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]].<ref>{{cite book | author=Olver, Peter J. |author-link=Peter J. Olver | title=Equivalents, Invariants, and Symmetry | ___location=Cambridge | publisher=[[Cambridge University Press]] | year=1995 | isbn=0-521-47811-1}}</ref> [[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|bibcode = 1965JMP.....6...94B }}</ref> and the first practical implementation was presented by {{ill|Anders Karlhede|sv}} 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|issue=9 |doi=10.1007/BF00771861|bibcode = 1980GReGr..12..693K |s2cid=120666569 }}</ref>

The main strategy of the algorithm is to take [[covariant derivative]]s 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 (symbolic computation system)|SHEEP]], but the size of the computations proved too challenging for early computer systems to handle.<ref>{{citation|first1=J. E.|last1=Åman|first2=A.|last2=Karlhede|title=A computer-aided complete classification of geometries in general relativity. First results|journal=Phys. Lett. A|volume=80|page=229|year=1980|issue=4 |doi=10.1016/0375-9601(80)90007-9|bibcode = 1980PhLA...80..229A }}</ref><ref>{{citation|first1=J. E.|last1=Åman|title=Manual for CLASSI: classification programs in general relativity|publisher=University of Stockholm Institute of Theoretical Physics}}</ref> Fortunately forFor 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.<ref>{{cite journal |author1=Pollney, D. |author2=Skea, J. F. |author3=d'Inverno, Ray | title=Classifying geometries in general relativity (three parts) | journal=Class. Quantum Grav. | year=2000 | volume=17 |issue=3 | pages=643–663, 2267–2280, 2885–2902 | doi=10.1088/0264-9381/17/3/306|bibcode = 2000CQGra..17..643P |s2cid=250907225 }}</ref>

==Physical applications==

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The Cartan–Karlhede algorithm has important applications in [[general relativity]]. One reason for this is that the simpler notion of [[curvature invariant]]s fails to distinguish spacetimes as well as they distinguish [[Riemannian manifold]]s. 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 [[definite bilinear form|positive definite]] [[metric tensor]]), have isotropy groups which are subgroups of the [[compact group|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.<ref>{{citation|first1=M. A. H.|last1=MacCallum|first2=J. E.|last2=Åman|title=Algebraically independent nth derivatives of the Riemannian curvature spinor in a general spacetime|journal=Classical and Quantum Gravity|volume=3|page=1133|year=1986|issue=6 |doi=10.1088/0264-9381/3/6/013|bibcode = 1986CQGra...3.1133M |s2cid=250892608 }}</ref> There are known models of spacetime requiring all 7 covariant derivatives.<ref>{{citation|first1=Robert|last1=Milson|first2=Nicos|last2=Pelavas|title=The type N Karlhede bound is sharp|journal=Class. Quantum Grav.|volume=25|year=2008|page=012001 |doi=10.1088/0264-9381/25/1/012001|arxiv=0710.0688|s2cid=15859985 }}</ref> For certain special families of spacetime models, however, often far fewer often suffice. It is now known, for example, that

*at most one differentiation is required to compare any two [[null dust solution]]s,

*at most two differentiations are required to compare any two Petrov '''D''' [[vacuum solution (general relativity)|vacuum solution]]s,

*at most three differentiations are required to compare any two perfect [[fluid solution]]s.<ref>{{cite book |author1=Stephani, Hans |author2=Kramer, Dietrich |author3=MacCallum, Malcolm |author4=Hoenselaers, Cornelius |author5=Hertl, Eduard | title=Exact Solutions to Einstein's Field Equations (2nd ed.) | ___location=Cambridge | publisher=Cambridge University Press | year=2003 | isbn=0-521-46136-7}}</ref>

*at most one differentiation is required to compare any two [[null dust solution]]s.<ref>{{cite book |author1=Stephani, Hans |author2=Kramer, Dietrich |author3=MacCallum, Malcolm |author4=Hoenselaers, Cornelius |author5=Hertl, Eduard | title=Exact Solutions to Einstein's Field Equations (2nd ed.) | ___location=Cambridge | publisher=Cambridge University Press | year=2003 | isbn=0-521-46136-7}}</ref>

==See also==