Swan Tower - a set of very interesting questions

Dec. 10th, 2012

01:14 am - a set of very interesting questions

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I could see giving this to physics students as a brain-bender. (In fact, I won't be surprised if it turns out somebody already has.)

In the first scenario, I believe -- operating on the remnants of my physics knowledge -- that it would accelerate downward. Gravity still acts on the rod; it will move, and the bits of it that pass through the blue portal re-emerge from the orange one with their momentum conserved, so it's (I think) functionally no different from letting it fall a really long distance. Probably it will achieve terminal velocity at some point.

In the second scenario, I think the rod would accelerate the other way? But I'm not sure. The falling orange portal would push some of the rod back out the blue portal, which pushes more into the orange portal, and you've basically got the same situation as in #1, except in the other direction. But the part I can't figure out is what happens when the orange portal comes to rest atop the blue one. (Or even not directly atop it -- you could stop any distance away that is less than the length of the rod.) Does the rod bend? There's no longer enough room for all its length between the portals, so I feel like it must, but I'm not sure how the force for that works out. (And actually, if the rod is allowed to move as in scenario #1, then I think you get this problem right away. Because then the rod is trying to come and go from both portals at once.)

In the third scenario, I think that if the portals are shot as depicted in the diagram, you've made a weak projectile. Move the orange portal, and now the rod falls through the floor and out the wall. (If you've let it build up momentum via the first scenario, then maybe it's not so weak.) But that assumption depends on what I think is an as-yet unanswered question in the games, namely, what happens if a portal goes away while something more solid than a beam of light is athwart the boundary. I'm presuming it severs the object in question, so that you've basically made an ordinary piece of pipe with a solder in the middle, which then falls through the blue portal. I'm not sure we ever saw that issue in action during the game, though, at least not as a puzzle. (Probably people have left turrets or cubes balanced on the portal boundary and then shot a new one; my guess is they fell to whichever side had the majority of their mass. But that may just be a coding default, rather than a conscious choice on the part of the designers to say that portals can't slice objects in half.)

It's been years since I thought about this stuff, though. Tell me, O internets: where have I got it wrong?

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From:mindstalk
Date:December 10th, 2012 03:14 pm (UTC)
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Case 1: the rod accelerating indefinitely creates a perpetual motion machine, violating conservation of energy. The rod moving at all requires converting potential energy into kinetic energy. Where's the potential energy coming from? Rod passing through the portal jumps up to a higher level of gravitational potential energy, which makes portals problematic to begin with.

So I'd assume no movement. Ditto for case 2. I don't know where the forces are coming from, because portal are magic science, but the conservation laws tell us the forces must exist. If conservation laws are violated then you can make up any answer.

Case 3: I was going to say I suspect violates conservation of angular momentum. On reading more carefully, assuming truncation seems safe. Any bending of the rod would violate conservation, or require compensation somewhere.
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From:swan_tower
Date:December 10th, 2012 08:51 pm (UTC)
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Rod passing through the portal jumps up to a higher level of gravitational potential energy, which makes portals problematic to begin with.

brooksmoses addresses that below. Based on the way the games work, I think you have to handwave it as being his Possibility 1, i.e. something undefined is supplying that additional potential energy. Since it's well-established in the games that you can fall through a portal multiple times to increase your velocity.

Leaving aside that part of the issue, however, and just looking at the rod as it is when Case A begins, I believe I'm correct in assuming the kinetic energy is coming from the same gravitational potential energy any rod balanced on one end would have -- except this one is not supported by the floor. Ergo, within the laws of the game world, I'm pretty sure it has to move.

In Case C, again, I'm analyzing it from the perspective of how it works in the games; if that violates conservation of angular momentum, then portals are even more handwave-y than I realized. :-) Because you can definitely shoot a hole in the floor beneath yourself and fall out a wall.
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From:brooksmoses
Date:December 11th, 2012 05:55 am (UTC)
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Case C doesn't violate conservation of angular momentum if the portals push on the wall that they're stuck to.

