Hertz Vector Research Articles

Metric reconstruction in Kerr spacetime

Abstract Metric reconstruction is the general problem of parameterizing GR in terms of its two “true degrees of freedom” e.g., by a complex scalar “potential”—in practice mostly with the aim of simplifying the Einstein equation (EE) within perturbative approaches. In this paper, we re-analyze the metric reconstruction procedure by Green, Hollands, and Zimmerman (GHZ) [Class. Quant. Grav. 37, 075001 (2020)], which is a generalization of the Chrzanowski-Cohen-Kegeles (CCK) approach. Contrary to the CCK method, that by GHZ is applicable not only to the vacuum, but also to the sourced linearized Einstein equation (EE) on Kerr. Our main innovation is a version of the GHZ integration scheme that is suitable for the initial value problem of the sourced linear EE. By iteration, our scheme gives the metric to as high an order in perturbation theory as one might wish, in principle. At each order, the metric perturbation is a sum of a corrector, obtained by solving a triangular system of transport equations, a reconstructed piece, obtained from a Hertz potential as in the CCK approach, and an algebraically special perturbation, determined by the ADM quantities. As a byproduct, we determine the precise relations between the asymptotic tail of the Hertz potential in the GHZ and CCK schemes, and the quantities relevant for gravitational radiation, namely, the energy flux, news- and memory tensors, and their associated BMS-supertranslations. We also discuss ways of transforming the metric perturbation to Lorenz gauge.

Gravitational waves on Kerr black holes: I. Reconstruction of linearized metric perturbations

Abstract The gravitational perturbations of a rotating Kerr black hole are notoriously complicated, even at the linear level. In 1973, Teukolsky showed that their physical degrees of freedom are encoded in two gauge-invariant Weyl curvature scalars that obey a separable wave equation. Determining these scalars is sufficient for many purposes, such as the computation of energy fluxes. However, some applications -- such as second-order perturbation theory -- require the reconstruction of metric perturbations. In principle, this problem was solved long ago, but in practice, the solution has never been worked out explicitly. Here, we do so by writing down the metric perturbation (in either ingoing or outgoing radiation gauge) that corresponds to a given mode of either Weyl scalar. Our formulas make no reference to the Hertz potential (an intermediate quantity that plays no fundamental role) and involve only the radial and angular Kerr modes, but not their derivatives, which can be altogether eliminated using the Teukolsky--Starobinsky identities. We expect these analytic results to prove useful in numerical studies and for extending black hole perturbation theory beyond the linear regime.

Investigation of the Anomalous Behavior of the Density and Thermal Expansion in a Two-Dimensional System with the Hertz Potential

Investigation of the Anomalous Behavior of the Density and Thermal Expansion in a Two-Dimensional System with the Hertz Potential

Diffraction Fields and Fields of Charges, Expressed in Terms of Hertz Vector in Electrodynamics

Diffraction Fields and Fields of Charges, Expressed in Terms of Hertz Vector in Electrodynamics

Generalized Hertz Vector in the Dissipative Electrodynamics

In the electromagnetic theory, the Hertz vector reduces the number of potentials in the free fields. The further advantage of this potential is that it is much easier to solve particular radiation processes. It indicates that the related method is sometimes more effective than the scalar and vector potential-based relations. Finally, the measurable field variables, the electric and magnetic fields, can be deduced by direct calculation from the Hertz vector. However, right now, the introduction of the Hertz vector operates if the conductive current density j = 𝜎E is neglected. We suggest a generalization by the conductive currents, i.e., when the electromagnetic field dissipates irreversibly to Joule heat. The presented procedure enables us to introduce also the Lagrangian formulation of the discussed dissipated electromagnetic waves. It paves a new way involving damping physical fields in a thermodynamic frame.

Theoretical macrobending loss in single-mode transmission through a uniform index fiber-optic cable

The macrobending loss of propagating waves in optical fibers is studied here by employing simple toroidal coordinates (r, θ, ϕ) for the governing Maxwell equations of electromagnetism, the solution for which is expressed by the central axial component Π ϕ of the Hertz vector Π. The refractive indices of the core and its cladding are supposed uniform all along the length of the cable. Effects of coating, Jacketing and the effect of elastic strain in the fiber are not considered. Knowing that the bending losses are of different nature when the radius of curvature R exceeds or is less than a certain critical value R c , the study is accordingly divided in to two parts. Firstly, when R > R c wave guide action holds, and it is shown through an example that it is small, varying linearly with the bending angle ϕ of the cable and practically independent of R. On the other hand, when R < R c , leaking waves are propagated and the problem is remodeled by a leaking ring source of EM waves in a medium consisting of the cladding only. The analysis of the model results in a bending loss formula that varies as ϕ3 and depends on R by a factor of the form e−αR/R.

