06 15 7 v 1 1 7 Ju n 20 03 M . W . Kalinowski ( Higher Vocational State School in Che lm , Poland ) A Warp

We consider in the paper some consequences of the Nonsymmetric Kaluza–Klein (Jordan–Thiry) Theory with spontaneous symmetry breaking connecting to the existence of the warp factor. We develop in the paper some applications and consequences of the Nonsymmetric Kaluza–Klein (Jordan–Thiry) Theory extensively presented in the first point of Ref. [1]. We refer for all the details to Ref. [1], especially to the book on nonsymmetric fields theory and its applications. We consider a possibility to take seriously additional dimensions in our theory in a framework similar to a Randall–Sundrum scenario. In our theory the additional dimensions connecting to the manifold M (a vacuum manifold) could be considered in such a way. They are not directly observable because the size of the manifold is very small. In order to see them it is necessary to excite massive modes of the scalar field (a tower of those fields). We find an interesting toy model (a 5-dimensional model) which can describe a possibility to travel with speed higher than a speed of light using the fifth dimension. This dimension has nothing to do with the fifth dimension in Kaluza–Klein theory. We do some analysis on an energy to excite a scalar field Ψ to get this special solution to the theory. We discuss also some quantitative relations involving travelling signals in the model via fifth dimension. We consider various posibilities to excite a warp factor due to fluctuations of a tower of scalar fields finding a density of an excitation energy. Eventually we find a zero energy (or almost zero) condition for such an excitation. We consider also a simple solution for hierarchy problem in the framework of the Nonsymmetric Kaluza–Klein (Jordan–Thiry) Theory. Let us consider the Eqs (5.3.17–19) of the first point of Ref. [1], p. 329, and let us release the condition that ρ is independent of y. Thus ρ = ρ(x, y), y ∈ G/G0. In Eq. (5.3.34) of the first point of Ref. [1], p. 333, we get in a place of λ 4 ρ ( Mg̃ρ,γρ,μ + n ggδμg̃ (δγ) ) ρ,ν · ρ,γ (1) the formula λ 4 ρ ( Mγ̃ρ,Cρ,M + n γ γDM γ̃ ρ,N · ρ,C ) . (2) This ρ has nothing to do with a density of energy considered below. 1 Now let us repeat the procedure from Section 5.5 of the first point of Ref. [1], i.e. the redefinition of gμν and ρ. We get the formula (5.5.5) of the first point of Ref. [1], p. 355, but in a place of Lscal(Ψ) we get the formula (5.14.3). Simultaneously Ψ = Ψ(x, y), x ∈ E, y ∈ G/G0 and the metric on a space-time E depends on y ∈ G/G0, i.e. gμν = gμν(x, y), x ∈ E, y ∈ G/G0. (3) Thus gμν is parametrized by a point of G/G0. Simultaneously we can interpret a dependence on higher dimensions as an existence of a tower of scalar fields ΨK (see Eqs (5.14.4–8), Eqs (5.14.11–14) from the first point of Ref. [1], p. 434–436, and a discussion below). The interesting point will be to find physical consequences of this dependence for gμν . This can be achieved by considering cosmological solutions of the theory. Thus let us come to Eq. (5.5.5) of the first point of Ref. [1], p. 355, supposing the lagrangian of matter fields is written as Lmatter, gμν = gνμ and depends on y ∈ G/G0. We get L √−g √ |g̃| = √−g √ |g̃| ( R(Γ̃ ) + λ 4 eLmatter + λ 4 Lscal(Ψ) + 1 λ2 e R̃(Γ̃ ) + 1 r2 e P̃ ) . (4) According to the standard interpretation of the constant λ we have