Killing invariants: an approach to the sub-classification of geometries with symmetry

Abstract

In principle, the local classification of spacetimes is always possible using the Cartan-Karlhede algorithm. However, in practice, the process of determining equivalence of two spacetimes relies on determining if a set of equations has a solution. Depending on the form of the equations this may be undecideable. Furthemore, if a solution does exist the equations may still be unsolvable in some way. In the case that spacetimes admit Killing vector fields with non-trivial orbits, we propose a new set of invariant quantities, called Killing invariants. These invariants will allow for the sub-classification of spacetimes admitting the same group of symmetries and will, in principle, be substantially less complicated than other sets of invariants. We apply this approach to the class of static spherically symmetric geometries as an illustrative example.