Kaluza-Klein Multidimensional Models with Ricci-Flat Internal Spaces: The Absence of the KK Particles

Abstract

We consider a multidimensional Kaluza-Klein (KK) model with a Ricci-flat internal space, for example, a Calabi-Yau manifold. We perturb this background metrics by a system of gravitating masses, for example, 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, that is, in KK modes. We also get formulae for the metric correction terms which enable us to calculate the gravitational tests: the deflection of light, the time-delay of the radar echoes, and the perihelion advance.

1. 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 Kaluza and 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. (In the brane-world models, the extra dimensions can be significantly larger [3, 4]. We do not consider such models in our paper.) 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, that is, excitations of fields in the given internal space. Such excitations were investigated in many articles (see, e.g., the classical papers [5–7]). Quite recently, KK particles were considered, for example, 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 paper that metric perturbations depend only on coordinates of the external spacetime; that is, the corresponding KK particles are absent. In our approach, the metric perturbations are caused by a system of gravitating masses, for example, 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, this 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, that is, in KK states.

The paper is structured as follows. In Section 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 Section 3, we provide the exact expressions for the metric correction terms. The main results are briefly summarized in concluding Section 4.

2. Smeared Extra Dimensions

We consider a block-diagonal background metrics, 𝑔^(𝒟)_𝑀𝑁𝑦=𝜂⁽⁴⁾_𝜇𝜈⊕𝑔^(𝑑)_𝑚𝑛𝑦, (1) which is determined on (𝒟=1+𝐷=4+𝑑)-dimensional product manifold 𝔐_𝒟=𝔐₄×𝔐_𝑑. Here, 𝜂⁽⁴⁾_𝜇𝜈(𝜇,𝜈=0,1,2,3) is the metrics of the four-dimensional Minkowski spacetime and 𝑔^(𝑑)_𝑚𝑛(𝑦)(𝑚,𝑛=4,…,𝐷) is the metrics of some 𝑑-dimensional internal (usually compact) Ricci-flat space with coordinates 𝑦≡𝑦^𝑚: 𝑅^(𝑑)_𝑚𝑛𝑔^(𝑑)_𝑚𝑛=0. (2)

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 𝔐₄, and 𝑦^𝑚=𝑋^𝑚(𝑚=4,5,…,3+𝑑) are coordinates on 𝔐_𝑑. We will also use the definitions for the spatial coordinates 𝑋∶𝑋𝑀(𝑀=1,2,…,𝐷) and 𝑥∶𝑥𝜇=𝑋𝜇(𝜇=1,2,3).

Now, we perturb the metrics (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 three-dimensional 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 following form for a perfect fluid [11, 12]: 𝑇^𝑀𝜈=𝜌𝑐²𝑑𝑠𝑑𝑥⁰𝑢^𝑀𝑢^𝜈,𝑢^𝑀=𝑑𝑋^𝑀,𝑇𝑑𝑠_𝑚𝑛=−𝑝_𝑚𝑔^𝑚𝑛+𝜌𝑐²𝑑𝑠𝑑𝑥⁰𝑢^𝑚𝑢^𝑛, (3) with the equation of state (EoS) in the internal space, 𝑝_𝑚=𝜔𝜌𝑐²𝑑𝑠𝑑𝑥⁰,𝑚=4,5,…,𝐷, (4) where 𝜔 is the parameter of EoS (we should note that the pressure (4) is the intrinsic pressure of a gravitating mass in the extra dimensions and is not connected with its motion; in other words, 𝜔 is the parameter of a particle) and the rest-mass density is 𝜌≡^𝑁_𝑝=1𝑔^(𝒟)−1/2𝑚_𝑝𝛿𝐗−𝐗_𝑝, (5) where 𝐗_𝑝 is a 𝐷-dimensional “radius vector” of the 𝑝th particle. In EoS, we include the factor 𝑑𝑠/𝑑𝑥⁰ just for convenience. We can also eliminate this factor from EoS and include it in the definition of the rest-mass density (5) (see, e.g., [11]). For our purpose, it is sufficient to consider a steady-state model. (For example, we can consider just one gravitating mass and place the origin of a reference frame on it.) 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: 𝑔_𝑀𝑁≈𝑔_𝑀𝑁+ℎ_𝑀𝑁,𝑔^𝑀𝑁≈𝑔^𝑀𝑁−ℎ^𝑀𝑁, (6) where ℎ_𝑀𝑁∼𝑂(1/𝑐²) and ℎ^𝑁_𝑀≡𝑔^𝑁𝑇ℎ_𝑀𝑇. Hereafter, the “hats” denote the unperturbed quantities. For example, the metric coefficients in (3)–(5) are already perturbed quantities.

