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So the idea is that the world has been destroyed, and an emergency plan to put the rest of humanity on floating cities/islands has been put in place. These islands subsist off of some sort of generator that has been in place since the beginning of this plan. The question that I have is what sort of generator would be needed to keep these cities afloat, but can be scaled down so that it would fit some sort of weapon, such as a laser or plasma gun or rail gun.

Ignoring the physical/material constraints of aforesaid weapons and generator, is it possible to create an infinite energy generator that can instantly output large amounts of energy with low to no human contact? I think that I am aiming for high voltage, only. Current doesn't matter as much.

An idea that I did have was to create a chamber containing a micro black hole, and have that micro black hole detonate, then harness the energy with near perfect efficiency, creating another micro black hole. The excess energy, should there be any, would be stored in battery cells with infinite capacity for charge. But I don't think that this would work, considering the mass required for even a micro blackhole.

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    $\begingroup$ Floating on what? If the world is destroyed, there is no ocean. IAC you didn't destroy the sun, so use solar power. $\endgroup$ – JDługosz Dec 28 '15 at 15:52
  • $\begingroup$ Do you really mean infinite in the mathematical sense or do you just mean large enough to never run out prior to the heat death of the universe? $\endgroup$ – The Anathema Dec 28 '15 at 15:56
  • $\begingroup$ You're asking "is there the ability to create a perpetual motion machine using black holes?" and the answer is still "no." You can't "detonate" a black hole, bleed off a little energy, and then create a new black hole with the same mass: you bled off some of the energy, the new black hole will be smaller and less massive. Also note that a black hole with a mass of 1000 kg still weighs 1000 kg (Even if it is $1.4 x 10^{-24}$ meters radius, smaller than a single neutrino). $\endgroup$ – Draco18s Dec 28 '15 at 15:56
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    $\begingroup$ An idea that I did have was to create a chamber containing a micro black hole, and have that micro black hole detonate, then harness the energy with near perfect efficiency, creating another micro black hole. Why not simple use the energy you took to create the black hole instead? Otherwise, you'll lose energy, unless the black hole absorbs a substantial amount of energy from the surroundings. $\endgroup$ – HDE 226868 Dec 28 '15 at 17:38
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    $\begingroup$ @Necessity - Ummm, no. Just no. The pieces will continue on their (earth's) orbit, subject to perturbations associated with the breakup of earth. Let me put it another way - if they "just fall out of earth's orbit", where would they fall to? And how would they get the velocities needed? You may be thinking of something like the Moon, but even that will continue to orbit the sun, although with a slightly different set of orbital parameters. $\endgroup$ – WhatRoughBeast Dec 31 '15 at 21:50
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This happens to be a favorite topic of mine, because I can bring up one of the niftiest things in mathematics: Noether's theorem. Noether's theorem is a theorem which can be applied to what are called Lagrangian systems, which make up nearly everything we believe the physics of the universe to be.

Noether's theorem states that, for every symmetry (i.e. a feature which doesn't change under some transform), there is a corresponding conserved value. As it happens, the time symmetry of the universe, which can be stated as "the laws of physics do not change over time," explicitly leads to the conservation of energy. Any system which does not conserve energy must either be non-Lagrangian or fundamentally requires the laws of physics to change over time!

Generally speaking, its not reasonable to play too many games with non-Lagrangian mechanics. There's some adjustment factors required to make general-relativity Lagrangian, but other than that, Lagrangian mechanics has dominated the physics field since it upset Newtonian motion (which is non-Lagrangian due to friction, until you properly adjust for the energy loss).

No reasonable scientist has ever suggested perpetual motion is possible, much less perpetual usable energy generation, using any fathomable means (note: Zero Point Energy is considered by science to be a source of energy, but not usable energy that can be harnessed, and the Casimir effect is predicted by many to be the result of QED alone, not any vacuum energy sources). Accordingly, if you need such a machine to power your islands, just walk away from science entirely. Make the system magical instead, and choose to constrict its magic to almost valid physics, except for the distasteful tendency to generate usable energy from nothing. Alternatively, identify an undiscovered way of storing great deals of energy in small volumes (like Heinlien's Shipstones). This cannot produce your infinite energy, but if they had a source of some arbitrarily large finite amount of energy to charge beforehand, they could last long enough.

And if you do the calculations yourself, you'll realize the amount of energy required to keep an island levitating, even in the most inefficient way possible, is not even a drop in a bucket compared to the energies that appear in stars. Heck, if the energy to levitate your islands was a drop in a bucket for some larger scale of energy, that entire bucket wouldn't even be a drop in the bucket for the energy budget of a sun.

