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.

原则上，使用 Cartan-Karlhede 算法始终可以对时空进行局部分类。然而，在实践中，确定两个时空等价的过程依赖于确定一组方程是否有解。根据方程的形式，这可能是无法确定的。此外，如果确实存在解，则方程在某些方面可能仍然无法解。在时空允许具有非平凡轨道的 Killing 矢量场的情况下，我们提出了一组新的不变量，称为 Killing 不变量。这些不变量将允许对承认同一组对称性的时空进行子分类，并且原则上将比其他不变量集简单得多。我们将这种方法应用于静态球对称几何形状类作为说明性示例。