerate.) One acquires positive false mass and the other negative false mass, so the total change is zero. How big will the false mass $m_f$ be? If we assume the gates are spherical and have radius $r_g$, one is higher by $h$ meters from the other and they are near Earth surface, the potential difference between them will simply be

$$ \Delta \Phi=gh\;, $$

where $g=9.81\;\mathrm{m\,s}^{-2}$ is the gravitational acceleration on Earth. To compensate the difference, an equal potential difference must be created by the false masses of the gates:

$$ \Delta \Phi=\frac{m_f G}{2d}\;, $$

where $d$ is the distance between the portal gates and $G$ is the gravitational constant. By putting the equations together, we yield $$ m_f=\frac{2dgh}{G}\approx dh\times 2.94\cdot 10^{11} \mathrm{kg\, m}^{-2}\;. $$

In order to obtain the acceleration at the gate, we use formula for gravitational acceleration using the gate false mass

$$ a_g=\frac{m_f G}{r_g^2}\;. $$

For example if the gates are 100 m above each other ($d=100\;\mathrm{m}$, $h=100\;\mathrm{m}$) and they are 1 m wide, they would acquire false mass $2.94\cdot 10^{11} \cdot 100 \cdot 100 = 3 \cdot 10^{15} \mathrm{kg}$, which is a mass of cube of water 14.5 km big compressed into a very small volume. This also means that the acceleration at the surface of gates would be $$ a_g=\frac{3 \cdot 10^{15} \cdot 6.672\cdot 10^{-11}}{1^2}=200,200 \;\mathrm{m\,s}^{-2}\;, $$ which is 20,000x stronger than Earth's gravity. They would probably tear anyone entering into pieces. Possible solution is to increase $r_g$ or to decrease $h$, even at cost that one of the portal gates will be levitating. (This should be possible, since there are forces between both gates trying to put them into the same heigth.)

Short summary:

• The portal gate that is higher acquires positive false mass, the lower gate acquires negative false mass. (Repels normal matter.) The mass is much smaller than mass of Earth, but since it is in a very small volume, it produces very big forces.
• It will not be harder to put the gates into motion because of this mass, but it will be very hard to move the gates into different heights. The gates are attracted - not necessarily towards each other, but always to the same heights. (For example if one is at sea level in Australia and the other in Austria, there will be no attraction.) Force of this attraction will be comparable with lifting weights of similar mass as is their false-mass in the Earth's gravity field.
• No energy can be gained by jumping through the gates or pouring water through the gate, since the forces induced by the false masses will cancel the energy gain.

Edit 1: Portals with nonzero inner length

The previously presented calculations assumed that the portals have zero inner length. In such case, the difference in potential energies has to enter through the false mass of the portal mouths, which often lead to extreme forces near the mouths. But if there is space inside the portal, an inner length $L$, the forces can be signifficantly smaller. Problem is, the forces are not uniquely defined and one has to assume some things to get particular numbers. Let us assume that

• In all points inside the portal, the acceleration (and thus also the force) acting on an object is the same.
• This acceleration is the same as the acceleration $a_g$ just outside the portal.

With these assumptions, we can calculate how does the portal behave now. If previously, we calculated the difference in potential $\Delta \Phi = g h$, now this difference is reduced by the potential gain inside the portal equal to $a_g L$. So we have

$$ \Delta \Phi=gh - a_g L\;. $$

With this assumption, now have to update the formula for $m_f$ to

$$ m_f=\frac{2d(gh-a_g L)}{G}\;, $$

which leads to

$$ a_g=\frac{2d(gh-a_g L)}{G} \cdot \frac{G}{r_g^2}\;. $$

By solving these two equations, we obtain the final formulas

$$ \begin{align} a_g &= \frac{2 d g h}{2 d L+r_g^2}\;,\\ m_f &= \frac{2 d g h r_g^2}{2 d G L+G r_g^2}\;. \end{align} $$

As we can see, with large inner length $L$, the forces near the portals can be made arbitrarily small. It is still true that if you are going from the lower portal mouth to the higher one, you are going "uphill" and you must exert work.

How does it look inside the portal? Interestingly, there would be no walls. The space wraps into itself, so it looks like if you are standing between four mirrors: all around, there are images of yourself and of other things inside the portal, just they are not images, but they are really there. You could see your own back.

Conservation of Momentum

The solution above conserves the energy properly. However, the energy is not the only thing that should be conserved. There is also momentum. With momentum, I do not have any nice idea how to put it into equations. But similarly to the previous case, I guess there would be forces acting on any entering object that change its momentum as it is passing through the portal.

