Is Relativity a specific instance of Riemannian geometry?

Question

If I am a mathematician and do not anything about Special/General Relativity, then should I study Riemannian geometry to learn Relativity? Is Relativity just an instance/example of some particular Riemannian manifold? Is Relativity completely contained with Riemannian geometry or are there distinct aspects to it? Thanks

Answer 1

General relativity takes place on a Lorentzian manifold, not a Riemannian manifold: the geometry is supposed to be "locally Minkowski" rather than "locally Euclidean". As such, the (pseudo)metric tensor is an indefinite bilinear form with signature $+---$, rather than a positive definite bilinear form.

I would say that general relativity is "a specific instance" of Lorentzian geometry in the same sense that classical mechanics is "a specific instance" of Euclidean geometry: that is, I wouldn't say such a thing at all. Although maybe you mean something different by the phrase than what I imagine.

Answer 2

Is Relativity completely contained with Riemannian geometry or are there distinct aspects to it?

Definitely not: the notion of "shortest path" (geodesic) behaves radically different in these theories.

So there are some VERY distinct aspects (because of vectors of negative length etc.) in relativity. But if you are new to the subject, then it is possibly useful to learn Riemannian Geometry first, because

1. A lot of technical stuff is the same

2. It is easier, more geometric, very "visual"

3. In General Relativity one often "slices" spacetime and talks about 3-dim Riemannian Manifolds - so you have to know that stuff anyway