Editing Triangle inequality (section)

==Reversal in Minkowski space==

The [[Minkowski space]] metric <math> \eta_{\mu \nu} </math> is not positive-definite, which means that <math> \|u\|^2 = \eta_{\mu \nu} u^\mu u^\nu</math> can have either sign or vanish, even if the vector {{mvar|u}} is non-zero. Moreover, if {{mvar|u}} and {{mvar|v}} are both timelike vectors lying in the future light cone, the triangle inequality is reversed:

: <math> \|u+v\| \geq \|u\| + \|v\|. </math>

A physical example of this inequality is the [[twin paradox]] in [[special relativity]]. The same reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in <math>n + 1</math> dimensions for any <math>n \geq 1</math>.  If the plane defined by <math>u</math> and <math>v</math> is space-like (and therefore a Euclidean subspace) then the usual triangle inequality holds.