The matrix theory of mathematical field and the motion of mathematical points in n-dimensional metric space

Abstract

A matrix theory of n-dimensional mathematical field and the motion of mathematical points in n-dimensional metric space is developed. Two spaces are considered: the n-dimensional space of an integrable coordinate vector $\{\mathit{x}\}$ with the integrable metric $\mathit{g}\mathrm{(}\mathit{x}\mathrm{)}$ and the n-dimensional space of a non-integrable but differentiable coordinate vector $\{\mathit{Q}\}$ , where $\mathit{d}\mathit{Q}=\mathit{e}\mathit{d}\mathit{x}$ , $\mathit{d}{\mathit{\tau }}^2=\mathit{d}\tilde{\mathit{Q}}\mathit{d}\mathit{Q}=\mathit{d}\tilde{\mathit{x}}\mathit{g}\mathrm{(}\mathit{x}\mathrm{)}\mathit{d}\mathit{x}$ , $\mathit{g}\mathrm{(}\mathit{x}\mathrm{)}=\tilde{\mathit{e}}\mathit{e}$ , $\partial \mathrm{(}\mathit{Q}\mathrm{)}={\mathit{e}}^{-1}\partial \mathrm{(}\mathit{x}\mathrm{)}$ . We call the coordinate space of the vector Q as the absolute space.

The derivatives of the non-integrable but differentiable matrix e are expressed through the elements of the Christoffel symbols and the elements of the Ricci and Riemann curvature matrices. The absolute velocity vector $\mathit{u}\mathrm{(}\mathit{Q}\mathrm{)}\equiv \mathit{d}\mathit{Q}\mathrm{/}\mathit{d}\mathit{\tau }$ and the absolute mathematical field matrix $\mathit{P}\equiv \mathit{u}\mathrm{(}\mathit{Q}\mathrm{)}\tilde{\partial }\mathrm{(}\mathit{Q}\mathrm{)}$ are introduced. We obtain two groups of the matrix field equations, the first of which is written in the two following forms: $\partial \mathrm{(}\mathit{Q}\mathrm{)}⟨\mathit{P}⟩-\tilde{\mathit{P}}\partial \mathrm{(}\mathit{Q}\mathrm{)}=\mathit{\rho }\mathit{u}\mathrm{(}\mathit{Q}\mathrm{)}$ , $\mathit{K}\mathit{u}\mathrm{(}\mathit{Q}\mathrm{)}=\mathit{\rho }\mathit{u}\mathrm{(}\mathit{Q}\mathrm{)}$ , where $⟨\mathit{P}⟩$ the trace of the matrix P, K is is the absolute Ricci matrix function, $\mathit{u}\mathrm{(}\mathit{Q}\mathrm{)}$ is the n-dimensional absolute velocity vector, ρ is a scalar function, which is the eigenvalue of K with the corresponding eigenvector $\mathit{u}\mathrm{(}\mathit{Q}\mathrm{)}$ . The interpretation of this pure mathematical theory in 4-dimensional space is the theory of the electromagnetic and gravitational fields and the motion of charged and neutral particles in the electromagnetic-gravitational field.