In geometry, given a pair of conjugate hyperbolas, two conjugate diameters are hyperbolically orthogonal. This relationship of diameters was described by Apollonius of Perga and has been modernized using analytic geometry. Hyperbolically orthogonal lines appear in special relativity as temporal and spatial directions that show the relativity of simultaneity. 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 reflections of each other over the asymptote of a given hyperbola. Two particular hyperbolas are frequently used in the plane:
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 a ‖ c , b ‖ d {\displaystyle a\rVert c,\ b\rVert d} , 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. 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 numbers z 1 = u + i v , z 2 = x + i y {\displaystyle z_{1}=u+iv,\quad z_{2}=x+iy} , the bilinear form is x u + y v {\displaystyle xu+yv} , while in the plane of hyperbolic numbers w 1 = u + j v , w 2 = x + j y , {\displaystyle w_{1}=u+jv,\quad w_{2}=x+jy,} the bilinear form is x u − y v . {\displaystyle xu-yv.}
The vectors z1 and z2 in the complex number plane, and w1 and w2 in the hyperbolic number plane are said to be respectively Euclidean orthogonal or hyperbolic orthogonal if their respective inner products [bilinear forms] are zero. The bilinear form may be computed as the real part of the complex product of one number with the conjugate of the other. Then
z 1 z 2 ∗ + z 1 ∗ z 2 = 0 {\displaystyle z_{1}z_{2}^{*}+z_{1}^{*}z_{2}=0} entails perpendicularity in the complex plane, while
w 1 w 2 ∗ + w 1 ∗ w 2 = 0 {\displaystyle w_{1}w_{2}^{*}+w_{1}^{*}w_{2}=0} 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. If g and g′ represent the slopes of the conjugate diameters, then g g ′ = − b 2 a 2 {\displaystyle gg'=-{\frac {b^{2}}{a^{2}}}} in the case of an ellipse and g g ′ = b 2 a 2 {\displaystyle gg'={\frac {b^{2}}{a^{2}}}} 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. 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 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
The second postulate of special relativity is that the speed of light does not depend on the inertial frame of reference in which the measurements are done. This postulate 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.
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