Algebraic unified theory of space-time and matter on the double variable plane
Using double numbers algebra we develop algebraic version of 2D relativity theory, which takes intermediate place between special and general relativity. In space-time free of matter the main object of the theory - hyperbolic potential F - is h-holomorphic function of double variable. Physically it is responsible for local splitting of space-time onto time and space directions in conformally deformed flat Minkowski space. It is shown that the effect of conformal deformation is in principle observable with the help of experiments concerning measurements with spatially separated clocks. Space-time with matter is described by relation F,h ≠ 0. The dynamics of hyperbolic field is described by special action, depending only on hyperbolic square of non-holomorphicity. It is shown, that field equation are non-linear h-conjugated wave equations with self-action. Specific properties of these equations are: 1) presence of the first integral; 2) compatibility (integrability) condition, defining class of admissible fields G(H2). The latter condition can be viewed as generalization of hyperbolic Cauchy-Riemannian condition and it is crucial for construction of consistent and reliable physical theory of space-time and matter in 2D. As an example we consider static 2D universe with 1D deformed bar. Some aspects of relations of SR and GR to the approach are discussed. We formulate super-extremum principle, allowing one to calculate fundamental constants and boundary conditions of the theory.