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I have an enormous civilization spanning a large amount of the universe. It is able to both travel and send information much faster than the speed of light without violating any laws of physics.
Resources are not really a large issue. This civilization, already millions of years old, plans not to reverse or stop the heat death -- they've decided it cannot be stopped -- they plan to slow it down as much as possible, even if they only get a few thousand extra years.

Is there any real way for them to accomplish this without traveling to other universes or breaking laws of physics?

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    $\begingroup$ Given that they can literally defy logic (send information faster than light without breaking the laws of physics, given that does break a law of physics), they could just use the principle of explosion. $\endgroup$ – PyRulez Jan 13 '16 at 2:01
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    $\begingroup$ As @PyRulez said, you've already broken a key law of physics. It's not really possible to answer the question with the laws of physics if you choose to break them. $\endgroup$ – HDE 226868 Jan 13 '16 at 3:05
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    $\begingroup$ @PyRulez Not necessarily, it breaks relativity or causality. While we can hardly imagine a world without causality (but still it's not impossible), we know that relativity is not the final physical theory (for starters, it's not reconciled with the quantum theories). $\endgroup$ – Radovan Garabík Jan 13 '16 at 7:15
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    $\begingroup$ What about using emotional energy of suffering young girls as they turn into human-killing suffering-eating eldritch monsters? $\endgroup$ – Euphoric Jan 13 '16 at 8:27
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    $\begingroup$ You seem to be asking the last question $\endgroup$ – Separatrix Jan 13 '16 at 12:12
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Assuming no big rip scenario, star husbandry is definitely the first step. Most of the energy produced by starts is just wasted, by careful controlled fusion (up to iron) the useful energy (and time) available to the civilization can be increased by many orders of magnitude.

Then collapse the iron first into "iron stars", harvesting its gravitational potential energy, then into black holes and harvest their Hawking radiation (if possible).

It's perhaps better to think of it as slowing down the inevitable entropy increase by throttling the wasteful energy production at the beginning of the universe history, during the stelliferous era.

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    $\begingroup$ Time to break out the Dyson Spheres. $\endgroup$ – Joe Bloggs Jan 13 '16 at 11:07
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    $\begingroup$ @JoeBloggs That might be the first step to the first step, but without moderating stellar fusion you'd get a lot of energy at the beginning and then a lot of cold sub-iron stellar remains that could have been burned more effectively in small fusion reactors. $\endgroup$ – Radovan Garabík Jan 13 '16 at 13:16
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    $\begingroup$ a decent Dyson sphere lets you use the waste energy from the star to perform star lifting operations, and gives you a nice controlled place to perform your fusion reactions before dumping the waste iron back into the cooling star to reclaim the gravitational potential energy (and hawking radiation if you get that far). Do it right and you've basically got your entire process in a nice self contained factory $\endgroup$ – Joe Bloggs Jan 13 '16 at 13:36
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Radovan Garabik already pointed out the most important thing, stopping the stars from wasting fusion fuel. Assuming, as he said no big rip or proton decay [1] civilisations can live of fusion, fisson, gravitational potential, black hole rotation and hawking radiation for a long time.

Going fully digital and living as uploaded minds, exploiting the temperature of the universe via Landauers Principle [2] and frame jacking (slowing the processing rate of the computers to keep the ultra cool and to slow down the time anyone in the simulation percives relative to the universe) will allow for optimal energy-conservation and makes even inter-supercluster communications practical without ftl.

Youtuber Isaac Arthur's Civilisations at the End of Time series might be of interest to you, as he explores hese concepts in detail. [3]

[1] https://en.m.wikipedia.org/wiki/Proton_decay

[2] https://en.m.wikipedia.org/wiki/Landauer%27s_principle

[3] https://www.youtube.com/playlist?list=PLIIOUpOge0LvHsTP5fm8oxB1qPS54sTMk

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Forge a new universe

It is quite possible that such a civilization could create an entire new universe to work with.

If such a universe turns out to be inhabitable by the race in question, then the heat death of the parent universe is irrelevant. Otherwise, the new universe can provide Universe-scale energy.

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Live faster

Instead of slowing the universe which would take huge resources - live faster. Thus people will experience a full and long life subjectively while time for celestial bodies continues as before.

How can this be achieved? Well firstly we know that time 'stands still' for people who are coping with a dangerous situation. This is thought to be an effect of adrenaline speeding up our reactions. Of course constant flooding with adrenaline would be deleterious to health but I'm sure by this time we would be masters of drugs.

Effectively humans live as long as a mayfly but to them it seems as though they experience a hundred years (or whatever lifetime they choose).

The main drawback would be that days/nights would seem to last for a hundred years. However by then holodecks or simply virtual reality would be so real that people would never need to see light of day.

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