Editing Hyperbolic orthogonality

{{short description|Relation of space and time in relativity theory}}
[[File:Orthogonality and rotation.svg|thumb|350px|Euclidean [[orthogonality]] is preserved by rotation in the left diagram; hyperbolic orthogonality with respect to hyperbola (B) is preserved by [[hyperbolic rotation]] in the right diagram.]]

In [[geometry]], the relation of '''hyperbolic orthogonality''' between two lines separated by the asymptotes of a [[hyperbola]] is a concept used in [[special relativity]] to define simultaneous events. Two events will be simultaneous when they are on a line hyperbolically orthogonal to a particular timeline. This dependence on a certain timeline is determined by velocity, and is the basis for the [[relativity of simultaneity]]. Furthermore, keeping time and space axes hyperbolically orthogonal, as in Minkowski space, gives a constant result when measurements are taken of the speed of light.

==Geometry==
Two lines are '''hyperbolic orthogonal''' when they are [[reflection (mathematics)|reflections]] of each other over the asymptote of a given [[hyperbola]].
Two particular hyperbolas are frequently used in the plane:
{{ordered list | list-style-type = upper-alpha
| 1 = ''xy'' = 1  with ''y'' = 0 as asymptote.

When reflected in the x-axis, a line ''y'' = ''mx'' becomes ''y'' = −''mx''.

In this case the lines are hyperbolic orthogonal if their [[slope]]s are [[additive inverse]]s.

| 2 = ''x''<sup>2</sup> − ''y''<sup>2</sup> = 1  with ''y'' = ''x'' as asymptote.

For lines ''y'' = ''mx'' with &minus;1 < ''m'' < 1, when ''x'' = 1/''m'', then ''y'' = 1.

The point (1/''m'' , 1) on the line is reflected across ''y'' = ''x'' to (1, 1/''m'').

Therefore the reflected line has slope 1/m and the slopes of hyperbolic orthogonal lines are [[multiplicative inverse|reciprocal]]s of each other.}}

The relation of hyperbolic orthogonality actually applies to classes of parallel lines in the plane, where any particular line can represent the class. Thus, for a given hyperbola and asymptote ''A'', a pair of lines (''a'', ''b'') are hyperbolic orthogonal if there is a pair (''c'', ''d'') such that <math>a \rVert c ,\  b \rVert d </math>, and ''c'' is the reflection of ''d'' across ''A''.

Similar to the perpendularity of a circle radius to the [[tangent]], a radius to a hyperbola is hyperbolic orthogonal to a tangent to the hyperbola.<ref name=L&W/><ref>Bjørn Felsager (2004), [https://www.dynamicgeometry.com/Documents/advancedSketchGallery/minkowski/Minkowski_Overview.pdf Through the Looking Glass – A glimpse of Euclid’s twin geometry, the Minkowski geometry] {{Webarchive|url=https://web.archive.org/web/20110716173907/http://www.dynamicgeometry.com/documents/advancedSketchGallery/minkowski/Minkowski_Overview.pdf |date=2011-07-16 }}, ICME-10 Copenhagen; pages 6 & 7.</ref>

A [[bilinear form]] is used to describe orthogonality in analytic geometry, with two elements orthogonal when their bilinear form vanishes. In the plane of [[complex number]]s <math>z_1 =u + iv, \quad z_2 = x + iy</math>, the bilinear form is <math>xu + yv</math>, while in the plane of [[hyperbolic number]]s <math>w_1 =  u + jv,\quad w_2 =  x +jy,</math> the bilinear form is <math>xu - yv .</math>
:The vectors ''z''<sub>1</sub> and ''z''<sub>2</sub> in the complex number plane, and ''w''<sub>1</sub> and ''w''<sub>2</sub> in the hyperbolic number plane are said to be respectively ''Euclidean orthogonal'' or ''hyperbolic orthogonal'' if their respective inner products [bilinear forms] are zero.<ref>Sobczyk, G.(1995) [https://garretstar.com/secciones/publications/docs/HYP2.PDF Hyperbolic Number Plane], also published in ''College Mathematics Journal'' 26:268–80.</ref>

The bilinear form may be computed as the real part of the complex product of one number with the conjugate of the other. Then
:<math>z_1 z_2^* + z_1^* z_2 = 0</math> entails perpendicularity in the complex plane, while
:<math>w_1 w_2^* + w_1^* w_2 = 0</math> implies the ''w'''s are hyperbolic orthogonal.

The notion of hyperbolic orthogonality arose in [[analytic geometry]] in consideration of [[conjugate diameters]] of ellipses and hyperbolas.<ref>Barry Spain (1957) [http://catalog.hathitrust.org/Record/000660610 Analytical Conics], ellipse §33, page 38 and hyperbola §41, page 49, from [[Hathi Trust]]</ref> If ''g'' and ''g''′ represent the slopes of the conjugate diameters, then <math>g g' = - \frac{b^2}{a^2}</math> in the case of an ellipse and <math>g g' =  \frac{b^2}{a^2}</math> in the case of a hyperbola. When ''a'' = ''b'' the ellipse is a circle and the conjugate diameters are perpendicular while the hyperbola is rectangular and the conjugate diameters are hyperbolic-orthogonal.

In the terminology of [[projective geometry]], the operation of taking the hyperbolic orthogonal line is an [[involution (mathematics)|involution]]. Suppose the slope of a vertical line is denoted ∞ so that all lines have a slope in the [[projectively extended real line]]. Then whichever hyperbola (A) or (B) is used, the operation is an example of a [[involution (mathematics)#Projective geometry|hyperbolic involution]] where the asymptote is invariant. Hyperbolically orthogonal lines lie in different sectors of the plane, determined by the asymptotes of the hyperbola, thus the relation of hyperbolic orthogonality is a [[heterogeneous relation]] on sets of lines in the plane.

==Constant light speed==
[[File:Minkowski-Diagramm_-_asymmetrisch.png|thumb|right|200px|''A'' represents an event connected by light to the origin. The hyperbolically-orthogonal blue axes have coordinates that measure light speed as the same ratio as the rectangular coordinates of the rest frame.]] 
One of the premises of relativity is that the [[speed of light]] does not depend on the [[inertial frame of reference]] in which the measurements are done. This premise has been associated with null results in the [[Michaelson–Morley experiment]]. As long as space and time axes are hyperbolically orthogonal, the measurement of the speed of light will give the same result. The seeming paradox of light speed invariance with respect to moving observers is resolved in special relativity by this feature of [[Minkowski space]].

==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>

==References==
{{Reflist}}

* [[G. D. Birkhoff]] (1923) ''Relativity and Modern Physics'', pages 62,3, [[Harvard University Press]].
* Francesco Catoni, Dino Boccaletti, & Roberto Cannata (2008) ''Mathematics of Minkowski Space'', [[Birkhäuser Verlag]], Basel. See page 38, Pseudo-orthogonality.
* [[Robert Goldblatt]] (1987) ''Orthogonality and Spacetime Geometry'', chapter 1: A Trip on Einstein's Train, Universitext Springer-Verlag {{ISBN|0-387-96519-X}} {{mr|id=0888161}}
* {{cite book|title=Gravitation|url=https://archive.org/details/gravitation00misn_003|url-access=limited|author1=J.A. Wheeler |author2=C. Misner |author3=K.S. Thorne |publisher=W.H. Freeman & Co|page=[https://archive.org/details/gravitation00misn_003/page/n82 58]|year=1973|isbn=0-7167-0344-0}}
{{Relativity}}

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