Maxwell-like Equations Research Articles

Constrain the time variation of the gravitational constant via the propagation of gravitational waves

The gravitational constant variation means the breakdown of the strong equivalence principle. As the cornerstone of general relativity, the validity of general relativity can be examined by studying the gravitational constant variation. Such variations have the potential to affect both the generation and propagation of gravitational waves. Here, our focus lies on the effect of gravitational constant variation specifically on the propagation of gravitational waves. In a previous paper (An et al., 2023 [13]) we have studied such effect based on a Maxwell-like equation. That work admits two drawbacks. The assumption that describing gravitational waves with Maxwell-like equation is not solid. And more such treatment can only deal with space dependent gravitational constant. In the current paper we will remedy these two issues. We employ two analytical methods, namely based on the Fierz-Pauli action and the perturbation of Einstein-Hilbert action around Minkowski spacetime, both leading to the same gravitational wave equation. By solving this equation, we find the effects of gravitational constant variation on gravitational wave propagation. The result is consistent with previous investigations based on Maxwell-like equations for gravitational waves for space dependent gravitational constant cases. In addition we can constrain the time variation of the gravitational constant via the propagation of gravitational waves based on GW170817 data. And more, we find that small variations in the gravitational constant result in an amplitude correction at the leading order and a phase correction at the sub-leading order for gravitational waves. These results provide valuable insights for probing gravitational constant variation and can be directly applied to gravitational wave data analysis.

Neoclassical models of charged particles

Classical electrodynamics (CED) has achieved great success in its domain of application, but despite this success, it has remained a theory that lacks complete self-consistency. It is worthwhile trying to make CED a self-consistent theory, because many important phenomena lie within its scope, and because modern field theories have been modelled on it. Alternative approaches to CED might help finding a definite formulation, and they might also lead to the prediction of new phenomena. Here we report two main results. The first one derives from standard CED. It is shown that the motion of a charged particle is ruled not only by the Lorentz equation, but also by equations that are formally identical to Maxwell equations. The latter hold for a velocity field and follow as a strict logical consequence of Hamilton’s action principle for a single particle. We construct a tensor with the velocity field in the same way as the electromagnetic tensor is constructed with the four potential. The two tensors are shown to be proportional to one another. As a consequence, and without leaving the realm of standard CED, one can envision new phenomena for a charged particle, which parallel those involving electromagnetic fields. The second result refers to a field-free approach to CED. This approach confirms the simultaneous validity of Maxwell-like and Lorentz equations as rulers of charged particle motion.

Retarded solutions for Weyl gravitoelectromagnetic fields

The retarded solutions to the Maxwell-like equations of Weyl gravitoelectromagnetism (GEM) are obtained in a similar way to those of electromagnetism. These solutions keep an analogous structure as their electromagnetic counterparts and they become the Jefimenko-type equations of GEM.

A geometrization on dual 1-jet spaces of the time-dependent Hamiltonian of electrodynamics

In this paper, we develop the distinguished Riemannian differential geometry (in the sense of d-connections, d-torsions, d-curvatures, and the geometrical Maxwell-like and Einstein-like equations) for the time-dependent Hamiltonian of momenta which governs the electrodynamics phenomena.

QUANTIZATION OF THE GRAVITATIONAL FIELD. THEORETICAL AND EXPERIMENTAL SUBSTANTIATION OF THE GRAVITATIONAL-ELECTROMAGNETIC RESONANCE. THE PHYSICAL NATURE OF THE QUANTUM OF THE GRAVITATIONAL FIELD.WHY THE SPEED OF LIGHT IN VACUUM IS CONSTANT

