Spacetime Encodings III - Second Order Killing Tensors

TL;DR

This paper investigates specific stationary axisymmetric vacuum spacetimes with second-order Killing tensors, providing explicit derivations, classifications, and insights into their geometric and dynamical properties, with implications for higher-order invariants.

Contribution

It offers a detailed derivation and classification of SAV spacetimes with second-order Killing tensors, and introduces methods that could extend to higher-order Killing tensors.

Findings

Revealed explicit relationships between Killing equations and metric functions.

Classified separable coordinate systems based on Killing equations.

Identified restrictive conditions for SAV spacetimes admitting second-order Killing tensors.

Abstract

This paper explores the Petrov type D, stationary axisymmetric vacuum (SAV) spacetimes that were found by Carter to have separable Hamilton-Jacobi equations, and thus admit a second-order Killing tensor. The derivation of the spacetimes presented in this paper borrows from ideas about dynamical systems, and illustrates concepts that can be generalized to higher- order Killing tensors. The relationship between the components of the Killing equations and metric functions are given explicitly. The origin of the four separable coordinate systems found by Carter is explained and classified in terms of the analytic structure associated with the Killing equations. A geometric picture of what the orbital invariants may represent is built. Requiring that a SAV spacetime admits a second-order Killing tensor is very restrictive, selecting very few candidates from the group of all possible SAV spacetimes. This restriction arises due to the fact that the consistency conditions associated with the Killing equations require that the field variables obey a second-order differential equation, as opposed to a fourth-order differential equation that imposes the weaker condition that the spacetime be SAV. This paper introduces ideas that could lead to the explicit computation of more general orbital invariants in the form of higher-order Killing Tensors.