On the geometry of Killing and conformal tensors

TL;DR

This paper studies second order Killing and conformal tensors through their spectral properties, providing conditions for their existence, canonical metric forms, and applications to space-time symmetries and geodesic integrals in general relativity.

Contribution

It characterizes Killing and conformal tensors of type I with two eigenvalues, offers canonical metric expressions, and links these tensors to quadratic first integrals in specific space-time geometries.

Findings

Characterization of type I tensors with two eigenvalues

Canonical forms for metrics with these symmetries

Geometric interpretation of geodesic integrals in Petrov-Bel type D space-times

Abstract

The second order Killing and conformal tensors are analyzed in terms of their spectral decomposition, and some properties of the eigenvalues and the eigenspaces are shown. When the tensor is of type I with only two different eigenvalues, the condition to be a Killing or a conformal tensor is characterized in terms of its underlying almost-product structure. A canonical expression for the metrics admitting these kinds of symmetries is also presented. The space-time cases 1+3 and 2+2 are analyzed in more detail. Starting from this approach to Killing and conformal tensors a geometric interpretation of some results on quadratic first integrals of the geodesic equation in vacuum Petrov-Bel type D solutions is offered. A generalization of these results to a wider family of type D space-times is also obtained. A generalization of these results to a wider family of type D space-times is also obtained.