Weyl Tensor Classification in Four-dimensional Manifolds of All Signatures

TL;DR

This paper develops a unified classification scheme for the Weyl tensor in all four-dimensional manifolds, extending the Petrov classification and boost weight approach to all signatures and complex cases, simplifying and unifying previous methods.

Contribution

It introduces a unified classification framework for the Weyl tensor applicable to all signatures and complex cases, generalizing the Petrov classification.

Findings

The classification encompasses all signatures and complex manifolds.

The Petrov classification is shown as a special case within the new scheme.

The boost weight classification is extended and proven equivalent to the bivector method.

Abstract

It is well known that the classification of the Weyl tensor in Lorentzian manifolds of dimension four, the so called Petrov classification, was a great tool to the development of general relativity. Using the bivector approach it is shown in this article a classification for the Weyl tensor in all four-dimensional manifolds, including all signatures and the complex case, in an unified and simple way. The important Petrov classification then emerges just as a particular case in this scheme. The boost weight classification is also extended here to all signatures as well to complex manifolds. For the Weyl tensor in four dimensions it is established that this last approach produces a classification equivalent to the one generated by the bivector method.