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The Universe, as far as we know, is all there is. Physics provides no real way for us to ever look anywhere but within it. We are cocooned from any possible communication from other dimensions or alternate realities. Other dimensions may exist and there may even be a boundary to the cosmos, but so far, we've never observed anything remotely resembling one.
On very large scales, the universe is actually a pretty simple place: it has been around for a finite amount of time, roughly 13.5 billion years; it looks pretty much the same everywhere - galaxies are almost evenly distributed across the cosmos - everywhere we look we see about the same number of galaxies;
... it is also big. Very, very big. And its getting bigger. The spacetime between all 100 billion galaxies in the universe is increasing. Like a roiling, seething froth, new, empty spacetime is being created as the universe ages, increasing the distances between the galaxies, pushing them apart.
Since the universe is expanding, it is a natural question to ask, what is it expanding into?
When we peer deep into the cosmos, we cannot see a boundary. So far, we have uncovered no evidence that a boundary exists. Space may extend to infinity or it may not, but in Einstein's universe things can be curved, and if things can be curved, they can be curved in on themselves, twisting and bending the shape of the universe into virutally anything imaginable. General relativity makes it possible to live in an infinite universe with no boundary at all.
Because of general relativity, spacetime is not a static entity. It is a dynamic and ever-changing fabric within which the locations of all galaxies are woven. Galaxies are not themselves moving very much, but they appear to move to us, because of new, cosmic real estate continually injected, increasing their distance from us.
It is this creation of new spacetime, and the rate at which it is being created, which determines how fast a galaxy appears to be moving away from us.
So what is the universe expanding into? When new spacetime is created, into what do the edges go? The answer depends on whether or not there are edges.
If we live in an infinite universe, then the answer has to be nothing. Adding more fabric to infinity doesn't make more infinity. An infinite universe would have no edges that expand and the question is meaningless. In such a universe, there would be no 'outside'.
On the other hand, if the universe is finite, with a boundary that we have not yet discovered, then the answer may be that we are expanding into something. If that is true however, then the boundary is so far away that we cannot see it and it can therefore never, ever affect us. We have already seen photons that have been travelling since the universe was only 500 million years old. Anything much further away lies beyond our detection forever. Given that our universe is expanding - if we cannot see the boundary now, this expansion guarantees we never will - it will forevermore get further and further away. it will always lie beyond our detection.
Only one hundred years ago, we had no idea there were other galaxies besides our own. It was thought that humanity and the galaxy we inhabit was an island, adrift in a universe of 100 billion stars. We now know that our universe is a vast, dynamic cauldron of activity: home to 100 billion galaxies, all racing away within a boiling ocean of spacetime. While we may yet find our universe is just an island, we have discovered it is much larger than we ever thought.
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