# Why model spacetime as a semi-Riemannian manifold?

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.

• Any general relativity textbook will provide all the arguments necessary. – Javier Mar 18 '18 at 12:44

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.