You can already decrease entropy locally, but net for the entire universe, entropy (as far as we can tell...) always increases monotonically. And any action you take to decrease local entropy still actually increases it in total.
So "The Last Question" is also basically asking - "how can we avoid the heat death of the universe?".
And phrased yet again differently: "Once all the stars burn out and all the uranium is fissioned and the coal burned and the universe is just a homogenous 5 degree kelvin soup of [I'm sure some physicist could tell me whatever fundamental particle it'll be that composes this soup] - what then? Is that just it?"
Once all stars die you will be able to live very long off of the heat of dead stars as they slowly cool off to background temperature.
And once those stars are completely cold, you can start converting their mass into energy by dropping them into black holes, piece by piece.
And once you dropped all matter you could into black holes you could live off merging black holes.
And once you merged black holes you could live off of the black hole radiation until all black holes evaporate.
But this would be if the Universe wasn't expanding at a growing rate. If the Universe is truly expanding at an ever accelerating pace there will come a Big Rip which will cause every fundamental particle to get further from all other particles at speeds faster than light. And then matter as we know it will cease to exist.
> While objects within space cannot travel faster than light, this limitation does not apply to the effects of changes in the metric itself. Objects that recede beyond the cosmic event horizon will eventually become unobservable, as no new light from them will be capable of overcoming the universe's expansion, limiting the size of our observable universe.
We can already observe this phenomenon happening at large scale. Our observable universe shrinks as galaxies effectively accelerate away from us faster than the speed of light (i.e. the space between us and them expands faster than the speed of light). Whether this phenomenon is inevitable at a local scale, especially an atomic scale, is unclear, AFAIU. See https://en.wikipedia.org/wiki/Big_Rip But inprinciple, IIUC, our own bodies are experiencing this effect right now. For now it's beyond miniscule, but slowly accelerating. This tiny local expansion and acceleration, summed over cosmic scales, is ostensibly how galaxies far enough apart can move away from each other faster than the speed of light. Fundamentally it's the continuation of the Big Bang.
It is just a mental shortcut. Of course, things cannot be moving faster than light because speed addition cannot lead to relative speeds faster than light.
What you will see is a slowing, red-shifting, darkening image of everything else as it speeds away from you asymptotically reaching speed of light until space between any two particles starts expanding so fast that no radiation can reach anything anymore and each particle forever lives in its own universe.
Of course we don't know that because we really don't understand how the universe is built and this is just our faulty formulas extrapolated way more than we can reasonably support with our experimental data.
The relevant quantity is "Boltzmann entropy", which is just the relationship between macroscopic observables and microscopic states such that entropy is lower when the former are very sensitive to the latter.
The entropy of your heart is very low: if we swap a cubic centimetre of heart containing the mitral valve for a cubic centimetre of the wall of the right atrium, you'll very quickly die. The entropy of the air around you is much higher: we can take a cubic centimetre of air from near your left nostril and swap it with a cubic centimetre of air from near your right nostril, and you probably would not notice.
We can repeat this volume-swapping process as a way of comparing entropy in various places and at various scales. For two comparable swaps, the one that causes the greater "damage" is the one with the lower entropy.
The entropy of empty space outside galaxy clusters is enormous: we can swap cubic megaparsecs of that around without making an appreciable difference to any astrophysical observables. The metric expansion of space keeps creating more such space. That's where most of the universe's entropy will be found: new empty space.
The entropy of black holes is also enormous. The "no-hair" condition means that a very short time after we throw things into them, black holes relax into a state where we cannot tell if what was thrown in had high entropy (gas, dust, other black holes) or low entropy (stars, space probes). All that's left for an outside observer of a black hole is (in order of difficulty in measuring) its position, linear and angular momentum, mass, one or more charges like the electric charge, Hawking radiation, and possibly some other radiation (all of which forms its macroscopic observables). A black hole would "hide" whether the 100kg of additional mass it recently gained was an astronaut or a blob of atomic hydrogen. A large black hole does not tell us how many black holes fell into it in the past, nor what their masses, momenta, or charges may have been. Its macroscopic observables are tremendously insensitive to its microscopic state. Black holes therefore have very high entropy.
So as black holes form, collide, and grow hierarchically, global entropy also increases over time.
> homogenous 5 degree kelvin soup
The metric expansion of space dilutes all matter. On current trends the energy density at every point will tend towards zero in the far future, and in the very far future will be dominated by radiation originating near the cosmic horizon through a process comparable to Hawking radiation.
Measuring temperature is tricky, but if we use some sort of thermometer made of ordinary matter which gets warmed up by electromagnetic radiation then gently floating in deep extragalactic space that thermometer would report about 2.7 kelvins, mostly driven by interactions with the cosmic microwave background. The reported temperature will drop over time, falling to less than 1 kelvin in about a hundred billion years and to yoctokelvins in about a trillion years. At about the trillion-years-from-now mark, on current trends, the cosmic microwave background will be colder than the horizon radiation, so there is a temperature floor for measurements of electromagnetic radiation.
(There are other things one could in principle use to generate a local temperature reading that continues to fall even further into the future).
> whatever fundamental particle it'll be that composes this soup
It's a very thin gas of very-long-wavelength photons.
This is all with the gently-floating-themometer being at rest in the comoving "cosmological" coordinates and consequently seeing the same temperature in every direction. An observer in constant linear motion against those coordinates ("boosted") will see half its sky warmer and the opposite half colder, and an ultraboosted observer could get quite toasty from the warm side of the "dipole anisotropy". Additionally, a strongly accelerated observer may see other particle species during the acceleration ("Unruh radiation").
Finally,
> And any action you take to decrease local entropy still actually increases it in total.
At cosmological scales, the impact of any sort of local structure or organization at the scale of even a large galaxy cluster ("let's keep our Virgo supercluster stuff from falling into black holes") is nearly nothing compared to the entropy generated by the metric expansion of space.
> ... increases monotonically ...
Because empty space continues to expand, and nothing from outside can enter into our Hubble volume.
Finally, sci-fi enthusiasts could think along physical lines like: what if new empty space is only apparently empty (false vacuum decay), what if we can import new matter into our Hubble volume (wormholes to elsewhere/elsewhen), or what if we can cause the universe to recollapse (dark energy as a fifth force that can drive expansion or contraction of space). Any of those might conflict with your "any action you take to decrease local entropy still actually increases it in total" without offending theoretical thermodynamicists.
