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Our universe has a finite size. It is often called the "radius of the universe", or "distance of the cosmic horizon".

If we would fly with relativistic speed at the position of our Earth, would this finite size Lorentz-contract? Or would it stay the same?

In simple words: Does the universe Lorentz-contract?

Would be observe the same size for the universe whatever our speed with respect to the reference frame defined by the average mass in the universe?

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We only talk about the Lorentz contraction in situations that may be described by special relativity and where the object to be Lorentz-contracted has a uniform well-defined speed $v$. Both of these conditions are violated in the case of the "whole Universe". We need to use general relativity to describe cosmology – changes of the size or speed of the whole Universe and/or many distant galaxies simultaneously – so what happens with them should be described by the full concepts of general relativity and "Lorentz contraction" isn't something that may be applied to the whole Universe. Moreover, different galaxies have a different $v$ so one can't attribute them a common Lorentz factor, either.

In general relativity, which is needed to describe cosmology, there are no inertial systems. There are more general "coordinate systems" and to describe which one you chose, you have to specify more than you did in the last sentence. In particular, in the FRW coordinates most sensibly connected with the "mass in the Universe", the coordinate distance between two points (two galaxies) remains constant in time despite the fact that their proper distance is increasing as the Universe is expanding. But even if one considers various changes of distances and/or contraction, it shouldn't be called Lorentz contraction because the speed of receding galaxies away from us isn't really an ordinary special relativistic speed; it is caused by the expansion of the whole Universe.

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Taylor and Wheeler in their book "Spacetime Physics" answer YES. They explain that you can cross the whole universe in a few seconds, if your speed is close enough to that of light.

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That's a few seconds of proper time. Even if cosmological distances could be discussed in terms of Lorentz transformations (they can't), you can't analyze this purely as a length contraction. Time dilation also plays a role. – Ben Crowell Jun 1 '13 at 18:56

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