Five-Dimensional Tangent Vectors in Space-Time

TL;DR

This paper explores five-dimensional tangent vectors in space-time, their mathematical properties, and potential applications in physics such as gravity and gauge theories, unifying physical quantities like stress-energy and angular momentum tensors.

Contribution

It introduces and analyzes the concept of five-dimensional tangent vectors, extending traditional tangent space concepts and proposing their use in advanced physical theories.

Findings

Five-dimensional tangent vectors can unify stress-energy and angular momentum tensors.

Mathematical definitions and properties of five-dimensional tangent vectors are established.

Potential applications in gravity and gauge theories are discussed.

Abstract

This article is a summary of a series of papers to be published where I examine a special kind of geometric objects that can be defined in space-time --- five-dimensional tangent vectors. Similar objects exist in any other differentiable manifold, and their dimension is one unit greater than that of the manifold. Like ordinary tangent vectors, the considered five-dimensional vectors and the tensors constructed out of them can be used for describing certain local quantities and in this capacity find direct application in physics. For example, such familiar physical quantities as the stress-energy and angular momentum tensors prove to be parts of a single five-tensor. In this paper I describe several different mathematical definitions of five-dimensional tangent vectors, discuss their basic algebraic and differential properties, and speak about their possible application in the theory of gravity and in gauge theories.