Well, when you study special relativity in a "covariant way" (using the formalism of Lorentzian geometry and so on...) you realize that the structure of the spacetime isn't a mere vector space, but rather, that of affine space $[1]$. But, a small conceptual gap is annoying me.

General Relativity introduces topological manifolds as the most fundamental stage and then you elevate them into smooth ones. In fact, we can say (due to the very structure of a manifold) that they are a huge collection of $\mathbb{R}^{n}$ vector spaces.

Concerning the context of physics and the whole realization and physical meaning of a affine space: can I say that a general manifold is locally an affine space?

$$* * *$$

$[1]$ M.Kriele. Spacetime.

  • $\begingroup$ If Euclidean space carries affine space structure surely you could treat a manifold as locally an affine space, but that seems like a slippery slope. The definition of a manifold isn't ambiguous. $\endgroup$ – Charlie Mar 1 at 14:46

The local space is more properly known as a tangent space. The tangent space is a vector space with each event in the manifold having its own tangent space and serving as the origin of its own tangent space. Of course, mathematically you can treat any vector space as an affine space since a vector space satisfies all of the axioms of an affine space, but generally it is not advisable to do so in the tangent spaces of GR. The reason it is not advisable to treat the local tangent space as an affine space is that it is important to remember the origin of the tangent space, i.e. the specific event of the manifold that the tangent space pertains to. Tangent spaces at different events are distinct spaces and that should not be forgotten.

In flat spacetime, the manifold itself is globally an affine space. In curved spacetime the generalization of the flat affine space is the manifold, not the local tangent spaces. Of course, a curved manifold does not satisfy all of the axioms of an affine space, so it is a more general structure. So the affine space is the manifold itself, but only for the special case of flat spacetime (no tidal gravity).

  • $\begingroup$ Could you explain more why is it generally... not advisable to do so in the tangent spaces of GR ? $\endgroup$ – Daddy Kropotkin Mar 1 at 15:19
  • $\begingroup$ So, we can define a particular structure for special relativity, but in order to generalize it we cannot become attached to affine spaces and therefore introduce manifolds as the underlying structure. $\endgroup$ – M.N.Raia Mar 1 at 15:24
  • $\begingroup$ Being a vector space is stronger than being an affine space in the sense that the former has an additional structure, namely the zero element. So why don't we exploit it? In fact, viewing the tangent spaces as vector spaces enable us to employ linear algebra to study the local nature of geometric structures. $\endgroup$ – Hyeongmuk LIM Mar 1 at 15:25
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    $\begingroup$ My comment was intended to be a reply to @DaddyKropotkin's question. Thank you for the clarification, Dale. $\endgroup$ – Hyeongmuk LIM Mar 1 at 15:36
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    $\begingroup$ Thanks to both of you +1 $\endgroup$ – Daddy Kropotkin Mar 1 at 20:31

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