Derivation of the Lorentz Boost from the Evans Wave Equation

The Lorentz boost is derived from the Evans wave equation of generally covariant unified field theory by constructing the Dirac spinor from the tetrad in the SU(2) representation space of non-Euclidean spacetime. The Dirac equation in its wave formulation is then deduced as a well-defined limit of the Evans wave equation. By factorizing the dAlembertian operator into Dirac matrices, the Dirac equation in its original first differential form is obtained from the Evans wave equation. Finally, the Lorentz boost is deduced from the Dirac equation using geometrical arguments. A self-consistency check of the Evans wave equation is therefore forged by deducing therefrom the Lorentz boost in the appropriate limit. This procedure demonstrates that the Evans wave equation governs the properties of matter and anti-matter in general relativity and unified field theory and leads both to Fermi-Dirac and Bose-Einstein statistics in general relativity.     

Derivation of the Evans Wave Equation from the Lagrangian and <Emphasis Type="Italic">Action</Emphasis>: Origin of the Planck Constant in General Relativity

No Heading The Evans wave equation is derived from the appropriate Lagrangian and action, identifying the origin of the Planck constant in general relativity. The classical Fermat principle of least time, and the classical Hamilton principle of least action, are expressed in terms of a tetrad multiplied by a phase factor exp(iS/), where S is the action in general relativity. Wave (or quantum) mechanics emerges from these classical principles of general relativity for all matter and radiation fields, giving a unified theory of quantum mechanics based on differential geometry and general relativity. The phase factor exp(iS/) is an eigenfunction of the Evans wave equation and is the origin in general relativity and geometry of topological phase effects in physics, including the Aharonov-Bohm class of effects, the Berry phase, the Sagnac effect, related interferometric effects, and all physical optical effects through the Evans spin field B⁽³⁾ and the Stokes theorem in differential geometry. The Planck constant is thus identified as the least amount possible of action or angular momentum or spin in the universe. This is also the origin of the fundamental Evans spin field B⁽³⁾, which is always observed in any physical optical effect. It originates in torsion, spin and the second (or spin) Casimir invariant of the Einstein group. Mass originates in the first Casimir invariant of the Einstein group. These two invariants define any particle.     

Development of the Evans Wave Equation in the Weak-Field Limit: The Electrogravitic Equation

The Evans wave equation [1-3] is developed in the weak-field limit to give the Poisson equation and an electrogravitic equation expressing the electric field strength E in terms of the acceleration g due to gravity and a fundamental scalar potential ⁽⁰⁾ with the units of volts (joules per coulomb). The electrogravitic equation shows that an electric field strength can be obtained from the acceleration due to gravity, which in general relativity is non-Euclidean spacetime. Therefore an electric field strength can be obtained, in theory, from scalar curvature R. This inference is supported by recent experimental data from the patented motionless electromagnetic generator [5].     

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.     

ON THE ADM EQUATIONS FOR GENERAL RELATIVITY

The Arnowitt–Deser–Misner (ADM) evolution equations for the induced metric and the extrinsic-curvature tensor of the spacelike surfaces which foliate the space-time manifold in canonical general relativity are a first-order system of quasi-linear partial differential equations, supplemented by the constraint equations. Such equations are here mapped into another first-order system. In particular, an evolution equation for the trace of the extrinsic-curvature tensor K is obtained whose solution is related to a discrete spectral resolution of a three-dimensional elliptic operator of Laplace type. Interestingly, all nonlinearities of the original equations give rise to the potential term in . An example of this construction is given in the case of a closed Friedmann–Lemaitre–Robertson–Walker universe. Eventually, the ADM equations are re-expressed as a coupled first-order system for the induced metric and the trace-free part of K. Such a system is written in a form which clarifies how a set of first-order differential operators and their inverses, jointly with spectral resolutions of operators of Laplace type, contribute to solving, at least in principle, the original ADM system.     

