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If an observer were to rotate around a point at near light speeds, what sort of length contraction would he observe the universe undergo?

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The formula for length contraction involves velocity only: $$L' = L\sqrt{1-v^2/c^2}.$$ Note that acceleration does not contribute. So the answer to your question is obtained by simply taking the instantaneous velocity.

As the object goes around the circle, the length contraction changes according to the change in direction of the velocity.

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