Noncompactified Kaluza–Klein Gravity
Abstract
:1. Introduction
2. Five-Dimensional Ricci–Flat Space and the Effective Field Equations in Four Dimensions
3. Modified Scalar–Tensor Theories
3.1. Modified Sáez–Ballester Theory in Arbitrary Dimensions
- Let us first obtain a dynamical equation for the scalar field , i.e., an extended version of (10), which, in turn, will be applied to retrieve other modified equations. Letting and in Equation (14), we obtain
- Let us now construct the Einstein tensor on the hypersurface. Concretely, we want to retrieve the counterpart of (14) on a D-dimensional hypersurface. In this regard, by taking the SB theory into account, we first relate the Ricci scalars and :Substituting and from relations (3) and (5), respectively (where we respect the expressions presented in the Remarks) into Equation (24), and then using Equation (23), after some manipulations, we obtainNow, we proceed as follows. By substituting and from relations (3) and (25) intoLet us also introduce the effective matter in Equation (27):(i) is the effective matter induced from the -dimensional ordinary energy momentum tensor:Obviously, assuming a bulk without a higher-dimensional ordinary matter, i.e., , then vanishes.(ii) is an induced energy momentum tensor associated with our herein MSBT framework, which, in turn, has three components:Here is exactly the same quantity introduced in the previous section, see Equation (12).
- Finally, let us obtain the last equation, which is an extended version of (9). In this sense, substituting and into Equation (14), we obtain
3.2. Modified Brans–Dicke Theory in Arbitrary Dimensions
- An equation for the scalar field is:
- In Equation (43), the induced scalar potential is obtained from a differential equation, see Equation (48); the effective energy-momentum tensor consists of two parts: withWe should note that in Equation (45) is exactly the same induced matter introduced in the IMT, but is given by
- A counterpart to the conservation Equation (9) introduced in the IMT is:
4. FLRW-MSBT Cosmology
4.1. Case (I) D-Dimensional Solutions with
4.2. Case (II) D-Dimensional Solutions with
- It is straightforward to show that the corresponding conservation equation is satisfied for our herein three matter fields. Namely,
- If we assume that the induced matter plays the role of an ordinary matter in the universe, it is better to check at least the satisfaction of the weak energy condition for it. Let us be more specific. Relations (112) imply that for satisfying , m must take negative values. Moreover, in order to satisfy , r and m must take positive and negative values, respectively. Therefore, if increases and decreases with cosmic time, then the weak energy condition will be satisfied.
- It is seen that , which can also be fulfilled under approximation conditions with and/or .
- Let us now focus on Equation (120). Before continuing our discussions, we want to mention that we will focus on the solutions with . Moreover, we assume that the constant coefficients and that appeared in relations (74) and (75) always take positive values. Furthermore, we do not need to write in front of the equations since they apply to all n.In what follows, we present two different cases.Case IIa: ,For this case, we obtainMoreover, for this case, we find , namely, f takes positive as well as negative values. However, in order to have an expanding universe, relation (74) yields . Therefore, from using (74) and (122), we find . We eventually conclude that must be constrained asThe above conditions lead to a cosmological model with and . However, the extra dimension in such a model shrinks with cosmic time, which is favorable within the Kaluza–Klein frameworks [7].Let us also determine the allowed ranges of the energy density, pressure, and density parameters. According to conditions (122) and (123) together with (that is satisfied for ), we findMoreover, in this case, both the potential and the kinetic term take negative values forever.Case IIb: ,In this case, we obtainMoreover, we find , which implies that f takes both positive and negative values. Using the similar procedure mentioned above, we eventually obtain:Admitting the above constraints together with assumption , we can easily show that the inequalities (124) hold among the corresponding physical quantities in this case.
5. Conclusions and Discussion
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
BD | Brans–Dicke |
FLRW | Friedmann–Lemaître–Robertson–Walker |
GR | General Relativity |
IMT | Induced–Matter Theory |
KK | Kaluza–Klein |
MBDT | Modified Brans–Dicke Theory |
MSBT | Modified Sáez–Ballester Theory |
SB | Sáez–Ballester |
STM | Space–Time–Matter |
1 | To keep in touch with the original works discussed in each (sub)section, let us apply the same units contained within. For example, in this section we use the same units as in [55]. |
2 | It is worth noting that noncompact extra dimensions have also been adopted within compactified KK theory as an approach to incorporate chiral fermions into the theory and to organize a vanishing four-dimensional cosmological constant, see e.g., [57,58]. However, these frameworks have adapted the Klein’s mechanism of harmonic expansion, i.e., a finite volume has been assumed for the compact manifold [7]. |
3 | It is worth mentioning that in the IMT [55], an apparent vacuum bulk was taken into account. (In this paper, a ‘vacuum’ spacetime means that ‘ordinary matter’ does not exist.) As mentioned earlier, however, we will consider a non-vanishing energy momentum tensor, , to construct an extended version of the corresponding reduced framework. One is free to impose higher-dimensional ordinary matter fields that, in turn, invoke an intricate Kaluza–Klein framework. It should be emphasized that with such a procedure there is a risk that introduces numerous degrees of freedom, so that the corresponding model can no longer be tested. In this respect, we usually assume a vacuum bulk in cosmological applications. |
4 | In the original SB theory, scalar potential was not added to the action [47]. |
5 | In [9], the general coordinate free framework has been used to construct the MSBT. However, in this paper, we employ a different particular approach. |
6 | In –dimensions, assuming , it will be possible to go from the Jordan frame to the Einstein frame by conformal transformations. |
7 | |
8 |
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Rasouli, S.M.M.; Jalalzadeh, S.; Moniz, P. Noncompactified Kaluza–Klein Gravity. Universe 2022, 8, 431. https://doi.org/10.3390/universe8080431
Rasouli SMM, Jalalzadeh S, Moniz P. Noncompactified Kaluza–Klein Gravity. Universe. 2022; 8(8):431. https://doi.org/10.3390/universe8080431
Chicago/Turabian StyleRasouli, Seyed Meraj Mousavi, Shahram Jalalzadeh, and Paulo Moniz. 2022. "Noncompactified Kaluza–Klein Gravity" Universe 8, no. 8: 431. https://doi.org/10.3390/universe8080431