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

22 pages, no figures

References in corpus (3)

• The non-integrability of the Zipoy-Voorhees metric
• Spacetime Encodings II - Pictures of Integrability
• Spacetime Encodings I- A Spacetime Reconstruction Problem

Cited by in corpus (5)

• Testing spacetime symmetry through gravitational waves from extreme-mass-ratio inspirals
• On integrability of certain rank 2 sub-Riemannian structures
• Killing Tensors in Koutras-McIntosh Spacetimes
• Killing tensors in stationary and axially symmetric space-times
• Non-Integrability of Geodesic dynamics of Chazy-Curzon space-time