The Simon and Simon-Mars Tensors for Stationary Einstein-Maxwell Fields

TL;DR

This paper explores the properties of Simon and Simon-Mars tensors in stationary Einstein-Maxwell spacetimes, showing their relations to spacetime symmetries and field alignments, especially in the Kerr-Newman-Taub-NUT solution.

Contribution

It extends the definition and properties of Simon and Simon-Mars tensors from vacuum to electrovacuum spacetimes, demonstrating their role in field and principal null direction alignments.

Findings

Simon tensors are redefined for nonvacuum cases to preserve their key properties.

In Kerr-Newman-Taub-NUT spacetime, the Simon tensor relates to the Ernst potential similarly as in vacuum.

Vanishing Simon tensor indicates alignment of principal null directions of various fields.

Abstract

Modulo conventional scale factors, the Simon and Simon-Mars tensors are defined for stationary vacuum spacetimes so that their equality follows from the Bianchi identities of the second kind. In the nonvacuum case one can absorb additional source terms into a redefinition of the Simon tensor so that this equality is maintained. Among the electrovacuum class of solutions of the Einstein-Maxwell equations, the expression for the Simon tensor in the Kerr-Newman-Taub-NUT spacetime in terms of the Ernst potential is formally the same as in the vacuum case (modulo a scale factor), and its vanishing guarantees the simultaneous alignment of the principal null directions of the Weyl tensor, the Papapetrou field associated with the timelike Killing vector field, the electromagnetic field of the spacetime and even the Killing-Yano tensor.