The embedding of the space–time in five dimensions: An extension of the Campbell–Magaard theorem

Abstract

We extend Campbell-Magaard embedding theorem by proving that any ndimensional semi-Riemannian manifold can be locally embedded in an (n+1)dimensional Einstein space. We work out some examples of application of the theorem and discuss its relevance in the context of modern higher-dimensional spacetime theories.

Gödel's universe and induced-matter theory

“…It is interesting to have a look at the components of the extrinsic curvature tensor Ω αβ of the hypersurface ψ = const of M 5 . In the coordinates of (5) it can easily be shown that Ω αβ is given by Ω αβ = − 1 2k ∂g αβ ∂ψ [18], so that the nonvanishing components of Ω αβ are…”

“…with ε 2 = 1, represents the line element of M n+1 in a coordinate neighbourhood V of M n+1 [6,26]. In the light of the above theorem let us take n = 4, ε = 1, φ = −k 2 , where k is a constant, and the set of analytical functions {g αβ (t, x, y, z, ψ} 2w(x + kψ)) = 0.…”

“…It states that any n-dimensional pseudo-Riemannian manifold (M n , g) can be locally, analytically and isometrically embedded in a Ricci-flat (n+1)-dimensional manifold (N n+1 , g). Since its "rediscovery" in the nineties [12] the theorem has found a number of applications and has been discussed in various contexts in the literature [13][14][15][16][17][18][19][20][21][22][23][24]. Therefore, in view of the Campbell-Magaard theorem one would like to look at Gödel's solution as a hypersurface embedded in a five-dimensional Ricci-flat space.…”

Seminal magnetic fields from inflato-electromagnetic inflation

“…The theory that motivates such scenario is the Induced Matter Theory (IMT) [11,12]. In this theory inflation can be recovered from the Campbell-Magaard theorem [13][14][15][16][17], which serves as a ladder to go between manifolds whose dimensionality differs by one. This theorem, which is valid in any number of dimensions, implies that every solution of the 4D Einstein equations with arbitrary energy momentum tensor can be embedded, at least locally, in a solution of the 5D Einstein field equations in vacuum.…”

Gravito-magnetic currents in the inflationary universe from WIMT

“…This formalism is inspired by the Induced Matter Theory (IMT), which is based on the assumption that ordinary matter and physical fields that we can observe in our 4D universe can be geometrically induced from a 5D Ricci-flat metric with a space-like noncompact extra dimension on which we define a physical vacuum [10][11][12]. The Campbell-Magaard theorem [13][14][15][16][17] serves as a ladder to move between manifolds whose dimensionality differs by one. This theorem, which is valid in any number of dimensions, implies that every solution of the 4D Einstein equations with arbitrary energy-momentum tensor can be embedded, at least locally, in a solution of the 5D Einstein field equations in vacuum.…”