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I'd like to collect arguments for the use of a semi-Riemannian manifold as a mathematical model for spacetime.
I think this is closely linked to the equivalence principle, but I can't shape this into a well formulated argument.
Other approaches are of course also welcome.

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    $\begingroup$ Any general relativity textbook will provide all the arguments necessary. $\endgroup$ – Javier Mar 18 '18 at 12:44
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The manifolds of general relativity are actually of a more restricted class---they are Lorentzian manifolds, meaning that they have a signature (1,n-1) for an n-dimensional space. This requirement is indeed due to the equivalence principle, which requires that free-falling observers exist in local inertial frames, that is, that special relativity applies. As a local frame, it coincides with the tangent space of the full manifold; and, we know the geometry of special relativity is Lorentzian. Therefore, all manifolds in general relativity must be locally diffeomorphic to Minkowski space.

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General relativity requires a manifold with the following properties: it must be 4 dimensional, it must have a time orientation and it must be connected. The requirement of a time orientation is so that any chosen manifold will be locally diffeomorphic to $\mathbb{R}^{1,3}$, i.e. that locally we recover special relativity.

Riemannian geometries don't fit the bill, since the metric tensor is positive definite: in other words, the line element $ds^2$ is always positive. However, to coincide with special relativity and satisfy the equivalence principle, we know that we need these line elements to be positive, zero or negative - corresponding to spacelike, lightlike and timelike. What does fit the bill are Semi-Riemannian geometries.

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    $\begingroup$ I don't think being simply connected is a constraint for general relativity. Also even non-time orientable manifolds are locally Minkowski space. $\endgroup$ – Slereah Mar 18 '18 at 13:43
  • $\begingroup$ Hrmm you're probably right that it might not be an initial consideration, but I'm not sure... I've edited it to say connected, but this is worth a read: arxiv.org/pdf/gr-qc/9305017v2.pdf $\endgroup$ – Akoben Mar 18 '18 at 14:02
  • $\begingroup$ Topological censorship is about the topology we can probe, not its actual topology (and it also relies on the ANEC being true) $\endgroup$ – Slereah Mar 18 '18 at 14:03

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