To get the metric correction terms, we should solve (in the corresponding order) the multidimensional Einstein equation 𝑅_𝑀𝑁=2𝑆_𝐷𝐺_𝒟𝑐⁴𝑇_𝑀𝑁−1𝐷−1𝑇𝑔_𝑀𝑁, (7) where 𝑆_𝐷=2𝜋^𝐷/2/Γ(𝐷/2) is the total solid angle (the surface area of the (𝐷−1)-dimensional sphere of the unit radius) and 𝐺_𝒟 is the gravitational constant in the (𝒟=𝐷+1)-dimensional spacetime. For our model and up to the 𝑂(1/𝑐²) terms, the Einstein equation (7) is reduced to the following system of equations (see [11] for details): 𝑅₀₀𝑔^(𝒟)1≈−2∇𝐿∇𝐿ℎ₀₀=2𝑆_𝐷𝐺_𝒟𝑐²𝐷−2+𝜔𝑑𝑅𝜇𝜈𝑔𝐷−1𝜌,𝐿=1,2,…,𝐷,_(𝒟)1≈−2∇𝐿∇𝐿ℎ𝜇𝜈=2𝑆_𝐷𝐺_𝒟𝑐²1−𝜔𝑑𝜇𝜈,𝑅𝐷−1𝜌𝛿𝜇,𝜈=1,2,3,_𝑚𝑛𝑔^(𝒟)1≈−2∇𝐿∇𝐿ℎ_𝑚𝑛=−2𝑆_𝐷𝐺_𝒟𝑐²𝜔(𝐷−1)+1−𝜔𝑑𝜌𝑔𝐷−1^𝑚𝑛, (8)(9)(10) where the Ricci-tensor components 𝑅_𝑀𝑁[𝑔^(𝒟)] are determined by (A.11).

Obviously, the solution of this system is determined by the rest-mass density 𝜌(𝑋). Let us suppose, for example, that the physically reasonable solution ℎ₀₀(𝑋) of (8) exists. (The exact form of such solution depends on the topology of the internal space. For example, in the case of toroidal internal spaces with the periodic boundary conditions in the directions of the extra dimensions, the corresponding solutions can be found in [13, 14]. However, as we will see later, the exact form of 𝐷-dimensional solutions is not important for our purpose. A similar situation took place in our paper [15] where this point was discussed in detail after Equation (2.56).) Then, we can write it in the form ℎ₀₀𝑋≡2𝜑𝑋𝑐². (11)

It is clear that the function 𝜑(𝑋) describes the nonrelativistic gravitational potential created by the system of masses with the rest-mass density (5). From (9) and (10) we get ℎ𝜇𝜈=1−𝜔𝑑ℎ𝐷−2+𝜔𝑑⁰⁰𝛿𝜇𝜈,ℎ_𝑚𝑛𝜔()=−𝐷−1+1−𝜔𝑑ℎ𝐷−2+𝜔𝑑⁰⁰𝑔_𝑚𝑛. (12)

It is worth noting that now we can easily determine (via the gravitational potential 𝜑(𝑋)) the functions 𝜉₁, 𝜉₂, and 𝜉₃ 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 ℎ₀₀=ℎ₁₁=ℎ₂₂=ℎ₃₃. (13)

This results in the black string/brane condition [17, 18]: 1𝜔=−2. (14)

Thus, to be in agreement with observations, this parameter should not be negligibly small; that is, 𝜔∼𝑂(1).

