Reducibility of Valence-3 Killing Tensors in Weyl's Class of Stationary and Axially Symmetric Space-Times

TL;DR

This paper investigates the structure of valence-3 Killing tensor fields in Weyl's class of stationary, axially symmetric space-times, showing they are generated by Killing vectors and quadratic tensors, with applications to Zipoy-Voorhees metrics.

Contribution

It proves that in Weyl's class, valence-3 Killing tensors are generated by Killing vectors and quadratic tensors, extending understanding of symmetries in these space-times.

Findings

Valence-3 Killing tensors are generated by Killing vectors and quadratic tensors in Weyl's class.

For Zipoy-Voorhees metrics, valence-3 Killing tensors are derived from Killing vectors and the metric.

The structure of Killing tensor fields simplifies in static, axially symmetric vacuum space-times.

Abstract

Stationary and axially symmetric space-times play an important role in astrophysics, particularly in the theory of neutron stars and black holes. The static vacuum sub-class of these space-times is known as Weyl's class, and contains the Schwarzschild space-time as its most prominent example. This paper is going to study the space of Killing tensor fields of valence 3 for space-times of Weyl's class. Killing tensor fields play a crucial role in physics since they are in correspondence to invariants of the geodesic motion (i.e. constants of the motion). It will be proven that in static and axially symmetric vacuum space-times the space of Killing tensor fields of valence 3 is generated by Killing vector fields and quadratic Killing tensor fields. Using this result, it will be proven that for the family of Zipoy-Voorhees metrics, valence-3 Killing tensor fields are always generated by Killing vector fields and the metric.