A Geometric Way to Obtain De-Sitter Subspaces Perpendicular to a Finite Size Extra-Dimension in Curved 5d Universes

Abstract

We define curved five-dimensional (5D) space–time from the embedding of 5D surfaces in a 6D flat space. Demanding that the 6D coordinates satisfy a separation of variables form and that the 5D metric is diagonal, we obtain that each curved 5D surface contains 4D hyperboloid de-Sitter subspaces with maximally symmetry SO (4,1). Therefore, we define a very special form for the curved 5D surface where the extra-dimension is perpendicular to the 4D hyperboloid de-Sitter spaces. By relating to a minimally coupled scalar field with a potential which depends on the extra-dimension only, the curved 5D surface's form is satisfied. A mechanism by means of which the extra-dimension can be of a finite size, is found. The borders of the finite extra-dimension are obtained when the scalar field potential goes to infinity for certain finite values of the scalar field. The geodesic lines' equations show that a particle cannot cross such borders.

Geometrical Correlation between Extra-dimensions and Universe Accelerated Expansion

“…In (E. Guendelman and H. Ruchvarger, 2004), we find the extra-dimensional metric component by a new equivalence principle that is an additional principle to Einstein's equivalence principle.…”

“…In our opinion, the best explanation for something that is unknown, is by using extra-dimensions. Therefore, we explained the universe accelerated expansion by using extra-dimensions (E. Guendelman and H. Ruchvarger, 2004).…”

“…In (E. Guendelman and H. Ruchvarger, 2004), we explain the universe accelerated expansion by defining a general orthogonal multidimensional variable separated coordinate system based on a three-sphere surface.…”

“…According to (40) ,for each last space-like extra-dimension comparing with (E. Guendelman and H. Ruchvarger, 2006) where 5  n ,we obtain a very similar solution as follows:…”

“…We assume that there is at least one geodesic which represent a straight line in the curved multidimensional space-time surface (this is a kind of "equivalence principle" of a new type (E. Guendelman and H. Ruchvarger, 2006)…”