Classification of Energy Momentum Tensors in $n \geq 5$ Dimensional Space-times: a Review

TL;DR

This paper reviews the algebraic classification of symmetric second order tensors in five and higher-dimensional space-times, providing canonical forms and Segre types relevant to string theory and brane-world models.

Contribution

It offers a comprehensive classification scheme for symmetric tensors in n-dimensional space-times, extending results from 5-D to higher dimensions with canonical forms and theorems.

Findings

Classified all Segre types for 5-D symmetric tensors.

Derived canonical forms for each Segre type in 5-D.

Extended classification to n-D space-times using induction.

Abstract

Recent developments in string theory suggest that there might exist extra spatial dimensions, which are not small nor compact. The framework of a great number of 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 review the main results on the algebraic classification of second order symmetric tensors in 5-dimensional space-times. All possible Segre types for a symmetric two-tensor are found, and a set of canonical forms for each Segre type is obtained. A limiting diagram for the Segre types of these symmetric tensors in 5-D is built. Two theorems which collect together some basic results on the algebraic structure of second order symmetric tensors in 5-D are presented. We also show how one can obtain, by induction, the classification and the canonical forms of a symmetric two-tensor on n-dimensional (n > 5) spaces from its classification in 5-D spaces, present the Segre types in n-D and the corresponding canonical forms. This classification of symmetric two-tensors in any n-D spaces and their canonical forms are important in the context of n-dimensional brane-worlds context and also in the framework of 11-D supergravity and 10-D superstrings.