The Subclassification Problem for Stationary Axisymmetric Spacetimes

Four-dimensional G₂ spacetimes are metrics with two abelian Killing vector fields and are of great interest of study. One method of examining such metrics is the Cartan-Karlhede (CK) algorithm which serves the purpose of classifying and comparing different spacetimes. It can be used to determine if two metrics are equivalent and thus that one can be transformed into the other via a coordinate transformation. While this method can be extremely powerful, the Cartan curvature invariants produced from each metric in the algorithm are often too complicated to compare. Equating the invariants from each may even be unsolvable. Due to this, alternative methods for comparing spacetimes have been proposed such as using differential invariants arising from the jet bundle over the spacetime in the place of curvature invariants. In this work, we outline a proposed method motivated by these alternative invariants and Papapetrou fields to define a new set of differential invariants derived from Killing vector fields that we hope will provide a simpler method of comparing spacetimes.