Fundamental Field Equations Research Articles

Developing novel wormhole metrics in finsler-randers geometry using the barthel connection and osculating-riemannian method

In this research, we focus on modelling a wormhole using Finsler-Randers geometry. By applying the Barthel connection from the range of Finslerian connections, we have developed a new wormhole metric, termed the osculating Barthel-Finsler-Randers metric, through the osculating-Riemannian method within the Finsler-Randers framework. The fundamental field equations are formulated in terms of the obtained metric tensor characterised by η(r), which plays a pivotal role in characterizing the anisotropy of the model. To explore the application of Finsler geometry in addressing the existence of a wormhole, we adopt an exponential shape function. Further, in order to streamline the computational aspects of our analysis, and to discuss the effect of anisotropy effect of Finsler Rander geometry, we adopt a specific functional form for the anisotropic parameter η(r)=arn, where α is a constant with implications for the anisotropy, among various possibilities for η(r) a non-zero analytical component of vector field in Randers metric. In addition, we consider three distinct values for the exponent n = −1, 0, and 1 and subsequently, we proceed to develop two distinct models for each of these three values of n with constant and non-constant redshift function. We analyse the physical implications of η(r) on the energy conditions. Our results demonstrate the crucial impact of anisotropy on the stability and structure of wormholes, emphasizing the requirement of exotic matter under specific conditions. This research deepens the understanding of wormhole physics within Finsler geometry, shedding light on the theoretical frameworks essential for the existence and stability of these cosmic structures.

Thermodynamics-based stability criteria for constitutive equations of isotropic hyperelastic solids

A generalized thermodynamic stability criterion for isotropic finite elastic solids is derived using the fundamental balance laws and field equations of continuum mechanics, which is then used to formulate constitutive inequalities for the polynomial form of hyperelastic constitutive equations. Individual thermodynamic constitutive inequalities (called T-C inequalities) are derived for the neo-Hookean, Mooney Rivlin, and three-parameter generalized Rivlin models under three pure homogeneous deformation modes, namely, uniaxial compression, uniaxial tension and shear (simple and pure), and are compared against two commonly used adscititious inequalities, the Baker-Ericksen (B-E) and E-inequalities. The range of stable model constants as defined by the T-C inequalities is represented by a region in an N-dimensional coordinate space (N is the total number of model constants), which is defined as the Region of Stability (ROS). It is shown that the ROS is a function of material deformation and evolves with the limiting strain, shrinking from an initially large region representing the necessary condition of thermodynamic stability to a converged region under infinite limiting strain that is equivalent to the ROS defined by the E-inequalities. By investigating the evolution of the ROS under different deformation modes, the implication of T-C inequalities on the selection of experimental routines and filtering of erroneous test data and model constants is discussed. It is also demonstrated that while the E-inequalities are over-restrictive for hyperelastic materials with small to moderate limiting strains, an observation supported by recent experimental evidence, the B-E inequalities are inaccurate under moderate to large limiting strain conditions. The applicability of the proposed mathematical framework to other hyperelastic strain energy density forms, such as exponential/logarithmic functions, is demonstrated by investigating the thermodynamic stability of the Fung-Demiray model. It is shown that the commonly assumed restriction that the Fung-Demiray model constants must be positive can be relaxed so that some typical material behaviors under small to moderate limiting strains can also be modeled.

On Discrete Physics (Digital Philosophy/Digital Cosmology) and the Cellular Automaton: A Perfect Mathematical Deterministic Structure for Reality as a Huge Computational Simulation

In this paper we provide an analysis and overview of some notable definitions, works and thoughts concerning discrete physics (digital philosophy) that mainly suggest a finite and discrete characteristic for the physical world, as well as, of the cellular automaton, which could serve as the basis of a (or the only) perfect mathematical deterministic model for the physical reality. In particular and as a confirmation, in the reference [37] also has been shown that all the fundamental field equations of physics, mathematically and uniquely, could be derived - where it is assumed that certain physical quantities are discrete.

