An alternative to the Simon tensor

TL;DR

This paper introduces a new symmetric, traceless tensor as an alternative to the Simon tensor for characterizing Kerr-NUT spacetimes, providing clearer geometric insights and a simple criterion involving the Ernst potential.

Contribution

It proposes a novel tensor in the quotient space that simplifies the geometric understanding and characterization of Kerr-NUT solutions, extending to Einstein-Maxwell systems.

Findings

The new tensor offers a more manifest geometric property of the quotient space.

A simple criterion for Kerr-NUT family based on the Ernst potential's gradient.

Extension of the obstruction tensor to Einstein-Maxwell systems with SU(1,2) covariance.

Abstract

The Simon tensor gives rise to a local characterization of the Kerr-NUT family in the stationary class of vacuum spacetimes. We find that a symmetric and traceless tensor in the quotient space of the stationary Killing trajectory offers a useful alternative to the Simon tensor. Our tensor is distinct from the spatial dual of the Simon tensor and illustrates the geometric property of the three dimensional quotient space more manifest. The reconstruction procedure of the metric for which the generalized Simon tensor vanishes is spelled out in detail. We give a four dimensional description of this tensor in terms of the Coulomb part of the imaginary selfdual Weyl tensor, which corresponds to the generalization of the three-index tensor defined by Mars. This allows us to establish a new and simple criterion for the Kerr-NUT family: the gradient of the Ernst potential becomes the non-null eigenvector of the Coulomb part of the imaginary selfdual Weyl tensor. We also discuss the ${\rm SU}(1,2)$ covariant extension of the obstruction tensor into the Einstein-Maxwell system as an intrinsic characterization of the Kerr-Newman-NUT family.