Four-Dimensional Geometric Quantities versus the Usual Three-Dimensional Quantities: The Resolution of Jackson’s Paradox

In this paper we present definitions of different four-dimensional (4D) geometric quantities (Clifford multivectors). New decompositions of the torque N and the angular momentum M (bivectors) into 1-vectors N_s, N_t and M_s, M_t, respectively, are given. The torques N_s, N_t (the angular momentums M_s, M_t), taken together, contain the same physical information as the bivector N (the bivector M). The usual approaches that deal with the 3D quantities etc. and their transformations are objected from the viewpoint of the invariant special relativity (ISR). In the ISR, it is considered that 4D geometric quantities are well-defined both theoretically and experimentally in the 4D spacetime. This is not the case with the usual 3D quantities. It is shown that there is no apparent electrodynamic paradox with the torque, and that the principle of relativity is naturally satisfied, when the 4D geometric quantities are used instead of the 3D quantities.     

Trouton–Noble Paradox Revisited

An apparent paradox is obtained in all previous treatments of the Trouton–Noble experiment; there is a three-dimensional (3D) torque T in an inertial frame S in which a thin parallel-plate capacitor is moving, but there is no 3D torque T′ in S′, the rest frame of the capacitor. Different explanations are offered for the existence of another 3D torque, which is equal in magnitude but of opposite direction giving that the total 3D torque is zero. In this paper, it is considered that 4D geometric quantities and not the usual 3D quantities are well-defined both theoretically and experimentally in the 4D spacetime. In analogy with the decomposition of the electromagnetic field F (bivector) into two 1-vectors E and B we introduce decomposition of the 4D torque N (bivector) into 1-vectors N _s , N _t . It is shown that in the frame of “fiducial” observers, in which the observers who measure N _s and N _t are at rest, and in the standard basis, only the spatial components and remain, which can be associated with components of two 3D torques T and T _t . In such treatment with 4D geometric quantities the mentioned paradox does not appear. The presented explanation is in complete agreement with the principle of relativity and with the Trouton–Noble experiment without the introduction of any additional torque.     

Homothetic motions in general relativity

Properties of homothetic or self-similar motions in general relativity are examined with particular reference to vacuum and perfect-fluid space-times. The role of the homothetic bivector with componentsH _[a;b] formed from the homothetic vectorH is discussed in some detail. It is proved that a vacuum space-time only admits a nontrivial homothetic motion if the homothetic vector field is non-null and is not hypersurface orthogonal. As a subcase of a more general result it is shown that a perfect-fluid space-time cannot admit a nontrivial homothetic vector which is orthogonal to the fluid velocity 4-vector.     

The Proof that Maxwell Equations with the 3D E and B are not Covariant upon the Lorentz Transformations but upon the Standard Transformations: The New Lorentz Invariant Field Equations

In this paper the Lorentz transformations (LT) and the standard transformations (ST) of the usual Maxwell equations (ME) with the three-dimensional (3D) vectors of the electric and magnetic fields, E and B, respectively, are examined using both the geometric algebra and tensor formalisms. Different 4D algebraic objects are used to represent the usual observer dependent and the new observer independent electric and magnetic fields. It is found that the ST of the ME differ from their LT and consequently that the ME with the 3D E and B are not covariant upon the LT but upon the ST. The obtained results do not depend on the character of the 4D algebraic objects used to represent the electric and magnetic fields. The Lorentz invariant field equations are presented with 1-vectors E and B, bivectors E_Hv and B_Hv and the abstract tensors, the 4-vectors E^a and B^a. All these quantities are defined without reference frames, i.e., as absolute quantities. When some basis has been introduced, they are represented as coordinate-based geometric quantities comprising both components and a basis. It is explicitly shown that this geometric approach agrees with experiments, e.g., the Faraday disk, in all relatively moving inertial frames of reference, which is not the case with the usual approach with the 3D bf E and B and their ST.     

Geometry of Lagrangian First-order Classical Field Theories

We construct a lagrangian geometric formulation for first-order field theories using the canonical structures of first-order jet bundles, which are taken as the phase spaces of the systems in consideration. First of all, we construct all the geometric structures associated with a first-order jet bundle and, using them, we develop the lagrangian formalism, defining the canonical forms associated with a lagrangian density and the density of lagrangian energy, obtaining the Euler-Lagrange equations in two equivalent ways: as the result of a variational problem and developing the jet field formalism (which is a formulation more similar to the case of mechanical systems). A statement and proof of Noether's theorem is also given, using the latter formalism.     

