Editing Hyperbolic orthogonality (section)

==Simultaneity==
Since [[Hermann Minkowski]]'s foundation for [[spacetime]] study in 1908, the concept of points in a spacetime plane being hyperbolic-orthogonal to a timeline (tangent to a [[world line]]) has been used to define '''simultaneity''' of events relative to the timeline, or [[relativity of simultaneity]]. In Minkowski's development the hyperbola of type (B) above is in use.<ref>{{Citation|author=Minkowski, Hermann|year=1909|title=[[s:de:Raum und Zeit (Minkowski)|Raum und Zeit]]|journal=Physikalische Zeitschrift|volume=10|pages=75–88}}
:*Various English translations on Wikisource: [[s:Space and Time|Space and Time]]</ref> Two vectors ({{var|x}}{{sub|1}}, {{var|y}}{{sub|1}}, {{var|z}}{{sub|1}}, {{var|t}}{{sub|1}}) and ({{var|x}}{{sub|2}}, {{var|y}}{{sub|2}}, {{var|z}}{{sub|2}}, {{var|t}}{{sub|2}}) are ''normal'' (meaning hyperbolic orthogonal) when
:<math>c^{2} \ t_1 \ t_2 - x_1 \ x_2 - y_1 \ y_2 - z_1 \ z_2 = 0.</math>
When {{var|c}} = 1 and the {{var|y}}s and {{var|z}}s are zero, {{var|x}}{{sub|1}} &ne; 0, {{var|t}}{{sub|2}} &ne; 0, then <math>\frac{c \ t_1}{x_1} = \frac{x_2}{c \ t_2}</math>.

Given a hyperbola with asymptote ''A'', its reflection in ''A'' produces the [[conjugate hyperbola]]. Any diameter of the original hyperbola is reflected to a [[conjugate diameters|conjugate diameter]]. The directions indicated by conjugate diameters are taken for space and time axes in relativity.
As [[E. T. Whittaker]] wrote in 1910, "[the] hyperbola is unaltered when any pair of conjugate diameters are taken as new axes, and a new unit of length is taken proportional to the length of either of these diameters."<ref>[[E. T. Whittaker]] (1910) [[A History of the Theories of Aether and Electricity]] Dublin: [[Longmans, Green and Co.]] (see page 441)</ref> On this [[principle of relativity]], he then wrote the Lorentz transformation in the modern form using [[rapidity]].

[[Edwin Bidwell Wilson]] and [[Gilbert N. Lewis]] developed the concept within [[synthetic geometry]] in 1912. They note "in our plane no pair of perpendicular [hyperbolic-orthogonal] lines is better suited to serve as coordinate axes than any other pair"<ref name=L&W>Edwin B. Wilson & Gilbert N. Lewis (1912) "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics" Proceedings of the [[American Academy of Arts and Sciences]] 48:387–507, esp. 415 {{doi|10.2307/20022840}}</ref>