In Riemannian geometry and semi-Riemannian geometry, the Cartan-Karlhede algorithm is a rather involved method of distinguishing two pseudo-Riemannian manifolds, up to local isometry. The method uses coframe fields and their covariant derivatives; it is originally due to Élie Cartan, but various later researchers have improved and refined it.
Physical Applications
The Cartan-Karlhede algorithm has important applications in general relativity. One reason for this is that the simpler notion of curvature invariants fails to distinguish spacetimes as well as they distinguish Riemannian manifolds, which possess a metric tensor having positive definite signature.
The method was implemented by Åman and Karlhede in special purpose symbolic computation engines such as SHEEP (symbolic computation system), for use in general relativity.
Cartan showed that for a four dimensional manifold, such as spacetime model in general realtivity, at most ten covariant derivatives are needed, but experience shows that far fewer are needed very often. Later researchers have lowered "ten". An important unsolved problem is to better predict how many differentiations are really neccessary for spacetimes having various properties. Faster implementations of the method are also desirable.