The embedding of space–times in five dimensions with nondegenerate Ricci tensor

Abstract

We discuss and prove a theorem which asserts that any n-dimensional semiRiemannian manifold can be locally embedded in a (n+1)-dimensional space with a non-degenerate Ricci tensor which is equal, up to a local analytic diffeomorphism, to the Ricci tensor of an arbitrary specified space. This may be regarded as a further extension of the Campbell-Magaard theorem.We highlight the significance of embedding theorems of increasing degrees of generality in the context of higher dimensional spacetimes theories and illustrate the new theorem by establishing the embedding of a general class of Ricci-flat spacetimes.

Harmonic maps and isometric embeddings of the spacetime

“…Campbell-Magaard's result may shed some light on our understanding of the mathematical structure of embedding theories, and applications of the theorem to the braneworld scenario, to five-dimensional non-compactified Kaluza-Klein gravity as well as to superstring theory have been found in recent years [8]. Extensions of the Campbell-Magaard theorem to more general cases have also been obtained [9,10].…”

“…Consider, for example, an extended version of the Campbell-Magaard theorem [9], which states that any n-dimensional semi-Riemannian manifold can be locally embedded in a (n + 1)-dimensional Einstein space. In this case the Einstein equations G µν = Λg µν are equivalent to the set…”

WIMT in Gullstränd–Painlevé and Reissner–Nordström metrics: induced stable gravito-magnetic monopoles

“…The vector − → n is normalized and orthogonal to the 4D tangent sub-space associated with the effective 4D 8 The dual tensor of a 2-form (characterized in this case by ( * ) F AB ) in 5D manifold is a 3-form defined by…”

“…We are interested in the cases where the extra dimension is non-compact, 1 and we define a physical vacuum supported by the Ricci-flatness condition [4]. This theory is founded in the Campbell-Magaard embedding theorem [5][6][7][8] as a par- 1 In contrast with the Kaluza-Klein Theory (KK) in which it is assumed that the extra dimension is compact, cyclic, and having a small radius. 2 In our convention the indices "a, b, c, .…”

Gravitating fluids with Lie symmetries

“…We point out that embedding of four-dimensional Riemannian manifolds in higher dimensional spacetimes with arbitrary Ricci tensors has been investigated by several authors. The physical motivation here is to understand the nature of physics in higher dimensions; the modern view is that the Riemannian manifold M 4 is a hypersurface in the higher dimensional bulk in the brane world scenario and other higher dimensional themes (Dahia and Romero 2002a, Dahia and Romero 2002b, Dahia et al 2008. Gupta and Sharma (1996) have generated a relativistic model in higher dimensions describing gravitating fluid plates.…”