Maxwell-like Equations Research Articles

Mixed-symmetry multiplets and higher-spin curvatures

We study the higher-derivative equations for gauge potentials of arbitrary mixed-symmetry type obtained by setting to zero the divergences of the corresponding curvature tensors. We show that they propagate the same reducible multiplets as the Maxwell-like second-order equations for gauge fields subject to constrained gauge transformations. As an additional output of our analysis, we provide a streamlined presentation of the Ricci-like case, where the traces of the same curvature tensors are set to zero, and we present a simple algebraic evaluation of the particle content associated with the Labastida and with the Maxwell-like second-order equations.

Reformulation of electromagnetic and gravito-electromagnetic equations for Lorentz system with octonion algebra

In this paper, the real, complex octonion algebra and their properties are defined. The electromagnetic and gravito-electromagnetic equations with monopoles in terms of S and $$\hbox {S}^{\prime }$$ reference systems are presented in vector notations. Additionally, the duality transformations of gravito-electromagnetic situation for two reference systems are also represented. Besides, it is explained that Maxwell-like equations for gravito-electromagnetism are also invariant under Lorentz transformations. By introducing complex octonionic differential operator, a new generalized complex octonionic field term consisting of electromagnetic and gravito-electromagnetic components has been firstly suggested for Lorentz system. Afterwards, a complex octonionic source equation is obtained as in basic way, more compact and elegant notation. By defining a new complex octonionic general potential term, the field equation is attained once again. The components of complex octonionic field and wave equations are written in detailed for S and $$\hbox {S}^{\prime }$$ reference systems.

Maxwellian theory of gravitational waves and their mechanical properties

We present a theory in Maxwellian form for gravitational waves in a flat background. This requires us to identify the gravitational analogues of the electric and magnetic fields for light. An important novelty, however, is that our analogues are not vector fields but rather rank-two tensor fields; in place of a three-component vector at each point in space, as in electromagnetism, our fields are three by three symmetric matrices at each point. The resulting Maxwell-like equations lead directly to a Poynting theorem for the local energy density associated with a gravitational wave and to associated local properties including densities of momentum and angular momentum.

Representation of electromagnetic and gravitoelectromagnetic Poynting theorems in higher dimensions

This study investigates whether the electromagnetic and gravitoelectromagnetic energy conservation equations are obtained together by using octonion algebra or not. Maxwell and Maxwell-like equations for linear gravity with magnetic monopole terms are used in the SI unit system. A new complex octonionic eld term is suggested for the rst time. The complex octonionic source equation is then obtained. Finally, Poynting theorems for both electromagnetism and gravitoelectromagnetism are dened for the rst time by using higher dimensional algebra.

Reformulation of Relativistic Quantum Mechanics Equations with Non-Commutative Sedeons

We present sixteen-component values “sedeons”, generating associative non-commutative space-time algebra. The generalized relativistic wave equations based on sedeonic wave function and space-time operators are proposed. We demonstrate that sedeonic second-order wave equation for massive field can be reformulated as the quasi-classical equation for the potentials of the field or in equivalent form as the Maxwell-like equations for the field intensities. The sedeonic first-order Dirac-like equations for massive and massless fields are also discussed.

SEDEONIC GENERALIZATION OF RELATIVISTIC QUANTUM MECHANICS

We represent sixteen-component values "sedeons," generating associative noncommutative space–time algebra. We demonstrate a generalization of relativistic quantum mechanics using sedeonic wave functions and sedeonic space–time operators. It is shown that the sedeonic second-order equation for the sedeonic wave function, obtained from the Einstein relation for energy and momentum, describes particles with spin 1/2. We showed that the sedeonic second-order wave equation can be reformulated in the form of the system of the first-order Maxwell-like equations for the massive fields. We proposed the sedeonic first-order equations analogous to the Dirac equation, which differ in space–time properties and describe several types of massive and massless particles. In particular we proposed four different equations, which could describe four types of neutrinos.

Do mirrors for gravitational waves exist?

