Kaluza-Klein theory and Dirac equation in higher dimensional Riemann-Cartan space

The higher dimensional Kaluza-Klein theory in Riemann-Cartan space is discussed. To clarify its implications, we investigate the simplest five-dimensional case of the theory in detail. The Einstein-like, Maxwell, and Dirac equations in four-dimensional space-time are obtained by reducing the corresponding five-dimensional field equations. The effect of spin-spin interaction induced by torsion is revealed by analyzing the Dirac equation in this case.     

Induced Matter and Particle Motion in Non-Compact Kaluza-Klein Gravity

We examine generalizations of the five-dimensional canonical metric by including a dependence of the extra coordinate in the four-dimensional metric. We discuss a more appropriate way to interpret the four-dimensional energy-momentum tensor induced from the five-dimensional space-time and show it can lead to quite different physical situations depending on the interpretation chosen. Furthermore, we show that the assumption of five-dimensional null trajectories in Kaluza-Klein gravity can correspond to either four-dimensional massive or null trajectories when the path parameterization is chosen properly. Retaining the extra-coordinate dependence in the metric, we show the possibility of a cosmological variation in the rest masses of particles and a consequent departure from four-dimensional geodesic motion by a geometric force. In the examples given, we show that at late times it is possible for particles traveling along 5D null geodesics to be in a frame consistent with the induced matter scenario.     

Stationary solutions in five-dimensional general relativity

Asymptotically Minkowskian stationary axisymmetric solutions of the five-dimensional Einstein equations are generated from particular classes of stationary axisymmetric solutions of the four-dimensional Einstein-Maxwell equations. First, the five-dimensional electrostatic and magnetostatic solutions are generated from the four-dimensional electrostatic solutions and a harmonic scalar field. Then, a new class of five-dimensional nonstatic solutions are generated from the four-dimensional class =+1. As an example, a three-parameter family of regular asymptotically flat rotating solutions is constructed. These solutions can be interpreted, after dimensional reduction to four dimensions, as extended elementary particles with mass, spin, electric charge, and magnetic dipole moment.     

Large-Scale Two-Dimensional Quantum Fluctuations of Five-Dimensional and Four-Dimensional Space-Time Metrics

Large-scale two-dimensional quantum fluctuations of five-dimensional space-time metric are constructed and the effect of the fluctuations on the nested four-dimensional worlds is studied. In doing so, the fluctuations affect not all four-dimensional worlds but only a part of them. The energy-momentum tensor of four-dimensional space-time has a physical form both in the absence and in the presence of fluctuations; it means that the fluctuations can be realized by real matter. A spatial region occupied by the fluctuations constructed in this work can be infinitely large and the fluctuations can occur during a long period of time. Therefore, we refer to these fluctuations as large-scale fluctuations.     

Various types of five-dimensional warp factor and effective Planck scale

Based on the assumption that the warp factor of four-dimensional spacetime and the one of fifth dimension are tied through a parameter α, we consider five-dimensional gravity with a 3-brane coupled to a bulk scalar field. For arbitrary value of α, the form of the warp factor is implicitly determined by hypergeometric function. Concretely, we show that the warp factor becomes explicit form for appropriate value of α, and study the relation between four-dimensional effective Planck scale and the brane tension. This setup allows the possibility of extending the diversity of brane world.     

The five-dimensional Dirac equation in the theory of algebraic spinors

The Dirac equation is considered in five-dimensional spaces with signatures (2,3), (4,1) and (0,5). The algebraic spinor formalism with the application of fermionic variables is used as the basis of real Clifford algebras and the module over this algebra. It is shown that solutions to the five-dimensional Dirac equation in spaces with signatures (2,3) and (4,1) can be expanded over solutions with zero value of the fifth component of the generalized momentum, and the equation is equivalent to an equation in four-dimensional spacetime.     

A possible unification of Newton's and Coulomb's forces

We have considered electric charge as the fourth component of the particle momentum in five-dimensional space–time. The fifth dimension has been compactified on a circle with an extremely small radius determined from the fundamental physics constants. First, we have given equations in the framework of five-dimensional special relativity and determined the corresponding reduction to four-dimensional space–time. Then, in order to obtain an appropriate charge-to-mass ratio and to avoid the Fourier modes problem, we have considered the propagation of an off-mass shell particle in the five-dimensional space–time which can be interpreted as the motion of an on-mass shell particle in the four-dimensional world we experience. As an example, we have discussed the five-dimensional kinematic equations associated with the electron-positron annihilation process into two photons. Finally, the consequences on the gravitational interaction between two elementary charged particles has been studied. As a main result, we have obtained a unification of Newton's gravitational and Coulomb's electrostatic forces.     

Five-Dimensional Warped Product Space-Time with Time-Dependent Warp Factor and Cosmology of the Four-Dimensional Universe

In this paper, we have studied a 5-dimensional warped product space-time with a time-dependent warp factor. This warp factor plays an important role in localizing matter to the 4-dimensional hypersurface constituting the observed universe and leads to a geometric interpretation of dynamical dark energy. The five-dimensional field equations are constructed and its solutions are obtained. The nature of modifications produced by this warp factor in the bulk geometry is discussed. The hypersurface is described by a flat FRW-type metric in the ordinary spatial dimension. It is found that the effective cosmological constant of the four-dimensional universe is a variable quantity monitored by the time-dependent warp factor. The universe is initially decelerated, but subsequently makes a transition to an accelerated phase at later times.     

Inducing the Cosmological Constant from Five-Dimensional Weyl Space

We investigate the possibility of inducing the cosmological constant from extra dimensions by embedding our four-dimensional Riemannian space-time into a five-dimensional Weyl integrable space. Following the approach of the space-time-matter theory we show that when we go down from five to four dimensions, the Weyl field may contribute both to the induced energy-tensor as well as to the cosmological constant Λ, or more generally, it may generate a time-dependent cosmological parameter Λ(t). As an application, we construct a simple cosmological model in which Λ(t) has some interesting properties.     

