Narrow Results

We discuss the question of whether the existence of singularities is an intrinsic property of 4D space–time. Our hypothesis is that singularities in 4D are induced by the separation of space–time from the other dimensions. We examine this hypothesis in the context of the so-called canonical or warp metrics in 5D. These metrics are popular because they provide a clean separation between the extra dimension and space–time. We show that the space–time section, in these metrics, inevitably becomes singular for some finite (nonzero) value of the extra coordinate. This is true for all canonical metrics that are solutions of the field equations in space–time-matter theory. This is a coordinate singularity in 5D, but appears as a physical one in 4D. At this singular hypersurface, the determinant of the space–time metric becomes zero and the curvature of the space–time blows up to infinity. These results are consistent with our hypothesis.

We study the relationship between space–time–matter (STM) and brane theories. These two theories look very different at first sight, and have different motivation for the introduction of a large extra dimension. However, we show that they are equivalent to each other. First we demonstrate that STM predicts local and nonlocal high-energy corrections to general relativity in 4D, which are identical to those predicted by brane-world models. Secondly, we point out that in brane models the usual matter in 4D is a consequence of the dependence of five-dimensional metrics on the extra coordinate. If the 5D bulk metric is independent of the extra dimension, then the brane is void of matter. Thus, in brane theory matter and geometry are unified, which is exactly the paradigm proposed in STM. Consequently, these two 5D theories share the same concepts and predict the same physics. This is important not only from a theoretical point of view, but also in practice. We propose to use a combination of both methods to alleviate the difficult task of finding solutions on the brane. We show an explicit example that illustrate the feasibility of our proposal.

We study null bulk geodesic motion in the brane world cosmology in the RS2 scenario and in the static universe in the bulk of the charged topological AdS black hole. We obtain equations describing the null bulk geodesic motion as observed in one lower dimension. We find that the null geodesic motion in the bulk of the brane world cosmology in the RS2 scenario is observed to be under the additional influence of extra non-gravitational force by the observer on the three-brane, if the brane universe does not possess the Z₂ symmetry. As for the null geodesic motion in the static universe in the bulk of the charged AdS black hole, the extra force is realized even when the brane universe has the Z₂ symmetry.

In brane-world theory in five dimensions, the bulk metric is usually written in Gaussian coordinates, where g_4μ = 0 and g₄₄ = -1. However, the choice g₄₄ = -1 is an external condition, not a requirement of the field equations. In this paper we study the consequences of having g₄₄ = εΦ², where ε = ±1 and Φ is a scalar function varying with time, . This varying field entails the possibility of variable fundamental physical "constants". These variations are different from those predicted in scalar–tensor and multidimensional theories. We solve the five-dimensional equations for a fixed brane and use the brane-world paradigm to determine the fundamental parameters in the theory, which are the vacuum energy σ, the gravitational coupling G and the cosmological term Λ₍₄₎. We present specific models where these physical quantities are variable functions of time. Different scenarios are possible but we discuss with some detail a model for which Ġ/G ~ H and Λ₍₄₎ ~ H², which seems to be favored by observations. Our results are not in contradiction to previous ones in the literature. In particular, to those where the brane is described as a domain wall moving in a static Sch–AdS bulk. Indeed these latter models in RS scenarios describe the same space–time as other solutions (with fixed brane) in Gaussian coordinates with . We conclude that the introduction of a time-varying Φ in brane-world theory yields a number of models that show variation in the fundamental physical "constants" and exhibit reasonable physical properties.

We consider the lightest Kaluza–Klein particle in universal extra dimensions (UED) to be a possible candidate for cold dark matter (CDM). We compute the relic densities for such Kaluza–Klein dark matter. This is done by first solving Boltzmann equation numerically to obtain freeze-out temperature and the ratio of number density and entropy density, for different masses. The relic densities (Ωh²) are then computed to find their variation with CDM masses. These calculations are performed considering two coannihilation channels namely B¹ B¹ → e⁺e^- and B¹ B¹ → qq.

Using the right representation in the evaluation of the creation probability of the universe, one can show that the external dimension of the Freund–Rubin model is min(s,n-s), where s is the rank of the antisymmetric field strength in the model.

