Derivation of x 5-Dependent Charged Metrics in 5-Dimensional General Relativity

New classes of charged-source 5D GR (Kaluza-Klein) metrics are derived here. These metrics, unlike prior Kaluza-Klein metrics, are both off-diagonal (representing charged solutions), andx ⁵-dependent. The x ⁵-dependence, in particular, is important as modern 5D GR (Induced Matter) theory has abandoned the cylindrical restriction on the 5th dimension (requiring it be curled up very small), and, thus, allows the metric to have x ⁵-dependence. x ⁵-dependence, therefore, represents a hitherto untapped potential for 5D GR investigation.     

Compactification of Two Dimensions in General Relativity

Motivated by Kaluza-Klein theory and modern string theories, the class of exact solutions yielding product manifolds M ₂ × S ² in general relativity is investigated. The compact submanifold homeomorphic to S ² is chosen to be a very small sphere. Choosing an anisotropic fluid as the particular physical model, it is proved that very large mass density and tension provide the mechanism of compactification. In case the transverse pressure is chosen to be zero, the corresponding spacetime is homeomorphic to ² × S ², and thus provides a tractable non-flat metric. In this simple metric, the geodesic equations are completely solved, yielding motions of massive test particles. Next, the corresponding wave mechanics (given by the Klein-Gordon equation) is explored in the same curved background. A general class of exact solutions is obtained. Four conserved quantities are explicitly computed. The scalar particles exhibit a discrete mass spectrum.     

On Degenerate Metrics and Electromagnetism

A theory of degenerate metrics is developed and applied to the problem of unifying gravitation with electromagnetism. The approach is similar to the Kaluza-Klein approach with a fifth dimension, however no ad hoc conditions are needed to explain why the extra dimension is not directly observable under everyday conditions. Maxwell's theory is recovered with differences only at very small length scales, and a new formula is found for the Coulomb potential that is regular everywhere.     

Extending the Redshift-Distance Relation in Cosmological General Relativity to Higher Redshifts

The redshift-distance modulus relation, the Hubble Diagram, derived from Cosmological General Relativity has been extended to arbitrarily large redshifts. Numerical methods were employed and a density function was found that results in a valid solution of the field equations at all redshifts. The extension has been compared to 302 type Ia supernova data as well as to 69 Gamma-ray burst data. The latter however do not truly represent a ‘standard candle’ as the derived distance moduli are not independent of the cosmology used. Nevertheless the analysis shows a good fit can be achieved without the need to assume the existence of dark matter. The Carmelian theory is also shown to describe a universe that is always spatially flat. This results from the underlying assumption of the energy density of a cosmological constant Ω_Λ=1, the result of vacuum energy. The curvature of the universe is described by a spacevelocity metric where the energy content of the curvature at any epoch is Ω _K =Ω_Λ−Ω=1−Ω, where Ω is the matter density of the universe. Hence the total density is always Ω _K +Ω=1.     

Quantum Mechanics and the Metrics of General Relativity

A one-to-one correspondence is established between linearized space-time metrics of general relativity and the wave equations of quantum mechanics. Also, the key role of boundary conditions in distinguishing quantum mechanics from classical mechanics, will emerge naturally from the procedure. Finally, we will find that the methodology will enable us to introduce not only test charges but also test masses by means of gauges.     

Hydrodynamics in 5-Dimensional Cosmological Special Relativity

The dynamics of a perfect fluid is studied in 5-dimensional special relativity, a framework which can be considered the 5-d generalization of cosmological special relativity as well as the flat specialization of 5-d brane world theory. This picture, as described in an earlier paper, directly includes a particle production mechanism. Here it is showed that the source of particle production vanishes if the fluid is isentropic. Moreover it is showed that the hydrodynamical equations can be interpreted in terms of a scale factor, giving rise to a set of equations which simulate in a sense Friedmann cosmology.     

Particle Production in 5-Dimensional Cosmological Relativity

The 5-dimensional extension of cosmological special and general relativity is considered. In this framework it is possible to define a 5-dimensional perfect fluid stress-energy tensor and to unify the equations of perfect hydrodynamics in a single 5-dimensional tensor conservation law. This picture in principle permits to interpret particle production phenomena as cosmological effects, in the spirit of open system cosmology.     

Axially Symmetric Inflationary Universe in General Relativity

We have investigated a simple axially symmetric inflationary universe in the presence of mass less scalar field with a flat potential. To get an inflationary universe, we have considered a flat region in which potential V is constant. Some physical properties of the universe are also discussed.     

