D. M. Volokitin | Searcharxiv

arXiv2013

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

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.

math.GM↗

arXiv2010

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

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.

physics.gen-ph↗