Classification of Second Order Symmetric Tensors in 5-Dimensional Kaluza-Klein-Type Theories

TL;DR

This paper provides an algebraic classification of second order symmetric tensors in 5D Kaluza-Klein Lorentzian spaces, identifying Segre types, canonical forms, and symmetry groups using Jordan matrices.

Contribution

It introduces a comprehensive classification scheme for these tensors, including canonical forms and symmetry group analysis, advancing understanding in higher-dimensional Lorentzian geometry.

Findings

Identified possible Segre types for tensors in 5D Kaluza-Klein spaces.

Derived canonical forms for each Segre type.

Analyzed symmetry groups associated with each canonical form.

Abstract

An algebraic classification of second order symmetric tensors in 5-dimensional Kaluza-Klein-type Lorentzian spaces is presented by using Jordan matrices. We show that the possible Segre types are $[1,1111]$, [2111], [311], [z,\bar{z},111], and the degeneracies thereof. A set of canonical forms for each Segre type is found. The possible continuous groups of symmetry for each canonical form are also studied.