I Physical meaning of a spacelike geodesic

[vanhees71]

That's also true, if you interpret the tangent space at the point under consideration as an affine point space. This affine space then indeed has the full Poincare symmetry, but for general GR spacetimes you usually don't have a translation symmetry.

 

[cianfa72]

That's also true, if you interpret the tangent space at the point under consideration as an affine point space.

As discussed in a recent thread, to endow the underlying set (i.e. the set of spacetime points) of an affine structure the set of tangent spaces at each point must met let me say a 'consistency condition'. Basically vectors belonging to different tangent spaces have to be understood/treated as elements of the same vector space (i.e. the translation vector space) in a such way that axioms of affine space are fullfilled.

 

[vanhees71]

We were talking about one tangent space at one fixed point in spacetime. An affine space is flat, but general-relativistic spacetime at presence of gravitational fields is not.