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By looking at this picture:

The earth is near the center of the universe. I've read that the universe look the same no matter where the observer is located. It is the same distance everywhere.

So I understand that for general relativity the universe need to be homogeneous and isotropic, so it will look the same no matter where I am.

But what if I'm on one of the planets near the right or left of the image, then if I draw the same picture of the universe, but from my perspective, then I would be also located in the center? If that's not the case (I'm actually near an edge), then part of my sky would be completely dark, and all the sky that way won't be isotropic?

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Earth is by definition the center of the observable universe. The observable universe is the region of space such that light from the beginning of the universe can reach the observer, i.e. Earth. – Alex Becker Jan 14 '13 at 19:30
up vote 7 down vote accepted

The universe has no edge so to speak. It is, however, finite in age, so light can only have traveled a given distance to get to us. Call this distance $R$, the "radius" of the universe. Any observer, anywhere, will see out to a distance $R$ in all directions from their location.

Now two different observers will have different origins for their respective observable universes, and so will see slightly (or vastly) different patches of the "full universe." (Be careful when talking about things outside your observable patch by the way - it is very easy to end up talking about impossible scenarios that produce nonsensical results.) This can happen even if the universe is closed (read: finite), so long as its size is bigger than something like $R$.

So no, no one is at the "edge" of the universe.

By the way, general relativity in no way requires homogeneity and isotropy. These are simply assumptions cosmologists make in order to take an utterly intractable problem (evolving the whole universe) and make it absurdly simple (see the FRW metric, which, although it may look complicated at first, is pretty much the must trivial thing you can do with general relativity). The homogeneous/isotropic assumptions, by the way, turn out to be justified on cosmological scales, though this was discovered only after the early days of GR-based cosmology, once we had very deep galaxy surveys.

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