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 possessis a ''noncompact'' [[metricLie tensorgroup]], while four-dimensional Riemannan manifolds (i.e., havingwith [[positive definite]] [[signaturemetric tensor]]), have isotropy groups which are subgroups of the [[compact]] Lie group SO(4).

The method was implemented by Åman and Karlhede in special purpose symbolic computation engines such as [[SHEEP (symbolic computation system)]], for use in general relativity.