On the classification of type D spacetimes

TL;DR

This paper classifies type D spacetimes using invariant properties of the Weyl tensor, providing explicit criteria and an algorithm to identify specific solutions like Reissner-Nordström within both vacuum and non-vacuum contexts.

Contribution

It introduces a new invariant-based classification scheme for type D spacetimes applicable to all such metrics, including vacuum and non-vacuum solutions, with explicit conditions and an operational detection algorithm.

Findings

Classification based on tensorial invariants of the Weyl tensor.

Explicit conditions for identifying specific metrics like Reissner-Nordström.

Analysis of subfamilies with constant Weyl eigenvalue argument.

Abstract

We give a classification of the type D spacetimes based on the invariant differential properties of the Weyl principal structure. Our classification is established using tensorial invariants of the Weyl tensor and, consequently, besides its intrinsic nature, it is valid for the whole set of the type D metrics and it applies on both, vacuum and non-vacuum solutions. We consider the Cotton-zero type D metrics and we study the classes that are compatible with this condition. The subfamily of spacetimes with constant argument of the Weyl eigenvalue is analyzed in more detail by offering a canonical expression for the metric tensor and by giving a generalization of some results about the non-existence of purely magnetic solutions. The usefulness of these results is illustrated in characterizing and classifying a family of Einstein-Maxwell solutions. Our approach permits us to give intrinsic and explicit conditions that label every metric, obtaining in this way an operational algorithm to detect them. In particular a characterization of the Reissner-Nordstr\"{o}m metric is accomplished.