Can universe be a closed manifold?
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In mathematics, there are definitions for a closed manifold (i.e., compactwithout boundary) and open manifold(i.e., one that is not compact and without boundary). A "closed universe" is necessarily a closed manifold. An "open universe" can be either a closedor open manifold.
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I had a question at MSE which gave a rise to another question.
Maxwell equations can be written in form
d⋆F=J
d⋆F=J
Then by Stokes theorem we have
∫UJ=∫Ud⋆F=∫∂U⋆F.
∫UJ=∫Ud⋆F=∫∂U⋆F.
If UU(universe) is closed then ∂U=∅∂U=∅ is empty. So ∫UJ=0∫UJ=0.
What does says about space time. Is it possible that ∫UJ=0∫UJ=0? Or is it proof that universe cannot be closed?
Maxwell equations can be written in form
d⋆F=J
d⋆F=J
Then by Stokes theorem we have
∫UJ=∫Ud⋆F=∫∂U⋆F.
∫UJ=∫Ud⋆F=∫∂U⋆F.
If UU(universe) is closed then ∂U=∅∂U=∅ is empty. So ∫UJ=0∫UJ=0.
What does says about space time. Is it possible that ∫UJ=0∫UJ=0? Or is it proof that universe cannot be closed?
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