(The easiest way to figure out some of this stuff is to imagine that there are very fast little arms on the surface of the portal that grab atoms of what's coming in, stop them, shuffle them through invisible tubes to the other portal, and the arms on the other portal reassemble them and throw them at the relevant speed. The basic laws of physics allow you to build something like that -- well, handwaving the details which may be impossible -- so if you can make that work, the portals can work the same way. And that can certainly grab a rod coming in one direction, eat it, reconstitute it, and throw it a different direction!)
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From:superversive
Date:December 10th, 2012 03:16 pm (UTC)
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In the first scenario, the rod should not accelerate at all. When the bottom bit of the rod falls through the portal, it appears at the other portal and therefore moves up. In so doing, it acquires potential energy equal to all the potential energy given up by the other bits of the rod moving downwards. The total potential energy of the rod remains the same (as you can see, because every bit of space occupied by the rod previously is still occupied by the rod). Since the potential energy remains the same, none of it is converted to kinetic energy and the rod remains in one place. If it was already in motion, it continues in motion until air resistance (or some other external force) causes it to stop. (Of course, this might not happen in the game, because the physics engine in the game might be making naive assumptions about gravity instead of calculating conversions of energy.)

In the second scenario, there is no longer enough space between portals for the length of the rod, so the rod will bend or break — just the same as if it were an ordinary rod jammed between two pistons and you moved the pistons closer together. There is also a possibility that the strength of the rod under compression is enough to resist the force that is moving the portals closer together, in which case the rod will actually hold the portal up and prevent it from moving. That, however, depends on the physics of the portals themselves, which is not specified.

In the third scenario, the rod simply has to come unstuck — it has to bend, break, or stretch, either at the solder joint or elsewhere. If it breaks, you do, in principle, have a weak projectile, since the rod (which is now an ordinary piece of metal, not joined to itself at both ends) is free to fall through the blue portal, and acquires kinetic energy which will make it shoot out of the orange portal with some velocity.

At least I think this is the case.

Edited at 2012-12-10 03:18 pm (UTC)
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From:swan_tower
Date:December 10th, 2012 08:55 pm (UTC)
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Yeah, the "increased potential energy" thing is obviously the major point of weirdness here, which brooksmoses dissects in detail below. I'm not sure if it's valid to say that the total potential energy of the rod remains the same, though; while that's true, I seem to remember also learning at one point that from the standpoint of physics, there is no "rod." There are only bits of rod, each one of which is acted upon by forces etc -- so the bits of rod that go through the blue portal first do indeed have more gravitational potential energy than they did before. I'm not sure the state of the rod as a whole is relevant, from that angle.
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From:superversive
Date:December 10th, 2012 09:13 pm (UTC)
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The reason the state of the rod as a whole is relevant is that every unit of volume occupied by a given mass of rod material before a given bit of rod ‘falls up’ through the portal is still occupied by the same mass of rod material after. The aggregate potential energy remains the same. Looking at the whole rod allows you to eliminate a lot of variables and simply rely on the commutative property of addition. If the potential energy in the ‘before’ state is A + B + C + D + E = X (with A, B, C, D, E representing portions of the rod), then the potential energy in the ‘after’ state is B + C + D + E + A, which still = X. You can thus treat the problem algebraically without knowing the actual energies involved.

brooksmoses did mention one thing I neglected, though, which is the possibility that the portals themselves work by adding energy to the system. That energy would then have to go somewhere. However, this is not specified in the problem as stated.
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From:brooksmoses
Date:December 11th, 2012 06:02 am (UTC)
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Yup, exactly -- to restate your point, if the rod is accelerating, it's not because the potential energy of the rod-as-a-whole is decreasing. For any given small motion, most of it decreases a little, and a little of it increases a lot, and it balances (assuming a uniform rod). So the energy to accelerate the rod has to come from somewhere else.

However, if your initial solution is right and no energy is being added by the portals, you can also do a force balance and find that the portal interface has to be applying enough force to the cross-section of the rod crossing it to hold up the rod. And it has to do that no matter how long the rod is (and how far apart the portals are), which means you can get some substantial forces applied to small cross-sections if the portals are far apart. I don't think you can fairly say that the problem-as-stated privileges the "no added energy" solution over the "no forces applied across the portal interface" solution; they both seem to add something notable that's neither explicitly stated nor explicitly prohibited.


Edited at 2012-12-11 06:25 am (UTC)
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From:brooksmoses
Date:December 10th, 2012 06:50 pm (UTC)
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Well, it all depends on the properties of the portals. I'll start with Case 1.

The first question is this: When we insert the rod through the bottom portal and some of it comes out the top, the part of the rod that comes out the top has been lifted up. That means it's increased in potential energy. Where does that energy come from?

There are many possibilities; I'll consider three.

