We investigate the classical gravitational tests for the six-dimensional Kaluza-Klein model with spherical (of a radius a) compactification of the internal space. The model contains also a bare multidimensional cosmological constant Λ6. The matter, which corresponds to this ansatz, can be simulated by a perfect fluid with the vacuum equation of state in the external space and an arbitrary equation of state with the parameter ω1 in the internal space. For example, ω1 = 1 and ω1 = 2 correspond to the monopole two-forms and the Casimir effect, respectively. In the particular case Λ6 = 0, the parameter ω1 is also absent: ω1 = 0. In the weak-field approximation, we perturb the background ansatz by a point-like mass. We demonstrate that in the case ω1 > 0 the perturbed metric coefficients have the Yukawa type corrections with respect to the usual Newtonian gravitational potential. The inverse square law experiments restrict the parameters of the model: a/ √ ω1 6 × 10 −3 cm. Therefore, in the Solar system the parameterized post-Newtonian parameter γ is equal to 1 with very high accuracy. Thus, our model satisfies the gravitational experiments (the deflection of light and the time delay of radar echoes) at the same level of accuracy as General Relativity. We demonstrate also that our background matter provides the stable compactification of the internal space in the case ω1 > 0. However, if ω1 = 0, then the parameterized post-Newtonian parameter γ = 1/3, which strongly contradicts the observations.
In the six-dimensional Kaluza-Klein model with the multidimensional cosmological constant Λ6, we obtain the black brane with spherical compactification of the internal space. The matter source for this exact solution consists of two parts. First, it is a fine-tuned homogeneous perfect fluid which provides spherical compactification of the internal space. Second, it is a gravitating massive body with the dustlike equation of state in the external space and tensionp1 = −(1/2)ε in the internal space. This solution exists both in the presence and absence of Λ6. In the weak-field approximation, we also get solutions of the linearized Einstein equations for the model with spherical compactification. Here, the gravitating matter source has the dustlike equation of state in the external space and an arbitrary equation of statep1 = Ωε in the internal space. In the case Λ6 > 0 and Ω = −1/2, these approximate solutions tend asymptotically to the weak-field limit of the exact black brane solution. Both the exact and asymptotic black branes satisfy the gravitational experiments at the same level of accuracy as general relativity.
In this paper, we consider a system of gravitating bodies in Kaluza-Klein models with toroidal compactification of the extra dimensions. To simulate the astrophysical objects (e.g., our Sun and pulsars) with energy density much greater than the pressure, we assume that these bodies are pressureless in the external space, i.e., the space we inhabit. At the same time, they may have nonzero parameters ω (ᾱ−3) (ᾱ = 4, . . . , D) in the equations of state in the extra dimensions. We construct the Lagrange function of this many-body system for any value of = ᾱ ω (ᾱ−3) . Moreover, the gravitational tests (PPN parameters, perihelion and periastron advances) require a negligible deviation from the latent soliton value = −(D −3)/2. However, the presence of pressure/tension in the internal space results necessarily in the smearing of the gravitating masses over the internal space and in the absence of KK modes. This looks very unnatural from the point of view of quantum physics.
In this paper we consider a multidimensional Kaluza-Klein (KK) model with a Ricci-flat internal space, e.g., a Calabi-Yau manifold. We perturb this background metrics by a system of gravitating masses, e.g., astrophysical objects such as our Sun. We suppose that these masses are pressureless in the external space but they have relativistic pressure in the internal space. We show that metric perturbations do not depend on coordinates of the internal space and gravitating masses should be uniformly smeared over the internal space. This means, first, that KK modes corresponding to the metric fluctuations are absent and, second, particles should be only in the ground quantum state with respect to the internal space. In our opinion, these results look very unnatural. According to statistical physics, any nonzero temperature should result in fluctuations, i.e. in KK modes. We also get formulae for the metric correction terms which enable to calculate the gravitational tests: the deflection of light, the time-delay of the radar echoes and the perihelion advance.
We consider Kaluza-Klein (KK) models where internal spaces are compact Einstein spaces. These spaces are stabilized by background matter (e.g., monopole form-fields). We perturb this background by a compact matter source (e.g., the system of gravitating masses) with the zero pressure in the external/our space and an arbitrary pressure in the internal space. We show that the Einstein equations are compatible only if the matter source is smeared over the internal space and perturbed metric components do not depend on coordinates of extra dimensions. The latter means the absence of KK modes corresponding to the metric fluctuations. Maybe, the absence of KK particles in LHC experiments is explained by such mechanism.
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