Unification of two dimensional special and general relativity by means of hypercomplex numbers

摘要

An extension of complex numbers and functions of complex variable is proposed through the properties of the related finite and infinite Lie groups. This is accomplished by hypercomplex number systems following the elementary algebra rules. More precisely, the functions of such systems define and infinite Lie group. The functional transformations of a particular two dimensional hypercomplex number system, holding the wave equation invariant (and then, the speed of light constant) are considered as a generalization of Lorentz group describing accelerated frames. According to General Relativity, such frames can represent physical fields. A physical interpretation of a theorem due to Bianchi, by which hypercomplex number systems are related to flat Riemann spaces, is shown to connect these systems to General Relativity. The investigations by generalized Special Relativity and by General Relativity give the same expected results allowing to speak of unification.