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I've been reading that Minkowski space is a real vector space; atleast in flat space-time. So that means that we should be able to define vector addition, scalar multiplication, etc. But since four-vectors in space-time are events, how does adding two events make sense? More importantly, what is the meaning of the resultant vector?

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    $\begingroup$ Let me ask you the same question, but for the plane. We setup Cartesian coordinates (x,y) which represents a point on the plane. Yet we add two points too! What are we doing there? $\endgroup$ – Prahar Sep 9 '16 at 16:57
  • $\begingroup$ "But since four-vectors in space-time are events" - Does spacetime position not form a four-vector? $\endgroup$ – Alfred Centauri Sep 9 '16 at 17:28
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Minkowski spacetime is an affine space not a vector space: vectors acts as group of translations on the spacetime moving events, but vectors are not events. If you arbitraily fix an origin $O$, there is exactly one vector $v(e)$ joining $O$ to every event $e$ of the spacetime. In this sense there is a one-to-one correspondence between events and four-vectors, but it strongly depends on the choice of the preferred origin.

If you also fix a pseudo-orthogonal basis, the components of $v(e)$ with respect to that basis are just the Minkowskian coordinates of $e$.

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