Narrow Results

Considering the evolution of the classical electromagnetic field inside a cavity containing a thin slab with time-dependent conductivity and dielectric permittivity, an approximate analytical solution for an infinite set of coupled ordinary differential equations for the mode amplitudes is found under certain simplifying assumptions. According to this solution, an amplification of the initial field cannot be possible, unless the dielectric permittivity can become strongly negative.

In this paper, a new model has been developed to reflect the similarity in the formulation between two-fluid plasma theory and the Maxwell equations of electromagnetism. For this aim, a mathematical tool based on complexified Cayley–Dickson octonions has been proposed to unify the behaviors of both the electromagnetic and fluidic character of plasma. Moreover, derived equations have been associated with their corresponding $8\times 8$ matrices simply and consistently.

For monochromatic solutions of D'Alembert's wave equation and Maxwell's equations, we obtain sharp bounds on the sup norm as a function of the far field energy. The extremizer in the scalar case is radial. In the case of Maxwell's equation, the electric field maximizing the value at the origin follows longitude lines on the sphere at infinity. In dimension d = 3, the highest electric field for Maxwell's equation is smaller by a factor 2/3 than the highest corresponding scalar waves.

The highest electric field densities on the balls B_R(0) occur as R → 0. The density dips to half max at R approximately equal to one third the wavelength. For these small R, the extremizing fields are identical to those that attain the maximum field intensity at the origin.

Let

The estimates of the spectral derivative near the thresholds are based on detailed trace estimates on the slowness surfaces. Two applications of these estimates are presented:

• Global spacetime estimates of the associated evolution unitary groups, that are also commonly viewed as decay estimates. In particular, the Dirac and Maxwell systems are explicitly treated.
• The finiteness of the eigenvalues (in the spectral gap) of the perturbed Dirac operator is studied, under suitable decay assumptions on the potential perturbation.

It is known that the classical variables are replaced by operators for the dynamics of the electromagnetic field, thus the second quantization is traditionally used. There is a significant question in modern theoretical physics area: “Why is there no fully developed first-quantized theory of photons?” In the present work, we argue that a photon wave function can be obtained if one uses the massless Duffin-Kemmer-Petiau (mDKP) equation. Thus, we focus on exact solutions of the mDKP equation the Robertson–Walker type spacetime via the Teleparallel Theory (TPT) of Gravity. The mDKP equation is generally known as the spinor formulation of the photons. We also focus on solutions of the Maxwell equations (MEs) to verify the equivalence between the mDKP and the ME formulations of the photons. It is shown that photons present an oscillatory behavior.

The aim of the present paper is to analyze some globally pathologic properties in worlds described by a class of static planary symmetric exact solutions with g₄₄ = -cosh²(αz), pointing out some unusual (pathological) features such as cosmic-time traps, non-spacelike imprisonments of test particle trajectories and structures very much like the ones of a BTZ black hole. Finally, we deal with the exactly solvable analytical behavior of the magnetostatic fields, stressing the significant difference from the Minkowskian case, namely a nonsingular, almost total, confinement of the field in a finite spacelike region, which we termed as a gravitomagnetic resonance, entirely due to the large scale pathology of such a Universe.

A value of the cosmological constant in a toy model of the five-dimensional universe is calculated in such a manner that it remains in agreement with both astronomical observations and the quantum field theory concerning the zero-point fluctuations of the vacuum. The (negative) cosmological constant is equal to the inverse of the Planck length squared, which means that in the toy model the vanishing of the observed value of the cosmological constant is a consequence of the existence of an energy cutoff exactly at the Planck level. In turn, a model for both a virtual and a real particle–antiparticle pair is proposed which describes properly some energetic properties of both the vacuum fluctuations and created particles, as well as it allows one to calculate the discrete "bare" values of an elementary particle's mass, electric charge and intrinsic angular momentum (spin) at the energy cutoff. The relationships between the discussed model and some phenomena such as the Zitterbewegung and the Unruh–Davies effect are briefly analyzed, too. The proposed model also allows one to derive the Lorentz transformation and the Maxwell equations while considering the properties of the vacuum filled with the sea of virtual particles and their antiparticles. Finally, the existence of a finite value of the vacuum-energy density resulting from the toy model leads us to the formulation of dimensionless Einstein equations which may be derived from the Lagrangian with a dimensionless (naively renormalized) coupling constant.

