Cartan–Karlhede algorithm: Difference between revisions

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The '''Cartan–Karlhede algorithm''' is a procedure for completely classifying and comparing [[Riemannian manifold]]s. Given two [[Riemannian manifold]]s of the same dimension, it is not always obvious whether they are [[local isometry|locally isometric]]. [[Élie Cartan]], using his [[exterior derivative|exterior calculus]] with his method of [[moving frames]], showed that it is always possible to compare the manifolds. [[Carl Brans]] developed the method further,<ref>{{citation|first1=Carl H.|last1=Brans|title=Invariant Approach to the Geometry of Spaces in General Relativity|journal=J. Math. Phys.|volume=6|page=94|year=1965|doi=10.1063/1.1704268}}</ref> and the first practical implementation was presented by A. Karlhede in 1980.<ref>{{citation|first1=A.|last1=Karlhede|title=A review of the geometrical equivalence of metrics in general relativity|journal=General Relativity and Gravitation|volume=12|page=693|year=1980|doi=10.1007/BF00771861}}</ref>

The main strategy of the algorithm is to take [[covariant derivative]]s of the [[Riemann tensor]]. In <math>''n</math>'' dimensions, at most <math>''n''(''n''+1)/2</math> differentiations suffice. If the Riemann tensor and its derivatives of the one manifold are algebraicalgebraically 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 [[tensor rank|rank 14 tensors]], each of which would have 163844¹⁴ = 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.<ref>{{citation|first1=J. E.|last1=Åman|first2=R. A.|last2=d'Inverno|first3=G. C.|last3=Joly|first4=M. A. H.|last4=MacCallum|title=Quartic Equations and Algorithms for Riemann Tensor Classification|journal=Lecture Notes in Computer Science|volume=174|page=47|year=1984|doi=10.1007/BFb0032829}}</ref> 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==