Covariant renormalizable modified and massive gravity theories on (non)commutative tangent Lorentz bundles

The fundamental field equations in modified gravity (including general relativity; massive and bimetric theories; Hořava–Lifshitz (HL); Einstein–Finsler gravity extensions etc.) possess an important decoupling property with respect to nonholonomic frames with 2 (or 3) + 2 + 2 + ⋯ spacetime decompositions. This allows us to construct exact solutions with generic off-diagonal metrics depending on all spacetime coordinates via generating and integration functions containing (un-)broken symmetry parameters. Such nonholonomic configurations/models have a nice ultraviolet behavior and seem to be ghost-free and (super-)renormalizable in a sense of covariant and/or massive modifications of HL gravity. The apparent noncommutativity and breaking of Lorentz invariance by quantum effects can be encoded into fibers of noncommutative tangent Lorentz bundles for corresponding "partner" anisotropically induced theories. We show how the constructions can be extended to include conjectured covariant renormalizable models with massive graviton fields and effective Einstein fields with (non)commutative variables.

High‐frequency radar cross‐sections of swell‐contaminated seas for a pulsed waveform

The first- and second-order monostatic cross-sections of swell-contaminated seas for high-frequency ground wave pulsed radar operation are derived from the fundamental electric field equations. In this case, the ocean surface, a mixture of both swell and wind wave components, is first represented with a Fourier series. In one of the two approaches used, the total Fourier coefficient is expanded as a linear sum of the contribution from swell and wind waves without considering the possible interaction between the two regimes; whereas in a second, more general situation, the analysis includes coupling effects between the wind waves and swell. Then, the corresponding electric fields received from the swell-contaminated sea surface are obtained. A Fourier transform of the autocorrelation of the electric fields gives the Doppler power spectral density (PSD), and a comparison between this PSD and monostatic radar range equation yields the radar cross-sections. Simulation shows that the assumption incorporating the coupling effects leads to results that better agree with historical field observations. Overall, the model proposed here not only lays a solid foundation for the development of future swell inversion algorithms, but also reveals the existence of non-linear energy transfer between swell and wind waves from a new perspective

Decoupling of Field Equations in Einstein and Modified Gravity

This paper is concerned with giving the proof that there is a general decoupling property of vacuum and nonvacuum gravitational field equations in Einstein gravity and f(R, T)-modifications. The constructions are possible in terms of geometric and physical objects adapted to certain nonholonomic 2+2 splitting with local fibred structure. This allows us to generate exact and/or parametric depending solutions with generic off–diagonal metrics and generalized, or Levi–Civita, connections. Different classes of modified spacetimes are determined by corresponding generating and integration functions depending, in general, on all space and time coordinates and may possess, or not, Killing symmetries. The initial data sets for the Cauchy problem and their global properties are analyzed. There are formulated the criteria of evolution with spacetime splitting and decoupling of fundamental field equations. Examples of exact solutions defining ellipsoid deformations of black hole metrics and solitonic configurations are provided.

The influence of measurement and relaxation time on flux jumps in high temperature superconductors

The influence of the magnetization and relaxation time on flux jumps in high temperature superconductors (HTSC) under varying magnetic field is studied using the fundamental electromagnetic field equations and the thermal diffusion equation; temperature variety corresponding to flux jump is also discussed. We find that for a low sweep rate of the applied magnetic field, the measurement and relaxation times can reduce flux jump and to constrain the number of flux jumps, even stabilizing the HTSC, since much heat produced by the motion of magnetic flux can transfer into coolant during the measurement and relaxation times. As high temperature superconductors are subjected to a high sweep rate or a strong pulsed magnetic field, magnetization undergoes from stability or oscillation to jump for different pause times. And the period of temperature oscillation is equal to the measurement and relaxation time.

