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June 3

What happens when an unbreakable box is filled?

Yet again my colleagues have stumped me. One of them asked the following: "what happens if an unbreakable box fills to its maximum capacity? For example, let us imagine an infinite water machine that teleports water in from another location, but water cannot teleport back out. This water machine is locked inside an absolutely unbreakable box with no way to access the outside. What would happen when the box gets filled to its maximum capacity with water?" Some proposed that the box would HAVE to break, but it is unbreakable after all. Others proposed the water would simply stop, but this proposed water machine cannot stop under any circumstances. And we mustn’t forget water cannot leave the box, ever. At the time I paid little attention, instead, focusing on my rather tasty Nutella sticks, yet now I cannot stop thinking about this problem. Is it solvable, or is it the next contender for the List of Paradoxes page? — Preceding unsigned comment added by 166.127.32.165 (talk) 14:42, 3 June 2022 (UTC)[reply]

A black hole happens. Card Zero (talk) 14:48, 3 June 2022 (UTC)[reply]

See Irresistible force paradox. Black holes aside, you seem to be creating a 'paradox' out of two mutually-incompatible premises. They can't both exist in the real world, or in any universe constrained by logic. Beyond the constraints of logic, anything can happen. AndyTheGrump (talk) 14:52, 3 June 2022 (UTC)[reply]

Three things could happen: A: Physics shows you that the box isn't really unbreakable; B: Physics shows you that the pump/teleporter cannot really provide water at infinite pressure; C: you compress the water (it isn't really incompressible) and force an ever-increasing amount of water into the box. After forcing a couple of Earth-masses of water into the one-litre box, it will turn into a black hole that grows to larger size than the box and your unbreakable box is now invisible in the black hole, so you cannot see whether it's still unbroken.

Physics doesn't really like infinities that cannot be explained away by balancing them with other infinities or are hidden behind some horizon. That is, infinitely strong boxes don't really exist. They would violate the laws of physics. PiusImpavidus (talk) 16:05, 3 June 2022 (UTC)[reply]

I think the answer is "anything you choose". You made up the universe in which such a thing as an unbreakable box can exist: you make up its rules of physics. ColinFine (talk) 16:17, 3 June 2022 (UTC)[reply]

The premise says that the box is unbreakable, but it doesn't say it's perfectly rigid, and (as PiusImpavidus points out) water isn't perfectly incompressible. So as the pressure grows, the water will compress according to its bulk modulus and the box will stretch according to its Young's modulus. To find out what happens, you'll need to know how stretchy the box is, and also whether it's in thermal contact with anything outside the box. It also depends on how large the box is initially. Assuming the box is not too large or too stretchy, and without something to remove heat very quickly, the water will soon become a supercritical fluid, which is very compressible, and you might get one of the high-pressure exotic phases of water ice. Eventually, nuclear fusion will start, and you'll need to know how opaque the box is to various forms of electromagnetic radiation. Almost any set of assumptions will eventually take you to a black hole of one size or another, but the path to get there may be very different. --Amble (talk) 20:49, 3 June 2022 (UTC)[reply]

If the box is very large, the water may collapse into a black hole before the box is full. Then assuming the rate of incoming water is constant in the frame of the pump, it will go to zero as seen by an outside observer because of gravitational time dilation. When the black hole eventually evaporates, the box will be filled with a thermal distribution of particles. The question of whether or not the particles remember that they used to be a water machine is itself an unresolved paradox. --Amble (talk) 22:46, 3 June 2022 (UTC)[reply]

If the box is rather small, but very rigid and very opaque, you'll go through supercritical steam and various exotic ices to a dense plasma that can undergo nuclear fusion. Because the box is opaque, the pressure and temperature will keep going up without establishing radiative equilibrium as stars do. If you hit some form of baryon number violation, the protons and electrons may be able to annihilate. Although it's stated that water cannot leave the box, there's no way that neutrinos will stay inside. You'll reach a new equilibrium and your unbreakable box plus unstoppable water machine will become a high-powered neutrino source. --Amble (talk) 22:52, 3 June 2022 (UTC)[reply]