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Just a short question: Is it valid to say that some event is space/time-like or is this term reserved for the distance between two events (the space-time interval) and four vectors?

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    $\begingroup$ an event is a point in an affine space. How would you define its length? $\endgroup$ – AccidentalFourierTransform Dec 5 '16 at 14:37
  • $\begingroup$ Ok, that's what I wanted to hear. On the other hand, a single event can be represented by a four-position, so that its lenght is the distance from the origin or am I wrong? $\endgroup$ – OD IUM Dec 5 '16 at 14:40
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    $\begingroup$ it is still the distance between two events: the event you are trying to measure and whatever event defines the origin $\endgroup$ – AccidentalFourierTransform Dec 5 '16 at 14:42
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The terms spacelike, timelike, and lightlike refer to space time intervals only. They are used when discussing the sign of the distance between two different events. Mathematically, this comes from the sign of the metric tensor acting on two different four-vectors, $g(\mathbf{x},\mathbf{y}) = g_{ij}x^i y^j$. The metric tensor applied to a single four vector (event) will not return a number, but rather a covector. Hence, it does not make sense to talk about the sign of the metric tensor applied to a single vector.

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