The present work focuses on the scalar and electromagnetic fields in a spatially closed FRW Universe with incoherent dust. First, as the complex scalar field is under consideration, we derive the general solutions of the Gordon equation, in terms of the positive-frequency-like parity modes, and the corresponding propagator. Secondly, we turn to the source-free Maxwell equations in order to find a (1 + 4)-parametric class of closed form solutions representing a non-propagating fundamental electric field and a burst of electromagnetic radiation.

The aim of the present work is to study, within an SO(3, 1)-gauge invariant formulation, the vertical dipole radiation perceived by an observer in Milne Universe. Using the explicit expressions of the radial component and the declination of the four potential, we compute the electric and magnetic field intensities and the effective radiated power, in terms of the excitation current amplitude feeding-in the radiating system.

These lecture notes accompany two classes given at the NRHEP2 school. In the first lecture I introduce the basic concepts used for analyzing well-posedness, that is the existence of a unique solution depending continuously on given data, of evolution partial differential equations. I show how strong hyperbolicity guarantees well-posedness of the initial value problem. Symmetric hyperbolic systems are shown to render the initial boundary value problem well-posed with maximally dissipative boundary conditions. I discuss the Laplace–Fourier method for analyzing the initial boundary value problem. Finally, I state how these notions extend to systems that are first-order in time and second-order in space. In the second lecture I discuss the effect that the gauge freedom of electromagnetism has on the PDE status of the initial value problem. I focus on gauge choices, strong-hyperbolicity and the construction of constraint preserving boundary conditions. I show that strongly hyperbolic pure gauges can be used to build strongly hyperbolic formulations. I examine which of these formulations is additionally symmetric hyperbolic and finally demonstrate that the system can be made boundary stable.

We present a systematic study of the noncommutative interaction of gravitational waves with matter at the quantum mechanical level in the long wavelength and small velocity limit. In particular, the noncommutative quantum dynamics of the Landau's problem is employed to measure the transmission amplitudes in terms of the Riemann's tensor elements by a gravitational wave in linearized approximation. It turns out that the Hamiltonian of the charged particles derived based on DeWitt's approach is valid by the deformed electromagnetic fields.

The field equations in a three-dimensional commutative space based on a set of commutation relations are derived. In this space, the commutation relation of the position and kinematic momentum of a particle is generalized to include a metric tensor field in addition to a vector field. The introduction of a metric tensor is a generalization of the commutation relation for Feynman’s proof of the Maxwell equations. In this paper, as the equations of motion and the field equations are classical, the Poisson bracket and not the commutation relation is used in the calculations.

As the commutative space is defined by the Poisson bracket, the equations of motion for the particle and the field equations for the metric tensor and vector are derived from the Poisson bracket in Hamiltonian mechanics. The Helmholtz conditions, which express the existence of a Lagrangian for a particle in the space, are also derived from the Poisson bracket. Then the field equations are calculated explicitly by two approaches. One is to calculate the Helmholtz conditions using the equations of motion. The other is to calculate the Jacobi identity for the kinematic momentum or velocity of the particle. In addition to the homogeneous Maxwell equations, the generalized field equations are obtained to define the generalized electric and magnetic fields of the tensor field. Just like the usual electric and magnetic fields, the generalized fields are invariant under a local gauge transformation and should play significant roles in physics.

Finally, the homogeneous Maxwell equations of the vector field are seen to exhibit similarities with the generalized field equations for the tensor field. This similarity provides a useful theoretical framework for constructing gravitoelectromagnetism, which is based on analogies between the equations for electromagnetism and relativistic gravitation. It remains to establish the usefulness of the theoretical framework with applications of the field equations.

It is well known that the Klein–Gordon equation in curved spacetime is conformally noninvariant, both with and without a mass term. We show that such a noninvariance provides nontrivial physical insights at different levels, first within the fully relativistic regime, then in the nonrelativistic regime leading to the Schrödinger equation, and then within the de Broglie–Bohm causal interpretation of quantum mechanics. The conformal noninvariance of the Klein–Gordon equation coupled to a vector potential is confronted with the conformal invariance of Maxwell’s equations in the presence of a charged current. The conformal invariance of the nonminimally coupled Klein–Gordon equation to gravity is then examined in light of the conformal invariance of Maxwell’s equations. Finally, the consequence of the noninvariance of the equation on the Aharonov–Bohm effect in curved space–time is discussed.