It is shown that gravitating objects that are at rest, or move without acceleration, create a standing gravitational wave in space. The length of this wave is a quantization step of the gravitational field. It is proportional to the mass of the gravitating object. The coefficient of proportionality is a value that is inverse to the linear density of the Planck mass, that is, proportional to the linear rarefaction of the Planck mass. A physically standing gravitational wave is a curvature, deformation of space under the influence of the gravitational field of a gravitating object. If we imagine a gravitating object as a material point, then the geometric picture of a standing gravitational wave can be represented as a set of hierarchical spherical equipotential surfaces embedded in each other, the radius of which changes away from the center of gravity by the value of the quantization step. It is shown that a standing gravitational wave has a quantum character. The quantum of the gravitational field is the square of the speed of light in a vacuum. The quantum of the gravitational field is equal to the gravitational potential of the gravitating object at a distance from it equal to the quantization step. Theoretical and experimental substantiation of the presence of gravitational-electromagnetic resonance (GER) in nature is given. This resonance arises when the wave vectors of a standing gravitational wave and an electromagnetic wave traveling in space are equal. GER is the basis for modulating the emission spectrum of stars and their clusters. The wavelength of the envelope of the spectrum is proportional to the mass of the radiating object. By measuring the wavelength of the envelope, one can accurately estimate the mass of the radiating object. The physical nature of the quantum gravitational field is the kinematic gravitational viscosity of the gravitational field of the baryonic matter of the observable Universe. Keywords: Maxwell-like equations of the gravitational field, the standing gravitational wave, the quantum of the gravitational field, the gravitational electromagnetic resonance, Planck constants, kinematic viscosity of the gravitational field.

A Note on the Gravitoelectromagnetic Analogy

We discuss the linear gravitoelectromagnetic approach used to solve Einstein’s equations in the weak-field and slow-motion approximation, which is a powerful tool to explain, by analogy with electromagnetism, several gravitational effects in the solar system, where the approximation holds true. In particular, we discuss the analogy, according to which Einstein’s equations can be written as Maxwell-like equations, and focus on the definition of the gravitoelectromagnetic fields in non-stationary conditions. Furthermore, we examine to what extent, starting from a given solution of Einstein’s equations, gravitoelectromagnetic fields can be used to describe the motion of test particles using a Lorentz-like force equation.

To the Theory of a Unified Field

Maxwell-like equations are offered that are valid not only for electromagnetic, but also for gravitational fields. They are based on an expanded form of the law of energy conversion, which explicitly takes into account the nonequilibrium of systems and the ability of energy to transfer from one energy carrier to another. This makes it possible to eliminate the distinction between electromechanics and the field theory, to find out the meaning of the magnetic vector potential as a function of the charge rotation speed, to reveal the tensor nature of the magnetic field and the presence of a divergent component of the scalar nature in it. The existence of the Lorentz magnetic forces and the presence of a work-performing moment therein are proved. Maxwell-like equations do not contain field operators and have a simpler form covering nevertheless a broader spectrum of phenomena due to taking into account the convective components of bias currents. Wave equations alternative to the Maxwell ones are offered and the non-electromagnetic nature of light is substantiated.

Dispersion relations in nonlinear electrodynamics and the kinematics of the Compton effect in a magnetic background

Non-linear electrodynamic models are re-assessed in this paper to pursue an investigation of the kinematics of the Compton effect in a magnetic background. Before considering specific models, we start off by presenting a general non-linear Lagrangian built up in terms of the most general Lorentz- and gauge-invariant combinations of the electric and magnetic fields. The extended Maxwell-like equations and the energy-momentum tensor conservation are presented and discussed in their generality. We next expand the fields around a uniform and time-independent electric and magnetic backgrounds up to second order in the propagating wave, and compute dispersion relations which account for the effect of the external fields. We obtain thereby the refraction index and the group velocity for the propagating radiation in different situations. In particular, we focus on the kinematics of the Compton effect in presence of external magnetic fields. This yields constraints that relate the derivatives of the general Lagrangian with respect to the field invariants and the magnetic background under consideration. We carry out our inspection by focusing on some specific non-linear electrodynamic effective models: Hoffmann-Infeld, Euler-Heisenberg, generalized Born-Infeld and Logarithmic.