The Evans Lemma of Differential Geometry

A rigorous proof is given of the Evans lemma of general relativity and differential geometry. The lemma is the subsidiary proposition leading to the Evans wave equation and proves that the eigenvalues of the d'Alembertian operator, acting on any differential form, are scalar curvatures. The Evans wave equation shows that the eigenvalues of the d'Alembertian operator, acting on any differential form, are eigenvalues of the index-contracted canonical energy momentum tensor T multiplied by the Einstein constant k. The lemma is a rigorous and general result in differential geometry, and the wave equation is a rigorous and general result for all radiated and matter fields in physics. The wave equation reduces to the main equations of physics in the appropriate limits, and unifies the four types of radiated fields thought to exist in nature: gravitational, electromagnetic, weak and strong.     

Polarizable-Vacuum (PV) Approach to General Relativity

Standard pedagogy treats topics in general relativity (GR) in terms of tensor formulations in curved space-time. An alternative approach based on treating the vacuum as a polarizable medium is presented here. The polarizable vacuum (PV) approach to GR, derived from a model by Dicke and related to the TH formalism used in comparative studies of gravitational theories, provides additional insight into what is meant by a curved metric. While reproducing the results predicted by GR for standard (weak-field) astrophysical conditions, for strong fields a divergence of predictions in the two formalisms (GR vs. PV) provides fertile ground for both laboratory and astrophysical tests to compare the two approaches.     

Massive Gauge Fields and the Planck Scale

No Heading The present work is devoted to massive gauge fields in special relativity with two fundamental constants- the velocity of light, and the Planck length, so called doubly special relativity (DSR). The two invariant scales are accounted for by properly modified boost parameters. Within above framework we construct the vector potential as the (1/2, 0) (0, 1/2) direct product, build the associated field strength tensor together with the Dirac spinors and use them to calculate various observables as functions of the Planck length.     

An Analytical Treatment of the Clock Paradox in the Framework of the Special and General Theories of Relativity

No Heading In this paper we treat the so called clock paradox in an analytical way by assuming that a constant and uniform force F of finite magnitude acts continuously on the moving clock along the direction of its motion assumed to be rectilinear (in space). No inertial motion steps are considered. The rest clock is denoted as (1), the to and fro moving clock is (2), the inertial frame in which (1) is at rest in its origin and (2) is seen moving is I and, finally, the accelerated frame in which (2) is at rest in its origin and (1) moves forward and backward is A. We deal with the following questions: (1) What is the effect of the finite force acting on (2) on the proper time interval ⁽²⁾ measured by the two clocks when they reunite? Does a differential aging between the two clocks occur, as it happens when inertial motion and infinite values of the accelerating force is considered? The special theory of relativity is used in order to describe the hyperbolic (in spacetime) motion of (2) in the frame I. (II) Is this effect an absolute one, i.e., does the accelerated observer A comoving with (2) obtain the same results as that obtained by the observer in I, both qualitatively and quantitatively, as it is expected? We use the general theory of relativity in order to answer this question. It turns out that _I = _A for both the clocks, ⁽²⁾ does depend on g = F/m, and = ⁽²⁾/⁽¹⁾ = (1 – ²atanhj)/ < 1. In it ; = V/c and V is the velocity acquired by (2) when the force is inverted.     

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.     

Yang-Mills Equation and Bures Metric

It is shown that the connection form (gauge field) related to the generalization of the Berry phase to mixed states proposed by Uhlmann satisfies the source-free Yang–Mills equation * D * D = 0, where the Hodge operator is taken with respect to the Bures metric on the space of finite-dimensional nondegenerate density matrices.     

Metric Expansion from Microscopic Dynamics in an Inhomogeneous Universe

Theories with ingredients like the Higgs mechanism, gravitons, and inflaton fields rejuvenate the idea that relativistic kinematics is dynamically emergent. Eternal inflation treats the Hubble constant H as depending on location. Microscopic dynamics implies that H is over much smaller lengths than pocket universes to be understood as a local space reproduction rate. We illustrate this via discussing that even exponential inflation in TeV-gravity is slow on the relevant time scale. In our on small scales inhomogeneous cosmos, a reproduction rate H depends on position. We therefore discuss Einstein-Strauss vacuoles and a Lindquist-Wheeler like lattice to connect the local rate properly with the scaling of an expanding cosmos. Consistency allows H to locally depend on Weyl curvature similar to vacuum polarization. We derive a proportionality constant known from Kepler＇s third law and discuss the implications for the finiteness of the cosmological constant.     

