Kaluza-Klein Metric Research Articles

Dark energy cosmological model with anisotropic fluid and time varying lambda in Kaluza-Klein metric

Dark energy cosmological model with anisotropic fluid and time varying lambda in Kaluza-Klein metric

Some calculations on Kaluza-Klein metric with respect to lifts in tangent bundle

In the present paper, a Riemannian metric on the tangent bundle, which is another generalization of Cheeger-Gromoll metric and Sasaki metric, is considered. This metric is known as Kaluza-Klein metric in literature which is completely determined by its action on vector fields of type X^{H} and Y^{V}. We obtain the covarient and Lie derivatives applied to the Kaluza-Klein metric with respect to the horizontal and vertical lifts of vector fields, respectively on tangent bundle TM.

Symmetrically Harmonic Kaluza-Klein Metrics on Tangent Bundles

Let (M, g) be a Riemannian manifold and G be a Kaluza-Klein metric on its tangent bundle TM. A metric H on TM is said to be symmetrically harmonic to G if the metrics G and H are harmonic w.r.t. each other; that is the identity maps id: (TM,G) → (TM,H) and id: (TM,H) → (TM,G) are both harmonic maps. In this work we study Kaluza-Klein metrics H on TM which are symmetrically harmonic to G. In particular, we characterize and determine horizontally and vertically conformal Kaluza-Klein metrics H on TM, which are symmetrically harmonic to G.

Boundedness of the number of nodal domains for eigenfunctions of generic Kaluza–Klein 3-folds

This article concerns the number of nodal domains of eigenfunctions of the Laplacian on special Riemannian $3$-manifolds, namely nontrivial principal $S^1$ bundles $P \to X$ over Riemann surfaces equipped with certain $S^1$ invariant metrics, the Kaluza-Klein metrics. We prove for generic Kaluza-Klein metrics that any Laplacian eigenfunction has exactly two nodal domains unless it is invariant under the $S^1$ action. We also construct an explicit orthonormal eigenbasis on the flat $3$-torus $\mathbb{T}^3$ for which every non-constant eigenfunction belonging to the basis has two nodal domains.

Dark energy–dark matter interactive model with time-varying lambda in Kaluza–Klein metric

The dark energy (DE)–dark matter (DM) interaction in cosmology is modeled considering Kaluza–Klein metric (five-dimensional FRW (Friedman–Robertson–Walker) metric) with time-varying cosmological constant (Λ). Λ is considered inversely proportional to square of the scalar factor (R−2). An interaction term (Q) introduced between DE and DM is proportional to dark matter density (ρm) and Hubble’s constant H, which facilitates analysis of energy transfer from DE to DM. Other physical parameters are also obtained and studied for flat and non-flat universe. The ratio of dark energy to dark matter density being a constant enables to deal with the cosmological coincidence problem. Phantom cross-behavior is also studied with this model.

Geometrical unification of gravitation and electromagnetism

A theory which achieves a complete geometrical unification of gravitation and electromagnetism (GUGE) is presented. This new theory is based on a recent proposal of proper time redefinition that leads to the construction of a Riemann metric, which naturally unifies gravity and electromagnetism. The 5--dimensional Riemann metric which arises looks exactly the same as the one postulated in the Kaluza-Klein (KK) 5-dimensional theory. Nevertheless, there are deep differences between GUGE and KK. The GUGE metric is deduced while the KK metric is postulated. In the GUGE field theory there is no need to impose either of the so called "cylinder" or "curling of coordinates" conditions, because they emerge as direct consequences of the construction of the GUGE metric. The GUGE field equations are fully equivalent to Einstein--Maxwell equations, while KK field equations are not. The GUGE 5--dimensional (geodesic) equations are equivalent to the 4--dimensional (non--geodesic) equations for a charged particle moving in the presence of gravitational and electromagnetic fields, unlike the KK 5--dimensional (geodesic) equations which are not and, in addition, yield non--gauge covariant geodesic equations. No extra scalar field appears in GUGE. The physical interpretation of the fifth dimension and of the role of the extra field in KK (internal coordinate in GUGE) are totally different in both approaches. Finally, GUGE results include electric charge conservation, electric charge quantization and electric charge contribution to the energy of charged particles even in the absence of electromagnetic fields, which implies (the observed fact) that there are no massless electrically charged particles in Nature, unlike the prevailing treatments of KK theories.