Unconventional Approach to Orbital-Free Density Functional Theory Derived from a Model of Extended Electrons

An equation proposed by Levy, Perdew and Sahni (Phys. Rev. A 30:2745, 1984) is an orbital-free formulation of density functional theory. However, this equation describes a bosonic system. Here, we analyze on a very fundamental level, how this equation could be extended to yield a formulation for a general fermionic distribution of charge and spin. This analysis starts at the level of single electrons and with the question, how spin actually comes into a charge distribution in a non-relativistic model. To this end we present a space-time model of extended electrons, which is formulated in terms of geometric algebra. Wave properties of the electron are referred to mass density oscillations. We provide a comprehensive and non-statistical interpretation of wavefunctions, referring them to mass density components and internal field components. It is shown that these wavefunctions comply with the Schrödinger equation, for the free electron as well as for the electron in electrostatic and vector potentials. Spin-properties of the electron are referred to intrinsic field components and it is established that a measurement of spin in an external field yields exactly two possible results. However, it is also established that the spin of free electrons is isotropic, and that spin-dynamics of single electrons can be described by a modified Landau-Lifshitz equation. The model agrees with the results of standard theory concerning the hydrogen atom. Finally, we analyze many-electron systems and derive a set of coupled equations suitable to characterize the system without any reference to single electron states. The model is expected to have the greatest impact in condensed matter theory, where it allows to describe an N-electron system by a many-electron wavefunction Ψ of four, instead of 3N variables. The many-body aspect of a system is in this case encoded in a bivector potential.     

Thermodynamical Behavior of Inhomogeneous Universe with Varying Λ in Presence of Electromagnetic Field

Thermodynamical behavior of inhomogeneous universe with varying Λ in presence of electromagnetic field is obtained. F ₁₂ is the non-vanishing component of electromagnetic field tensor. To get a deterministic solution, it is assumed that the free gravitational field is Petrov type-II non-degenerate. The value of cosmological constant is found to be small and positive supported by recent results from the supernovae observations recently obtained by High-Z Supernovae Ia Team and Supernovae Cosmological Project. A relation between cosmological constant and thermodynamical quantities is established. Some physical and geometric properties of the model are also discussed.     

To Consider the Electromagnetic Field as Fundamental,and the Metric Only as a Subsidiary Field

In accordance with an old suggestion of Asher Peres (1962), we consider the electromagnetic field as fundamental and the metric as a subsidiary field. In following up this thought, we formulate Maxwell’s theory in a diffeomorphism invariant and metric-independent way. The electromagnetic field is then given in terms of the excitation and the field strength F = (E,B). Additionally, a local and linear “spacetime relation” is assumed between H and F, namely H ~ κ F, with the constitutive tensor κ. The propagation is studied of electromagnetic wave fronts (surfaces of discontinuity) with a method of Hadamard. We find a generalized Fresnel equation that is quartic in the wave covector of the wave front. We discuss under which conditions the waves propagate along the light cone. Thereby we derive the metric of spacetime, up to a conformal factor, by purely electromagnetic methods.     

Geometric classification of the torsion tensor of space‐time

Torsion appears in literature in quite different forms. Generally, spin is considered to be the source of torsion, but there are several other possibilities in which torsion emerges in different contexts. In some cases a phenomenological counterpart is absent, in some other cases torsion arises from sources without spin as a gradient of a scalar field. Accordingly, we propose two classification schemes. The firstone is based on the possibility to construct torsion tensors from the product of a covariant bivector and a vector and their respective space‐time properties. The secondone is obtained by starting from the decomposition of torsion into three irreducible pieces. Their space‐time properties again lead to a complete classification. The classifications found are given in a U ₄, a four dimensional space‐time where the torsion tensors have some peculiar properties. The irreducible decomposition is useful since most of the phenomenological work done for torsion concerns four dimensional cosmological models. In the second part of the paper two applications of these classification schemes are given. The modifications of energy‐momentum tensors are considered that arise due to different sources of torsion. Furthermore, we analyze the contributions of torsion to shear, vorticity, expansion and acceleration. Finally the generalized Raychaudhuri equation is discussed.     

The Difference Between the Standard and The Lorentz Transformations of the Electric and the Magnetic Fields

In this paper it is shown by using the Clifford algebra formalism that the usual Lorentz transformations of the three-dimensional (3D) vectors of the electric and magnetic fields E and B (which will be named as standard transformations (ST)) are different than the Lorentz transformations (LT) of well-defined quantities from the 4D spacetime. This difference between the ST and the LT is obtained regardless of the used algebraic objects (1-vectors or bivectors) for the representation of the electric and magnetic fields in the usual observer dependent decompositions of F. The LT correctly transform the whole 4D quantity, e.g., E_f : F · γ₀, whereas the ST are the result of the application of the LT only to the part of E_f, i.e., to F, but leaving γ₀ unchanged. The new decompositions of F in terms of 4D quantities that are defined without reference frames, i.e., the absolute quantities, are introduced and discussed. It is shown that the LT of the 4D quantities representing electric and magnetic fields correctly describe the motional electromotive force (emf) for all relatively moving inertial observers, whereas it is not the case with the ST of the 3D E and B.     