Cosmological bounces, cyclic universes, and effective cosmological constant in Einstein-Cartan-Dirac-Maxwell theory

Einstein-Cartan theory is an extension of the standard formulation of general relativity characterized by a nonvanishing torsion. The latter is sourced by the matter fields via the spin tensor, and its effects are expected to be important at very high spin densities. In this work, we analyze in detail the physics of Einstein-Cartan theory with Dirac and Maxwell fields minimally coupled to the spacetime torsion. This breaks the $U(1)$ gauge symmetry, which is suggested by the possibility of a torsion-induced phase transition in the early Universe. The resulting Dirac-like and Maxwell-like equations are nonlinear with self-interactions as well as having fermion-boson nonminimal couplings. We discuss several cosmological aspects of this theory under the assumption of randomly oriented spin densities (unpolarized matter), including bounces, acceleration phases, and matter-antimatter asymmetry in the torsion era, as well as late-time effects such as the generation of an effective cosmological constant, dark energy, and future bounces within cyclic solutions.

Coupling of a biquaternionic Dirac field to a bosonic field

We extend the biquaternionic Dirac equation to include interactions with a background bosonic Geld. The obtained biquaternionic Dirac equation yields Maxwell-like equations that hold for both a matter Geld and an electromagnetic Geld. We establish that the electric Geld is perpendicular to the matter magnetic Geld and the magnetic Geld is perpendicular to the matter inertial Geld. We show that the inertial and magnetic masses are conserved separately. The magnetic mass density arises as a result of the coupling between the vector potential and the matter inertial Geld. The presence of the vector and scalar potentials and also the matter inertial and magnetic Gelds modify the Standard form of the derived Maxwell equations. The resulting interacting electrodynamics equations are generalizations of the equations of Wilczek or Chern-Simons axion-like Gelds. The coupled Geld in the biquaternioic Dirac Geld reconstructs the Wilczek axion Geld. We show that the electromagnetic Geld vector \(\overrightarrow{F}=\overrightarrow{E}+ic\overrightarrow{B}\), where \(\overrightarrow{E}\) and \(\overrightarrow{B}\) are the respective electric and magnetic Gelds, satisGes the massive Dirac equation and, moreover, \(\overrightarrow{\triangledown}\cdot\overrightarrow{F}=0\).

Relativistic generalization of the Schrödinger-Newton model for the wavefunction reduction

We consider the model of the self-gravity driven spontaneous wavefunction reduction proposed by L. Diosi, R. Penrose et al. and based on a self-consistent system of Schrödinger and Poisson equations. An analogous system of coupled Dirac and Maxwell-like equations is proposed as a relativization. Regular solutions to the latter form a discrete spectrum in which all the “active” gravitational masses are always positive, and approximately equal to inertial masses and to the mass [Formula: see text] of the quanta of Dirac field up to the corrections of order [Formula: see text]. Here [Formula: see text] is the gravitational analogue of the fine structure constant negligibly small for nucleons. In the limit [Formula: see text] the model reduces back to the nonrelativistic Schrödinger-Newton one. The equivalence principle is fulfilled with an extremely high precision. The above solutions correspond to various states of the same (free) particle rather than to different particles. These states possess a negligibly small difference in characteristics but essentially differ in the widths of the wavefunctions. For the ground state the latter is [Formula: see text] times larger the Compton length, so that a nucleon cannot be sufficiently localized to model the reduction process.

Fundamental Fields as Eigenvectors of the Metric Tensor in a 16-Dimensional Space-Time

An alternative approach to the usual Kaluza-Klein way to field unification is presented which seems conceptually more satisfactory and elegant. The main idea is that of associating each fundamental interaction and matter field with a vector potential which is an eigenvector of the metric tensor of a multidimensional space-time manifold (n-dimensional “vierbein”). We deduce a system of field equations involving both Einstein and Maxwell-like equations for the fundamental fields. Confinement of the fields within the observable 4-dimensional space-time and non-vanishing particles’ rest mass problem are shown to be related to the choice of a scalar boson field (Higgs boson) appearing in the theory as a gauge function. Physical interpretation of the results, in order that all the known fundamental interactions may be included within the metric and connection, requires that the extended space-time is 16-dimensional. Fermions are shown to be included within the additional components of the vector potentials arising because of the increased dimensionality of space-time. A cosmological solution is also presented providing a possible explanation both to space-time flatness and to dark matter and dark energy as arising from the field components hidden within the extra space dimensions. Suggestions for gravity quantization are also examined.

