Spacetime Encodings IV - The Relationship between Weyl Curvature and Killing Tensors in Stationary Axisymmetric Vacuum Spacetimes

TL;DR

This paper investigates the link between Weyl curvature and Killing tensors in stationary axisymmetric vacuum spacetimes, proposing an algebraic test for the existence of a fourth invariant of geodesic motion.

Contribution

It introduces a new algebraic method to determine the existence of fourth-order Killing tensors using a symmetric non-covariant formulation and 2D manifold analysis.

Findings

Established a constructive test for fourth-order Killing tensors.

Linked Weyl curvature properties to geodesic invariants.

Reformulated Killing equations in a novel 2D manifold framework.

Abstract

The problem of obtaining an explicit representation for the fourth invariant of geodesic motion (generalized Carter constant) of an arbitrary stationary axisymmetric vacuum spacetime generated from an Ernst Potential is considered. The coupling between the non-local curvature content of the spacetime as encoded in the Weyl tensor, and the existence of a Killing tensor is explored and a constructive, algebraic test for a fourth order Killing tensor suggested. The approach used exploits the variables defined for the B\"{a}ckland transformations to clarify the relationship between Weyl curvature, constants of geodesic motion, expressed as Killing tensors, and the solution generation techniques. A new symmetric non-covariant formulation of the Killing equations is given. This formulation transforms the problem of looking for fourth-order Killing tensors in 4D into one of looking for four interlocking two-manifolds admitting fourth-order Killing tensors in 2D.