Spacetime in Newtonian kinematics

[jcsd]

I've just had a silly thought. Newtonian spacetime wrt to postion space is basically a vector bundle.

If you want to assign each postion vector to a point in on a Riemannian/Pseudo-Riemannian manifold you have to take the following two things into account:

a) Each vector in any given vector space in the bundle can be idenitifeid another vector in any given vector space in the bundle so you need a geometry that is flat

b) The metric should reduce to the metrics on postion space and time when appropiate.

This only allows E⁴ or M⁴.Though in Newtonian kinematics there is no way to make sense of the extra structure, I find it interesting that Newtonian kinematics can limit your choice to two spaces, one of which is the correct space to use for special relativity (even if in general relativity a) should be disregarded and b) is not really ture).

 

[jcsd]

This thread is partly the product of a little alcohol, but does anyone agree with me that there are only two (psuedo) Riemannian manifolds were you can get Newtonian kinetmatics in the limit of when a) and b) (which seem sensible proscriptions in the context of Newtonian kienmatics)?