Campbell's theorem

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. ["Campbell's Theorem." Webster's Dictionary. Ed. Philip M. Parker. Websters-online-dictionary.com. 2008. 30 June 2008 .] The theorem guarantees that any n-dimensional Riemannian manifold can be locally embedded in an ("n" + 1)-dimensional Ricci-flat Riemannian manifold. [Romero, Carlos, Reza Tavakol, and Roustam Zalaltedinov. "The Embedding of General Relativity in Five Dimensions". N.p.: Springer Netherlands, 2005.]

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 R_{a 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. [Lindsey, James E., et al. "On Applications of Campbell's Embedding Theorem." On Applications of Campbell's Embedding Theorem 14 (1997): 1 17. Abstract.]

References