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Can spacetime be non-orientable?

Apart from the constraints put on the topology of spacetime by QFT (Parity For example), if the global topology of a universe is that of a Mobius band and at one point there is a gravitational field pointing towards a star and a passenger travels along the band and returns to the same point. Then he will find the gravitational field pointing away from the star causing an anti-gravity effect. Does this mean that space-time must be orientable?

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med_student
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I think Can spacetime be non-orientable? is a duplicate as qmechanic says.

One of the points not made in the answers to the previous question is that General Relativity is a local theory and does not dictate the topology of the universe at all. Whether the universe is simply connected, multiply connected or indeed non-orientable will determine what solutions are possible, but it's down to experiment to determine which solution we live in and therefore what the topology of the universe is.

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John Rennie
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