Homothetic motions with null homothetic bivectors in general relativity

It is shown that if a nonflat vacuum space-time admits a homothetic vector field with a null homothetic bivector then that space-time is algebraically special. If that homothetic vector field is a nontrivial one (not a Killing one) then the space-time is Petrov type III orN.     

Spherically symmetric solution in higher-dimensional teleparallel equivalent of general relativity

A theory of(N+1)-dimensional gravity is developed on the basis of the teleparallel equivalent of general relativity(TEGR).The fundamental gravitational field variables are the(N+1)-dimensional vector fields,defined globally on a manifold M,and the gravitational field is attributed to the torsion.The form of Lagrangian density is quadratic in torsion tensor.We then give an exact five-dimensional spherically symmetric solution(Schwarzschild(4+1)-dimensions).Finally,we calculate energy and spatial momentum using gravitational energy-momentum tensor and superpotential 2-form.     

Mass spectra of light and heavy mesons in the-Dirac equation with power-law potential

The mass spectra of both light and heavy mesons are studied in the Dirac equation with an equally mixed 4-vector and scalar power-law potential model. This potential provides an excellent fit not only to the mass spectra ofp°, ϕ, Ψ and υ families but also to those ofD, F andB mesons. The light quark masses inp° and ϕ as well as in atom-like mesons are very close to the current quark masses.     

“True Transformations Relativity” and Electrodynamics

Different approaches to special relativity (SR) are discussed. The first approach is an invariant approach, which we call the true transformations (TT) relativity. In this approach a physical quantity in the four-dimensional spacetime is mathematically represented either by a true tensor (when no basis has been introduced) or equivalently by a coordinate-based geometric quantity comprising both components and a basis (when some basis has been introduced). This invariant approach is compared with the usual covariant approach, which mainly deals with the basis components of tensors in a specific, i.e., Einstein's coordinatization of the chosen inertial frame of reference. The third approach is the usual noncovariant approach to SR in which some quantities are not tensor quantities, but rather quantities from 3+1 space and time, e.g., the synchronously determined spatial length. This formulation is called the apparent transformations (AT) relativity. It is shown that the principal difference between these approaches arises from the difference in the concept of sameness of a physical quantity for different observers. This difference is investigated considering the spacetime length in the TT relativity and spatial and temporal distances in the AT relativity. It is also found that the usual transformations of the three-vectors (3-vectors) of the electric and magnetic fields E and B are the AT. Furthermore it is proved that the Maxwell equations with the electromagnetic field tensor F^ab and the usual Maxwell equations with E and B are not equivalent, and that the Maxwell equations with E and B do not remain unchanged in form when the Lorentz transformations of the ordinary derivative operators and the AT of E and B are used. The Maxwell equations with F^ab are written in terms of the 4-vectors of the electric E^a and magnetic B^a fields. The covariant Majorana electromagnetic field 4-vector ^a is constructed by means of 4-vectors E^a and B^a and the covariant Majorana formulation of electrodynamics is presented. A Dirac like relativistic wave equation for the free photon is obtained.     

Hestenes' Tetrad and Spin Connections

Defining a spin connection is necessary for formulating Dirac's bispinor equation in a curved space-time. Hestenes has shown that a bispinor field is equivalent to an orthonormal tetrad of vector fields together with a complex scalar field. In this paper, we show that using Hestenes' tetrad for the spin connection in a Riemannian space-time leads to a Yang-Mills formulation of the Dirac Lagrangian in which the bispinor field Ψ is mapped to a set of SL(2,R)× U(1) gauge potentials F_α^K and a complex scalar field ρ. This result was previously proved for a Minkowski space-time using Fierz identities. As an application we derive several different non-Riemannian spin connections found in the literature directly from an arbitrary linear connection acting on the tensor fields (F_α^K, ρ). We also derive spin connections for which Dirac's bispinor equation is form invariant. Previous work has not considered form invariance of the Dirac equation as a criterion for defining a general spin connection.     

Methods of Quantum Field Theory with Rapidly Increasing Spectral Functions

A review of the methods of constructing the scattering amplitudes and Green functions in the non-renormalizable quantum field theories with rapidly increasing spectral functions is given. Such theories as the theories with the chiral Lagrangians, those which describe the interactions with gravitational fields and the theories of weak interactions can be taken as examples. Three big groups of these methods are presented here. In particular, these are as follows: construction of the scattering amplitude in x-space (the function F(x)), construction of the scattering amplitude in momentum space (the function F˜ (p)) and definition of the scattering amplitudes by solving the appropriate equations (in x- or p-Space). This works was reported at the Second International Symposium on Non-local Quantum Field Theory in 1970 at Azau, USSR.