Structure-property relationships of Kaolin-nanocomposite beads decorated with tertiary amines: Influence of shape on network elasticity and multi-responsivity

Aluminosilicate mineral Kaolin-loaded nanocomposite beads containing cationic 2-(dimethylamino)ethyl methacrylate (DMAEMA) and thermoresponsive N-isopropylacrylamide (NIPA) have been successfully synthesized by combined method of dropwise freezing into cryogenic liquid and sedimentation cryogelation. Hertzian elasticity, thermal stability, and responsive-swelling properties were investigated. The nanocomposite gels exhibited better mechanical properties and thermal stability than neat copolymer gels. The stress and displacement distribution on surface of Kaolin-loaded nanocomposite beads were evaluated with classical Hertz theory, and the results were compared with the results obtained from Rubber elasticity theory. The scaling relationship for contact stress and surface displacement was obtained assuming Hertz's small contact area, and the compression force was scaled with displacement with an exponent of 1.34 ± 0.08. The elastic properties directly attributed to effective potential showed that Hertz potential is valid for small deformations and the elastic modulus decreased with the swollen bead diameter. Addition of Kaolin and NIPA to PDMAEMA-based network has opposite effect on the mechanical properties. The elastic modulus increased, but the swelling ratio decreased with addition of Kaolin. The opposite behavior was observed when NIPA was added, the swelling increased and the mechanical properties weakened. All gels showed significant temperature sensitivity as a result of combination of PDMAEMA and PNIPA, while the swelling was higher at acidic pH than alkaline pH. An overshooting effect attributed to the dynamic conformational changes of tertiary amine of DMAEMA was observed during swelling in pH 3.6. The increase in network density of nanocomposite gel with addition of Kaolin leads to slower relaxation of molecules, and water absorption tends to become relaxation-controlled. An increase in diffusion exponent is observed as diffusion changes from Fickian to non-Fickian. Influence of single salts; NaCl, KCl and K2CO3 on swelling of nanocomposites was investigated. The swelling of Kaolin-loaded nanocomposites decreased with increasing ionic strength, and a “salting-out” effect was observed. The use of naturally occurring nanoscale particles in such structural material design is an important method for biomedical applications and separation technology.

Metastable states of microgel fluids with Hertzian interaction potentials

We consider here a coarse-grained model for a soft matter system in which the particles interact with the Hertzian potential uH(r)=ϵ01−r/σμ, characterized by parameters {ϵ0,σ,μ}. Using the coarse-grained one-particle density ρ(x) as an order parameter, the free energy functional typical of the classical density functional theory (DFT) is studied. The static correlations of the uniform density state, required as an input in the DFT model, is calculated using the Bridge function method for the Hertzian fluid. The free energy functional analysis obtains new minima signifying metastable states between uniform liquid and the crystal. In particular, using classical DFT methods, a metastable amorphous state characterized by a low degree of mass localization is predicted. We obtain the transformation line on the density-temperature plot showing the possible liquid–liquid transition between the uniform liquid state and the inhomogeneous liquid state. With the increase of ϵ0 for the Hertzian interaction, the model reproduces the hard-sphere results.

Vector Bessel beams: General classification and scattering simulations

Apart from a lot of fundamental interest, vector Bessel beams are widely used in optical manipulation, material processing, and imaging. However, the existing description of such beams remains fragmentary, especially when their scattering by small particles is considered. We propose a new general classification of all existing vortex Bessel beam types in an isotropic medium based on the superposition of transverse Hertz vector potentials. This theoretical framework contains duality and coordinate rotations as elementary matrix operations and naturally describes all relations between various beam types. This leads to various bases for Bessel beams and uncovers the novel beam type with circularly symmetric energy density. We also discuss quadratic functionals of the fields (such as the energy density and Poynting vector) and derive orthogonality relations between various beam types. Altogether, it provides a comprehensive reference of all properties of Bessel beams that may be relevant for applications. Next we generalize the formalism of the Mueller scattering matrices to arbitrary Bessel beams accounting for their vorticity. Finally, we implement these beams in the ADDA code - an open-source parallel implementation of the discrete dipole approximation. This enables easy and efficient simulation of Bessel-beams scattering by particles with arbitrary shape and internal structure.

Spectrally-Accurate Analysis of EM Scattering by Material and Metamaterial, Isotropic or Anisotropic, Radially Inhomogeneous Spherical Objects

We study electromagnetic (EM) plane wave scattering by material and metamaterial radially inhomogeneous spherical objects that are either isotropic or uniaxially anisotropic with radial anisotropy. Using field representations in terms of scalar Hertz potentials, we reformulate Maxwell’s equations into two systems of first-order ordinary differential equations, which we then convert into equivalent second-kind linear Volterra matrix integral equations. The discretization scheme uses a high-order, entire domain, Nystrom method that employs the Gauss–Legendre–Radau quadrature rule. The proposed algorithms have spectral convergence and apply to either single-layer or multilayer structures with continuous or sectionally continuous profiles.