Possibility 1: The energy is supplied by the portal, either because it has an internal energy source (which will eventually run out) or it has an external one. (I disagree with mindstalk; this is not inherently problematic. It just means that portals themselves can provide energy and thus must get it from somewhere. There are many possible "somewhere"s.) If this is the case, then the rod will start falling and accelerate until it's balanced by air friction -- which will be a fair bit faster than terminal velocity in still air, because there will be a column of air dragged through the portals with it and that lowers the air resistance it sees. You'll get a sizable wind downward around the set of portals from all this, and the portals will be supplying quite a lot of energy to make it go. If they have internal energy sources, eventually they'll run out and do whatever it is they do when they run out.

Possibility 1a: Same as 1, but the portals violate conservation of energy by effectively having infinite internal reservoirs.

Possibility 2: The energy is supplied by pushing the rod through the portals. This is actually a fairly tidy solution -- you can assume that moving something through the portals moves it through a force-field so that if you move an object through the portals and then back to where it started, it ends up with the net work (where work is the product of force and distance, which makes it equivalent to energy) equalling zero. This way the portals don't need to supply any energy on their own. If you do this, the rod will fall until it gets as low as it can go, which is resting on the bottom portal. (It will bounce a bit in the process. And then, in the real world, it will probably fall over.)

There is a problem, though -- the forces can get fairly high. If you have the portals 100 feet away from each other, say, then moving the rod through the portal takes as much work as lifting it 100 feet in the air. Doing the math, which I'll leave out, that means a force as if you were lifting a 100-foot column of the same cross-section as what's going through the portal. Anything you throw at it will just bounce, unless you throw it really hard. Try to push an egg through it, and the pressure on the inside of the egg will cause the downhill end to explode if it doesn't just shatter.

Possibility 3: Like possibility 1, but the energy comes from the thermal energy of the object. When the rod falls through the bottom portal and comes out the top one, it comes out a little colder. Aside from being a handwavey cop-out, it violates the second law of thermodynamics. And, even though the second law is really more a statement about the average of bulk probabilities that seems made to be broken, nature seems to frown heavily on things that break it; you probably have something like Maxwell's demon where quantum effects require you to have an energy input to make things run. Also, what happens if you try to put something through that's too cold to provide the energy? This doesn't actually seem like it would work without magic.

Possibility 4: Like 3, but the energy comes by converting some of the mass of whatever you put through. Again a bit of a cop-out but it could work. The downside is that you probably get all sorts of hard radiation from the portals, especially when things go in the top portal and out the bottom one (note: air molecules will be doing this all the time). Here the result is like 1 except the mass of the rod and surrounding air is slowly being converted to radiation until it's all gone.

For the sake of argument, let's discard possibilities that violate the basic conservation laws (1a and 3). That leaves 1, 2, and 4, and the choice between them is up to aesthetic preference. 1 is the most useful, but 2 has the advantage that it doesn't need an energy source and doesn't tend to cause cancer.

Edited at 2012-12-10 06:52 pm (UTC)
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From:brooksmoses
Date:December 10th, 2012 07:03 pm (UTC)
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Actually, I guess the picture called that "Case A", not "Case 1" -- and it's also considering the "continuous" rod with the ends soldered together.

There, for my possibility 2, what happens is that it will act sort of like a wheel on a frictionless bearing -- it will fall a bit and oscillate back and forth and eventually stop with a cross-section of locally-maximum density sitting in the portal.
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From:brooksmoses
Date:December 10th, 2012 07:46 pm (UTC)
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So, Case B.

Second question: Can the portals move relative to each other?

Possibility 1: Nope. This is sort of a null experiment, though there's an interesting question of whether they sit floating in space (immobile relative to what, exactly?) or fall to the ground. Or maybe they can only be placed on solid surfaces.

Third question, assuming they can move relative to each other: Does a compressive force across the things going through the portal result in a balancing force outward on the portals? What about a shear force?

This turns out to be an interesting question.

Possibility 2: No, compressive forces across the things going through the portal don't result in balancing forces outward on the portals. In Case 2 of the question, that means we can just gently push the portals together, and squish the metal rod into a pancake with no effort. That means we're not doing much work to push the portals together (and thus not putting in much energy) but a lot of work is being done on the rod (and thus much energy is going into it). Thus, we return to the question from Case A of where the energy comes from; the likely answer would be that it's coming from the portals either through an internal or external source.

