93:, but the size of the computations proved too challenging for early computer systems to handle. 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.
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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. There are known models of spacetime requiring all 7 covariant
82:+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
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The potentially large number of derivatives can be computationally prohibitive. The algorithm was implemented in an early symbolic computation engine,
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MacCallum, M. A. H.; Åman, J. E. (1986), "Algebraically independent nth derivatives of the
Riemannian curvature spinor in a general spacetime",
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derivatives. For certain special families of spacetime models, however, far fewer often suffice. It is now known, for example, that
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324:Åman, J. E.; Karlhede, A. (1980), "A computer-aided complete classification of geometries in general relativity. First results",
113:. This difference in behavior is due ultimately to the fact that spacetimes have isotropy subgroups which are subgroups of the
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Pollney, D.; Skea, J. F.; d'Inverno, Ray (2000). "Classifying geometries in general relativity (three parts)".
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Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius; Hertl, Eduard (2003).
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Karlhede, A. (1980), "A review of the geometrical equivalence of metrics in general relativity",
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Brans, Carl H. (1965), "Invariant
Approach to the Geometry of Spaces in General Relativity",
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developed the method further, and the first practical implementation was presented by
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includes some data derived from an implementation of the Cartan–Karlhede algorithm.
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Milson, Robert; Pelavas, Nicos (2008), "The type N Karlhede bound is sharp",
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at most three differentiations are required to compare any two perfect
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at most two differentiations are required to compare any two Petrov
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of the same dimension, it is not always obvious whether they are
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Manual for CLASSI: classification programs in general relativity
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fails to distinguish spacetimes as well as they distinguish
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The Cartan–Karlhede algorithm has important applications in
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at most one differentiation is required to compare any two
124:, while four-dimensional Riemannian manifolds (i.e., with
362:, University of Stockholm Institute of Theoretical Physics
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is a procedure for completely classifying and comparing
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Exact
Solutions to Einstein's Field Equations (2nd ed.)
105:. One reason for this is that the simpler notion of
131:), have isotropy groups which are subgroups of the
62:The main strategy of the algorithm is to take
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538:Mathematical methods in general relativity
506:. Cambridge: Cambridge University Press.
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221:Equivalents, Invariants, and Symmetry
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381:(3): 643–663, 2267–2280, 2885–2902.
175:Vanishing scalar invariant spacetime
284:General Relativity and Gravitation
185:Frame fields in general relativity
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197:Interactive Geometric Database
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482:10.1088/0264-9381/25/1/012001
418:Classical and Quantum Gravity
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225:Cambridge University Press
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70:. Cartan showed that in
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180:Computer algebra system
97:Physical applications
84:Petrov classification
64:covariant derivatives
460:Class. Quantum Grav.
117:SO(1,3), which is a
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107:curvature invariants
26:Riemannian manifolds
22:Riemannian manifolds
533:Riemannian geometry
430:1986CQGra...3.1133M
387:2000CQGra..17..643P
375:Class. Quantum Grav
338:1980PhLA...80..229A
296:1980GReGr..12..693K
262:1965JMP.....6...94B
145:null dust solutions
74:dimensions at most
40:with his method of
304:10.1007/BF00771861
103:general relativity
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135:Lie group SO(4).
126:positive definite
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119:noncompact
46:Carl Brans
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312:120666569
122:Lie group
59:in 1980.
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219:(1995).
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133:compact
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