Cartan–Karlhede algorithm: Difference between revisions

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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,