But there's more, if we allow the portals to be non-parallel. Consider if we put the two halves of the portal on the same wall of a spaceship -- we throw a tennis ball in one, and it comes back to us out the other. Now, put a long pole through the one, and the end of it comes back to us out the other. We can lean on the two ends of the rod; we're pushing against ourselves, and the rod can't move without getting compressed. But, to lean against it, we have to also push the other way on the floor of the spaceship (by balance of forces). And that means that there's an unbalanced force on the floor of the spaceship, pushing it away from the rod. We've just built a reactionless drive! Not to mention that we're not putting any energy in, but the spacecraft will start accelerating, and there's no clear way the portals could be supplying energy for that.

Even if we require the portals to be parallel, if they're some distance apart we can make similar arguments about shear force and rotational momentum, if the portals are any distance apart.

(There is, interestingly, a solution if the portals are parallel and zero distance apart; this doesn't violate conservation of momentum. And, because the distance is zero, it also doesn't require a "force field" to fix the energy balance the energy balance from Case A! This should be physically possible! And indeed it is; an ordinary hula hoop painted orange on one side and blue on the other will suffice quite nicely.)

So, we conclude if we want interesting portals, they have to get pushed on by things going through them, which means that they need to be able to push on something, in order not to accelerate away.

Possibility 3: They push on whatever surface they're applied to (or, if not applied to a surface, they wave about until they push up against one, as pushed by the things sticking through them, air currents, etc.). This is sort of unsatisfying unless you require them to be stuck to a surface, in which case it works pretty well.

Possibility 4: They push against the surrounding matter in general with a large force-field, so that they can appear to be stuck in mid-air when they're really pushing unnoticably against everything for a half-mile around. This keeps up appearances without violating physical laws, but you've got "spooky action at a distance" going on, and can probably use this for other force-field effects in weird ways. On the other hand, portals are all about spooky action at a distance anyway, so this may be entirely fair.

Here, I'd again discard possibility 2 as violating basic conservation laws, and 1, 3, and 4 remain as reasonable options that are up to aesthetic preference. They don't actually make much difference to the situation of Case B, except for whether the assembly sits in midair or falls to the floor.
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From:brooksmoses
Date:December 10th, 2012 08:15 pm (UTC)
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And, Case C:

Fourth question: What happens to something sticking through the portal when the portal is shot somewhere else? (Can you even do that?)

Possibility 1: You can't do that if cutting the thing would take more energy than cutting air -- i.e., if the molecules on one side have notable attractive force to the ones on the other. This is sort of a hazy limit, though -- what if there's a piece of lint floating through when you want to reshoot the portal? And, for that matter, what about liquids, which have some amount of tensile strength and thus require energy to break (which is why surface tension exists)? And why molecules and not atoms?

Possibility 1a: Like possibility 1, but the portal has finite thickness, and anything coherent that's within that region will end up on one side or the other when the portal is re-shot. That solves the issues with molecules and dust motes, but you still have a problem with liquids, though.

For either of these, the answer to Case C is easy: You can't do that; you can't get the portal off the rod until you cut the rod and pull it out.

Possibility 2: Reshooting the portal cuts things. Or possibly reconnects them if you are very accurate with it. This takes energy; we assume the portal-gun supplies it. You can cut anything by sticking it through a portal and then breaking the portal. Portals are dangerous, yo.

Possibility 2a: It cuts things, but only up to a certain energy level. It will cut off your finger, but not through a steel rod, maybe.

Here, the rod gets cut, so the question about whether it extends off to infinity is moot.

Possibility 3: You can't break and re-form portals; you can only move them. This gets us basically back to things like the Case B discussion; you can only create Case C by dragging the portal hard enough to cut the rod. Otherwise, the rod holds the portal in place and you can't get it off until you cut the rod.

I can't think of a plausible possibility that gets rods extending off to infinity even with violating conservation laws, so I'll leave that out.

All of these seem to amount to roughly the same two options; either you cut the rod or you can't get the portal off it. "Shooting" a "new" portal isn't meaningfully different from shrinking the existing one to zero size, moving it very fast, and re-expanding it.

Aesthetically, though, I prefer possibility 3 with shrinking to zero (or not) and re-expanding.

