Five-Dimensional Tangent Vectors in Space-Time: VI. Bivector Derivative and Its Application

TL;DR

This paper introduces a bivector derivative for higher-dimensional vector fields in Riemannian geometry, explores its implications for curvature, Einstein, and Maxwell equations, and extends classical concepts to five-vector frameworks.

Contribution

It develops a new bivector derivative for four- and five-vector fields, generalizes curvature and Einstein equations, and proposes a five-vector extension of Maxwell's equations.

Findings

Defined bivector derivative for higher-dimensional vector fields.

Introduced five-vector curvature tensor and generalized Einstein equations.

Proposed a five-vector form of Maxwell's equations.

Abstract

This is the sixth, concluding part of a series of papers the first five of which have been submitted to the present archive in mid 1998 and published as INR preprints in 1999. The present paper was printed as an INR preprint, too, but for nonscientific reasons was never made public in any form, electronic or hard-copy. In it I define the bivector derivative for four- and five-vector fields in the case of arbitrary Riemannian geometry; examine a more general case of five-vector affine connection; introduce the five-vector analog of the curvature tensor; and consider a possible five-vector generalization of the Einstein and Kibble-Sciama equations. In conclusion, I define the bivector derivative for the fields of nonspacetime vectors and tensors and derive a possible five-vector generalization of Maxwell's equation.