Classical Kaluza-Klein Theory Research Articles

Quantum Kaluza-Klein theory with M2(ℂ)

Following steps analogous to classical Kaluza-Klein theory, we solve for the quantum Riemannian geometry on C∞(M) ⊗ M2(ℂ) in terms of classical Riemannian geometry on a smooth manifold M , a finite quantum geometry on the algebra M2(ℂ) of 2 × 2 matrices, and a quantum metric cross term. Fixing a standard form of quantum metric on M2(ℂ), we show that this cross term data amounts in the simplest case to a 1-form Aμ on M, which we regard as like a gauge-fixed background field. We show in this case that a real scalar field on the product algebra with its noncommutative Laplacian decomposes on M into two real neutral fields and one complex charged field minimally coupled to Aμ. We show further that the quantum Ricci scalar on the product decomposes into a classical Ricci scalar on M, the Ricci scalar on M2(ℂ), the Maxwell action ||F||2 of A and a higher order ||A.F||2 term. Another solution of the QRG on the product has A = 0 and a dynamical real scalar field ϕ on M which imparts mass-splitting to some of the components of a scalar field on the product as in previous work.

Kinematic quantities and Raychaudhuri equations in a 5D universe

Based on some ideas that have emerged from the classical Kaluza–Klein theory, we present a 5D universe as a product bundle over the 4D spacetime. This enables us to introduce and study two categories of kinematic quantities (expansions, shear, vorticity) in a 5D universe. One category is related to the fourth dimension (time), and the other one comes from the assumption of the existence of the fifth dimension. The Raychaudhuri type equations that we obtain in the paper, lead us to results on the evolution of both the 4D expansion and the 5D expansion in a 5D universe.

A NEW APPROACH FOR SPACE-TIME-MATTER THEORY

This is the first paper in a series of three papers on a new approach for space-time-matter (STM) theory. The main purpose of this approach is to replace the Levi-Civita connection on the space-time from the classical Kaluza–Klein theory by what we call the Riemannian horizontal connection on the general Kaluza–Klein space. This is done by a development of a 4D tensor calculus whose geometrical objects live in a 5D space. The 4D tensor calculus and the Riemannian horizontal connection enable us to define in a 5D space some 4D differential operators: horizontal differential, horizontal gradient, horizontal divergence and horizontal Laplacian, which have a great role in the presentation of the STM theory in a covariant form. Finally, we introduce and study the horizontal electromagnetic tensor field, the horizontal Ricci tensor and the horizontal Einstein gravitational tensor field, which replace the well-known tensor fields from the classical Kaluza–Klein theory.

Five-dimensional PPN formalism and experimental test of Kaluza–Klein theory

The parametrized post-Newtonian formalism for 5-dimensional metric theories with a compact extra dimension is developed. The relation of the 5-dimensional and 4-dimensional formulations is then analyzed, in order to compare the higher dimensional theories of gravity with experiments. It turns out that the value of post-Newtonian parameter γ in the reduced 5-dimensional Kaluza–Klein theory is two times smaller than that in 4-dimensional general relativity. The departure is due to the existence of an extra dimension in the Kaluza–Klein theory. Thus the confrontation between the reduced 4-dimensional formalism and Solar system experiments raises a severe challenge to the classical Kaluza–Klein theory.