So, there you have it -- combining them all makes about 3 by 3 by 2, or about 18, different options for portal physics that are physically consistent with Newtonian physics.
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From:swan_tower
Date:December 10th, 2012 08:58 pm (UTC)
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To match the world as observed in the game, it pretty much has to be your Possibility 1. Possibility 2 is contradicted by actual gameplay (several puzzles are solved by increasing your velocity through multiple continuous falls), and there's no evidence for 3 or 4. So let's handwave it as 1: there's an unknown energy source explaining what we observe.

(Editing because it makes more sense to put all my replies in one comment.)

Or maybe they can only be placed on solid surfaces.

This is indeed true (in the game -- I'm answering all of this according to the physics demonstrated in the game), and furthermore can only be placed on solid surfaces painted with a particular substance. So maybe that's where "portal energy" comes from, and if you shoot a portal onto a particular spot often enough, you'll tap it out. (It's either that or something in the portal gun itself.)

But there's more, if we allow the portals to be non-parallel.

Which you are indeed allowed to do.

the portal has finite thickness

Arguably this is true, since there's a point in the game where you basically balance on the edge of a portal for a moment.

You can't break and re-form portals; you can only move them.

This one gets complicated. When you have two portals and relocate one of them, it appears to shrink to nothing and expand in the new location. You cannot, however, "turn off" one or both sides; you can only shoot to a new spot. Once you start placing portals, you will have both blue and orange at all times.

. . . unless you pass through one of the force-fields in the game, which does indeed turn off your portals. So clearly "breaking" a portal in the truest sense can happen, but it requires the intervention of other special technology.

Edited at 2012-12-10 09:11 pm (UTC)
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From:brooksmoses
Date:December 11th, 2012 06:06 am (UTC)
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Ooh, the "shrink-to-nothing and expand in the new location" version is nice. And, as far as considering the behavior of what you can do with individual portals, really the key piece is "you can't relocate a portal without shrinking it to zero so it's not connecting anything first."

And it seems pretty obvious to me that the answer to Case C is: When you relocate the portal, assuming you can do that, it first shrinks to nothing and cuts the bar in half by doing so.
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From:brooksmoses
Date:December 11th, 2012 06:17 am (UTC)
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Also, as perhaps a side note: There have been many nights when I've pondered at the "how do portals work with physical conservation laws?" as something to ponder while falling asleep. So I didn't just think all this up in response to your question!

There's also a related fun magical object, which is a portable hole (or tunnel) -- you've got the two ends of the hole, but they're not holes in anything physical. Imagine something that's basically a couple of flat disks (like portals!) where if you look "in" one you see a tunnel going to the other, but the outside of the tunnel doesn't exist and you can wave your hand between the disks or stick them on opposite sides of a wall or something without harming the wall. Unlike portals, though, the two ends are permanent and act like they're physically connected, so moving one will move the other (though the length part may be elastic), and the distance from one end to the other is the same whether you go through or around. Those can be entertaining to play with too, and the physical implications are a lot simpler.
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From:davidgoldfarb
Date:December 10th, 2012 11:58 pm (UTC)
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Portals as shown in the game definitely violate conservation not just of angular momentum but straight-out of momentum -- there are any number of puzzles that are solved by falling to speed up and then getting to change direction without having to accelerate. Momentum is a vector quantity: the momentum you have when falling down is not the same momentum as when you are zooming up and to the left.

Depending on how exactly they work, they probably also violate conservation of energy and the Second Law of Thermodynamics.

The answer to question A seems clear to me: if you put the rod in place and then dropped it, it would fall downward -- the boundary between the two ends would move downwards at an accelerating rate. Welding the ends together shouldn't change that.

Question B is a lot less clear. I haven't played the games, but do the portals ever go on to moving surfaces? That may not be possible. But question C is something that clearly ought to be possible, from what we see in the games -- what happens is basically whatever the programmers thought would be cool.
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From:brooksmoses
Date:December 11th, 2012 06:03 am (UTC)
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They don't violate conservation of momentum if they push on the wall when you fall out of them, any more than a bouncing ball violates conservation of momentum! :)
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From:davidgoldfarb
Date:December 11th, 2012 07:25 pm (UTC)
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Well, okay, yeah, but I can't recall seeing any indication that they do.

Some more thoughts on question C:
Having the bar stretched to infinite outward extent is silly and obviously wrong. What has to happen is that the portal closing cuts the bar; then the new portal opening on the wall spits out the bar (still at its original length) moving at speed. The final result is as though you had cut the bar in the middle, then took the original two ends and welded them together.
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