How might one create a room in which time flows faster?
And what would be needed to expand this effect, scaling up to the size of a city, or maybe a country?
I've seen this question, and it seems that while one could do it with gravity, the effectiveness and side-effects would be rather undesirable. Please note that while I'm looking for a "plausible" mean to achieve this, I'd like to discard unfeasibility due to the lack of energy or material resources.
So, let's just assume we have an infinite stash of all materials, and an infinite amount of energy. But we still need to get humans inside and they must be able to live exactly as we do: same gravity, same pace and all: it would just be faster from the view of outsiders.
In short terms, the goal is to create a room, or even a city, where time flows faster, at the rate of at least 1 year per second. So, outside this place, I could see what would happen in hundreds or thousands of years.
Another restriction would be that this room cannot be too far from the outside world. I'm talking about maybe some kilometers at most.
Are there any "plausible" ways for this to happen? Please forgive me if this question threads on the lines of nuisance, I'm no physicist.
Also, if there aren't any known ways to distort time like this, are there any unproven theories I could use?
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So time dilation isn't the way to go, because that just makes time slow down, not speed up.
I think the only way you're going to be able to do this is by getting into String Theory (M Theory?) a little bit. Which is tricky, but I believe that it does allow for pocket dimensions where time doesn't flow the same way that it does in our perception of reality.
Maybe check with the people over in Physics?
At this point you're getting out beyond anything we can test, so you may have to go to the sci-fi equivalent of magic...
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Here's one method that doesn't rely on either gravitational time dilation or simulation:
If you look at the basic equations of quantum mechanics, you notice that if you scale all masses by a common factor, you'll get essentially the same physics, just at different length and time scales. In particular, when all masses are multiplied by a factor $\lambda$, all lengths get scaled by a factor $1/\lambda$ and all processes go faster by a factor $\lambda$, The latter being exactly what you want. (Caveat: I don't know enough about nuclear physics/QCD to say for sure that this is the case there, too, but I suspect it is; indeed I seem to remember having read somewhere that the mass scaling is a requirement for the standard renormalization procedures "” the way how infinities are avoided in quantum field theories "” to work.)
Now the question is of course how you could scale all masses. Well, according to Higgs theory, the fundamental particles get their rest masses from interaction with the Higgs field (all other mass contributions are ultimately just kinetic/potential energies which automatically scale the right way). So one might think that if you find a way to manipulate the Higgs field, you should be able to scale masses that way.
How to apply it to your specific requirements
A speed-up to one second per year would be a factor of about $3\cdot 10^8$, that is, you'd have to scale masses up by that amount. This would also scale down sizes by the same amount, so a human would shrink to about half a dozen nanometers. That is, you could easily put a complete city into a matchbox.
However there's a caveat: Your human would now weight about $10^{10}\,\rm kg$. I guess that would locally give quite strong gravitational fields (gravitation does not follow the scaling law; correspondingly quantizing it the standard way does not work). So quite possibly you'd not be able to achieve that large acceleration without adverse gravitational effects, unless you're also able to locally manipulate the gravitational constant so that also the gravitation plays nice with it. Note also that a strong gravitational field would add a slowdown counteracting your speedup; this is another reason to decrease it.
So in summary: You'd need to be able to manipulate the Higgs field to get the speedup (combined with a shrinking), and additionally to locally manipulate the gravitational constant (to avoid the gravitational effects of the increased masses). But if you can do both, you'll get a sped-up city in a matchbox.
Further caveats to the speed-up
The speed-up inevitably means that any radiation coming from outside will be seen as very low frequency in your town. That means you'll have to ensure sufficient thermal insulation in order to prevent your town from freezing. Similarly, any radiation coming from your town will be shifted up in energy; with one second per year, even visible light will turn into gamma radiation. So to protect yourself, you better surround that city with appropriate radiation shielding.
If you settle with less extreme speed-ups (say, just one day per second), you'll reduce the problems with the radiation from the city; however it will still be very hot when viewed from outside; better make that matchbox out of fireproof material.
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