Non-Euclidean Geometry - Kinematic Geometries

Kinematic Geometries

Hyperbolic geometry found an application in kinematics with the cosmology introduced by Herman Minkowski in 1908. Minkowski introduced terms like worldline and proper time into mathematical physics. He realized that the submanifold, of events one moment of proper time into the future, could be considered a hyperbolic space of three dimensions. Already in the 1890s Alexander Macfarlane was charting this submanifold through his Algebra of Physics and hyperbolic quaternions, though Macfarlane didn’t use cosmological language as Minkowski did in 1908. The relevant structure is now called the hyperboloid model of hyperbolic geometry.

The non-Euclidean planar algebras support kinematic geometries in the plane. For instance, the split-complex number z = eaj can represent a spacetime event one moment into the future of a frame of reference of rapidity a. Furthermore, multiplication by z amounts to a Lorentz boost mapping the frame with rapidity zero to that with rapidity a.

Kinematic study makes use of the dual numbers to represent the classical description of motion in absolute time and space: The equations are equivalent to a shear mapping in linear algebra:

With dual numbers the mapping is

Another view of special relativity as a non-Euclidean geometry was advanced by E. B. Wilson and Gilbert Lewis in Proceedings of the American Academy of Arts and Sciences in 1912. They revamped the analytic geometry implicit in the split-complex number algebra into synthetic geometry of premises and deductions.