Formal Solution of the Fourth Order Killing equations for Stationary Axisymmetric Vacuum Spacetimes

TL;DR

This paper derives an analytic expression for the fourth order Killing tensor in stationary axisymmetric vacuum spacetimes, aiding the understanding of geodesic structures around compact objects and potentially improving spacetime mapping via gravitational waves.

Contribution

It provides the first explicit analytic form of the fourth order Killing tensor for SAV spacetimes, extending the understanding of conserved quantities in these geometries.

Findings

Derived a closed-form integral expression for the fourth order Killing tensor.

Identified conditions under which the solution is exact or approximate.

Discussed implications for spacetime mapping using gravitational wave data.

Abstract

An analytic understanding of the geodesic structure around non-Kerr spacetimes will result in a powerful tool that could make the mapping of spacetime around massive quiescent compact objects possible. To this end, I present an analytic closed form expression for the components of a the fourth order Killing tensor for Stationary Axisymmetric Vacuum (SAV) Spacetimes. It is as yet unclear what subset of SAV spacetimes admit this solution. The solution is written in terms of an integral expression involving the metric functions and two specific Greens functions. A second integral expression has to vanish in order for the solution to be exact. In the event that the second integral does not vanish it is likely that the best fourth order approximation to the invariant has been found. This solution can be viewed as a generalized Carter constant providing an explicit expression for the fourth invariant, in addition to the energy, azimuthal angular momentum and rest mass, associated with geodesic motion in SAV spacetimes, be it exact or approximate. I further comment on the application of this result for the founding of a general algorithm for mapping the spacetime around compact objects using gravitational wave observatories.