Recent observations of Type Ia supernovae provide evidence for the acceleration of our universe, which leads to the possibility that the universe is entering an inflationary epoch. We simulate it under a "big bounce" model, which contains a time variable cosmological "constant" that is derived from a higher dimension and manifests itself in 4D spacetime as dark energy. By properly choosing the two arbitrary functions contained in the model, we obtain a simple exact solution in which the evolution of the universe is divided into several stages. Before the big bounce, the universe contracts from a Λ-dominated vacuum, and after the bounce, the universe expands. In the early time after the bounce, the expansion of the universe is decelerating. In the late time after the bounce, dark energy (i.e. the variable cosmological "constant") overtakes dark matter and baryons, and the expansion enters an accelerating stage. When time tends to infinity, the contribution of dark energy tends to two thirds of the total energy density of the universe, qualitatively in agreement with observations.

The Einstein field equations can be derived in n dimensions (n>2) by the variations of the Palatini action. The Killing reduction of five-dimensional Palatini action is studied on the assumption that pentads and Lorentz connections are preserved by the Killing vector field. A Palatini formalism of four-dimensional action for gravity coupled to a vector field and a scalar field is obtained, which gives exactly the same field equations in Kaluza–Klein theory.

A five-dimensional Ricci-flat cosmological solution is studied by assuming that the induced 4D matter contains two components: the usual fluid for dark matter as well as baryons and a scalar field with an exponential potential for dark energy. With use of the phase-plane analysis it is shown that there exist two late-time attractors one of which corresponds to a universe dominated by the scalar field alone and the other is a scaling solution in which the energy density of the scalar field remains proportional to that of the dark matter. It is furthermore shown that for this 5D scaling solution the universe expands with the same rate as in the 4D FRW models and not relies on which 4D hypersurface the universe is located in the 5D manifold.

In Spacetime–Matter theory we assume that the 4D induced matter of the 5D Ricci-flat bouncing cosmological solutions contains a perfect fluid as well as an induced scalar field. Then we show that the conventional 4D quintessence and phantom models of dark energy could be recovered from the 5D cosmological solutions. By using the phase-plane analysis to study the stability of evolution of the 5D models, we find that the conventional 4D late-time attractor solution is also recovered. This attractor solution shows that the scale factors of the phantom dominated universes in both the 4D and 5D theories will reach infinity in a finite time and the universes will be ended at a new kind of spacetime singularity at which everything will be annihilated. We also find that the repulsive force of the phantom may provide us with a mechanics to explain the bounce.

We study the accelerating expansion and the induced dark energy of the 5D Ricci-flat universe which is characterized by having a big bounce as opposed to a big bang. We show that the arbitrary function μ(t) contained in the 5D solutions can be rewritten in terms of the redshift z as a new arbitrary function f(z), and we find that there is a correspondence between this f(z) and the potential V(ϕ) of the 4D quintessence models. Using this correspondence, the arbitrary function f(z) and the 5D solution could be specified for a given form of the potential V(ϕ).

In braneworld models, Spacetime-Matter and other Kaluza–Klein theories, our spacetime is devised as a four-dimensional hypersurface orthogonal to the extra dimension in a five-dimensional bulk. We show that the FRW line element can be "reinvented" on a dynamical four-dimensional hypersurface, which is not orthogonal to the extra dimension, without any internal contradiction. This hypersurface is selected by the requirement of continuity of the metric and depends explicitly on the evolution of the extra dimension. The main difference between the "conventional" FRW, on an orthogonal hypersurface, and the new one is that the latter contains higher-dimensional modifications to the regular matter density and pressure in 4D. We compare the evolution of the spacetime in these two interpretations and find that a wealth of "new" physics can be derived from a five-dimensional metric if it is interpreted on a dynamical (non-orthogonal) 4D hypersurface. In particular, in the context of a well-known cosmological metric in 5D, we construct a FRW model which is consistent with the late accelerated expansion of the universe, while fitting simultaneously the observational data for the deceleration parameter. The model predicts an effective equation of state for the universe, which is consistent with observations.

We suggest a novel extension to the Kaluza–Klein scheme that allows us to obtain consistently all SU(n) Einstein–Yang–Mills theories. This construction is based on allowing the five-dimensional spacetime to carry some nonvanishing torsion; however, the four-dimensional spacetime remains intrinsically torsion-free.