Mass and Charge in Brane-World and Non-Compact Kaluza-Klein Theories in 5 Dim

In classical Kaluza-Klein theory, with compactified extra dimensions and without scalar field, the rest mass as well as the electric charge of test particles are constants of motion. We show that in the case of a large extra dimension this is no longer so. We propose the Hamilton-Jacobi formalism, instead of the geodesic equation, for the study of test particles moving in a five-dimensional background metric. This formalism has a number of advantages: (i) it provides a clear and invariant definition of rest mass, without the ambiguities associated with the choice of the parameters used along the motion in 5D and 4D, (ii) the electromagnetic field can be easily incorporated in the discussion, and (iii) we avoid the difficulties associated with the “splitting” of the geodesic equation. For particles moving in a general 5D metric, we show how the effective rest mass, as measured by an observer in 4D, varies as a consequence of the large extra dimension. Also, the fifth component of the momentum changes along the motion. This component can be identified with the electric charge of test particles. With this interpretation, both the rest mass and the charge vary along the trajectory. The constant of motion is now a combination of these quantities. We study the cosmological variations of charge and rest mass in a five-dimensional bulk metric which is used to embed the standard k = 0 FRW universes. The time variations in the fine structure “constant” and the Thomson cross section are also discussed.

Nematic Structure of Space-Time and Its Topological Defects in 5D Kaluza-Klein Theory

We show, that classical Kaluza-Klein theory possesses hidden nematic dynamics. It appears as a consequence of 1+4-decomposition procedure, involving 4D observers 1-form λ. After extracting of boundary terms the, so called, “effective matter” part of 5D geometrical action becomes proportional to the square of the anholonomicity 3-form λ ∧ dλ. It can be interpreted as twist nematic elastic energy, responsible for elastic reaction of 5D space-time on presence of anholonomic 4D submanifold, defined by λ. We derive both 5D covariant and 1+4 forms of the 5D nematic equilibrium equations, consider simple examples and discuss some 4D physical aspects of generic 5D nematic topological defects.

Einstein and the Kaluza–Klein particle

In his search for a unified field theory that could undercut quantum mechanics, Einstein considered five-dimensional classical Kaluza–Klein theory. He studied this theory most intensively during the years 1938–1943. One of his primary objectives was finding a non-singular particle solution. In the full theory this search got frustrated, and in the x 5-independent theory Einstein, together with Pauli, argued it would be impossible to find these structures.

Geodesic equations for a charged particle in the unified theory of gravitational and electromagnetic interactions

A new model of gravitational and electromagnetic interactions is constructed as a version of the classical Kaluza-Klein theory based on a five-dimensional manifold as the physical space-time. The velocity space of moving particles in the model remains four-dimensional as in the standard relativity theory. The spaces of particle velocities constitute a four-dimensional distribution over a smooth five-dimensional manifold. This distribution depends only on the electromagnetic field and is independent of the metric tensor field. We prove that the equations for the geodesics whose velocity vectors always belong to this distribution are the same as the charged particle equations of motion in the general relativity theory. The gauge transformations are interpreted in geometric terms as a particular form of coordinate transformations on the five-dimensional manifold.

The nonsymmetric Kaluza-Klein (Jordan-Thiry) theory in the electromagnetic case

We present the nonsymmetric Kaluza-Klein and Jordan-Thiry theories as interesting propositions of physics in higher dimensions. We consider the five-dimensional (electromagnetic) case. The work is devoted to a five-dimensional unification of the NGT (nonsymmetric theory of gravitation), electromagnetism, and scalar forces in a Jordan-Thiry manner. We find “interference effects” between gravitational and electromagnetic fields which appear to be due to the skew-symmetric part of the metric. Our unification, called the nonsymmetric Jordan-Thiry theory, becomes the classical Jordan-Thiry theory if the skew-symmetric part of the metric is zero. It becomes the classical Kaluza-Klein theory if the scalar fieldρ=1 (Kaluza's Ansatz). We also deal with material sources in the nonsymmetric Kaluza-Klein theory for the electromagnetic case. We consider phenomenological sources with a nonzero fermion current, a nonzero electric current, and a nonzero spin density tensor. From the Palatini variational principle we find equations for the gravitational and electromagnetic fields. We also consider the geodetic equations in the theory and the equation of motion for charged test particles. We consider some numerical predictions of the nonsymmetric Kaluza-Klein theory with nonzero (and with zero) material sources. We prove that they do not contradict any experimental data for the solar system and on the surface of a neutron star. We deal also with spin sources in the nonsymmetric Kaluza-Klein theory. We find an exact, static, spherically symmetric solution in the nonsymmetric Kaluza-Klein theory in the electromagnetic case. This solution has the remarkable property of describing “mass without mass” and “charge without charge.” We examine its properties and a physical interpretation. We consider a linear version of the theory, finding the electromagnetic Lagrangian up to the second order of approximation with respect toh μv =g μv −n μv . We prove that in the zeroth and first orders of approximation there is no skewonoton interaction. We deal also with the Lagrangian for the scalar field (connected to the “gravitational constant”). We prove that in the zeroth and first orders of approximation the Lagrangian vanishes.

