Kaluza-Klein multidimensional models with Ricci-flat internal spaces: the absence of the KK particles

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.


Introduction
It may turn out that our Universe is multidimensional and has more than three spatial dimensions. This fantastic idea was first proposed in the twenties of the last century in the papers by T. Kaluza and O. Klein [1,2], and since then it has consistently attracted attention of researchers. Obviously, to be compatible with our observable four-dimensional spacetime, the extra dimensions should be compact and sufficiently small. According to the recent accelerator experiments, their size should be of the order or less than the Fermi length 10 −17 cm 1 . If the internal space is very small, the hope to detect them is connected with Kaluza-Klein (KK) particles which correspond to appropriate eigenfunctions of the internal manifold, i.e. excitations of fields in the given internal space. Such excitations were investigated in many articles (see, e.g., the classical papers [5][6][7]). Quite recently, KK particles were considered, e.g., in the papers [8,9].
The internal compact space can have different topology. The Ricci-flat space is of special interest because Calabi-Yau manifolds, which are widely used in the superstring theory [10], belong to this class. A particular case of toroidal compactification was considered in our previous paper [11]. We demonstrate in this article that metric perturbations depend only on coordinates of the external spacetime, i.e. the corresponding KK particles are absent. In our approach, the metric perturbations are caused by a system of gravitating masses, e.g., by astrophysical objects. This is the main difference between our investigation and studies in [8,9] where the metric and form-field perturbations are considered without taking into account the reason of such fluctuations. Our analysis in [11] clearly shows that the inclusion of the matter sources, being responsible for the perturbations, imposes strong restrictions on the model. In the present paper, we investigate this important phenomenon in the case of Ricci-flat internal spaces and arrive again at the same conclusions. That is, KK modes corresponding to the metric fluctuations are absent. We also show that gravitating masses should be uniformly smeared over the internal space. From the point of quantum mechanics, it means that particles must be only in the ground state with respect to the internal space. Of course, it looks very unnatural. According to statistical physics, any nonzero temperature should result in fluctuations, i.e. in KK states.
The paper is structured as follows. In Sec. 2, we consider the multidimensional model with the Ricci-flat internal space. This background spacetime is perturbed by the system of gravitating masses. We demonstrate that the metric perturbations do not depend on the coordinates of the internal space and gravitating masses are uniformly smeared over the internal space. That is, KK modes are absent. In Sec. 3, we provide the exact expressions for the metric correction terms. The main results are briefly summarized in concluding Sec. 4.

