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Hi,

I was wondering, which spacetime model do you prefer for Newtonian dynamics? VI Arnold constructs it on an affine space [itex]\mathbb{A}^4[/itex] with an Euclidean space [itex]\mathbb{E}^3[/itex] defined on each time cross-section. The construction of time is somewhat cumbersome, involving defining [itex]\mathbb{R}^4[/itex] to be a translation group of the affine spacetime and then defining time as a mapping from this onto [itex]\mathbb{R}[/itex] (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 [itex]\mathbb{R}^3[/itex] on [itex]\mathbb{R}[/itex] and reference frames to be cross-sections of this fibre bundle. This formalism seems to be less tedious.

Which one do you prefer?

Molu

I was wondering, which spacetime model do you prefer for Newtonian dynamics? VI Arnold constructs it on an affine space [itex]\mathbb{A}^4[/itex] with an Euclidean space [itex]\mathbb{E}^3[/itex] defined on each time cross-section. The construction of time is somewhat cumbersome, involving defining [itex]\mathbb{R}^4[/itex] to be a translation group of the affine spacetime and then defining time as a mapping from this onto [itex]\mathbb{R}[/itex] (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 [itex]\mathbb{R}^3[/itex] on [itex]\mathbb{R}[/itex] and reference frames to be cross-sections of this fibre bundle. This formalism seems to be less tedious.

Which one do you prefer?

Molu

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