Can black holes in classical Kaluza-Klein theory have scalar hair?

It is shown that in the context of classical Kaluza-Klein theory there are no neutral black hole solutions with non-trivial scalar coupling between the four-dimensional spacetime and the n-dimensional (curved or flat) internal space. Only electrically or magnetically charged black holes can exhibit such a coupling.

Classical Kaluza-Klein description of the hydrogen atom

The vacuum field equations of the classical five-dimensional Kaluza-Klein theory are solved for a static, spherically symmetric system. It is shown that the projection of the solution into four-dimensional space-time represents the Reissner-Nordstrom solution and that the five-dimensional geodesic equation of a properly chosen free particle represents the equation of motion of an electron in the field of a proton, as given by the Einstein-Maxwell equations. The electrical field of a charged particle is interpreted as the rotational inertial-dragging field outside a rotating particle in five-dimensional space-time.

Suppression of singularities by the g55 field with mass and classical vacuum polarization in a classical Kaluza-Klein theory.

It is shown that a straightforward version of a classical Kaluza-Klein theory predicts the existence of classical vacuum polarization and that if the ${g}_{55}$ field carries a mass than the theory admits centrally symmetric solutions for which the Schwarzschild singularity becomes a naked point singularity with no event horizon and for which the electric field is everywhere finite.

On unifying electromagnetism and gravitation without curvature

This paper is devoted to a five-dimensional unification of the gravitational theory of Hayashi and Shirafuji with electromagnetism. Interference effects are found between gravitational contributions of matter spin and electromagnetism. This unification becomes the classical Kaluza–Klein theory if contributions of the torsion tensor related with spin are neglected.

The nonsymmetric Kaluza–Klein theory

This paper is devoted to a five-dimensional unification of Moffat’s theory of gravitation and electromagnetism. We found ‘‘interference effects’’ between gravitational and electromagnetic fields which appear to be due to the skew-symmetric part of the metric of Moffat’s theory. Our unification called the nonsymmetric Kaluza–Klein theory becomes the classical Kaluza–Klein theory if the skew-symmetric part of the metric is zero. The possible generalization to an arbitrary gauge group is discussed.

On the nonsymmetric Jordan–Thiry theory

This paper is devoted to a five-dimensional unification of Moffat's theory of gravitation and electromagnetism. We found "interference effects" between gravitational and electromagnetic fields which appear to be due to the skew-symmetric part of the metric of Moffat's theory. Our unification, called the nonsymmetric Jordan–Thiry theory, becomes the classical Kaluza–Klein theory if the skew-symmetric part of the metric is zero. The possible generalization to an arbitrary gauge group is discussed.

The nonsymmetric-nonabelian Kaluza-Klein theory

The author considers an (n+4)-dimensional unification of Moffat's theory of gravitation and Yang-Mills field theory with nonabelian gauge group G. The author found 'interference effects' between gravitational and Yang-Mills (gauge) fields which appear to be due to the skewsymmetric part of the metric of Moffat's theory and the skewsymmetric part of the metric on the group G. The unification, called the nonsymmetric-nonabelian Kaluza-Klein theory, becomes classical Kaluza-Klein theory if the skewsymmetric parts of both metrics are zero.