Thin superconducting films are predicted to be highly reflective mirrors for gravitational waves at microwave frequencies. The quantum-mechanical non-localizability of the negatively charged Cooper pairs, which is protected from the localizing effect of decoherence by an energy gap, causes the pairs to undergo non-picturable, non-geodesic motion in the presence of a gravitational wave. This non-geodesic motion, which is accelerated motion through space, leads to the existence of mass and charge supercurrents inside the superconducting film. On the other hand, the decoherence-induced localizability of the positively charged ions in the lattice causes them to undergo picturable, geodesic motion as they are carried along with space in the presence of the same gravitational wave. The resulting separation of charges leads to a virtual plasma excitation within the film that enormously enhances its interaction with the wave, relative to that of a neutral superfluid or any normal matter. The existence of strong mass supercurrents within a superconducting film in the presence of a gravitational wave, dubbed the “Heisenberg–Coulomb effect,” implies the specular reflection of a gravitational microwave from a film whose thickness is much less than the London penetration depth of the material, in close analogy with the electromagnetic case. The argument is developed by allowing classical gravitational fields, which obey Maxwell-like equations, to interact with quantum matter, which is described using the Bardeen–Cooper–Schrieffer (BCS) and Ginzburg–Landau theories of superconductivity, as well as a collisionless plasma model. Several possible experimental tests of these ideas, including mesoscopic ones, are presented alongside comments on the broader theoretical implications of the central hypothesis.

DERIVATION OF THE SOURCE-FREE MAXWELL AND GRAVITATIONAL RADIATION EQUATIONS BY GROUP THEORETICAL METHODS

We derive source-free Maxwell-like equations in flat space–time for any helicity j by comparing the transformation properties of the 2(2j+1) states that carry the manifestly covariant representations of the inhomogeneous Lorentz group with the transformation properties of the two helicity j states that carry the irreducible representations of this group. The set of constraints so derived involves a pair of curl equations and a pair of divergence equations. These reduce to the free-field Maxwell equations for j = 1 and the analogous equations coupling the gravito-electric and the gravito-magnetic fields for j = 2.

Differential operators in terms of Clebsch-Gordon (C-G) coefficients and the wave equation of massless tensor fields

The quantum theory of angular momentum affords a treatment of tensors and vectors in a spherical basis. By using this theory we define the tensor differential operators: divergence, curl and gradient which act on a tensor of any rank, in terms of C-G coefficients. With these definitions we obtain a matrix representation and useful properties for those operators. An interesting application of this formalism is to find the wave equation of a tensor of any rank in a linear theory. This provides a new common way to look at the wave equations associated with both Maxwell's equations and the Maxwell-like equations for the linearized Weyl curvature tensor in gravitoelectromagnetism describing gravitational radiation on a Minkowski spacetime background.

Electromagnetic Herglotz dyadics in chiral media

In this paper electromagnetic Herglotz dyadics in chiral media are considered. The basic equations for dyadic electromagnetic fields in chiral media are presented and the direct scattering problem for the perfect conductor is formulated as a dyadic one. We define Beltrami Herglotz dyadics using the corresponding vector fields and the Herglotz norm as well. It is proved that a Beltrami dyadic satisfies the Herglotz condition if and only if it is a Herglotz dyadic. A representation theorem for Beltrami Herglotz dyadics is proved and a density theorem is extended to the dyadic case. A dyadic electromagnetic Herglotz pair is defined and a growth property for it is derived. It is proved that the set of all dyadic electromagnetic Herglotz pairs is dense within the set of dyadic electromagnetic pairs which solve the Maxwell-like equations for electromagnetic scattering in chiral media.

