Simply Amusing Algebra and Analysis or Electromagnetic and Gravitational Fields in the Single System of Equations

TL;DR

This paper develops algebraic and analytical frameworks in 4D pseudo-euclidean and pseudo-Riemannian spaces, deriving analogues of Cauchy-Riemann conditions, solutions, and restrictions that suggest a link to electromagnetic fields.

Contribution

It introduces new algebraic and differential analysis methods in 4D spaces, deriving Cauchy-Riemann analogues and solutions, and proposes a hypothesis connecting these to electromagnetic fields.

Findings

Derived Cauchy-Riemann analogues in 4D spaces

Obtained general and special wave solutions for metric components

Suggested a link between differentiable functions and electromagnetic fields

Abstract

In this article the algebra and the basis of corresponding analysis in 4-dimensional spaces are constructed, in pseudoeuclidean with signature (1, -1, -1, -1) and pseudo-Riemannian corresponding to the real space-time. In both cases the analogues of Cauchy-Riemann conditions are obtained. They are the systems of 1-st order partial differential equations, linear for the pseudoeuclidean and quasi-linear for the pseudo-Riemannian space (linear as about the components of differentiable function ant its derivatives so about the derivatives of metric tensor). The general solution for pseudoeuclidean space which is the flat waves of components of dependent function, and special (spherical-symmetric) wave-like (as for the components of differentiable function so for the components of metric tensor) solution for the pseudo-Riemannian space are got. In the last case the absence of central singularity for the components of metric tensor is interesting. From the Cauchy-Riemann condition follows that the differentiable function is constant along some isotropic curves given by 1-st order differential equations. The demand these curves to be geodetic lines leads to the differential restrictions for the metric tensor itself. The special kind of these restrictions is obtained. The hypothesis that the differentiable function can be interpreted as an electromagnetic field is expressed.