Methods of constructive synthesis of antenna system of radio communication building on low-lying vibrators

In recent years, the radio communication system has undergone significant modernization. This concerns both the design and development of new means of communication and the improvement and modernization of existing ones and the development of new antenna-feeder devices for them. A general feature of radio communication networks is the significant dependence of the main characteristics of communication objects on the type and parameters of antenna systems and on decisions regarding their location. The developed technique of constructive synthesis of the antenna system of the radio communication device built on low-lying emitters is offered, with the help of which it is possible to synthesize the antenna system which will improve the technical characteristics of the mobile radio communication means. The method of constructive synthesis of the antenna system of radio communication equipment built on low-lying emitters is based on the mathematical model of calculating the directional characteristics of the speaker from low-located emitters for radio communication facilities. This model was presented in the article “Mathematical model for calculating the directional characteristics of the system of vibrators of radio communication equipment” [1]. Research methods – synthesis of antenna systems with the necessary pattern is based on the Hertz vector method, the method of induced electromotive forces, the method of mirror images and analysis of the results. The influence of the shape and number of elements of the antenna system on the formation of its radiation pattern is determined and the calculation for the selection of the optimal design with the required electrical characteristics is established. That is, the proposed technique helps to improve and simplify the technical performance of transmitting and receiving speakers. Based on the results of the research, a mathematical model for calculating the directional characteristics of the AS, built on low-lying emitters for radio communication equipment and methods of constructive synthesis of AS radio communication equipment built on low-located emitters, which is possible in the absence of with realized gain up to 16 dB. It is expedient to apply the developed technique: during the development of new AS and antenna-feeder devices for modern field radio communication equipment and AS for unmanned aerial vehicles; during the organization of communication by a field special radio communication node.

Scalar Pre-potentials for Spinor and Tensor Fields on Spacetime

We review a technique for solving a class of classical linear partial diferential systems of relevance to physics in Minkowski spacetime. All the equations are amenable to analysis in terms of complex solutions in the kernel of the scalar Laplacian and a complexifed Hertz potential. The complexifcation prescription ensures the existence of regular physical solutions with chirality and propagating, non-singular, pulse-like characteristics that are bounded in all three spatial dimensions. The technique is applied to the source-free Maxwell, Bopp-Landé-Podolsky and linearised Einstein feld systems, and particular solutions are used for constructing classical models describing single-cycle laser pulses and a mechanism is discussed for initiating astrophysical jets. Our article concludes with a brief introduction to spacetime Cliford algebra ideals that we use to represent spinor felds. We employ these to demonstrate how the same technique used for tensor felds enables one to construct new propagating, chiral, non-singular, pulse-like spinor solutions to the massless Dirac equation in Minkowski spacetime.

Smoke rings re-visited: an analytic solution for the transient electric field produced by a vertical magnetic dipole on or in a homogeneous conductive half-space

For a vertical magnetic dipole (VMD) on or under the surface of a conductive half-space, explicit analytic formulae, not Hankel transform expressions, have been derived for the transient electric field induced below the ground, and the corresponding Hertz potential. These formulae permit rapid calculation of surface and underground dB/dt and B-field components, as well as of the induced current distribution. The solution allows the current produced by the surface reflection to be isolated, revealing how the interaction with the ground surface is responsible for the downward diffusion of the smoke ring. The mathematical form of the solution has also permitted confirmation of a prediction that the current distribution from a buried dipole in the limit of late time is a scaled version of the current distribution produced by a VMD on the ground surface.

Spontaneous Emission Rate and the Density of States Inside a One Dimensional Photonic Crystal

Different densities of electromagnetic states inside a one dimensional photonic crystal (1D PC) are studied. Hertz vector formalism is used to calculate Green’s tensor inside a layered structure, semi-analytically. Based on the obtained Green’s tensor, the local density of electromagnetic states (LDOS) and the density of states (DOS) inside a 1D PC are calculated and discussed. The Green’s tensor is also used to approximate the density of Bloch states inside the 1D PC and is compared with its exact calculation based on the 1D PC dispersion relations. Using a practical 1D PC parameters in the visible range, the aforementioned quantities are calculated and verified with a full-wave solver based on finite element method (FEM). The formulations and the results are aimed to be helpful in thermal and spontaneous radiation studies.