Asymmetric Perfectly Matched Layer for the Absorption of Waves

The perfectly matched layer (PML) has become a standard for comparison in the techniques that have been developed to close the system of Maxwell equations (more generally, wave equations) when simulating an open system. The original Berenger PML formulation relies on a split version of Maxwell equations, with numerical electric and magnetic conductivities. We present here an extension of this formulation, which introduces counterparts of the electric and magnetic conductivities affecting the term which is spatially differentiated in the equations. The phase velocity along each direction is also multiplied by an additional coefficient. We show that under certain constraints on the additional numerical coefficients, this ‘medium’ does not generate any reflection at any angle or any frequency and is thus a perfectly matched layer. Technically it is a superset of Berenger's PML to which it reduces for a specific set of parameters, and like it, it is anisotropic. However, unlike the PML, it introduces some asymmetry in the absorption rate and is therefore labeled an APML, for asymmetric perfectly matched layer. We present here the numerical considerations that have led us to introduce such a medium as well as its theory. Several finite-difference numerical implementations are derived (in one, two, and three dimensions) and the performance of the APML is contrasted with that of the PML in one and two dimensions. Using plane wave analysis, we show that our APML implementations lead to higher absorption rates than the considered PML implementations. Although we have considered in this paper the finite-difference discretization of Maxwell-like equations only, the APML system of equations may be used with other discretization schemes, such as finite elements, and may be applied to other equations, for applications beyond electromagnetics.

THE EFFECTIVE ENERGY-MOMENTUM TENSOR IN KALUZA–KLEIN GRAVITY WITH LARGE EXTRA DIMENSIONS AND OFF-DIAGONAL METRICS

We consider a version of Kaluza–Klein theory where the cylinder condition is not imposed. The metric is allowed to have explicit dependence on the "extra" coordinate(s). This is the usual scenario in brane-world and space-time-matter theories. We extend the usual discussion by considering five-dimensional metrics with off-diagonal terms. We replace the condition of cylindricity by the requirement that physics in four-dimensional space-time should remain invariant under changes of coordinates in the five-dimensional bulk. This invariance does not eliminate physical effects from the extra dimension but separates them from spurious geometrical ones. We use the appropriate splitting technique to construct the most general induced energy-momentum tensor, compatible with the required invariance. It generalizes all previous results in the literature. In addition, we find two four-vectors, [Formula: see text] and [Formula: see text], induced by off-diagonal metrics, that separately satisfy the usual equation of continuity in 4D. These vectors appear as source-terms in equations that closely resemble the ones of electromagnetism. These are Maxwell-like equations for an antisymmetric tensor [Formula: see text] that generalizes the usual electromagnetic one. This generalization is not an assumption, but follows naturally from the dimensional reduction. Thus, if[Formula: see text] could be identified with the electromagnetic tensor, then the theory would predict the existence of classical magnetic charge and current. The splitting formalism used allows us to construct 4D physical quantities from five-dimensional ones, in a way that is independent from how we choose our space-time coordinates from those of the bulk.

Schwarz–Sen duality made fully local

Duality symmetric electromagnetic action à la Schwarz–Sen is shown to appear naturally in a chain of equivalent actions which interchange equations of motion with Bianchi identities. Full symmetry of the electromagnetic stress tensor is exploited by generalizing this duality symmetric action to allow for a space-time dependent mixing angle between electric and magnetic fields. The rotated fields are shown to satisfy Maxwell-like equations which involve the mixing angle as a parameter, and a generalized gauge invariance of the new action is established.

Quaternionic formulation of the classical fields

In this article we present the quaternionic formulation of classical Maxwell-like field equations using complex quaternions and show that in this formalism they get an elegant, economical and compact form. In the quaternionic description of classical fields two quantities are essential: the field quaternion embracing all field variables and the differential quaternionic operator embracing the time and space differential operators. Applying the quaternionic operator on the field quaternion one gets an equation which contains many individual equations. In this formalism the eight equations of the Maxwell equations can be written as one quaternionic. We describe different types of quaternionic fields and discuss their possible occurrence in nature.

Describing Weyl neutrinos by a set of Maxwell-like equations

In this note we show that Weyl’s theory of massless neutrinos can be formulated in a form closely parallel to Maxwell’s electromagnetism. The neutrino spinor wave function is explicitly represented in terms of its real and imaginary components. The Weyl equation then takes a form similar to a set of inhomogeneous Maxwell equations involving a particularlongitudinal source. We relate plane-wave solutions of these equations to those corresponding to actual neutrinos and anti-neutrinos, identifying the spinor components in each case. We briefly comment on the possibility of constructing simple supersymmetric models relating spin-(l/2) and spin-1 fields.