The solutions (11)-(12) should satisfy the gauge condition (A.7). Taking into account that ℎ=𝑔^𝑀𝑁ℎ_𝑀𝑁=[2(𝜔𝑑−1)/(𝐷−2+𝜔𝑑)]ℎ₀₀, we can easily get that for 𝑁=𝜈(𝜈=0,1,2,3) this condition is automatically satisfied. However, in the case 𝑁=𝑛(𝑛=4,5,…,𝐷) we arrive at the condition ∇_𝐿𝑔^𝐿𝑇ℎ_𝑇𝑛−12𝜕_𝑛𝜔()ℎ=−𝐷−1𝜕𝐷−2+𝜔𝑑^𝑛ℎ₀₀=0. (15) Because 𝜔≠0, to satisfy this equation, we must demand that 𝜕_𝑛ℎ₀₀=2𝑐²𝜕_𝑛𝜑𝑋=0. (16)

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. In 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 coordinates 𝑋=(𝑥,𝑦), and we can expand it in appropriate eigenfunctions of the compact internal space, that is, 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 coordinates 𝑥 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 viewpoint of quantum and statistical physics.

3. Metric Components

Now, we want to get the expression for the metrics up to 1/𝑐² terms. Since 𝑥⁰=𝑐𝑡, the component 𝑔₀₀ should be calculated up to 𝑂(1/𝑐⁴): 𝑔₀₀≈𝜂₀₀+ℎ₀₀+𝑓₀₀, (17) where ℎ₀₀∼𝑂(1/𝑐²) and 𝑓₀₀∼𝑂(1/𝑐⁴). For the rest metric coefficients, we still have (6): 𝑔𝑀𝑁≈𝑔𝑀𝑁𝑀𝑁+ℎ, ℎ𝑀𝑁∼𝑂(1/𝑐²). We recall that we consider the steady-state case: 𝑔₀𝑀=0.

According to the results of the previous section, the gravitating masses are uniformly smeared over the internal space. Therefore, the rest-mass density (5) now reads 1𝜌=𝑉_𝑑^𝑁_𝑝=1𝑚_𝑝𝛿𝐱−𝐱_𝑝−𝑔⁽⁴⁾=1𝑉_𝑑𝜚₃−𝑔⁽⁴⁾, (18) where 𝑉_𝑑=∫_𝔐𝑑𝑑^𝑑𝑦|𝑔^(𝑑)| is the volume of the internal space and 𝜚₃(𝑥)≡^𝑁_𝑝=1𝑚_𝑝𝛿𝐱−𝐱_𝑝. (19) Here, 𝐱 is a three-dimensional radius vector in the external space. Additionally, it is clear that all prefactors 𝜉₁, 𝜉₂, and 𝜉₃ in (A.10) are also functions of the external spatial coordinates 𝑥. For convenience, we redetermine the function 𝜉₃∶𝜉₃(𝑥)≡𝛼(𝑥). Then, ℎ_𝑚𝑛(𝑥,𝑦)=𝛼(𝑥)𝑔_𝑚𝑛(𝑦), where 𝛼(𝑥)∼𝑂(1/𝑐²).

To get the metric correction terms, we should solve the Einstein equation (7). First, we consider the 𝑂(1/𝑐²) terms ℎ_𝑀𝑁. These terms satisfy the system (8)–(10), where we must perform the following replacements: 𝜌→𝜚₃/𝑉_𝑑, where 𝑉_𝑑=∫_𝔐𝑑𝑑^𝑑𝑦|𝑔^(𝑑)| is the volume of the unperturbed internal space and −∇𝐿∇𝐿𝜇𝜈→Δ=𝛿𝜕²/𝜕𝑥𝜇𝜕𝑥𝜈. Introducing the definition ℎ₀₀≡2𝜑/𝑐², we can easily get from this system 𝜑=−𝐺_𝑁^𝑁_𝑝=1𝑚_𝑝𝐱−𝐱_𝑝, (20) where 𝐺_𝑁=𝑆_𝐷(𝐷−2+𝜔𝑑)2𝜋𝑉_𝑑()𝐺𝐷−1^𝒟 (21) is the Newtonian gravitational constant and 𝜑(𝑥) is the nonrelativistic gravitational potential created by 𝑁 masses. At the same time ℎ𝜇𝜈=1−𝜔𝑑ℎ𝐷−2+𝜔𝑑⁰⁰𝛿𝜇𝜈,ℎ_𝑚𝑛𝑥,𝑦=𝛼(𝑥)𝑔_𝑚𝑛𝑦,𝜔()𝛼=−𝐷−1+1−𝜔𝑑ℎ𝐷−2+𝜔𝑑⁰⁰. (22)

