Lorentz Transformations Expressed in Complex and Split-Complex Form

E. J. Zampino (2010), JBIS, 63, pp.282-288

Refcode: 2010.63.282
Keywords: Lorentz Transformations, super-luminal speeds, relativity, complex numbers, split-complex numbers

Abstract:
Lorentz transformations of Special Relativity are expressed in terms of complex and split-complex variables. These representations involve a complex or split-complex conjugate such that “negative time” becomes a part of the Lorentz transformation. The Lorentz transformations can be expressed as two equations instead of the usual four. Use of split-complex numbers, results in a particularly simple transformation equation expressed in terms of an exponential function. The complex and split-complex variable representations are also used to express the square of the four dimensional spacetime interval. Graphical representations of the Lorentz transformation on the complex plane are shown, and in particular for a superluminal transformation.