Scattering of generalized Bessel beams simulated with the discrete dipole approximation

We consider the simulation of scattering of the high-order vector Bessel beams in the discrete dipole approximation framework (DDA). For this purpose, a new general classification of all existing Bessel beam types was developed based on the superposition of transverse Hertz vector potentials. Next, we implemented these beams in ADDA code – an open-source parallel implementation of the DDA. The code enables easy and efficient simulation of Bessel beams scattering by arbitrary-shaped particles. Moreover, these results pave the way for the following research related to the Bessel beam scattering near a substrate and optical forces.

Tiered structure and symmetry of the electromagnetic equations

The structure of Maxwell's equations is reproduced for the potentials introducing a second vector potential , in addition to the usual vector potential . These in turn can be described in terms of the so-called potpotentials and the process continued ad infinitum. A unified scheme is expounded, where the relationship between contiguous and alternate tiers is presented. The Heaviside Larmor (HL) symmetry is fulfilled in any one tier. In the presence of matter, the electric–magnetic symmetry is broken by electric monopoles but preserved for the electric and magnetic dipoles. The Hertz potentials will be shown to be the second tier potentials. The obtention of vector solutions from a scalar solution will be shown to be a direct consequence of linear superposition, the HL symmetry and the structure of the electromagnetic equations for all tiers. The Debye potentials provide such solutions for spherical symmetry. The physical reality of the quantities in any one tier is discussed, in particular regarding the question of gauge transformations and physical observables.

Unraveling the vector nature of generalized space-fractional Bessel beams

We introduce an exact analytical solution of the homogeneous space-fractional Helmholtz equation in cylindrical coordinates. This solution, called the vector space-fractional Bessel beam (SFBB), has been established from the Lorenz gauge condition and Hertz vector transformations. We perform scalar and vector wave analysis focusing on electromagnetics applications, especially in cases where the dimensions of the beam are comparable to its wavelength $({k}_{r}\ensuremath{\approx}k)$. The propagation characteristics such as the diffraction and self-healing properties have been explored with particular emphasis on the polarization states and transverse propagation modes. Due to continuous order orbital angular momentum dependence, this beam can serve as a bridge between the ordinary integer Bessel beam and the fractional Bessel beam and, thus, can be considered as a generalized solution of the space-fractional wave equation that is applicable in both integer and fractional dimensional spaces. The proposed SFBBs provide better control over the beam characteristics and can be readily generated using digital micromirror devices, spatial light modulators, metasurfaces, or spiral phase plates. Our findings offer insights on electromagnetic wave propagation, thus paving a route towards applications in optical tweezers, refractive index sensing, optical trapping, and optical communications.

Time-domain metric reconstruction for hyperbolic scattering

Self-force methods can be applied in calculations of the scatter angle in two-body hyperbolic encounters, working order by order in the mass ratio (assumed small) but with no recourse to a weak-field approximation. This, in turn, can inform ongoing efforts to construct an accurate model of the general-relativistic binary dynamics via an effective-one-body description and other semi-analytical approaches. Existing self-force methods are to a large extent specialised to bound, inspiral orbits. Here we develop a technique for (numerical) self-force calculations that can efficiently tackle scatter orbits. The method is based on a time-domain reconstruction of the metric perturbation from a scalar-like Hertz potential that satisfies the Teukolsky equation, an idea pursued so far only for bound orbits. The crucial ingredient in this formulation are certain jump conditions that (each multipole mode of) the Hertz potential must satisfy along the orbit, in a 1+1-dimensional multipole reduction of the problem. We obtain a closed-form expression for these jumps, for an arbitrary geodesic orbit in Schwarzschild spacetime, and present a full numerical implementation for a scatter orbit. In this paper we focus on method development, and go only as far as calculating the Hertz potential; a calculation of the self-force and its physical effects on the scatter orbit will be the subject of forthcoming work.

Assessment about Luneberg integrals and application to digital in-line holography

In this publication, the Luneberg integrals are revisited and the conditions of the using of such integrals have been recalled. Additivity law of Luneberg’s integrals and the link with the Frenel kernel for the propagation are discussed. By means of the definition of the Luneberg’s integrals, the propagation of a vectorial electromagnetic field (Hertz potentials) is developed and a new approach of the computation have been proposed based on Zernike polynomials. With this new approach simulations of holograms is illustrated in the case of the digital in-line holography with an opaque disk.

Structural transition in two-dimensional Hertzian spheres in the presence of random pinning.

Using molecular dynamics simulation we have investigated the influence of random pinning on the phase diagram and melting scenarios of a two-dimensional system with the Hertz potential for α=5/2. It has been shown that random pinning can cardinally change the mechanism of first-order transition between the different crystalline phases (triangular and square) by virtue of generating hexatic and tetratic phases: a triangular crystal to hexatic transition is of the continuous Berezinskii-Kosterlitz-Thouless (BKT) type, a hexatic to tetratic transition is of first order, and finally, there is a continuous BKT-type transition from tetratic to the square crystal.