Now, we intend to determine the 𝑓₀₀∼𝑂(1/𝑐⁴) correction term. Obviously, to do it, we need to solve only the 00-component of the Einstein equation (7). Keeping in mind the prefactor 1/𝑐⁴ in the right hand side of (7), it is clear that the energy-momentum tensor component 𝑇₀₀ and the trace 𝑇 must be determined up to 𝑂(1). Therefore, similar to the paper [11], we get 𝑇₀₀≈𝜚₃𝑐²𝑉_𝑑1+3𝐷−4+𝜔𝑑2()ℎ𝐷−2+𝜔𝑑⁰⁰,𝜚𝑇≈³𝑐²𝑉_𝑑1+𝐷−𝜔𝑑2()ℎ𝐷−2+𝜔𝑑⁰⁰(1−𝜔𝑑). (23)

To get the Ricci-tensor component 𝑅₀₀ up to 𝑂(1/𝑐⁴), we can use (B.2) and Equations (2.29), (2.38) and (2.39) in [15] and arrive at the following formula: 𝑅₀₀≈12Δ𝑓₀₀−2𝑐⁴𝜑²+1𝑐²+Δ𝜑𝐷−12𝐷−2+𝜔𝑑𝑐⁴𝜑Δ𝜑, (24) where Δ=𝛿𝜇𝜈𝜕²/𝜕𝑥𝜇𝜕𝑥𝜈 is the 3-dimensional Laplace operator. Therefore, the 00-component of the Einstein equation reads 12Δ𝑓₀₀−2𝑐⁴𝜑²=1𝑐⁴=𝜑Δ𝜑4𝜋𝐺_𝑁𝑐^4𝑁_𝑝=1𝑚_𝑝𝜑^′𝐱_𝑝𝛿𝐱−𝐱_𝑝, (25) where we used the equation Δ𝜑=4𝜋𝐺_𝑁∑^𝑁_𝑝=1𝑚_𝑝𝛿(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 𝑝th: 𝜑^′𝐱_𝑝=_𝑞≠𝑝𝜑_𝑞𝐱_𝑝−𝐱_𝑞,Δ𝜑_𝑝𝐱−𝐱_𝑝=4𝜋𝐺_𝑁𝑚_𝑝𝛿𝐱−𝐱_𝑝. (26)

Therefore, the solution of (25) is 𝑓₀₀(𝐱)=2𝑐⁴𝜑²(𝐱)+2𝑐^4𝑁_𝑝=1𝜑_𝑝𝐱−𝐱_𝑝𝜑^′𝐱_𝑝. (27)

Obviously, in the case of the toroidal internal space, this section reproduces the results of our paper [11]. The found metric correction terms enable us 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].

4. 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, for example, astrophysical objects such as our Sun. It is well known that pressure inside 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) [16–18]. 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, that is, 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/𝑐⁴). We found other components up to 𝑂(1/𝑐²). This enables us 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]).

Appendices

A. Ricci Tensor in the Weak-Field Approximation

In our paper we use the signature of the metrics (+,−,−,−,…) and determine the Ricci-tensor components as follows: 𝑅_𝑀𝑁≡𝑅^𝑆_𝑀𝑆𝑁=𝜕_𝐿Γ^𝐿_𝑀𝑁−𝜕_𝑁Γ^𝐿_𝑀𝐿+Γ^𝐿_𝑀𝑁Γ^𝑆_𝐿𝑆−Γ^𝑆_𝑀𝐿Γ^𝐿_𝑁𝑆, (A.1) where the Christoffel symbols are Γ^𝐿_𝑀𝑁=12𝑔^𝐿𝑇−𝜕_𝑇𝑔_𝑀𝑁+𝜕_𝑀𝑔_𝑁𝑇+𝜕_𝑁𝑔_𝑀𝑇,𝑀,𝑁,𝐿,…=0,1,2,…,𝐷. (A.2)

The metrics 𝑔^(𝒟)_𝑀𝑁≡𝑔_𝑀𝑁 is determined on (𝒟=1+𝐷)-dimensional manifold 𝔐_𝒟.

