Editing World line (section)

===Simultaneous hyperplane===
Since a world line <math> w(\tau) \isin R^4</math> determines a velocity 4-vector <math> v = \frac {dw}{d\tau}</math> that is time-like, the Minkowski form <math> \eta(v,x)</math> determines a linear function <math> R^4 \rarr R</math> by <math>  x \mapsto \eta( v , x ) .</math> Let ''N'' be the [[kernel (linear algebra)|null space]] of this linear functional. Then ''N'' is called the '''simultaneous hyperplane''' with respect to ''v''. The [[relativity of simultaneity]] is a statement that ''N'' depends on ''v''. Indeed, ''N'' is the [[orthogonal complement]] of ''v'' with respect to &eta;. 
When two world lines ''u'' and ''w'' are related by <math> \frac {du}{d\tau} = \frac {dw}{d\tau}, </math> then they share the same simultaneous hyperplane. This hyperplane exists mathematically, but physical relations in relativity involve the movement of information by light. For instance, the traditional electro-static force described by [[Coulomb's law]] may be pictured in a simultaneous hyperplane, but relativistic relations of charge and force involve [[retarded potential]]s.