Cartan–Karlhede algorithm: Difference between revisions

Revision as of 07:28, 16 September 2005

Revision as of 08:11, 16 September 2005

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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.

Cartan showed that for''at amost fourten dimensionalcovariant manifold,derivatives suchare asneeded [[spacetime]]to modelcompare inany generaltwo realtivity,Lorentzian atmanifolds'' mostby tenhis covariant derivatives are neededmethod, but experience shows that far fewer areoften neededsuffice, veryand often. Laterlater researchers have lowered "ten".his upper Anbound importantconsiderably. unsolved problemIt is tonow better predict how many differentiations are really neccessaryknown, for spacetimesexample, having various properties. Faster implementations of the method are also desirable.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 wny two [[null dust solution]]s.

An important unsolved problem is to better predict how many differentiations are really neccessary 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 are also desirable, since even with fast computers running symbolic manipulation packages using the latest differential algebra algorithms, these computations tend become very resource intensive, often pushing the limit of modern computer systems (or exceeding them).

==See also==

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*[[Computer algebra system]]

*[[Frame fields in general relativity]]

*[[Petrov classification]]

==External links==

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==References==

*{{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.

*{{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 classfications 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.

[[Category:Riemannian geometry]]