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If you mean matter, yes, it is moving (unless it is in the center of the universe).
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Can the Universe have a centre ... A centre would imply that there are other places that are not central .. eventually you would arrive at an edge.
Space is accelerating, and as it is accelerating away from every observer .. I don't think there can be a central point.
If there was, however, i don't think Einsteins theory of relativity would ever have become fact. Everything would be measured from the centre, it would be the one 'relative' point ... But Einstein said "Every observers view is as legitimate as any other."
Greg
I agree with number 3 ... I don't think there would be a method of detecting number 2 unless the energy of the particles were changing ... Number 1.... hmmm... I don't really get it ... if the whole universe was expanding, why would there be more energy available to create new particles??