Cartan–Karlhede algorithm: Difference between revisions

Revision as of 08:40, 16 September 2005

Revision as of 16:49, 18 September 2005

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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 RiemannanRiemannian manifolds (i.e., with [[positive definite]] [[metric tensor]]), have isotropy groups which are subgroups of the [[compact]] Lie group SO(4).

Cartan's method was adapted and improved for general relativity by A. Karlhede, and

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*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 neccessarynecessary for spacetimes having various properties. For example, somewhere two and five differentiations, at most, are required to compare any two Petrov '''III''' vacuum solutions. Overall, it seems to safe to say that at most six differentiations are required to compare any two spacetime models likely to arise in general relativity.

Faster implementations of the method running under a modern symbolic computation system available for modern [[operating system]]s in common use, such as [[Linux]], would also be highly desirable. It has been suggested that the power of this algorithm has not yet been realized, due to insufficient effort to take advantage of recent improvements in [[differential algebra]]. The appearance in the "near future" of a proper on-line database of known solutions has been rumored for decades, but this has not yet come to pass. This is particularly regretableregrettable since it seems very likely that a powerful and convenient database is well within the capability of modern software.

==See also==

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*{{Book reference | Author=Stephani, Hans; Kramer, Dietrich; MacCallum, Malcom; Hoenselaers, Cornelius; Hertl, Eduard| Title=Exact Solutions to Einstein's Field Equations (2nd ed.) | Publisher=Cambridge: Cambridge University Press | Year=2003 | ID=ISBN 0-521-46136-7}} Chapter 9 offers an excellent overview of the basic idea of the Cartan method and contains a useful table of upper bounds, more extensive than the one above.

*{{Journal_reference | Author=Pollney, D.; Skea, J. F.; and d'Inverno, Ray | Title=Classifying geometries in general relativity (three parts) | Journal=Class. Quant. Grav. | Year=2000 | Volume=17 | Pages=643-663, 2267-2280, 2885-2902}} A research paper describing the authors' database holding classficationsclassifications of exact solutions up to local isometry.

*{{Book reference | Author=Olver, Peter J. | Title=Equivalents, Invariants, and Symmetry | Publisher=Cambridge:Cambridge University Press | Year=1995 | ID=ISBN 0-521-47811-1}} An introduction to the Cartan method, which has wide applications far beyond general relativity or even Riemannian geometry.