Cartan–Karlhede algorithm: Difference between revisions

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==Physical applications==

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⁺(3,'''R'''), 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).

There are known models of spacetime requiring as many as 7 covariant derivatives of the Riemann tensor.<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|doi=10.1088/0264-9381/25/1/012001|arxiv=0710.0688}}</ref>. For certain special families of spacetime models, however, often far fewer often suffice. It is now known, for example, that

Cartan showed that ''at most ten covariant derivatives are needed to compare any two Lorentzian manifolds'' by his method, but experience shows that far fewer often suffice, and later researchers have lowered his upper bound considerably. It is now known, for example, that

*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,

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

An important unsolved problem is to better predict how many differentiations are really necessary for spacetimes having various properties. For example, somewhere between two and five differentiations, at most, are required to compare any two Petrov '''III''' vacuum solutions. Overall, it seems to safe to say{{weasel-word|date=November 2012}} that at most six differentiations are required to compare any two spacetime models likely to arise in general relativity.

==See also==