Projective invariance and the fifth coordinate

The paper considers the consequences of the principle of parametric (projective) invariance in the context of the general variational problem for stationary curves in a four-dimensional space-time. The necessity is shown of introducing a parameter along the trajectory as a fifth coordinate. The condition of cylindricality along it then acquires an obvious significance. The projective invariants are calculated for the trajectories of charged test particles in a gravielectromagnetic field. It is shown that one of these coincides with the density of the electromagnetic-field Lagrangian.     

Five-dimensional space-time: Mass and the fundamental length

Relativistic physics is described by a sixteen-component hypercomplex number system which reduces to the eight-component complex quaternion system when rest mass → 0. Rest mass is identified with a scale setting, cyclic, fifth dimension. The Lorentz group is generalized to rotations in five dimensions, a ten-parameter group. Special Relativity and General Relativity are tentatively welded into one unified covariance scheme in five-dimensional space-time.     

Spherically Symmetric Solutions of Einstein's Vacuum Equations in Five Dimensions

Multi-dimensional spherically symmetric spacetimes are of interest in the study of higher-dimensional black holes (and solitons) and higher-dimensional cosmological models. In this paper we shall present a comprehensive investigation of solutions of the five-dimensional spherically symmetric vacuum Einstein field equations subject only to the condition of separability in the radial coordinate (but not necessarily in the remaining two coordinates). A variety of new solutions are found which generalize a number of previous results. The properties of these solutions are discussed with particular attention being paid to their possible astrophysical and cosmological applications. In addition, the four-dimensional properties of matter can be regarded as geometrical in origin by a reduction of the five-dimensional vacuum field equations to Einstein's four-dimensional theory with a non-zero energy-momentum tensor constituting the material source; we shall also be interested in the induced matter associated with the new five-dimensional solutions obtained.     

Multisupergravity from latticized extra dimension

The construction of the linearized four-dimensional multisupergravity from five-dimensional linearized supergravity with discretized fifth dimension is presented. The one-loop vacuum energy is evaluated when (anti)periodic boundary conditions are chosen for (bosons) fermions, respectively, or vice versa. It is proposed that the relation between discretized M-theory and strings may be found in the same fashion.     

Exact wave solutions of the five-dimensional einstein equations

Exact wave solutions are found for the five-dimensional Einstein equations in a vacuum. The solutions describe the consistent gravitational scalar wave processes [6] and generalize to five dimensions the known four-dimensional Einstein-Rosen and Bondi-Pirani-Robinson wave metrics and the Rosen metric. By means of the method of rotations involving the coordinates of x⁵, the solutions obtained are generalized to the case in which there is an electromagnetic field present. For each of the solutions, the three-dimensional electric and magnetic field strengths are found. Moscow University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 112–117, October, 1996.     

Algebraic classification of gravitational fields in five-dimensional space-time

We consider the algebraic classification of five-dimensional empty space-time (Kalutsa type) with one time-like direction as a generalization of the Petrov algebraic classification of gravitational fields in four-dimensional space-time. We study two special cases: a) zero electromagnetic field and zero scalar field; b) nonzero electromagnetic field and zero scalar field. For the (1+4) separated Kalutsa five-metric we introduce the pentad metric of a tangent five-space, which is mapped together with the curvature tensor into a ten-dimensional real flat vector space. The classification is constructed in local geodesic coordinates for the above two cases. In both cases the characteristic equation can be reduced to a sixth-order equation that can be simplified when certain requirements are satisfied. Our results demonstrate the nontrivial nature of algebraic classification in five dimensions.This work was performed within the framework of the State Science and Technology Astronomy program.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 73–78, March, 1995.     

A simple derivation of the overlap Dirac operator

We derive the vector-like four-dimensional overlap Dirac operator starting from a five-dimensional Dirac action in the presence of a delta-function space–time defect. The effective operator is obtained by first integrating out all the fermionic modes in the fixed gauge background, and then identifying the contribution from the localized modes as the determinant of an operator in one dimension less. We define physically relevant degrees of freedom on the defect by introducing an auxiliary defect-bound fermion field and integrating out the original five-dimensional bulk fields.     

A general integral of the Jordan-Brans-Dicke field equations for static spherically symmetric perfect fluid distributions

The Jordan-Brans-Dicke field equations [1] contain the four-dimensional field equations of the five-dimensional projective unified theory. As it should be, Einstein's theory is incorporated as a limiting case. In this paper we present a method to determine explicitly for every static spherically symmetric solution of Einstein's theory with perfect fluid an analogous solution of Jordan-Brans-Dicke theory. As a particular example a “generalized interior Schwarzschild solution” is given.     

Geometro-Differential Conception of Extended Particles and the Semigroup of Trajectories in Minkowski Space-Time

The semigroup of trajectories in Minkowski space-time and its induced representations are constructed as a generalization of the Galilei case. They describe relativistic pointlike particles and yield the free propagator as a path integral in the space of trajectories parametrized by a fifth parameter. This non physical propagator in a five-dimensional space is integrated over the fifth parameter to yield the physical propagator in Minkowski space. Thereafter, this notion is applied to a model of extended particles with internal Poincaré symmetry and moving in an external Minkowski space. The geometrical structure is of Hilbert bundles and the interaction is introduced as a connection. The propagator is a path integral with respect to either the internal and external trajectories and reduces to a product of an internal and an external propagator when the interaction is ignored, just as has been found in a previous work with representations of the group rather than those of the semigroup.