Geometric Interpretation of the Mixed Invariants of the Riemann Spinor

TL;DR

This paper explores the geometric meaning of mixed invariants of the Riemann spinor, revealing their role in classifying Einstein-Maxwell and perfect fluid spacetimes and analyzing the Bel-Robinson tensor's properties.

Contribution

It provides a geometric interpretation of mixed invariants of the Riemann spinor and their application to classifying specific spacetime types and tensor behaviors.

Findings

Mixed invariants classify Einstein-Maxwell fields by principal null directions.

Results on the Bel-Robinson tensor's behavior as a quartic form.

Insights into the relative orientation of gravitational and electromagnetic fields.

Abstract

Mixed invariants are used to classify the Riemann spinor in the case of Einstein-Maxwell fields and perfect fluids. In the Einstein-Maxwell case these mixed invariants provide information as to the relative orientation of the gravitational and electromagnetic principal null directions. Consideration of the perfect fluid case leads to some results about the behaviour of the Bel-Robinson tensor regarded as a quartic form on unit timelike vectors.