Smeared extra dimensions
We consider a block-diagonal background metrics µν (µ, ν = 0, 1, 2, 3) is the metrics of the four-dimensional Minkowski spacetime, and g In what follows, X M (M = 0, 1, 2, . . . , D) are coordinates on M D , x µ = X µ (µ = 0, 1, 2, 3) are coordinates on M 4 and y m = X m (m = 4, 5, . . . , 3 + d) are coordinates on M d . We will also use the definitions for the spatial coordinatesX : XM (M = 1, 2, . . . , D) and x : xμ = Xμ (μ = 1, 2, 3). Now, we perturb the metrics (2.1) by a system of N discrete massive (with rest masses m p , p = 1, . . . , N ) bodies. We suppose that the pressure of these bodies in the external threedimensional space is much less than their energy density. This is a natural approximation for ordinary astrophysical objects such as our Sun. For example, in general relativity, this approach works well for calculating the gravitational experiments in the Solar system [12]. In other words, the gravitating bodies are pressureless in the external/our space. On the other hand, we suppose that they may have pressure in the extra dimensions. Therefore, nonzero components of the energy-momentum tensor of the system can be written in the form for a perfect fluid [11,12]: with the equation of state (EoS) in the internal space where ω is the parameter of EoS 2 and the rest-mass density is whereX p is a D-dimensional "radius-vector" of the p−th particle. In EoS, we include the factor ds/dx 0 just for convenience. We can also eliminate this factor from EoS and include it in the definition of the rest-mass density (2.5) (see, e.g., [11]). For our purpose, it is sufficient to consider a steady-state model 3 . That is, we disregard the spatial velocities of the gravitating masses.
In the weak-field approximation, the perturbed metric components can be written in the following form: Hereafter, the "hats" denote the unperturbed quantities. For example, the metric coefficients in (2.3)-(2.5) are already perturbed quantities.
To get the metric correction terms, we should solve (in the corresponding order) the multidimensional Einstein equation  .7) is reduced to the following system of equations (see [11] for details): where the Ricci-tensor components R M N [g (D) ] are determined by the equation (A.11).
Obviously, the solution of this system is determined by the rest-mass density ρ(X). Let us suppose, e.g., that the physically reasonable solution h 00 (X) of (2.8) exists 4 . Then, we can write it in the form It is clear that the function ϕ(X) describes the nonrelativistic gravitational potential created by the system of masses with the rest-mass density (2.5). From Eqs. (2.9) and (2.10) we get and It is worth noting that now we can easily determine (via the gravitational potential ϕ(X)) the functions ξ 1 , ξ 2 and ξ 3 introduced in (A.10). It is well known (see, e.g., [16]) that to be in agreement with the gravitational tests (the deflection of light and the Shapiro time-delay experiment) at the same level of accuracy as general relativity, the metric correction terms should satisfy the equalities (2.14) This results in the black string/brane condition [17,18] Because ω = 0, to satisfy this equation, we must demand that Hence, the gravitational potential is the function of the spatial coordinates of the external/our space: ϕ(X) = ϕ(x). Obviously, it takes place only if the rest-mass density also depends only on the spatial coordinates of the external space: ρ(X) = ρ(x). In other words, the gravitating masses should be uniformly smeared over the internal space. For short, we usually call such internal spaces "smeared" extra dimensions. This conclusion has the following important effect. Suppose that we have solved for the considered particle a multidimensional quantum Schrödinger equation and found its wave function Ψ(X). In general, this function depends on all spatial coordinatesX = (x, y), and we can expand it in appropriate eigenfunctions of the compact internal space, i.e. in the Kaluza-Klein (KK) modes. The ground state corresponds to the absence of these particles. In this state the wave function may depend only on the coordinatesx of the external space. The classical rest-mass density is proportional to the probability density |Ψ| 2 . Therefore, the demand, that the rest-mass density depends only on the coordinates of the external space, means that the particle can be only in the ground quantum state, and KK excitations are absent. This looks very unnatural from the point of quantum and statistical physics.

Metric components
Now, we want to get the expression for the metrics up to 1/c 2 terms. Since x 0 = ct, the component g 00 should be calculated up to O(1/c 4 ): . We remind that we consider the steady-state case: g 0M = 0. According to the results of the previous section, the gravitating masses are uniformly smeared over the internal space. Therefore, the rest-mass density (2.5) now reads | is the volume of the internal space and Here,x is a three-dimensional radius-vector in the external space. Additionally, it is clear that all prefactors ξ 1 , ξ 2 and ξ 3 in (A.10) are also functions of the external spatial coordinates x. For convenience, we redetermine the function ξ 3 : To get the metric correction terms, we should solve the Einstein equation (2.7). First, we consider the O(1/c 2 ) terms h M N . These terms satisfy the system (2.8)-(2.10) where we must perform the following replacements: volume of the unperturbed internal space, and −∇L∇L → △ = δμν∂ 2 /∂xμ∂xν. Introducing the definition h 00 ≡ 2ϕ/c 2 , we can easily get from this system is the Newtonian gravitational constant and ϕ(x) is the nonrelativistic gravitational potential created by N masses. At the same time Now, we intend to determine the f 00 ∼ O(1/c 4 ) correction term. Obviously, to do it, we need to solve only the 00-component of the Einstein equation (2.7). Keeping in mind the prefactor 1/c 4 in the right hand side of (2.7), it is clear that the energy-momentum tensor component T 00 and the trace T must be determined up to O(1). Therefore, similar to the paper [11], we get:  [15] and arrive at the following formula: (3.10) where △ = δμν∂ 2 /∂xμ∂xν is the 3-dimensional Laplace operator. Therefore, the 00-component of the Einstein equation reads where we used the equation △ϕ = 4πG N N p=1 m p δ(x −x p ). The function ϕ ′ (x p ) is the potential of the gravitational field at a point with the radius vector x p produced by all particles, except the p-th: and Therefore, the solution of Eq. (3.11) is (3.14) Obviously, in the case of the toroidal internal space, this section reproduces the results of our paper [11]. The found metric correction terms enable to calculate the gravitational tests (the deflection of light, the time-delay of the radar echoes and the perihelion advance) in full analogy with the paper [18].