The geometry of generalized Cheeger–Gromoll metrics on the total space of transitive Euclidean Lie algebroids

Natural metrics (Sasaki metric, Cheeger–Gromoll metric, Kaluza–Klein metrics etc. ) on the tangent bundle of a Riemannian manifold is a central topic in Riemannian geometry. Generalized Cheeger–Gromoll metrics is a family hp,q of natural metrics on the tangent bundle depending on two parameters with p∈R and q≥0. This family possesses interesting geometric properties. If p=q=0 we recover the Sasaki metric and when p=q=1 we recover the classical Cheeger–Gromoll metric. A transitive Euclidean Lie algebroid is a transitive Lie algebroid with a Euclidean product on its total space. In this paper, we show that the analogous of natural metrics 11Natural metrics on the tangent bundles come from a classification of natural transformation of Riemannian metrics on manifolds to metrics on tangent bundles. We use the some terminology in this paper to designate metrics obtained on the total space of a Euclidean Lie algebroid by imitating the expression of g-natural metrics on the tangent bundle.can be built in a natural way on the total space of transitive Euclidean Lie algebroids. Then we study the properties of generalized Cheeger–Gromoll metrics on this new context. We show a rigidity result of these metrics which generalizes some rigidity results known in the case of the tangent bundle. We show also that considering natural metrics on the total space of transitive Euclidean Lie algebroids opens new interesting horizons. For instance, Atiyah Lie algebroids constitute an important class of transitive Lie algebroids and we will show that natural metrics on the total space of Atiyah Euclidean Lie algebroids have interesting properties. In particular, if M is a Riemannian manifold of dimension n, then the Atiyah Lie algebroid associated to the O(n)-principal bundle of orthonormal frames over M possesses a family depending on a parameter k>0 of transitive Euclidean Lie algebroids structures say AO(M,k). When M is a space form of constant curvature c, we show that there exist two constants Cn<0 and K(n,c)>0 such that (AO(M,k),h1,1) is a Riemannian manifold with positive scalar curvature if and only if c>Cn and 0<k≤K(n,c).

Remarks about the Kaluza-Klein metric on tangent bundle

The paper is concerned with the Kaluza–Klein metric on the tangent bundle over a Riemannian manifold. All kinds of Riemann curvature tensors are computed and some curvature properties are given. The compatible almost complex structure is defined on the tangent bundle, and necessary and sufficient conditions for such a structure to be integrable are described. Then, the condition is given under which the tangent bundle with these structures is almost Kähler. Finally, almost golden complex structures are defined on this setting and some results related to them are presented.

Some applications on tangent bundle with Kaluza-Klein metric

Some applications on tangent bundle with Kaluza-Klein metric

On Hamiltonians with position-dependent mass from Kaluza–Klein compactifications

In a recent paper (Morris (2015) [1]), an inhomogeneous compactification of the extra dimension of a five-dimensional Kaluza–Klein metric has been shown to generate a position-dependent mass (PDM) in the corresponding four-dimensional system. As an application of this dimensional reduction mechanism, a specific static dilatonic scalar field has been connected with a PDM Lagrangian describing a well-known nonlinear PDM oscillator. Here we present more instances of this construction that lead to PDM systems with radial symmetry, and the properties of their corresponding inhomogeneous extra dimensions are compared with the ones in the nonlinear oscillator model. Moreover, it is also shown how the compactification introduced in this type of models can alternatively be interpreted as a novel mechanism for the dynamical generation of curvature.

Kaluza-Klein Bulk Viscous Cosmological Model with Time Dependent Gravitational Constant and Cosmological Constant

Cosmological models with time varying gravitational constant G and cosmological constant Λ in the presence of viscous fluid in Kaluza-Klein metric were investigated. The solutions to Einstein Field Equation were obtained for different types of G, with bulk coefficient ξ = ξ 0 ρ d (where ρ is density of the Universe, d is some constant) and lambda Λ = α H 2 + β R −2 where H and R are Hubble parameter and scale factor respectively. Two possible models are suggested, one where G is proportional to H and, the other where G is inversely proportional to H. While the former leads to a non-singular model, the latter results in an inflationary model. Both Cosmological models show that the Universe is accelerating; but at the early stage of the Universe the behaviour of both models is quite different,which has been studied through the variation of decelerating parameter q with time.

Minimal unit vector fields with respect to Riemannian natural metrics

Let (M,g) be a Riemannian manifold. We denote by G˜ an arbitrary Riemannian g-natural metric on the unit tangent sphere bundle T1M, such metric depends on four real parameters satisfying some inequalities. The Sasaki metric, the Cheeger–Gromoll metric and the Kaluza–Klein metrics are special Riemannian g-natural metrics. In literature, minimal unit vector fields have been already investigated, considering T1M equipped with the Sasaki metric G˜S[12]. In this paper we extend such characterization to an arbitrary Riemannian g-natural metric G˜. In particular, the minimality condition with respect to the Sasaki metric G˜S is invariant under a two-parameters deformation of the Sasaki metric. Moreover, we show that a minimal unit vector field (with respect to G˜) corresponds to a minimal submanifold. Then, we give examples of minimal unit vector fields (with respect to G˜). In particular, we get that the Hopf vector fields of the unit sphere, the Reeb vector field of a K-contact manifold, and the Hopf vector field of a quasi-umbilical hypersurface with constant principal curvatures in a Kähler manifold, are minimal unit vector fields (with respect to G˜).

