Affine space or fibre bundle: spacetime formalism in Newtonian mechanics

[loom91]

Hi,

I was wondering, which spacetime model do you prefer for Newtonian dynamics? VI Arnold constructs it on an affine space $\mathbb{A}^4$ with an Euclidean space $\mathbb{E}^3$ defined on each time cross-section. The construction of time is somewhat cumbersome, involving defining $\mathbb{R}^4$ to be a translation group of the affine spacetime and then defining time as a mapping from this onto $\mathbb{R}$ (why time is also not an affine rather than vector space, I don't know).

I found Roger Penrose's construction to far more natural, where he defines Newtonian spacetime to be the fibre bundle $\mathbb{R}^3$ on $\mathbb{R}$ and reference frames to be cross-sections of this fibre bundle. This formalism seems to be less tedious.

Which one do you prefer?

Molu

 

[loom91]

Any comments?

 

[genneth]

Why does it matter? Not meaning to be rude, just actually intrigued. Is there an application where such a formal definition is necessary?

 

[loom91]

Why does it matter? Not meaning to be rude, just actually intrigued. Is there an application where such a formal definition is necessary?

Well, if someone asked you what the spacetime structure was in classical mechanics, how would you answer?

Molu

 

[loom91]

Also, physics does not exist to help the engineers.

Molu

 

[genneth]

Well, if someone asked you what the spacetime structure was in classical mechanics, how would you answer?

Molu

I'd describe the uses of space and time in classical mechanics. However, that's not the same as giving an explicit differential geometry construction. I'm fairly certain that spacetime does not really care about what brand of differential geometry we use to describe it -- only we do. And we should pick whatever is useful to the problems we are solving. Frankly, I almost never consider the problem of an actual construction for a spacetime when I'm doing classical mechanics -- it's all in the background.

 

[Hurkyl]

Why does it matter? Not meaning to be rude, just actually intrigued. Is there an application where such a formal definition is necessary?

I think the point is that physicists studying Newtonian dynamics already think this way (or something similar) -- so there is good reason to try to explicitly capture this mode of thought.

(IMO this sort of thing is essential to bridging the communication gap between physicists and mathematicians)

Physicists will often speak of the position r of a particle. But, of course, they really mean that they want to express position as a function r(t) of time, except not really because they don't think of it that way.

Well, time can be represented by the Euclidean line, which I will denote as T. Let E denote Euclidean 3-space. Let X be the trivial E-bundle on T.

If we take the right perspective (the topos of sheaves on T if you really care), then in this context, X is the right notion of Euclidean 3-space, and a section of X is the right (global) notion of a point. If U is an open subset of T, then a section of X restricted to U is the right local notion of a point -- e.g. the position of a particle that only exists during U.



1: It turns out that in this context, there is a difference between the Dedekind construction of the reals and the Cauchy construction of the reals. Here, I mean the Dedekind construction.

 

[robphy]

One way to compare two formalisms is to see which one extends more naturally to other situations... or running things backwards... which arises as the limit of another situation (like Special Relativity, or General Relativity).

One might also judge the formalism on how it may be related to operational definitions. In other words, using well-defined measurement procedures and their interpretations, how can one deduce the claimed structures [without introducing any spurious structures which lead to predictions not found in the physical world].

 

[Hurkyl]

The advantage to Arnold's construction, if I understand the underlying idea, is that he assumes less -- Penrose builds the projection onto time into his definition of spacetime, but Arnold, I think, allows the projection to be just another parameter that can be determined experimentally, and probably generalizes more easily to SR.

 

[robphy]

Why does it matter? Not meaning to be rude, just actually intrigued. Is there an application where such a formal definition is necessary?

In some introductions to foundational aspects of relativity, the authors motivate our understanding of the structure of space and time with various models of spacetime. Further structures are defined based on such models... motivated physically and geometrically.

Some examples are found in:
Ehlers (Survey of General Relativity Theory, The Nature and Structure of Spacetime)
Geroch (General Relativity from A to B)
Heller (The Science of Space-Time with Raine) and (Theoretical Foundations of Cosmology)
Penrose (Structure of Spacetime)
Trautman (Theory of Gravitation)
Yaglom (A Simple Non-Euclidean Geometry and Its Physical Basis)

The above may provide a way of thinking how to possibly generalize physics and our models of spacetime as we know them today.

 

[loom91]

The advantage to Arnold's construction, if I understand the underlying idea, is that he assumes less -- Penrose builds the projection onto time into his definition of spacetime, but Arnold, I think, allows the projection to be just another parameter that can be determined experimentally, and probably generalizes more easily to SR.

That makes sense. When going from Newton to SR, Penrose abandons the fibre bundle and switches to the Minkowski space, while Arnold would only need to replace the Galilean structure by the Minkowski structure. Anyway, it's refreshing to see time 'constructed' :-)

Molu

 

[loom91]

I think the point is that physicists studying Newtonian dynamics already think this way (or something similar) -- so there is good reason to try to explicitly capture this mode of thought.

(IMO this sort of thing is essential to bridging the communication gap between physicists and mathematicians)

Physicists will often speak of the position r of a particle. But, of course, they really mean that they want to express position as a function r(t) of time, except not really because they don't think of it that way.

Well, time can be represented by the Euclidean line, which I will denote as T. Let E denote Euclidean 3-space. Let X be the trivial E-bundle on T.

If we take the right perspective (the topos of sheaves on T if you really care), then in this context, X is the right notion of Euclidean 3-space, and a section of X is the right (global) notion of a point. If U is an open subset of T, then a section of X restricted to U is the right local notion of a point -- e.g. the position of a particle that only exists during U.



1: It turns out that in this context, there is a difference between the Dedekind construction of the reals and the Cauchy construction of the reals. Here, I mean the Dedekind construction.

I'm afraid all that went way over my head. Could you explain with less terminology?

Molu

 

[Hurkyl]

Let T denote the Euclidean line. (it represents the set of points of time)
Let E denote Eucliden 3-space.
Let X denote the trivial E-bundle over T.

If we step out of the set-theory world and step into the world over T, we have analogues of many familiar notions.

Roughly speaking, we have the following correspondences:

sets == bundles
elements of a set == sections of a bundle
the (Dedekind) real numbers == the trivial R-bundle over T
Euclidean 3-space == X


By considering the "world over T" perspective, we get closer to how (I think) people think about classical space-time.

e.g. consider the energy of a system. In the more elementary perspective, you have to treat energy is a function from time into R. But in the world over T, energy really is just a real number; the notion of a "time-varying number" makes literal sense in this perspective... but in the elementary perspective it's just a figurative notion.