Linearized electromagnetic and gravito-Heavisidian chirality in split octonion space-time

In this paper, we construct a split octonionic mathematical approach to generalized electromagnetic and gravito-Heavisidian chirality of dyons by modification of the Drude–Born–Fedorov constitutive relations. In this context, we describe dual Euclidean space-times structure associated with [Formula: see text] Zorn’s vector matrix realization of split octonion. As such, using the Zorn’s vector matrix realization, an alternative form of generalized Proca–Maxwell equations of massive dyons is obtained in chiral media. It is well known that in weak unified gravito-Heavisidian field, the Einstein’s equations become Maxwell-like equations under the first approximation. Thus, we study the gravito-Heavisidian analogous theory to electromagnetic theory, and discuss the Drude–Born–Fedorov constitutive relations, gravito-Heavisidian field, Proca–Maxwell equations and gravito-Heavisidian wave equations for linear gravitational chiral field of gravito-dyons in flat split octonion space-time.

On Maxwell Equations for Gravitational Field

For explicitly time depending mass density which satisfies a continuity equation, it is shown that Maxwell-like equations for gravitational field follow naturally without any need of General Relativity Theory approximation or related assumptions. As a consequence, it is shown that several features already known in Electrodynamics (Poynting vector, density of energy, tensor stress, and radiation) are totally reproduced for gravitational field.

On the Relativistic Canonical Quantum Mechanics and Field Theory of Arbitrary Spin

The new relativistic equations of motion for the particles with arbitrary spin and nonzero mass, suggested by author in years 2014–2016, are under consideration. The complete version of brief paper in J. Phys: Conf. Ser., 670 (2016) 012047(1-16) is given. The axiomatic level description of the relativistic canonical quantum mechanics of an arbitrary mass and spin has been given. The 64-dimensional ClR(0,6) algebra in terms of Dirac gamma matrices has been suggested. The interpretation of the 28-dimensional gamma matrix representation of SO(8) algebra over the field of real numbers is given. The link between the relativistic canonical quantum mechanics of the arbitrary spin and the covariant local field theory in the form of extended Foldy–Wouthuysen transformation has been found. Different methods of the Dirac equation derivation have been reviewed. The manifestly covariant field equation for an arbitrary spin that follows from the corresponding equation of relativistic canonical quantum mechanics, has been considered. The found equations are without redundant components. The partial examples for spin s=3/2 and s=2 are presented. The covariant local field theory equations for spin s = (3/2,3/2) particle-antiparticle doublet and spin s = (2,2) particle-antiparticle doublet have been introduced. The Maxwell and slightly generalized Maxwell-like equations containing mass member have been considered as well.

Vorticity and helicity in relativistic heat-conducting fluid

The evolution of heat-conducting fluid described by a pair of Maxwell-like equations is used to construct thermal-fluid helicity and thermal-helicity currents. These currents are found to be dissipative. It is shown that the magnetic part of the particle vorticity two-form is a thermal-fluid vorticity flux vector field composed of a linear combination of the fluid’s vorticity and a spacelike twist of heat flow lines. Heat flow lines are non-geodesic because of the interplay between gravitation and the entropy entrainment in a system composed of a heat-conducting fluid which is in state of rapid differential rotation and far from equilibrium. In general, alignment of the heat flux vector with that of the fluid’s vorticity leads to non-conservation of thermal-fluid vorticity flux in both a thermal-fluid flux tube and a stream tube. It is demonstrated that the twist of the fluid’s vortex lines is caused by the heat flow along the fluid’s vorticity vector in the case of an axisymmetric stationary differentially rotating heat-conducting fluid configuration. In this case, dissipation of thermal-fluid vorticity flux along the flux tube is caused by coupled effects of the fluid’s vorticity magnitude, thermal resistivity and entropy entrainment.

