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.^[1] 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,^[2] and the first practical implementation was presented by Anders Karlhede in 1980.^[3]

The main strategy of the algorithm is to take covariant derivatives of the Riemann tensor. Cartan showed that 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. The algorithm was implemented in an early symbolic computation engine, SHEEP, but the size of the computations proved too challenging for early computer systems to handle.^[4][5] Fortunately for most problems considered, far fewer derivatives than the maximum are actually required, and the algorithm is more manageable on modern computers. On the other hand, no publicly available version exists in more modern software.^[6]

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⁺(1,3), 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).

In 4 dimensions, Karlhede's improvement to Cartan's program reduces the maximal number of covariant derivatives of the Riemann tensor needed to compare metrics to 7. In the worst case, this requires 3156 independent tensor components.^[7] There are known models of spacetime requiring all 7 covariant derivatives.^[8] For certain special families of spacetime models, however, often far fewer often suffice. 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.^[9]

See also

• Vanishing scalar invariant spacetime
• Computer algebra system
• Frame fields in general relativity

External links

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

References

1. Olver, Peter J. (1995). Equivalents, Invariants, and Symmetry. Cambridge: Cambridge University Press. ISBN 0-521-47811-1.
2. Brans, Carl H. (1965), "Invariant Approach to the Geometry of Spaces in General Relativity", J. Math. Phys., 6: 94, Bibcode:1965JMP.....6...94B, doi:10.1063/1.1704268
3. Karlhede, A. (1980), "A review of the geometrical equivalence of metrics in general relativity", General Relativity and Gravitation, 12: 693, Bibcode:1980GReGr..12..693K, doi:10.1007/BF00771861
4. Åman, J. E.; Karlhede, A. (1980), "A computer-aided complete classification of geometries in general relativity. First results", Phys. Lett. A, 80: 229, Bibcode:1980PhLA...80..229A, doi:10.1016/0375-9601(80)90007-9
5. Åman, J. E., Manual for CLASSI: classification programs in general relativity, University of Stockholm Institute of Theoretical Physics
6. Pollney, D.; Skea, J. F.; 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.
7. MacCallum, M. A. H.; Åman, J. E. (1986), "Algebraically independent nth derivatives of the Riemannian curvature spinor in a general spacetime", Classical and Quantum Gravity, 3: 1133, Bibcode:1986CQGra...3.1133M, doi:10.1088/0264-9381/3/6/013
8. Milson, Robert; Pelavas, Nicos (2008), "The type N Karlhede bound is sharp", Class. Quantum Grav., 25, arXiv:0710.0688, doi:10.1088/0264-9381/25/1/012001
9. Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius; Hertl, Eduard (2003). Exact Solutions to Einstein's Field Equations (2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-46136-7.