Kaluza-Klein models: Can we construct a viable example?

In Kaluza-Klein models with toroidal compactification of the extra dimensions, we investigate soliton solutions of Einstein equation. The nonrelativistic gravitational potential of these solitons exactly coincides with the Newtonian one. We obtain the formulas for perihelion shift, deflection of light, time delay of radar echoes and post-Newtonian (PPN) parameters. Using the constraint on PPN parameter $\gamma$, we find that the solitonic parameter $k$ should be very big: $|k|\ge 2.3\times {10}^4$. We define a soliton solution which corresponds to a pointlike mass source. In this case the soliton parameter $k=2$, which is clearly contrary to this restriction. A similar problem with the observations takes place for static spherically symmetric perfect fluid with the dustlike equation of state in all dimensions. The common for both of these models is the same (dustlike) equations of state in our three dimensions and in the extra dimensions. All dimensions are treated at equal footing. This is the crucial point. To be in agreement with observations, it is necessary to break the symmetry (in terms of equations of state) between the external/our and internal spaces. It takes place for black strings which are particular examples of solitons with $k\to \infty$. For such $k$, black strings are in concordance with the observations. Moreover, we show that they are the only solitons which are at the same level of agreement with the observations as in general relativity. Black strings can be treated as perfect fluid with dustlike equation of state $p_0=0$ in the external/our space and very specific equation of state $p_1=-(1/2)\epsilon$ in the internal space. The latter equation is due to negative tension in the extra dimension. We also demonstrate that dimension 3 for the external space is a special one. Only in this case we get the latter equation of state. We show that the black string equations of state satisfy the necessary condition of the internal space stabilization. Therefore, black strings are good candidates for a viable model of astrophysical objects (e.g., Sun) if we can provide a satisfactory explanation of negative tension for particles constituting these objects.