The axisymmetric problem with electrostatic fields and some spherically symmetric black holes in EMD theory

The fundamental axisymmetric field equations of Einstein–Maxwell–dilation (EMD) theory with electric fields are simplified to Ernst-like and Laplace equations; all the solutions in the low-energy limit of string theory are generalized to the current case. On the basis of the work of Wei et al (2002 Class. Quantum Grav. 19 6469), the TS-like class of solutions and the class of solutions given in terms of two harmonics are further analysed and discussed, particularly the spherically symmetric solutions.

DYNAMIC THERMOELASTIC PROCESSES IN MICROPERIODIC COMPOSITES

The aim of this article is to present (i) a formulation of the field equations characterizing a "refined averaged model" describing the dynamical thermoelastic processes in microperiodic composite materials, (ii) an analysis of particular models, and (iii) an application of an "effective modulus model" to solve a boundary initial value problem for a microperiodic layered semispace. Using a modeling procedure proposed in papers by Wo niak et al. [1,2], the fundamental field equations of coupled thermoelasticity for general microperiodic composite materials are derived and a one-dimensional layered composite model is obtained. The fundamental field equations involve the functions \\eqalign{{\\bf u}({\\bf x}, t) = {\\bf U} ({\\bf x}, t)+h^a({\\bf x}){\\bf V}^a({\\bf x}, t)\\cr \\vartheta({\\bf x}, t) =\\Theta({\\bf x}, t)+h^a({\\bf x})\\Phi^a({\\bf x}, t)\\qquad a=1,\\ldots ,n} where h a are the microshape functions; u and are the displacement and temperature fields, respectively; U , V a , , and | a are the basic unknown V-macrofunctions (V means a representative volume element). In a general case, a refined averaged model is described by a system of four partial differential equations for the displacement and temperature macrofunctions U and , coupled with a system of 4 n partial differential equations for the displacement and temperature microcorrectors V a and | a . In this case, the microstructure length-scale effect on the macrobehavior ofthecompositeistaken into account. In the framework of an effective modulus model the dynamical problem of thermal stresses for a composite semispace, corresponding to thermal shock on the surface x =0, has been formulated and solved in a closed form by a Laplace transform technique proposed by Ignaczak and the author [3]. The solutions obtained for the total temperature and stress fields are analyzed numerically and graphically. A survey of papers devoted to modeling layered thermoelastic composites including various homogenization theories is presented, among others, by Wo niak and Wierzbicki [2].

Nonlocal elasticity and dislocation image force due to cracks

Fundamental field equations of nonlocal elasticity are presented. With these equations, the image force on a screw dislocation due to a crack is analyzed using the conformal mapping technique. Two cases are considered: one is for a finite-length crack, the other is for an infinite one. All classical singularities of the dislocation image force are eliminated when the dislocation tends to the crack tip. The maximum of the force is obtained at the crack tip.

Eddy current and hysteresis losses in ferromagnetic media

This paper is a contribution to the evaluation of losses in ferromagnetic media, due to eddy currents and magnetic hysteresis, the field source being the time harmonic induced e.m.f. The fundamental field equations are solved numerically by using a finite element formulation, corresponding to the space discretization, and a second order trapezoidal (Crank-Nickolson) method for the time integration. The method is applied to a soft ferromagnetic sample, being the magnetic hysteresis simulated by using a classical Preisach model.

On Einstein’s equations for space–times admitting a non-null Killing field

We consider the three-dimensional formulation of Einstein’s theory for space–times possessing a non-null Killing field ξa. It is known that for the vacuum case some of the basic field equations are deducible from the others. It will be shown here how this result can be generalized for the case of essentially arbitrary matter fields. The systematic study of the structure of the fundamental field equations is carried out. In particular, the existence of geometrically preferred reference systems is shown. Using local coordinates of this type two approaches are presented resulting in resolvent systems of partial differential equations for the basic field variables. Finally, the above results are applied for perfect fluid space–times describing possible equilibrium configurations of relativistic dissipative fluids.

A thermodynamical approach to the structure of weiss domains in deformable ferroelectric crystals

This work is the continuation of previous studies and deals with a somewhat general theory for the domain structure of deformable ferroelectric crystals. In contrast to earlier works which are valid only at equilibrium the balance equations of electromagnetic continuum are employed after certain simplifications appropriate to the present situation to derive the fundamental field equations and jump conditions. The second principle of thermodynamics is then fully exploited to determine the constitutive relations for various volume and surface fields.