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    $\begingroup$ Or 2 other ways to look at it. In order to use energy, we require it to flow from a high energy point to a low energy point. Even if the Zero Point Energy has "lots of energy", since that's the minimum for our Universe, there's no where for it to flow to and no way for us to use it. Looked at another way, if we can simply add energy to our Universe (not conserve energy), we'll eventually all roast to death. $\endgroup$ – Jim2B Dec 29 '15 at 3:58
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What you are asking for is related to perpetual motion machine of second kind and impossible under any physical laws.

Simply put, energy can not be created, nor destroyed, it can only be moved from place with high density of energy to place with low energy density, while taking some of the moved energy to make other work in the process.

Above simple explanation refers to every and all processes. Imagine having a river, you build turbines in the stream. Water on high ground has higher energy density (potential energy), water on low ground has lower energy density, normally difference is dissipated through friction and other processes. Your turbines, however, take some of the energy and produce the electricity, but you can't use this energy to power pumps that would move water back to high ground, energy produced is much lower than energy difference between height levels. You need to rely on other processes to refill your high-ground reservoir. In this case you rely on sun to evaporate water which later condenses and falls as rain, filling rivers and lakes. Same applies to every and all processes.

In theory, using reversible processes, you can have machine that can move forever, but without producing any useful work - orbiting planets come very close, they lose very little kinetic energy and seemingly move forever, but if you took any of that energy to do useful work, it would slow the planet down.

That being said, there are some solutions, depending on what your story requires and how "soft" you are willing to go with sci-fi aspect.

First: floating. Floating requires actively counteracting gravitation, that's bad as it requires huge energy expenditure if you are actively using engines for that. It's great if you can replace that with orbiting. Douglas Adams wrote that flying is an art of falling and missing the planet. It's a great joke, but when referring to orbiting, it's completely true - you constantly fall, but your forward speed is high enough that you "miss" the earth and stay in space. If they have to float in atmosphere, make them Zeppelins - obtaining hydrogen to replace lost gas is trivial compared to other problems of self-contained city.

Second: energy. You need energy reservoir, usually called "fuel", and you need to "refuel" your generators from time to time, that's unavoidable if you want to go the semi-hard sci-fi route. Best you can do to avoid maintenance is automation, here are some wild ideas that can be scaled and are relatively easily refuelled by robotic ships.

  1. Implosion based fusion may be hugely scalable - take plastic pellet filled with deuterium, heat it with laser to explosively vaporise it, imploding deuterium and forcing fusion. It's proven to work and is theorised to be workable as portable power source. Assume that same can be done with hydrogen, power your islands by tokamaks or just hundreds of scaled up generators and let your humans mine outer layers of sun for hydrogen. Not infinite, but effectively a non-concern unless you start spending that energy on matter+antimatter creation to build entire planets.
  2. There are some unknowns about CPT symmetry where you could probably hide errors and small lies leading to direct matter->energy conversion, without necessitating antimatter (antimatter can't be mined, so using antimatter for fuel is like pumping water upstream). This isn't even remotely theorised so you can go wild with scalability, either way, you are constrained by E=m*c^2.
  3. Evaporation of black holes. BH the mass of Earth would have event horizon only 3 cm in diameter, and slowly evaporate. The smaller the BH, the faster it emits energy. Claim that there's "sweet-spot" size where BHs live relatively long and emit enough energy to be useful as energy sources. Use fusion energy or something like that to force collapse of progressively larger objects until you reach sweet-spot size. Make sweet-spot mass to be in the order of magnitude of asteroid masses for islands and few kilograms for personal generators, that way you have a lot of fuel on your hands. Warning: I haven't researched exact values and you are still constrained by E=m*c^2 anyway.

If nothing of the above fancies you, just go "the black box route" and have aliens or insane genius invent it. Humans in your world don't have to understand how it works to replicate it - you don't need understanding of thermodynamics or chemistry to build car engine if you are given blueprints, but you need thermodynamics, chemistry and a lot more to improve it or scale it in any way other than adding more cylinders. Impart some limits on characteristics so that this device can't be an answer to all and every need, because that would make a boring story, and you can have some stories where people try to figure out how to solve problems with more energy, or have to rework their solutions when one of the devices malfunctions.

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The traditional hand-wave to get you effectively unlimited energy (but not infinite energy, which would destroy the universe) is to say that it's based on extracting zero-point energy. Since no one has any idea how this might be done, no one can say that you're doing it wrong.

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    $\begingroup$ That's like saying that a body of water at the lowest point on Earth still has gravitational potential energy from being 4000 miles from the center of the Earth. If you could harness that with a waterfall you can get energy! $\endgroup$ – JDługosz Jan 1 '16 at 21:10

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