In a particular variant of Kaluza–Klein theory, the so-called induced-matter theory (IMT), it is shown that any configuration of matter may be geometrically induced from a five-dimensional vacuum space. By using a similar approach we show that any distribution of charges and currents may also be induced from a five-dimensional space. Although in the case of IMT the geometry is Riemannian and the fundamental equations are the five-dimensional Einstein equations in vacuum, here we consider a Minkowskian geometry and five-dimensional Maxwell equations in vacuum.

It is well known that, in the five-dimensional scenario of braneworld and space–time-mass theories, geodesic motion in 5D is observed to be non-geodesic in 4D. Usually, the discussion is purely geometric and based on the dimensional reduction of the geodesic equation in 5D, without any reference to the test particle whatsoever. In this work we obtain the equation of motion in 4D directly from the principle of least action. So our main thrust is not the geometry but the particle observed in 4D. A clear physical picture emerges from our work. Specifically, that the deviation from the geodesic motion in 4D is due to the variation of the rest mass of a particle, which is induced by the scalar field in the 5D metric and the explicit dependence of the space–time metric on the extra coordinate. Thus, the principle of least action not only leads to the correct equations of motion, but also provides a concrete physical meaning for a number of algebraic quantities appearing in the geometrical reduction of the geodesic equation.

We obtain the most general static cylindrically symmetric vacuum solutions of the Einstein field equations in (4 + N) dimensions. Under the assumption of separation of variables, we construct a family of Levi–Civita–Kasner vacuum solutions in (4 + N) dimensions. We discuss the dimensional reduction of the static solutions. Depending on the reduction procedure, they can be interpreted either as a scalar-vacuum generalization of Levi–Civita spacetimes, or as the effective 4D vacuum spacetime outside of an idealized string in braneworld theory.

Here we consider a variant of the five-dimensional Kaluza–Klein (KK) theory within the framework of Einstein–Cartan formalism that includes torsion. By imposing a set of constraints on torsion and Ricci rotation coefficients, we show that the torsion components are completely expressed in terms of the metric. Moreover, the Ricci tensor in 5D corresponds exactly to what one would obtain from torsion-free general relativity on a 4D hypersurface. The contributions of the scalar and vector fields of the standard KK theory to the Ricci tensor and the affine connections are completely nullified by the contributions from torsion. As a consequence, geodesic motions do not distinguish the torsion free 4D spacetime from a hypersurface of 5D spacetime with torsion satisfying the constraints. Since torsion is not an independent dynamical variable in this formalism, the modified Einstein equations are different from those in the general Einstein–Cartan theory. This leads to important cosmological consequences such as the emergence of cosmic acceleration.

In this brief review, we discuss the viability of a multi-dimensional geometrical theory with one compactified dimension. We discuss the case of a Kaluza–Klein (KK) fifth-dimensional theory, addressing the problem by an overview of the astrophysical phenomenology associated with this five-dimensional (5D) theory. By comparing the predictions of our model with the features of the ordinary (four-dimensional (4D)) Relativistic Astrophysics, we highlight some small but finite discrepancies, expectably detectible from the observations. We consider a class of static, vacuum solutions of free electromagnetic KK equations with three-dimensional (3D) spherical symmetry. We explore the stability of the particle dynamics in these spacetimes, the construction of self-gravitating stellar models and the emission spectrum generated by a charged particle falling on this stellar object. The matter dynamics in these geometries has been treated by a multipole approach adapted to the geometric theory with a compactified dimension.

We use a Kaluza–Klein model with variable cosmological and gravitational terms to discuss the nature of galactic entropy function. For this purpose, we assume a universe filled with dark fluid and consider five-dimensional (5D) field equations using the Gamma law equation. We mainly discuss the validity of the first and generalized second laws of galactic thermodynamics for viable Kaluza–Klein models.

Through the Kaluza–Klein theory, we investigate the quantum dynamics of a Klein–Gordon particle under the Aharonov–Bohm effect for bound states, where it is subject to the linear and Coulomb-type central potentials inserted in the Klein–Gordon equation by modification of the mass term. Then, we determine analytically solutions of bound states and the energy profile of the scalar particle in this background.