A higher-dimensional string cosmological model in a scalar-tensor theory of gravitation

A string cosmological model is obtained and presented by considering a five-dimensional Kaluza-Klein metric in the scalar-tensor theory of gravitation proposed by Brans and Dicke without taking any relation between rest energy density ρ and string tension density λ. Some physical and geometrical properties of the model are also discussed.

Geometry of Kaluza–Klein metrics on the sphere $${\mathbb{S}^3}$$

In this paper, we introduce a new family of Riemannian metrics \({\tilde g_{\lambda\mu\nu}}\) on the three-sphere and study its geometric properties, starting from the description of their curvature. Such metrics, which include the standard metric g0 and Berger metrics on \({\mathbb{S}^3}\) as special cases, are called “of Kaluza–Klein type”, because they are induced in a natural way by the corresponding metrics defined on the tangent sphere bundle \({T_1 \mathbb{S}^2(\kappa)}\). Each sphere \({(\mathbb{S}^3, \tilde g_{\lambda\mu\nu})}\) is a homogeneous space, and we obtain a full classification of its homogeneous structures. Moreover, we introduce and study a natural almost contact structure \({(\varphi,\xi,\eta)}\), for which \({(\varphi,\xi,\eta,\tilde g_{\lambda\mu\nu})}\) is a (homogeneous) almost contact metric structure on the three-sphere. Finally, we see that for a suitable family of Kaluza–Klein type metrics \({\tilde g_{ac}}\) on \({\mathbb{S}^3}\), it is possible to construct a two-parameter family of harmonic morphisms from \({(\mathbb{S}^3, \tilde g_{ac})}\) to \({\mathbb{S}^2(\kappa)}\).

On the geometrical representation of the Jacobian in the path integral reduction problem

By using the formula for the scalar curvature of the manifold with the Kaluza–Klein metric we obtain the geometrical representation of the Jacobian resulted from the path integral reduction problem in Wiener path integrals for a scalar particle on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple Lie group.

The conformal Penrose limit: back to square one

We show that the conformal Penrose limit is an ordinary plane wave limit in a higher dimensional framework which resolves the spacetime singularity. The higher dimensional framework is provided by Ricci-flat manifolds which are of the form MD = Md × B, where Md is an Einstein spacetime that has a negative cosmological constant and admits a spacelike conformal Killing vector, and B is a complete Sasaki–Einstein space with constant sectional curvature. We define the Kaluza–Klein metric of MD through the conformal Killing potential of Md and prove that Md has a conformal Penrose limit if and only if MD has an ordinary plane wave limit. Further properties of the limit are discussed.

A higher-dimensional string cosmological model in Brans–Dicke theory of gravitation

Field equations in the presence of cosmic string source are obtained in a scalar tensor theory of gravitation proposed by Brans and Dicke (Phys. Rev. 124, 925 (1961)) with the aid of a five-dimensional Kaluza–Klein metric. An exact string cosmological model is presented which represents a five-dimensional Reddy string (Astrophys. Space Sci. 286, 2003b) in Brans–Dicke theory. Some physical properties of the model are also discussed

Cosmological Model with a Viscous Fluid in a Kaluza-Klein Metric

A Cosmological model with a viscous fluid in Kaluza-Klein metric is obtained assuming a time-dependent equation of state. The solution is in fact a generalization of an earlier work by Hajj and Boutros for a perfect fluid. It is also found that dimensional reduction of the extra space takes place such that the five-dimensional universe naturally evolves into an effective four-dimensional one. The dynamical behavior of the model is examined and it is also found that with a decrease in extra space the observable 3D space entropy increases thus accounting for the large value of entropy observable at present.

Cosmological Model with a Time-Dependent Equation of State in a Kaluza–Klein Metric

A homogeneous cosmological model in Kaluza–Klein metric is obtained assuming a time-dependent equation of state. The solution is in fact generalization of an earlier work by Hajj and Boutros for a perfect fluid. It is also found that dimensional reduction of the extra space takes place such that the five-dimensional universe naturally evolves into an effective four-dimensional one. The dynamical behaviour of the model is examined and it is also found that with a decrease in extra space the observable three-dimensional space entropy increase thus accounting for the large value of entropy observable at present.

Causal properties of kaluza-klein metrics

We present some global properties of Kaluza-Klein metrics. We state that the Kaluza-Klein metrics associated to globally hyperbolic spacetimes are globally hyperbolic. Applications to the solutions of the relativistic Lorentz equation are shown.