Now, we suppose that the metric coefficients 𝑔_𝑀𝑁 correspond to the perturbed metrics, and in the weak-field approximation they can be decomposed as 𝑔_𝑀𝑁≈𝑔_𝑀𝑁+ℎ_𝑀𝑁,𝑔^𝑀𝑁≈𝑔^𝑀𝑁−ℎ^𝑀𝑁, (A.3) where 𝑔_𝑀𝑁 is the background unperturbed metrics and the metric perturbations ℎ_𝑀𝑁∼𝑂(1/𝑐²) (𝑐 is the speed of light). As usual, ℎ^𝑁_𝑀≡𝑔^𝑁𝑇ℎ_𝑇𝑀. Here and after, the “hats” denote unperturbed quantities. Then, up to linear terms ℎ, the Christoffel symbols read Γ^𝐿_𝑀𝑁≈Γ^𝐿_𝑀𝑁+12𝑔^𝐿𝑇−𝜕_𝑇ℎ_𝑀𝑁+𝜕_𝑀ℎ_𝑁𝑇−Γ^𝑃_𝑀𝑁ℎ_𝑃𝑇+𝜕_𝑁ℎ_𝑀𝑇−Γ^𝑃_𝑁𝑀ℎ_𝑃𝑇=Γ^𝐿_𝑀𝑁+12𝑔^𝐿𝑇−∇_𝑇ℎ_𝑀𝑁+∇_𝑀ℎ_𝑁𝑇+∇_𝑁ℎ_𝑀𝑇, (A.4) and for the Ricci tensor we get 𝑅_𝑀𝑁≈𝑅_𝑀𝑁+12−∇_𝐿∇^𝐿ℎ_𝑀𝑁+𝑄_𝑀𝑁, (A.5) where we introduced a new tensor: 𝑄_𝑀𝑁≡∇_𝐿∇_𝑀ℎ^𝐿_𝑁+∇_𝐿∇_𝑁ℎ^𝐿_𝑀−∇_𝑁∇_𝑀ℎ^𝐿_𝐿=∇_𝑀∇_𝐿ℎ^𝐿_𝑁−12𝜕_𝑁ℎ^𝐿_𝐿+∇_𝑁∇_𝐿ℎ^𝐿_𝑀−12𝜕_𝑀ℎ^𝐿_𝐿−𝑅^𝐿_𝑁𝑃𝑀+𝑅^𝐿_𝑀𝑃𝑁ℎ^𝑃_𝐿+𝑅_𝑃𝑀ℎ^𝑃_𝑁+𝑅_𝑃𝑁ℎ^𝑃_𝑀. (A.6)

To get the last expression, we used the relation (91.8) in [12]. The gauge conditions (see, e.g., the paragraph  105 in [12]) ∇_𝐿ℎ^𝐿_𝑁−12𝜕_𝑁ℎ^𝐿_𝐿=0 (A.7) simplify considerably this expression: 𝑄_𝑀𝑁=−𝑅^𝐿_𝑁𝑃𝑀+𝑅^𝐿_𝑀𝑃𝑁ℎ^𝑃_𝐿+𝑅_𝑃𝑀ℎ^𝑃_𝑁+𝑅_𝑃𝑁ℎ^𝑃_𝑀. (A.8)

Now, we suppose that the manifold 𝔐_𝒟 is a product manifold 𝔐_𝒟=𝔐₄×𝔐_𝑑 with a block-diagonal background metrics of the form (1). However, in this appendix we do not assume that the internal manifold 𝔐_𝑑 is a Ricci-flat one. For the metrics (1) and an arbitrary internal space 𝔐_𝑑, the only nonzero components of the Riemann tensor 𝑅_{𝐿𝑁𝑀𝑃}[𝑔^(𝒟)] and the Ricci tensor 𝑅_𝑁𝑀[𝑔^(𝒟)] are 𝑅_{𝑙𝑛𝑚𝑝}𝑔^(𝒟)=𝑅_{𝑙𝑛𝑚𝑝}𝑔^(𝑑),𝑅_𝑚𝑛𝑔^(𝒟)=𝑅_𝑚𝑛𝑔^(𝑑). (A.9)

