Segre Types of Symmetric Two-tensors in n-Dimensional Spacetimes

TL;DR

This paper classifies the algebraic types of Ricci tensors in n-dimensional Kaluza-Klein spacetimes using Jordan matrix propositions, providing canonical forms for various Segre types.

Contribution

It introduces a classification scheme for Ricci tensors in higher-dimensional spacetimes based on Jordan matrix analysis, with explicit canonical forms.

Findings

Identifies possible Segre types in n-dimensional spacetimes.

Provides canonical forms for each Segre type.

Establishes a link between Jordan matrices and Ricci tensor classification.

Abstract

Three propositions about Jordan matrices are proved and applied to algebraically classify the Ricci tensor in n-dimensional Kaluza-Klein-type spacetimes. We show that the possible Segre types are [1,1...1], [21...1], [31\ldots 1], [z\bar{z}1...1] and degeneracies thereof. A set of canonical forms for the Segre types is obtained in terms of semi-null bases of vectors.