A Note on Segre Types of Second Order Symmetric Tensors in 5-D Brane-world Cosmology

TL;DR

This paper classifies second order symmetric tensors in five-dimensional brane-world cosmology, providing foundational results relevant for higher-dimensional theories like supergravity and superstrings.

Contribution

It reexamines the algebraic classification of symmetric tensors in 5D and establishes theorems that facilitate extending these classifications to higher dimensions.

Findings

Proves key theorems on the algebraic structure of tensors in 5D.

Provides a method to extend classifications from 4D to n-dimensional spaces.

Highlights relevance for supergravity and superstring theories.

Abstract

Recent developments in string theory suggest that there might exist extra spatial dimensions, which are not small nor compact. The framework of most brane cosmological models is that in which the matter fields are confined on a brane-world embedded in five dimensions (the bulk). Motivated by this we reexamine the classification of the second order symmetric tensors in 5--D, and prove two theorems which collect together some basic results on the algebraic structure of these tensors in 5-dimensional space-times. We also briefly indicate how one can obtain, by induction, the classification of symmetric two-tensors (and the corresponding canonical forms) on n-dimensional spaces from the classification on 4-dimensional spaces. This is important in the context of 11--D supergravity and 10--D superstrings.