Conclusion
In our paper, we have studied the multidimensional KK model with the Ricci-flat internal space. Such models are of special interest because Calabi-Yau manifolds, which are widely used in the superstring theory [10], belong to this class. In the absence of matter sources, the background manifold is a direct product of the four-dimensional Minkowski spacetime and the Ricci-flat internal space. Then, we perturbed this background metrics by a system of gravitating masses, e.g., astrophysical objects such as our Sun. It is well known that pressure inside of these objects is much less than the energy density. Therefore, we supposed that gravitating masses are pressureless in the external space. However, they may have relativistic pressure in the internal space. Moreover, the presence of such pressure is the necessary condition for agreement with the gravitational tests (the deflection of light, the time-delay of the radar echoes and the perihelion advance) [16][17][18]. We clearly demonstrated that metric perturbations do not depend on the coordinates of the internal space and gravitating masses are uniformly smeared over the internal space. This means, first, that KK modes corresponding to the metric fluctuations are absent and, second, the particles corresponding to the gravitating masses should be only in the ground quantum state with respect to the internal space.
Of course, it looks very unnatural. According to statistical physics, any nonzero temperature should result in fluctuations, i.e. in KK modes. On the other hand, the conclusion about the absence of KK particles is consistent with the results of [19,20], where a mechanism for the symmetries formation of the internal spaces was proposed. According to this mechanism, due to the entropy decreasing in the compact subspace, its metric undergoes the process of symmetrization during some time after its quantum nucleation. A high symmetry of internal spaces actually means suppression of the extra degrees of freedom. We also obtained the formulae for the metric correction terms. For the 00-component, we got it up to O(1/c 4 ). We found other components up to O(1/c 2 ). This enables to calculate the gravitational tests: the deflection of light, the time-delay of the radar echoes and the perihelion advance. It also gives a possibility to construct the Lagrange function for the considered many-body system (see, e.g., [11]).
and for the Ricci tensor we get where we introduced a new tensor To get the last expression, we used the relation (91.8) in [12]. The gauge conditions (see, e.g., the paragraph 105 in [12])∇ simplify considerably this expression: Typically, the matter, which perturbs the background metrics, is taken in the form of a perfect fluid with isotropic pressure. In the case of the product manifold, the pressure is isotropic in each factor manifolds (see, e.g., the energy-momentum tensor (2.3)). Obviously, such perturbation does not change the topology of the model, e.g., the topologies of the factor manifolds in the case of the product manifold. Additionally, it preserves also the block-diagonal structure of the metric tensor. In the case of a steady-state model (our case) the non-diagonal perturbations h 0M (M = 1, 2, 3, . . . , D) are also absent. Therefore, the metric correction terms are conformal to the background metrics: whereL = 1, 2, . . . , D and we took into account that X 0 = ct. 5 It is worth noting that, as it follows from Eq. (A.8), QMN is also equal to zero when the following condition holds:R L NP M = −R L M P N , resulting in the Ricci-flat spaceRMN = 0 (but not vice versa).