Convergence in Galilean space-time

[Wox]

To talk about differentiable vector fields in Galilean space-time, one needs to define convergence. Galilean space-time is an affine space and its associated vector space is a real 4-dimensional vector space which has a 3-dimensional subspace isomorphic to Euclidean vector space.

There is no norm defined in Galilean space-time so how does one define convergence in this space?

 

[Fredrik]

Galilean spacetime is really just the set $\mathbb R^4$ with a set of preferred coordinate systems, so you would need a good reason to not use the standard topology on $\mathbb R^4$.

To even talk about "differentiable vector fields" on Galilean spacetime, it must be defined as a manifold. A manifold is a Hausdorff and 2nd countable topological space with other stuff defined on it. In this case, I see no reason to not take the 2nd countable Hausdorff space to be $\mathbb R^4$ with the standard topology.

 

[lugita15]

Fredrik, it is possible, and in some way quite fruitful for the purposes of generalization, to treat Galilean spacetime not as ℝ⁴, but as an affine structure which better embodies the principle of relativity. See the attached excerpt from Roger Penrose's Road to Reality. He associates Aristotelian physics with ℝ⁴, but he feels Galilean physics deserve something greater. (Then after the part I excerpted, he goes even grander, developing what he calls the "Newtonian" spacetime, which embodies the principle of equivalence. It's really an amazing book.)

 

[Fredrik]

Yes, I skimmed that section a few years ago, so I'm familiar with the idea. I'm also familiar with the basics of fiber bundles. But I have to admit that I don't really see the point. It just seems to make things more complicated.

 

[lugita15]

Yes, I skimmed that section a few years ago, so I'm familiar with the idea. I'm also familiar with the basics of fiber bundles. But I have to admit that I don't really see the point. It just seems to make things more complicated.

The point is to put the theory in such a form that you just need to make a trivial change in order to get to other theories like SR and GR. It's just like how we express classical mechanics in terms of Hamiltonians and Poisson brackets, so we can easily move into quantum mechanics by just tweaking the theory slightly.

 

[Fredrik]

The point is to put the theory in such a form that you just need to make a trivial change in order to get to other theories like SR and GR. It's just like how we express classical mechanics in terms of Hamiltonians and Poisson brackets, so we can easily move into quantum mechanics by just tweaking the theory slightly.

I would agree that it makes things more similar to GR, but I would say that it makes things less similar to SR, which is the logical next step after Galilean/Newtonian theories of motion. I'm a big fan of the "nothing but relativity" approach to theories with $\mathbb R^4$ as the set of events. You make the very natural assumption that there's a group of smooth bijections on $\mathbb R^4$ that take straight lines to straight lines, and you prove that there are only two such groups: The Galilean group and the Poincaré group. (Actually I think the requirement of "smoothness" is unnecessarily strong). So to make the move to SR, all you have to do is to choose the other possible group. Then you have to do some work to generalize to arbitrary coordinate systems (basically just learn the definition of a manifold), and once you have done that, you can make the move to GR simply by saying that the metric isn't specifically the Minkowski metric, but a metric to be determined from an equation.