Analysis of simple constitutive equations for viscoelastic liquids

The behavior of general Maxwell-like constitutive equations is analyzed. The total set of the constitutive equations includes the evolution equations for the “configuration tensor” c and an expression for the stress tensor through the tensor c. Several problems are considered. (i) It is shown that the general form for these equations derived recently from the Poissonbracket formalism, is more easily obtained by using a local approach of irreversible thermodynamics for both compressible and incompressible cases. Also, general Maxwell-like constitutive equations, with mixed upper and lower convected time derivatives, are derived using this method, (ii) The problem of Hadamard stability for these constitutive equations is then studied by using the method of “frozen coefficients”. The necessary and sufficient conditions for evolution are established. These conditions demonstrated the common results of 3D instability for the Johnson-Segalman and Phan-Thien-Tanner generalized constitutive equations, with exceptions only for equations with upper and lower convected derivatives. Therefore only Maxwell liquids with a “perfect elastic limit” are further studied. For these liquids, the evolution conditions are expressed explicitly as some constraints imposed on possible forms of free energy (elastic potential). It is also demonstrated that, for the latter liquids, the evolution conditions are the same as those for thermodynamic stability. For the incompressible case, with det c = 1, simple sufficient conditions for global evolutionarity are also found. (iii) It is shown that the configuration tensor c is positive definite for any piecewise smooth strain history. Also, simple sufficient conditions are found for the boundedness of the stress tensor at a given strain history. (iv) The results are applied to some viscoelastic constitutive equations proposed in the literature. (v) The importance of det c in the formulations of the constitutive equations and the relations between the instabilities in rheological equations and instabilities observed in flows of polymer liquids are then discussed.

Electromagnetic Picture of Inertial Forces

It is found from the equation of motion of a massive particle in a general noninertial frame that inertial forces on the particle can be regarded equivalently as the electromagnetic Lorentz force. Here the vector potential corresponds to the local velocity of the noninertially moving frame and the scalar potential to the local kinetic energy of the frame. The above consideration is applied to a rotating system with time-dependent angular velocity. As another application, on regarding a locally disturbed medium which carries an elastic wave as a noninertial frame, we derive Maxwell-like equations for the fields.

Some astrophysical consequences of the extended Maxwell-like gravitational field equations

The Maxwell-like gravitational field equations have been generalized and coupled through the gravitational four-potential on the electromagnetic Maxwell's equations. It is shown that this has several astrophysical consequences, of which the main are the following (i) the gravitational instability of a system of mass bodies manifesting itself by a Hubble-like motion on cosmological scales, (ii) the possible change of light intensity propagating through a large distance (and so a possible change of the real energy output of some very distant objects, e.g., quasars), (iii) non-stability of a planetary system on the cosmological time scales, due to the momentum increase of the moving bodies in a generalized gravitational field.

The generalized Einstein-Maxwell theory of gravitation

A new Lagrangian theory of gravitation in which the metric and the arbitrary affine connection are regarded as independent field variables has been considered. Making use of the pure geometrical objects only from the variational principle the empty field equations are derived. It is shown that the metric obeys the ordinary Einstein equations of general relativity. However, the covariant derivative of the metric tensor does not vanish, so that the vector's length is generally nonintegrable under the parallel displacement. The torsion trace vector turns out to be the natural dynamical variable, satisfying the Maxwell-like equations with tensor of homothetic curvature as the Maxwell tensor. The equations of motion are explored; they are shown to be identical to the motion of electric charge under the Lorentz force. The conservation laws are discussed.

A maxwellian tetrad theory of gravity with preferred frames: A

A theory of gravity is proposed which seeks to mimic maxwellian electromagnetism whilst maintaining the principle of equivalence. The curls of tetrad potentials are taken as field strengths and Maxwell-like free-field equations are set up which contain a non-linear gravitational current term and which are generally covariant but not locally Lorentz covariant. The weak field approximation is solved for static metrical spherical symmetry and solutions constructed which agree with the GR predictions.