In this paper, we propose a quaternion form of equations describing electromagnetic field in a homogeneous isotropic medium without dispersion. It is shown that by renormalizing the values of field inductions and sources, one can transform the asymmetrical Maxwell equations to a highly symmetric form. This provides a possibility to introduce the scalar and vector field potentials and represent the generalized equation for electromagnetic field in the form of a single second-order quaternionic wave equation. We demonstrate that this equation reduces to the system of ordinary hyperbolic wave equations for the field potentials and on the other hand, the same equation is equivalent to the system of Maxwell equations for renormalized field intensities. The symmetry of the renormalized Maxwell equations allows one to obtain the second-order relations for energy and momentum, as well as for Lorentz invariants of the renormalized fields, which formally have the same form as for the fields in a vacuum. In addition, the generalization of renormalized equations to the case of magnetic sources corresponding to the models of Dirac magnetic monopoles and Schwinger dyons is discussed.

We show, both semi-classically as well as quantum–mechanically that the vacuum becomes birefringent in the presence of a counterpropagating light beam. For the KJ laser energy, this effect is not observable. However, with several Mega joule laser pulses, this effect will be detectable.

In recent years, an entirely new class of nanofluids has been developed, namely hybrid nanofluids. Their efficiency when it comes to heat transfer has been demonstrated to be much higher than the traditional nanofluids. In this study, the flow of hybrid nanofluid is analyzed subjected to the flow geometry of oscillatory disk that also rotates with the constant angular speed. Mainly, we investigated hybrid nanofluid flow subjected to an oscillatory disk that rotates at a constant angular speed as an incompressible, laminar and time-dependent electrically conducting flow. The implication of similarity functions normalizes the flow phenomenon. Furthermore, we solved the governing system by implementing the finite differences, and then the iterative procedure is carried out with the help of Successive Over-Relaxation (SOR) technique. In the later part of the paper, graphical representation is given to discuss dimensionless factors in both scenarios of nanofluid and hybrid nanofluid.

Starting from a first-order approximate field description function for laser pulses, the method currently used to approximate chirped laser pulse (CLP) substitutes frequency and wave vector related variables with spatiotemporally varying functions. We investigated the error involved by calculating the relative deviation from Maxwell equations. Errors for the electric and magnetic fields are analyzed separately, and behaviors related to parameter changes (that is, in laser width, pulse duration and chirp parameter) were studied. Results show that aberration associated with currently used field-description functions for CLP increases monotonically with chirp parameter, and the deviation introduced by chirping is proportional to the relative frequency span of the laser. Simulations based on these functions will lead to considerable error, especially for laser pulses with large chirping.

We study the role of the homological topological property of a space-time with holes (a multiple connected manifold) on the formal solution of the electromagnetic irradiation problem taking place on these “holed” space-times. In this paper, in addition to the main focus of study, we present as well important studies on this irradiation problem on other mathematical frameworks.

The Aharonov–Bohm electrodynamics is a generalization of Maxwell theory with reduced gauge invariance. It allows to couple the electromagnetic field to a charge which is not locally conserved, and has an additional degree of freedom, the scalar field $S={\partial }_{\alpha }{A}^{\alpha }$, usually interpreted as a longitudinal wave component. By reformulating the theory in a compact Lagrangian formalism, we are able to eliminate S explicitly from the dynamics and we obtain generalized Maxwell equation with interesting properties: they give ${\partial }_{\mu }{F}^{\mu \nu }$ as the (conserved) sum of the (possibly non-conserved) physical current density $j^{\nu }$, and a “secondary” current density $i^{\nu }$ which is a nonlocal function of $j^{\nu }$. This implies that any non-conservation of $j^{\nu }$ is effectively “censored” by the observable field $F^{\mu \nu }$, and yet it may have real physical consequences. We give examples of stationary solutions which display these properties. Possible applications are to systems where local charge conservation is violated due to anomalies of the Adler–Bell–Jackiw (ABJ) kind or to macroscopic quantum tunnelling with currents which do not satisfy a local continuity equation.

We consider a splitting approximation of the magnetohydrodynamic equations which consists of decoupling the Navier–Stokes and Maxwell equations which form the magnetohydrodynamic system, on small time intervals. To compute the approximate solutions we start with a pair of initial velocity and magnetic fields and alternately solve Navier–Stokes and dynamo equations. The main result of the paper states that for sufficiently smooth initial data, the proposed scheme converges in L^p norm with p > 3. Besides the applications in numerical analysis, this scheme can also be a valuable tool in understanding the physical and mathematical structure of the magnetohydrodynamic equations.