Cartan–Karlhede algorithm
The Cartan–Karlhede algorithm is a procedure for completely classifying and comparing Riemannian manifolds. Given two Riemannian manifolds of the same dimension, it is not always obvious whether they are locally isometric. Élie Cartan, using his exterior calculus with his method of moving frames, showed that it is always possible to compare the manifolds. Carl Brans developed the method further,[1] and the first practical implementation was presented by A. Karlhede in 1980.[2]
The main strategy of the algorithm is to take covariant derivatives of the Riemann tensor. In n dimensions, at most n(n+1)/2 differentiations suffice. If the Riemann tensor and its derivatives of the one manifold are algebraically compatible with the other, then the two manifolds are isometric. The Cartan–Karlhede algorithm therefore acts as a kind of generalization of the Petrov classification.
The potentially large number of derivatives can be computationally prohibitive. For example in 4 dimensions, the algorithm may in the worst case require the tenth derivative of the Riemann tensors, which results in a pair of rank 14 tensors, each of which would have 414 = 268435456 components (though not all independent). The algorithm was implemented in an early symbolic computation engine, SHEEP (symbolic computation system), but the size of the computations proved too challenging for early computer systems to handle.[3] Fortunately for most problems considered, far fewer derivatives are actually required, and the algorithm is more manageable on modern computers. On the other hand, no publicly available modern version exists.
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. This difference in behavior is due ultimately to the fact that spacetimes have isotropy subgroups which are subgroups of the Lorentz group SO+(3,R), which is a noncompact Lie group, while four-dimensional Riemannian manifolds (i.e., with positive definite metric tensor), have isotropy groups which are subgroups of the compact Lie group SO(4).
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 any two null dust solutions.
An important unsolved problem is to better predict how many differentiations are really necessary for spacetimes having various properties. For example, somewhere between two and five differentiations, at most, are required to compare any two Petrov III vacuum solutions. Overall, it seems to safe to say[weasel words] that at most six differentiations are required to compare any two spacetime models likely to arise in general relativity.
See also
External links
- Interactive Geometric Database includes some data derived from an implementation of the Cartan–Karlhede algorithm.
References
- ^ Brans, Carl H. (1965), "Invariant Approach to the Geometry of Spaces in General Relativity", J. Math. Phys., 6: 94, doi:10.1063/1.1704268
- ^ Karlhede, A. (1980), "A review of the geometrical equivalence of metrics in general relativity", General Relativity and Gravitation, 12: 693, doi:10.1007/BF00771861
- ^ Åman, J. E.; d'Inverno, R. A.; Joly, G. C.; MacCallum, M. A. H. (1984), "Quartic Equations and Algorithms for Riemann Tensor Classification", Lecture Notes in Computer Science, 174: 47, doi:10.1007/BFb0032829
- Stephani, Hans; Kramer, Dietrich; MacCallum, Malcom; Hoenselaers, Cornelius; Hertl, Eduard (2003). Exact Solutions to Einstein's Field Equations (2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-46136-7.
{{cite book}}: CS1 maint: multiple names: authors list (link) 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. - Pollney, D.; Skea, J. F.; and d'Inverno, Ray (2000). "Classifying geometries in general relativity (three parts)". Class. Quant. Grav. 17: 643–663, 2267–2280, 2885–2902. Bibcode:2000CQGra..17..643P. doi:10.1088/0264-9381/17/3/306.
{{cite journal}}: CS1 maint: multiple names: authors list (link) A research paper describing the authors' database holding classifications of exact solutions up to local isometry. - Olver, Peter J. (1995). Equivalents, Invariants, and Symmetry. Cambridge: Cambridge University Press. ISBN 0-521-47811-1. An introduction to the Cartan method, which has wide applications far beyond general relativity or even Riemannian geometry.