Path integral formulation of general diffusion processes

A method is given for the derivation of a path integral representation of the Green's function solutionP of equationsP/t=L P,L being some Liouville operator. The method is applied to general diffusion processes.Feynman's path integral representation of the Schrödinger equation and Stratonovich's path integral representation of multivariate Markovian processes are obtained as special cases if the metric of the general diffusion process is flat. For curved phase spaces our result is a nontrivial generalization and new. New applications e.g. to quantized motion in general relativity, to transport processes in inhomogeneous systems, or to nonlinear non-equilibrium thermodynamics are made possible. We expect applications to be fruitfull in all cases where (continuous) macroscopic transport processes in Riemann geometries have to be considered.     

Planck—Wheeler Quantum Foam as White Noise: Metric Diffusion and Congruence Focussing for Fluctuating Spacetime Geometry

It is expected that quantum effects endow spacetime with stochastic properties near the Planck scale as exemplified by random fluctuations of the metric, usually referred to as spacetime foam or geometrodynamics. In this paper, a methodology is presented for incorporating Planck scale stochastic effects and corrections into general relativity within the ADM formalism, by coupling the Riemann 3-metric to white noise. The ADM—Cauchy evolution of a Riemann 3-metric h _ij (t) induced on spacelike hypersurface C(t) can be interpreted within pure general relativity as a smooth geodesic flow in superspace, whose points consist of equivalence classes of 3-metrics. Coupling white noise to h _ij gives Langevin stochastic differential equations for the Cauchy evolution of h _ij, which is now a Brownian motion or diffusion in superspace. A fluctuation h_ij away from h _ij is considered to be related to h _ij by elements of the diffeomorphism group diff(C). Hydrodynamical Fokker—Planck continuity equations are formulated describing the stochastic Cauchy evolution of h _ij as a probability flow. The Cauchy invariant or equilibrium solution gives a stationary probability distribution of fluctuations peaked around the deterministic metric. By selecting a physically viable ansatz for the scale dependent diffusion coefficient, one reproduces the Wheeler uncertainty relation for the metric fluctuations of quantum geometrodynamics. Treating h _ij as a random variable, a non-linear Raychaudhuri—Langevin equation is derived describing the geometro-hydrodynamics of a congruence of fluid or dust matter propagating on the stochastic spacetime. For an initially converging congruence >0 at s the singularity =– at future proper time s=3/||$, which is expected in general relativity, is now smeared out near the Planck scale. Proper time s can be extended indefinitely (s) so that intrinsic metric fluctuations can restore geodesic completeness although the geodesics remain trapped for all time: although a singularity can be removed the collapsing matter still creates a black hole. A Fokker—Planck formulation also gives zero probability that – for s. Essentially, the short distance stochastic corrections to the deterministic equations of general relativity can remove pathologies such as singularities, conjugate points and geodesic incompleteness.     

Probability Metrics and Uniqueness of the Solution to the Boltzmann Equation for a Maxwell Gas

We consider a metric for probability densities with finite variance on ^d , and compare it with other metrics. We use it for several applications both in probability and in kinetic theory. The main application in kinetic theory is a uniqueness result for the solution of the spatially homogeneous Boltzmann equation for a gas of true Maxwell molecules.     