The Particle as a Statistical Ensemble of Events in Stueckelberg–Horwitz–Piron Electrodynamics

In classical Maxwell electrodynamics, charged particles following deterministic trajectories are described by currents that induce fields, mediating interactions with other particles. Statistical methods are used when needed to treat complex particle and/or field configurations. In Stueckelberg-Horwitz-Piron (SHP) electrodynamics, the classical trajectories are traced out dynamically, through the evolution of a 4D spacetime event $x^\mu (\tau)$ as $\tau$ grows monotonically. Stueckelberg proposed to formalize the distinction between coordinate time $x^0 = ct$ (measured by laboratory clocks) and chronology $\tau$ (the temporal ordering of event occurrence) in order to describe antiparticles and resolve problems of irreversibility such as grandfather paradoxes. Consequently, in SHP theory, the elementary object is not a particle (a 4D curve in spacetime) but rather an event (a single point along the dynamically evolving curve). Following standard deterministic methods in classical relativistic field theory, one is led to Maxwell-like field equations that are \hbox{$\tau$-dependent} and sourced by a current that represents a statistical ensemble of instantaneous events distributed along the trajectory. The width $\lambda$ of this distribution defines a correlation time for the interactions and a mass spectrum for the photons emitted by particles. As $\lambda$ becomes very large, the photon mass goes to zero and the field equations become $\tau$-independent Maxwell's equations. Maxwell theory thus emerges as an equilibrium limit of SHP, in which $\lambda$ is larger than any other relevant time scale. Thus, statistical mechanics is a fundamental ingredient in SHP electrodynamics, and its insights are required to give meaning to the concept of a particle.

Gravitoelectromagnetism in teleparallel equivalent of general relativity: A new alternative

Spaniol and Andrade introduced gravitoelectromagnetism in TEGR by considering superpotentials, times the determinant of tetrads, as the gravitoelectromagnetic fields. However, since this defined gravitoelectromagnetic field strength does not give rise to a complete set of Maxwell-like equations, we propose an alternative definition of the gravitoelectromagnetic field strength: instead of superpotentials, torsions are taken as the gravitoelectromagnetic field strengths. Based on this new proposal, we are able to derive a complete set of Maxwell-like equations. We then apply them to obtain explicit expressions of the gravitoelectromagnetic fields both in Schwarzschilds spacetime and for gravitational waves.

Maxwellian quantum mechanics

Expanding the ordinary Dirac's equation in quaternionic form yields Maxwell-like field equations. As in the Maxwell's formulation, the particle fields are represented by a scalar, ψ0 and a vector ψ→. The analogy with Maxwell's equations requires that the inertial fields are E→D=c2α→×ψ→, and B→D=α→ψ0+cβψ→ and that ψ0=−cβα→·ψ→, where β, α→ and c are the Dirac matrices and the speed of light, respectively. An alternative solution suggests that magnetic monopole-like behavior accompanies Dirac's field. In this formulation, a field-like representation of Dirac's particle is derived. It is shown that when the vector field of the particle, ψ→, is normal to the vector α→, Dirac's field represents a medium with maximal conductivity. The energy flux (Poynting vector) of the Dirac's fields is found to flow in opposite direction to the particle's motion. A system of equivalently symmetrized Maxwell's equations is introduced. A longitudinal (scalar) wave traveling at speed of light is found to accompany magnetic charges flow. This wave is not affected by presence of electric charges and currents. The Lorentz boost transformations of the matter fields are equivalent to cψ→′=cψ→±βα→ψ0,ψ0′=ψ0∓cβα→·ψ→.

Link between the relativistic canonical quantum mechanics of arbitrary spin and the corresponding field theory

The new relativistic equations of motion for the particles with arbitrary spin and nonzero mass have been introduced. The axiomatic level description of the relativistic canonical quantum mechanics of the arbitrary mass and spin has been given. The 64-dimensional ClR(0,6) algebra in terms of Dirac gamma matrices has been suggested. The link between the relativistic canonical quantum mechanics of the arbitrary spin and the covariant local field theory has been found. Different methods of the Dirac equation derivation have been reviewed. The manifestly covariant field equations for an arbitrary spin that follow from the quantum mechanical equations have been considered. The covariant local field theory equations for spin s = (1,1) particle-antiparticle doublet, spin s = (1,0,1,0) particle antiparticle multiplet, spin s = (3/2,3/2) particle-antiparticle doublet, spin s = (2,2) particle-antiparticle doublet, spin s = (2,0,2,0) particle-antiparticle multiplet and spin s = (2,1,2,1) particle-antiparticle multiplet have been introduced. The Maxwell-like equations for the boson with spin s = 1 and nonzero mass have been introduced as well.