On long enough cosmological time scales, hydrogen and helium nucleii will become scarce in the Universe. It seems to me that any advanced civilisations that might exist in that epoch would have the motivation to try and prevent the stars from using them up, in order to burn the fuel more slowly and extract a greater proportion of the energy as usable work.
One might say that, on time scales measured in trillions of years, the stars are an unsustainable use of the universe's fuel. This question is about whether such civilisations would have the means to do something about it. My questions are:
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What would be the most energy-efficient way (using known physics) to blow apart a star or otherwise prevent or greatly slow the rate at which it performs fusion? We're assuming this civilisation has access to vast amounts of energy but doesn't want to waste it unnecessarily, since the aim is to access energy from the hydrogen the star would have burned. In order for this to be worthwhile, the energy gained from doing this would have to be substantially more than the energy the process takes.
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What would be the astronomical signature of such an activity? If it was happening in a distant galaxy, would we be able to detect it from Earth?
Update
I'm still interested in this question. There are some great answers below, but currently I don't feel that any of them approach the most efficient way to solve the problem, i.e. a way likely to be attempted by a civilisation with the resources and motivation to do so. Luboš' answer makes it clear that the energy requirements are not too stringent in themselves (you just need to skim off a small fraction of a nearby gas giant and fuse the hydrogen into heavier elements; this question already presupposes the ability to extract power from fusion on a literally astronomical scale) but points out that no "bomb" made of atoms can enter a star and blow it up. However, this does not rule out other methods, such as increasing the star's angular momentum through some means (as in AlanSE's answer) or somehow (perhaps electromagnetically) removing plasma from its surface rather than blowing it up from the inside.
If removing plasma from the surface is possible then ideas based on this have a certain appeal, because the resulting hydrogen could be fused, releasing energy that could be used to skim off more, and so on exponentially. (Kind of like the opposite of that bit in the film 2010 when a self-replicating monolith turns Jupiter into a star.) Some of the resulting energy could be put into increasing the star's angular momentum as well. However, I don't know enough about plasma physics to know whether there's a way to suck the plasma up from the star's surface. (It can be done gravitationally, of course, but then the hydrogen just ends up being depleted by runaway fusion on the surface of another star, so this isn't very useful.)
Given this, my current specific questions are:
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Is there a plausible way to remove material from the surface of a star at an appreciable rate, either electromagnetically or through some other means? If so, is there anything other than the (virtually unlimited) availability of energy that limits the rate at which this could be done?
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Given a sufficiently huge supply of energy, is there an effective way to rapidly increase a star's angular momentum? It seems (per AlanSE's answer) that redirecting the star's radiation would take too long, but perhaps firing massive objects or particle beams at it would accomplish the same effect in a shorter time.
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Aside from these ideas, would adding heavier elements than hydrogen to the star suppress fusion by absorbing neutrons? What quantities would be required in order to affect the star's evolution by this method? (A civilisation that eats stars will generate a huge supply of stable nucleii as a waste product, so maybe it makes sense to simply dump some of them into the next star.)
Best Answer
Burning (and fusion) is "unsustainable" by definition because it means to convert an increasing amount of fuel to "energy" plus "waste products" and at some moment, there is no fuel left.
I am not sure whether the word "unsustainable" was used as a joke, a parody of the same nonsensical adjective that is so popular with the low-brow media these days, but I have surely laughed (because it almost sounds like you are proposing to extinguish the Sun to be truly environment-friendly). The thermonuclear reaction in the Sun has been "sustained" for 4.7 billion years and about 7.5 billion years are left before the Sun goes red giant. That's over 10 billion years – many other processes are much less sustainable than that. More importantly, there is nothing wrong about processes' and activities' being "unsustainable". All the processes in the real world are unsustainable and the most pleasant ones are the least sustainable, too.
But back to your specific project.
When it comes to energy, it is possible to blow a star apart without spending energy that exceeds the actual thermonuclear energy stored in the star. Just make a simple calculation for the Sun. Try to divide it to 2 semisuns whose mass is $10^{30}$ kilograms, each. The current distance between the two semisuns is about $700,000$ kilometers, the radius of the Sun. You want to separate them to a distance where the potential energy is small, comparable to that at infinity.
It means that you must "liberate" the semisuns from a potential well. The gravitational potential energy you need to spend is $$ E = \frac{G\cdot M\cdot M}{R} = \frac{6.67\times 10^{-11}\times 10^{60}}{700,000,000} = 10^{41}\,{\rm Joules} $$ That's equivalent to the energy of $10^{24}$ kilograms (the mass of the Moon or so) completely converted to energy via $E=mc^2$, or thermonuclear energy from burning the whole Earth of hydrogen (approximately).
You may force the Sun to do something like the "red giant" transition prematurely and save some hydrogen that is unburned. To do so, you will have to spend the amount of energy corresponding to the Earth completely burned via fusion.
But of course, the counting of the energy which was "favorable" isn't the only problem. To actually tear the Sun apart, you would have to send an object inside the Sun that would survive the rather extreme conditions over there, including 15 million Celsius degrees and 3 billion atmospheres of pressure. Needless to say, no solid can survive these conditions: any object based on atoms we know will inevitably become a plasma. A closely related fact is that ordinary matter based on nuclei and electron doesn't allow for any "higher-pressure" explosion than the thermonuclear one so there's nothing "stronger" that could be sent to the Sun as an explosive to counteract the huge pressure inside the star.
One must get used to the fact that plasma is what becomes out of anything that tries to "intervene" into the Sun – and any intruder would be quickly devoured and the Sun would restore its balance. The only possible loophole is that the amount of this stuff is large. So you may think about colliding two stars which could perhaps tear them apart and stop the fusion. This isn't easy. The energy needed to substantially change the trajectory of another star is very, very large, unless one is lucky that the stars are already going to "nearly collide" which is extremely unlikely.
Physics will not allow you to do such things. You would need a form of matter that is more extreme than the plasma in the Sun, e.g. the neutron matter, but this probably can't be much lighter (and easier to prepare, e.g. when it comes to energy) than the star itself. A black hole could only drill a hole (when fast enough) or consume the Sun (which you don't want).
However, if you allow the Sun to be eaten by a black hole, you will actually get a more efficient and more sustainable source of energy. Well, too sustainable. ;-) A black hole of the mass comparable to the solar mass would have a radius about 3 miles. It would only send roughly one photon of the 3-mile-long wavelength every nanosecond or so in the Hawking radiation and it would only evaporate after $10^{60}$ years or so. It would be so sustainable that no one could possibly observe the energy it is emitting. However, the black hole would ultimately emit all the energy $E=mc^2$ stored in the mass.
If there are powerful civilizations ready to do some "helioengineering", they surely don't suffer from naive and primitive misconceptions about the world such as the word "sustainable" and many other words that are so popular in the mentally retarded movement known as "environmentalism". These civilizations may do many things artificially but they surely realize that the thermonuclear reaction in the stars is a highly efficient and useful way to get the energy from the hydrogen fuel. Even some of us realize that almost all the useful energy that allowed the Earth to evolve and create life and other things came from the Sun.
The Sun may become unsustainable in 7.5 billion years but according to everything we know about Nature, it's the optimum device to provide large enough civilizations – whole planets – with energy.