Classification of the traceless Ricci tensor in 4-dimensional pseudo-Riemannian spaces of neutral signature

TL;DR

This paper classifies the algebraic types of the traceless Ricci tensor in 4D neutral signature pseudo-Riemannian spaces, using matrix properties and Petrov-Penrose types, and compares it with the complex case.

Contribution

It provides a new algebraic classification scheme for the traceless Ricci tensor in neutral signature spaces, linking it with Petrov-Penrose types and complex case comparisons.

Findings

Classification based on matrix properties of $C_{ab}$

Identification of Petrov-Penrose types for Plebański spinors

Comparison with complex case classification

Abstract

The traceless Ricci tensor $C_{ab}$ in 4-dimensional pseudo-Riemannian spaces equipped with the metric of the neutral signature is analyzed. Its algebraic classification is given. This classification uses the properties of $C_{ab}$ treated as a matrix. The Petrov-Penrose types of Pleba\'nski spinors associated with the traceless Ricci tensor are given. Finally, the classification is compared with a similar classification in the complex case.