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I'm studying Minkowski space time (M4) and they say it's a 4 dimensions real affine space. M4 is an affine space so there is a non-empty set A, a 4 dimension real vector space V, and there is a function f: AxA-->V (with the proper property). The elements of A are the events.

The doubt: Have I to think that I’m in certain inertial frame of reference with a certain coordinate system so that every element of A is identify with 4 numbers? Or have I to think of A as a very abstract set of points and I set the coordinates of A after setting the basis for V?

I'm asking this because I have read that the affine coordinate system comes after the definition of the affine space, but in this case, I don't understand how it's possible to identify an event before setting a coordinate system. Are coordinate system and affine coordinate system two different things?

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  • $\begingroup$ I think you might have made a mistake when you wrote that there should be a function f: A -> V. Normally, affine spaces are equipped with an action of a vector space giving translations in the affine space, but such an action would be a map g: A + V -> A: a point in A can be translated by a vector in V to give another point in A. $\endgroup$ – Stijn B. Sep 11 '18 at 13:43
  • $\begingroup$ Actually it was a typing error, I should have written f:AxA-->V with proper property that are: for every element P of A and for every vector v of V it exist one and only one Q such that f(P,Q)=v, for every P, Q, S elements of A it is true that f(P,Q)+f(Q,S)=f(P,S). This definition is equivalent to yours. Now that we solved this misunderstandig my doubts is still there. $\endgroup$ – SimoBartz Sep 11 '18 at 13:56
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The elements of A (the events in Minkowski space) exist indepently of your choice of coordinate system on A. Similarly, one can define translations (the action by a vector) in a purely abstract fashion, without referring to any set of coordinates at all. Hence the abstract notion of an affine space (or vector space, or manifold) is more fundamental than the structure provided by a coordinate system.

Of course, it is often convenient to think about affine spaces like Minkowski space entirely in terms of coordinates, but keep in mind that there is a more abstract structure underneath. Furthermore, Minkowski space is a rather harmless kind of space, but in more complicated settings, not being careful enough in making the distinction between a space and its coordinates can lead to unwanted side-effects, which is why mathematicians often prefer coordinate-free notation in the first place.

Think of the real world around you. The universe is a kind of space as well (which we are supposed to describe as a Minkowski space), but it does not come with any canonical coordinate system at all: you are free to choose any set of points and give them the coordinates you like, as long as your coordinate system makes sense.

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  • $\begingroup$ Thanks for the answer, it's really useful. About the last part of my question, is the coordinate system (for example spherical coordinate system for the space and a 'clock' for the time) a different concept respect to the affine coordinate system? $\endgroup$ – SimoBartz Sep 11 '18 at 14:08
  • $\begingroup$ An affine coordinate system is a special case of a coordinate system, namely one that respects the affine structure of the space. This structure is not manifestly clear in spherical coordinates, for instance, so spherical coordinates do not form an affine coordinate system. $\endgroup$ – Stijn B. Sep 11 '18 at 15:51

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