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Campbell's theorem, also known as Campbell’s embedding theorem and the Campbell-Magaarrd theorem, is a mathematical theorem that evaluates the asymptotic distribution of random impulses acting with a determined intensity on a damped system.[1] The theorem guarantees that any n-dimensional Riemannian manifold can be locally embedded in an (n + 1)-dimensional Ricci-flat Riemannian manifold.[2]
Mathematical Definition
Campbell's theorem states that any n-dimensional Riemannian manifold can be embedded locally it an (n+1) manifold with a Ricci curvature of Ra b=0. The theorem also states, in similar form, that an n-dimensional pseudo-Riemann manifold can be both locally and isometrically embedded in an n(n+1)/2 pseudo-Eucladian space.
Applications
Campbell’s theorem can be used to produce the embedding of numerous 4-dimensional spacetimes in 5-dimensional Ricci-flat spaces. It is also used to embed a class of n-dimensional Einstein spaces.[3]
References
- ^ "Campbell's Theorem." Webster's Dictionary. Ed. Philip M. Parker. Websters-online-dictionary.com. 2008. 30 June 2008 <http://www.websters-online-dictionary.org/Ca/Campbell's_theorem.html>.
- ^ Romero, Carlos, Reza Tavakol, and Roustam Zalaltedinov. The Embedding of General Relativity in Five Dimensions. N.p.: Springer Netherlands, 2005.
- ^ Lindsey, James E., et al. "On Applications of Campbell's Embedding Theorem." On Applications of Campbell's Embedding Theorem 14 (1997): 1 17. Abstract.
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- This page was last modified on 4 July 2008, at 08:59.
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