# Affine space for Minkowski space time

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?

• 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. Sep 11, 2018 at 13:43
• 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. Sep 11, 2018 at 13:56