Hyperbolic orthogonality

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.

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

The bilinear form may be computed as the real part of the complex product of one number with the conjugate of the other.

Then The notion of hyperbolic orthogonality arose in analytic geometry in consideration of conjugate diameters of ellipses and hyperbolas.

In the terminology of projective geometry, the operation of taking the hyperbolic orthogonal line is an involution.

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.

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.

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.

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.

[5] Two vectors (x1, y1, z1, t1) and (x2, y2, z2, t2) are normal (meaning hyperbolic orthogonal) when When c = 1 and the ys and zs are zero, x1 ≠ 0, t2 ≠ 0, then

The directions indicated by conjugate diameters are taken for space and time axes in relativity.

"[6] 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.

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.
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.