Cartan–Karlhede algorithm

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 at most ten covariant derivatives are needed to compare any two Lorentzian manifolds by his method, but experience shows that far fewer often suffice, and later researchers have lowered his upper bound considerably. It is now known, for example, that

• at most two differentiations are required to compare any two Petrov D vacuum solutions,
• at most three differentiations are required to compare any two perfect fluid solutions,
• at most one differentiation is required to compare wny two null dust solutions.

An important unsolved problem is to better predict how many differentiations are really neccessary for spacetimes having various properties. For example, somewhere two and five differentiations, at most, are required to compare any two Petrov III vacuum solutions. Overall, it seems to safe to say that at most six differentiations are required to compare any two spacetime models likely to arise in general relativity.

Faster implementations of the method are also desirable, since even with fast computers running symbolic manipulation packages using the latest differential algebra algorithms, these computations tend become very resource intensive, often pushing the limit of modern computer systems (or exceeding them).

See also

• Computer algebra system
• Frame fields in general relativity
• Petrov classification

External links

• Interactive Geometric Database includes some data derived from an implementation of the Cartan-Karlhede algorithm.

References

• . ISBN 0-521-46136-7. {{cite book}}: Missing or empty |title= (help); Unknown parameter |Author= ignored (|author= suggested) (help); Unknown parameter |Publisher= ignored (|publisher= suggested) (help); Unknown parameter |Title= ignored (|title= suggested) (help); Unknown parameter |Year= ignored (|year= suggested) (help) Chapter 9 offers an excellent overview of the basic idea of the Cartan method and contains a useful table of upper bounds, more extensive than the one above.
• Template:Journal reference A research paper describing the authors' database holding classfications of exact solutions up to local isometry.
• . ISBN 0-521-47811-1. {{cite book}}: Missing or empty |title= (help); Unknown parameter |Author= ignored (|author= suggested) (help); Unknown parameter |Publisher= ignored (|publisher= suggested) (help); Unknown parameter |Title= ignored (|title= suggested) (help); Unknown parameter |Year= ignored (|year= suggested) (help) An introduction to the Cartan method, which has wide applications far beyond general relativity or even Riemannian geometry.