In the book Geometrical Methods of mathematical physics, in the section 2.10, talking about fiber bundles, gives the example of the Newtonian physics fiber-bundle structure:

The view of spacetime taken by Newtonian physics has a natural fiber-bundle structure. [...] One effect of Einstein's relativity was to destroy this bundle structure and to substitute something else, a metric structure.

I understand that I cannot describe a relativistic space in terms of fiber-bundles because of I cannot split the $\mathbb{R}^{4}$ manifold into $\mathbb{R}^{3}$ and $\mathbb{T}$. How can I show it mathematically?


1 Answer 1


By $\mathbb{T}$ I assume you are referring to an $\mathbb{R}$ factor which has been picked out as corresponding to time.

In fact, you will note that $\mathbb{R}^4$ is indeed an $\mathbb{R}^3$ bundle over $\mathbb{R}$ since we can very easily write down a projection to make this happen (a simple $\pi:\langle t,x,y,z\rangle\mapsto t$ would do the trick). It's also an $\mathbb{R}$ bundle over $\mathbb{R}^3$ which we can realize by the projection $\pi:\langle t,x,y,z\rangle\mapsto\langle x,y,z\rangle$. It's also an $\mathbb{R}^2$ bundle over $\mathbb{R}^2$. The space $\mathbb{R}^4$ is so simple you could probably come up with an arbitrary number of ways to split the space into a bundle structure.

These are statements that are simply true about $\mathbb{R}^4$, no physics needed. I point this out because there is more to the story. It's the $\mathbb{R}^3$ bundle over $\mathbb{R}$ structure which is important in Galilean mechanics (which is presumably what your reference means when they refer to "Newtonian" physics), and this is what I presume your reference means when they mention a bundle structure.

The key thing to understand is spacetime symmetry. The Galilean group of spacetime transformations, which is what defines Galilean mechanics, respect the $\mathbb{R}^3$ over $\mathbb{R}$ structure I described above. Specifically, Galilean transformations are fiber-preserving diffeomorphisms on $\mathbb{R}\times\mathbb{R^3}$. The difference once you go to special relativity is that boosts are able to mix together space and time. As such, the group of spacetime symmetries we want to talk about and work with will no longer be fiber-preserving and hence there's really no good reason to think about the bundle structure anymore since it's not invariant under the collection of transformations we want to talk about.

The bundle structure is still there in the sense that this is simply a property of $\mathbb{R}^4$, it just no longer has any utility. And frankly, I can't think, off the top of my head, of anything particularly useful that the bundle structure in Galilean relativity allows us to say that the statement "Galilean symmetry" doesn't already imply.


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