The forces would act both on the passing object and on the portal gates. If it is easier to rotate the portal gate so that the object would not change its direction of motion (yes, it could be easy - the portal gates could have negligible inertial mass), they would rotate to a proper position. If they are somehow fixed, the passing object would get strong kick from the portal and its momentum would be transferred to the portal gates. This would definitely not be pleasant experience if you are entering quickly - probably like hitting a concrete wall.

Note

I do not know if this is the solution you wanted, but it seems quite self-consistent to me. I definitely do not guarantee that it is consistent with the general relativity - it is probably not. (On one side because the portal gates are curvature o space and require extreme exotic mass, which brings additional effects. On the other side because the proposed model is not the only one - I know at least about one more consistent possibility, that is probably the case for wormholes. If you want to go this deep, you should be asking about wormholes, not portals.

Answer 2

This is a really cool question, and I think I can (partially) answer it. Here goes.

If two portals are placed a different heights, someone approaching the higher portal should perceive a gravitational pull towards the portal, right?

I'll assume that you're talking about an object in between the two portals. In this situation, it would be as if the point in space where the upper portal lies were actually where the lower portal is - closer to the planet. The lower portal, on the other hand, would essentially be farther from the planet. But would anything happen because of this? Well, the portal is supposedly a "hole" in space. It's a two-dimensional opening. That means that there is no way for a three-dimensional object to only be at that point. The three dimensional object would have to be partly outside the portal, and so would feel the same pull of the planet as it would feel if the portal wasn't there. In the case of the higher portal, there would still be a pull towards the planet. Whether or not the object would move towards the portal would depend on how far it is away from it - in other words, whether or not the portal's gravity balances out the normal gravity.

Would someone approaching the lower portal also perceive a 'push' away from the portal, essentially some sort of anti-gravity force? After all, when passing through the portal they would gain potential energy, which cannot be gained for free.

This depends. The "anti-gravity pull" would come from the upper portal. Again, it depends on the distance between the portals, and where the object is in between them.

If one portal is placed in the ocean, would a bubble of air form around that portal, or a bubble of water around the other portal?

Here's an experiment that can give you the answer. Take a 2-liter bottle of soda (already empty). Fill it with water, so that it is entirely full. Screw the cap on tight, so no air can get in. Now take a toothpick and make a small hole midway down the bottle. What happens? (I'll hide the answer if you want to do it for yourself)

The water will not flow out! Now put a second hole in it. Here, the water will flow out.

This represents what would happen if the ocean and atmosphere were not connected - which they are. Now do the experiment with the bottle open.

The water will flow out.

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I highly doubt such a portal could exist. As you pointed out, it would violate conservation of energy (continued increase in kinetic energy), as well as special relativity (the object would move instantaneously - thus faster than the speed of light). If you're really pressed, you have to invoke magic. There's no realistic way for this to work.

Note: I know about the theorized Einstein-Rosen bridges, but they're highly speculative and have some key problems. See the Wikipedia page.

Answer 3

As a simple solution you could say that any differences in potential energy between a low and a high portal are compensated by the portal system itself.

When an object is teleported from a lower portal to a higher portal, the portal gun provides the necessary energy difference from its internal power supply. When that supply runs out, the portal either collapses or the preservation of momentum doesn't work anymore.

Another way to solve this is by converting the potential energy difference into thermal energy. An object moving to a higher portal becomes colder, one moving to a lower portal becomes hotter. Some early science fiction stories written by Larry Niven which deal with teleportation solve the thermodynamic problem this way.

Regarding gravity passing through the portals: Gravitation is by far the weakest among the four fundamental forces of nature. The smaller the scale, the more negligible does it become. At subatomic levels it doesn't play a role anymore, so when the portals would not transfer gravity, it likely would have no effect on objects which travel through it, as long as the portals aren't so far away that there are differences in gravity which would be so large that they cause sheer forces on objects which are half through the portals.

Answer 4

Greg Egan has some notes on this subject accompanying his novel, The Book of All Skies, here: http://www.gregegan.net/ALLSKIES/01/Gravity.html

The math gets... rather complicated, but he shows how it is possible to model the effect of a portal on the electric field with a layer of electric charges and dipoles (and you can do something similar with the gravitational field). Seems like it would be possible, if difficult, to work out how the gravitational field of Earth would behave around a portal with its endpoints at different elevations based on the info there.