Brane-World Models Emerging from Collisions of Plane Waves in 5D

We consider brane-world models embedded in a five-dimensional bulk spacetime with a large extra dimension and a cosmological constant. The cosmology in 5D possesses wave-like character in the sense that the metric coefficients in the bulk are assumed to have the form of plane waves propagating in the fifth dimension. We model the brane as the plane of collision of waves propagating in opposite directions along the extra dimension. This plane is a jump discontinuity which presents the usual Z ₂ symmetry of brane models. The model reproduces the generalized Friedmann equation for the evolution on the brane, regardless of the specific details in 5D. Model solutions with spacelike extra coordinate show the usual big-bang behavior, while those with timelike extra dimension present a big bounce. This bounce is an genuine effect of a timelike extra dimension. We argue that, based on our current knowledge, models having a large timelike extra dimension cannot be dismissed as mathematical curiosities in non-physical solutions. The size of the extra dimension is small today, but it is increasing if the universe is expanding with acceleration. Also, the expansion rate of the fifth dimension can be expressed in a simple way through the four-dimensional deceleration and Hubble parameters as – q H. These predictions could have important observational implications, notably for the time variation of rest mass, electric charge and the gravitational constant. They hold for the three (k = 0, + 1, – 1) models with arbitrary cosmological constant, and are independent of the signature of the extra dimension.     

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.     

Bianchi Type-V Inflationary Universe in General Relativity

A Bianchi type-V space-time is considered in the presence of massless scalar field with a flat potential. To get an inflationary universe, we have considered a flat region in which potential V is constant. Some physical and kinematical properties of the model are also discussed.     

Noncommutative Unification of General Relativity and Quantum Mechanics. A Finite Model

We construct a model unifying general relativity and quantum mechanics in a broader structure of noncommutative geometry. The geometry in question is that of a transformation groupoid given by the action of a finite group on a space E. We define the algebra of smooth complex valued functions on , with convolution as multiplication, in terms of which the groupoid geometry is developed. Owing to the fact that the group G is finite the model can be computed in full details. We show that by suitable averaging of noncommutative geometric quantities one recovers the standard space-time geometry. The quantum sector of the model is explored in terms of the regular representation of the algebra , and its correspondence with the standard quantum mechanics is established.     

A Higher Dimensional Inflationary Universe in General Relativity

A five dimensional Kaluza-Klein inflationary universe is investigated in the presence of massless scalar field with a flat potential. To get an inflationary universe a flat region in which potential V is constant is considered. Some physical and kinematical properties of the universe 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 . 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 d. 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 . 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.     

Derivation of O(3) Electrodynamics from the Einstein–Sachs Theory of General Relativity

General relativity is reduced to O(3) electrodynamics by consideration of the irreducible representations of the Einstein group and through a particular choice of basis. The photon is shown always to possess a scalar curvature R, and so the origin of quantization is found in general relativity.     

Plane Symmetric String Cosmological Model in Modified Theory of General Relativity

It is shown that homogeneous plane symmetric string cosmological model for Takabayasi string i.e. ρ=(1+ω)λ does not exist in Barber’s second self creation theory. Further it is found that the string cosmological model in this theory exist only when ω=0. Therefore model for ρ=λ (geometric string) is constructed. Some physical and geometrical properties of the model are discussed.     

A Possible Modification of Einstein's Theory of General Relativity

This article suggests a new metric theory of gravitation, in which metric field is determined not only by matter and nongravitational field but also by vector graviton field, and in principle there is no need to introduce the Einstein's tensor. In order to satisfy automatically the geodesic postulate, an additional coordinate condition is needed.For the spherically symmetric static field, it leads us to quite different conclusions from those of Einstein's general relativity in the interior region of the surface of infinite redshift. Accurate to the first order of GM/r, it obtains the same results about the four experimental tests of general relativity.     

The Principle of Covariance and the Hamiltonian Formulation of General Relativity

The implications of the general covariance principle for the establishment of a Hamiltonian variational formulation of classical General Relativity are addressed. The analysis is performed in the framework of the Einstein-Hilbert variational theory. Preliminarily, customary Lagrangian variational principles are reviewed, pointing out the existence of a novel variational formulation in which the class of variations remains unconstrained. As a second step, the conditions of validity of the non-manifestly covariant ADM variational theory are questioned. The main result concerns the proof of its intrinsic non-Hamiltonian character and the failure of this approach in providing a symplectic structure of space-time. In contrast, it is demonstrated that a solution reconciling the physical requirements of covariance and manifest covariance of variational theory with the existence of a classical Hamiltonian structure for the gravitational field can be reached in the framework of synchronous variational principles. Both path-integral and volume-integral realizations of the Hamilton variational principle are explicitly determined and the corresponding physical interpretations are pointed out.