On Some Algebra and the Corresponding Analysis in the Pseudo-Riemannian Space with Signature 1,-1,-1,-1

TL;DR

This paper constructs a specific 4-dimensional algebra in a pseudo-Riemannian space with signature (1,-1,-1,-1), develops a theory of 4D functions based on it, and solves related equations, including wave solutions propagating at light speed.

Contribution

It introduces a new algebraic framework in pseudo-Riemannian space and applies it to analyze 4D functions, including solving Cauchy-Riemann equations under various symmetries.

Findings

Wave solutions propagate at speed c along null geodesics.

The algebra enables solving Cauchy-Riemann equations in symmetric cases.

Properties of the metric tensor are physically postulated.

Abstract

In this paper the certain 4-dimensional algebra in 4-dimensional pseudo-Riemannian space with signature (1, -1, -1, -1) is constructed. On the basis of this algebra the elements of the analysis, i.e. the theory of 4-dimensional functions of the 4-dimensional variable are built up. In the process of designing the analysistwo additional assumptions aboutthe properties of functions are made. Obtained under different assumptionsabout the properties of functions, the Cauchy-Riemann equations are solved in flat, spherically and cylindrically symmetric cases. In the cylindrically symmetric case the wave solutions were obtained both for the metric tensor and for the 4-dimensional function. Both waves spreadwith the speed equal to unity along null geodesics. The properties of the metric tensor as physically real object are postulated, but the idea what the 4-dimensional function (vector field) is remains unclear.