Theeventual heat death of the universe is an awful time to be alive considering you wouldn't have long left after the last star finally fades away.
In this scenario there is a small group of about a dozen human survivors who live in a colony on an earth-like planet that orbits the last star to burn out. The group know of their impending doom and have time to prepare for it, about 10 years (but feel free to adjust this time if you think of something cool).
Now I fully expect answers that lead to the eventual death of everything from this scenario but what would this group need to do in an attempt to continue existing for as long as possible in a universe with no heat?
Note: This would be trillions of years into the future so feel free to go a little crazy with future science, as long as it makes sense. It would also be interesting to see how a race of humans with current-day technology could survive in the same conditions.
I mean, literally. As in "get out of the universe". The heat death of our universe is by definition not a survivable event. That's the bad news. The good news is that it takes a very, very long time to happen. So a technological society has time to develop technological solutions to the problem.
Cross over to an alternative universe. Multiverse theory is endorsed by a surprisingly large number of prominent physicists. So let's assume the physicists are right, that multiple universes exist, that at least some of them still have a few trillion years left in them when our universe is grinding to a halt, and that science has advanced to the point where it's possible to travel between universes at will. Why struggle to survive in a dying universe when you can just move next door and start over on a new planet?
Return to a previous time in the current universe. So multiverse theory didn't pan out despite its prominent endorsements; who cares? Perhaps wormholes are real and capable of traversing both space and time. If they are, it's plausible that a technological society will learn to control them before the universe's heat death occurs. And if they do that, they can use them to travel to any desired place and time within the current universe. So they can just pick a more hospitable moment in time, and go there. But might lead to interesting cyclical dilemmas as repeated generations keep jumping back in time to avoid facing the heat death of the universe (only to find some subsequent generation already camped out in their chosen real-estate).
Jump in a black hole. So both multiverses and wormholes turned out to be fake; guess we're screwed. But no need to wait around to die an icy death. Instead find the nearest black hole (or use your stockpile of doomsday devices to create one) and jump in. At least you get to go out on your own terms. And who knows, maybe black holes are actually survivable or will transport you to somewhere a little less doomed. If nothing else, it won't be boring and perhaps relativistic time dilation will give you a good view of the universe's final moments.
Kind of an inversion of the previous approach, but maybe multiverses do exist but we're not able to safely transport living things from our universe to another. That doesn't necessarily mean we can't snag a fresh sun or two every now and again. You know, just enough to keep a tiny ember burning in our universe.
Heat death doesn't occur until you reach maximum entropy, which is something that you can hold off indefinitely if you're able to pull in even relatively small (say, on the order of an M-class star) clumps of matter every few billion years. If you've got the technology, and a power source for it (the big question is whether it can be possible to pull a star from one universe to another using less energy than the star will generate over its useful lifetime), you can wait it out.
Come to think of it, you can wait it out regardless since the heat death of the universe can't occur until well after any warm-blooded life forms have ceased functioning. The most you can ever observe is the prelude.
I think the question is fundamentally ill-posed. As long as there are humans alive somewhere, the universe is millions to the power of millions of years away from the "heat death" (which is not at all an agreed-upon or well-understood thing).
The heat death (if we assume it is well-defined) is definitely not "when the last star fades away". It's waaaay later than that, at least after the final supermassive black hole has evaporated due to Hawking radiation, which would take on the order of 10$^100$ years after the last star dies.
By definition then, there's no way of 'surviving' the event, because you'd have to be dead in the first place for the condition to have been reached. Dead, decomposed and the energy you released into the ground dissipated universally. You get the idea.
But lets assume the question is: can a civilization survive the extinguishing of the last star. Well, yes. Aside from any local reserves of fissile material, the planets themselves are still orbiting, so there's still masses (sic) of energy sitting in gravitational wells, waiting to be harvested. And that dead star is still rotating the galactic center, and that's still caught up in some amazingly long range interactions with other galaxies. And so on. Extracting usable work from all this energy will be a significant challenge however, but if the motivation is there...
The classical physics definition of "the heat death of the universe" is the moment when the universe reaches "thermodynamic equilibrium (AKA maximum entropy)." which means that there are "no net macroscopic flows of matter or of energy."
Note that there are numerous theories and wild ideas in physics that suggest that either this point would literally never be reached due to other events or that it would decrease again afterwards and thus multiple heat deaths might be possible.
Note above the word "macroscopic". There is, in practice, a lot happening below that scale and if you accept the concept that life can be realised as dynamic information, then it is possible to assume that after many, many trillions of years of technological advances we might figure out a way to live down there at that scale.
While another poster suggested "jumping back" to previously stable points in time, it is worth mentioning that we don't really know what time is. Therefore, there are properties and functions of time with applications that we aren't even aware of yet.
For example, once you can control time, you can probably exert some control over space-time. Infinite control means you could craft physical laws that recirculate energy throughout the universe in a predictable format, like a river with walls. This could be a corollary to Einstein's "Cosmological Constant," where the universe expands as it does now, contracts a bit to harvest and recirculate the energy, expand again, and so on.
2: Space-time control could have "leaks" of its own, resulting in the same eventual heat death. You could still get a few trillion-trillion years or so of existence, which ought to help a sufficiently advanced civilization come up with a more permanent solution.
As a final note: we are considering options based on what we know today. Thomas Malthus predicted the end of civilization as population exceeds resources, however he lived in a time when the primary economy was making food and the primary fuel was food and muscle (livestock, slaves, etc). It is unlikely he could have foreseen a future where robots harvest crops from year-round solar-powered greenhouses. It is therefore reasonable to expect that our limited understanding of the universe denies us the vision to see solutions beyond the tools we are aware of.
Given how long humans have existed and how little the time, in geologic terms, it took for us to go from hunter-gatherers to the Information Age, I suspect that if we can avoid nuclear, environmental, or biological apocalypse, we will figure it out in time.
If you can accurately measure how much usable energy you have (lif you have to err, err on the lower side), and you can spend energy proportionally to that amount, you end up with an exponential curve of dwindling energy, which mathematically never ends.
The tricky part is when you are dependent on processes which are not proportional like that. For example, we are currently very dependent on activities which rely on quantized behaviors, such as the emission of photons. Those events will have to be more and more rare as the energy levels decrease. You would also likely choose to concentrate your energy in a smaller and smaller portion of your space, in order to permit at least a small portion to be using such quantized energy. In fact, this has lead to two competing extremes as to how to accomplish this goal. There is the continuous process, where you try to keep a fluidly decreasing amount of energy usage, and the discrete approach, where you subtly collect energy for as long as needed to permit one quick burst of a finite length of energy. Presumably whatever the final solution will be will involve a cross between both of these approaches.
The hard part is knowledge of the heat death: you don't have it. There is no way for science to know that heat death will occur, as opposed to us discovering that our mathematical models which proved heat death will occur were wrong. Sure in our 359 years or so of modern science and thermodynamics, we're pretty sure that's the direction things go. We have another 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 years before galaxy sized black holes vanish, and I'm not even going to write out all the zeroes in the $10^10^56$ years before quantum tunneling might start poking holes that let us into other universes, by our theories. We have a long time to find out that we have an incomplete theory. The hard part will be not developing a survival scheme that assumes we've seen it all, and starts down the path of exponential decay to live forever like Nietzche's Last Man. We would need to continue to reach out and observe the dying universe, looking for hope. If we find it, we need to be about to harness it to build the best world we possibly can.
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