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 manifold (see, e.g., the energy-momentum tensor (3)). Obviously, such perturbation does not change the topology of the model, for example, 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 nondiagonal perturbations ℎ₀𝑀  (𝑀=1,2,3,…,𝐷) are also absent. Therefore, the metric correction terms are conformal to the background metrics: ℎ₀₀=𝜉₁𝑔^(𝒟)₀₀=𝜉₁𝜂₀₀,ℎ𝜇𝜈=𝜉₂𝑔𝜇𝜈^(𝒟)=−𝜉₂𝛿𝜇𝜈,ℎ𝜇,𝜈=1,2,3,_𝑚𝑛=𝜉₃𝑔^(𝒟)_𝑚𝑛=𝜉₃𝑔^(𝑑)_𝑚𝑛, (A.10) where 𝜉_1,2,3 are some scalar functions of all spatial coordinates 𝑋. It can be easily verified that under these conditions (it is worth noting that, as it follows from (A.8), 𝑄_𝑀𝑁 is also equal to zero when the following condition holds: 𝑅^𝐿_𝑁𝑃𝑀=−𝑅^𝐿_𝑀𝑃𝑁, resulting in the Ricci-flat space 𝑅_𝑀𝑁=0 (but not vice versa)) the tensor 𝑄_𝑀𝑁 in (A.8) is equal to zero: 𝑄_𝑀𝑁=0. Therefore, up to 𝑂(1/𝑐²) terms, we get 𝑅_𝑀𝑁𝑔^(𝒟)≈𝑅_𝑀𝑁𝑔^(𝒟)−12∇_𝐿∇^𝐿ℎ_𝑀𝑁=𝑅_𝑀𝑁𝑔^(𝒟)−12∇𝐿∇𝐿ℎ_𝑀𝑁1+𝑜𝑐², (A.11) where 𝐿=1,2,…,𝐷 and we took into account that 𝑋⁰=𝑐𝑡.

B. Ricci Tensor for a Block-Diagonal Metrics

In this appendix we consider a block-diagonal metrics: 𝑔^(𝒟)_𝑀𝑁𝑥,𝑦=𝑔^(𝑑1)_𝜇𝜈(𝑥)⊕𝑔^(𝑑2)_𝑚𝑛=𝑔𝑥,𝑦_{(𝑑1)𝜇𝜈}(𝑥)⊕𝑒^𝛼(𝑥)𝑔^(𝑑2)_𝑚𝑛𝑦. (B.1)

Then, for the Ricci-tensor components we get the following expressions: 𝑅_𝜇𝜈𝑔_𝑀𝑁=𝑅_𝜇𝜈𝑔^(𝑑1)_𝜆𝜏−𝑑₂2∇^(𝑑1)_𝜇𝜕_𝜈𝛼+12𝜕_𝜇𝛼𝜕_𝜈𝛼,𝑅_𝑚𝜈𝑔_𝑀𝑁𝑅=0,_𝑚𝑛𝑔_𝑀𝑁=𝑅_𝑚𝑛𝑔^(𝑑2)_𝑘𝑙−12𝑒^𝛼(𝑥)𝑔^(𝑑2)_𝑚𝑛𝑔^{(𝑑1)𝜆𝜏}×∇^(𝑑1)_𝜆𝜕_𝜏𝛼+𝑑₂2𝜕_𝜆𝛼𝜕_𝜏𝛼. (B.2)(B.3)(B.4)

Acknowledgments

This work was supported by the “Cosmomicrophysics-2” programme of the Physics and Astronomy Division of the National Academy of Sciences of Ukraine. The work of Maxim Eingorn was supported by NSF CREST Award HRD-0833184 and NASA Grant NNX09AV07A.