Embedding of Particle Waves in a Schwarzschild Metric Background

The special and general relativity theories are used to demonstrate that the velocity of an unradiative particle in a Schwarzschild metric background, and in an electrostatic field, is the group velocity of a wave that we call a particle wave, which is a monochromatic solution of a standard equation of wave motion and possesses the following properties. It generalizes the de Broglie wave. The rays of a particle wave are the possible particle trajectories, and the motion equation of a particle can be obtained from the ray equation. The standing particle wave equation generalizes the Schrödinger equation of wave amplitudes. The particle wave motion equation generalizes the Klein–Gordon equation; this result enables us to analyze the essence of the particle wave frequency. The equation of the eikonal of a particle wave generalizes the Hamilton–Jacobi equation; this result enables us to deduce the general expression for the linear momentum. The Heisenberg uncertainty relation expresses the diffraction of the particle wave, and the uncertainty relation connecting the particle instant of presence and energy results from the fact that the group velocity of the particle wave is the particle velocity. A single classical particle may be considered as constituted of geometrical particle wave; reciprocally, a geometrical particle wave may be considered as constituted of classical particles. The expression for a particle wave and the motion equation of the particle wave remain valid when the particle mass is zero. In that case, the particle is a photon, the particle wave is a component a classical electromagnetic wave that is embedded in a Schwarzschild metric background, and the motion equation of the wave particle is the motion equation of an electromagnetic wave in a Schwarzschild metric background. It follows that a particle wave possesses the same physical reality as a classical electromagnetic wave. This last result and the fact that the particle velocity is the group velocity of its wave are in accordance with the opinions of de Broglie and of Schrödinger. We extend these results to the particle subjected to any static field of forces in any gravitational metric background. Therefore we have achieved a synthesis of undulatory mechanics, classical electromagnetism, and gravitation for the case where the field of forces and the gravitational metric background are static, and this synthesis is based only on special and general relativity.     

Primary Matter Creation in a Weyl–Dirac Cosmological Model

In the framework of an integrable Weyl–Dirac (W–D) theory a cosmological model is proposed. It describes a universe that began its expansion from a primary pre-Planckian geometric entity containing no matter. During the pre-Planckian period, from R ₀ =5.58×10 ^–36 cm to R_I=5.58×10 ^–34 cm, this embryonic universe has undergone a very rapid expansion and cosmic matter was created by geometry. At R_I the universe was already filled with matter having the Planckian density _P and being in the state of prematter (P=–), while the Weylian geometric elements were insignificant. This state is the Planckian egg that has served as the initial state of the singularity-free cosmological model ⁽¹⁾ considered in the framework of Einstein's general theory of relativity. The W–D character of the geometry and the cosmological constant are significant in the pre-Planckian period during the matter creation. In the dust-dominated period a relic of the W–D geometry causes a global dark matter effect. In between the pre-Planckian and dust period one has Einstein's framework and is negligible.     

On the inertial mass concept in special and general relativity

Inertial mass in relativity theory is discussed from a conceptual view. It is shown that though relativistic dynamics implies a particular dependence of the momentum of a free particle on its velocityin special relativity, which diverges as v approaches c, the inertial mass itself of a moving body remains constant, from any frame of observation. However, extension to general relativity does conceptually introduce variability of the inertial mass of a body, through a necessarily generally covariant field theory of inertia, when the Mach principle is incorporated into the theory of general relativity, as a theory of matter.     

Levi–Civita effect in the polarizable vacuum (PV) representation of general relativity

The polarizable vacuum (PV) representation of general relativity (GR), derived from a model by Dicke and related to the TH formalism used in comparative studies of gravitational theories, provides for a compact derivation of the Levi–Civita effect (both magnetic and electric), herein demonstrated.This revised version was published online in April 2005. The publishing date was inserted.     

Kinematics and dynamics off(R) theories of gravity

We generalise the equations governing relativistic fluid dynamics given by Ehlers and Ellis for general relativity, and by Maartens and Taylor for quadratic theories, to generalisedf(R) theories of gravity. In view of the usefulness of this alternative framework to general relativity, its generalisation can be of potential importance for deriving analogous results to those obtained in general relativity. We generalise, as an example, the results of Maartens and Taylor to show that within the framework of generalf(R) theories, a perfect fluid spacetime with vanishing vorticity, shear and acceleration is Friedmann-Lemaître-Robertson-Walker only if the fluid has in addition a barotropic equation of state. It then follows that the Ehlers-Geren-Sachs theorem and its almost extension also hold forf(R) theories of gravity.