Development of an advanced industrial metal detector instrumentation

In the normal three-coil balanced detection system for metal contaminants in products, a centrally excited coil provides a symmetric field linking two outer detector coils which are appropriately balanced to produce a zero voltage output for the detector circuit. The product to be inspected is passed on a conveyor belt through the centre of the triple coil system and the presence of any metal causes an imbalance initially on the input coil which is eventually mirrored by an imbalance in the output coil, with the signal being appropriately identified by phase-sensitive detection circuitry. The output signal is a function of many variables, including signal frequency, permeability and resistance of the sample, geometrical position of the sample within the coil cross-sectional area, and coil spacing. The reference phase angle is a crucial adjustment factor which, inter alia, compensates for product factors such as moisture content and other parameters which affect the electrical characteristics. Utilising the work of Poritsky (1960) it was not difficult to solve the fundamental field equations and optimise the various design parameters to maximise the output signal for any coil shape or sample type, and a simple CAD program was written to automate the design process. The performance specifications of new designs could be easily generated without the need for prototype construction and testing and, in addition, the claims of competitive manufacturers could be countered. Another problem addressed was that of long-term drift in the balanced coil system caused by thermal and other effects.

Analysis of Multi-Pin Modular Daughterboard-to-Backplane Connectors at High Bit Rate Signals

A theoretical model for the electrical characterization of multi-pin modular daughterboard-to-backplane connectors at high bit rate signals is developed. The fundamental field equations are transformed into a linear system of equations for the currents and voltages at the edges of the pins of the connector.Efficient formulaes for the calculation of self and mutual inductances and capacitances between the pins which are involved in the linear system of equations are obtained. Using the previous developed theoretical model the connector's performance can be examined in both the frequency and the time domain.Numerical calculations are performed for three types of existing connectors. Furthermore, they are compared to each other for bit-rates in the range of 100 Mbits/sec up to 500 Mbits/sec.

Many-Body-Systems and Exciton Model in New-Tamm-Dancoff- Formulation

The NTD-method is a procedure to compute differences of eigenvalues in quantum mechanical problems: ωαβ=λα-λβ. It is an instruction to transform and truncate an infinite linear system of eigenvalue equations ω τk= Ak m τm which is derived with the aid of fundamental field equations or corresponding Hamilton-operators, as e.g. with Heisenberg's nonlinear spinor equation. In this paper we want to test the NTD-method for a many-body-model in solid state physics. We elaborate on the physical and mathematical aspects by choosing a suitable transformation τ → φ = C τ to get a new linear 1 system ω φk= Bkk+2iφk+2i which permits a truncation to evaluate approximation of states. The efficiency of this method is demonstrated by treating a two-body-system in presence of polarisation quanta, known as exciton model

A theory of gravitation incorporating the quadratic action principle of relativity

The hypothesis adopted in this work is that any permissible metric field whatsoever must satisfy some self-contained set of field equations for the metric tensor gϱσ. The adoption of such a hypothesis is motivated by the precariousness of the general-relativistic interpretation of Mach’s principle, which is often used to justify a phenomenological approach to the theory. The fundamental equations of the gravitational field are provided in this instance by an action principle in which the Lagrangian is quadratic in the components of the Riemann tensor,vie. RϱσRϱσ+ βR2, thus ensuring consistency with standard relativity theory based on Einstein’s vacuum equations. The set of relationships Rϱσ-1/2gϱσR = -ϰTϱσ is retained as the definition of the matter tensor Tϱσ. Attention is concentrated on the solutions of the (generally fourth-order) fundamental field equations in the static, spherically symmetric case. It is shown that under these symmetry conditions there are no asymptotically flat solutions with nonzero Riemann scalar, and no regular solutions with nonconstant Riemann scalar. Sets of exact solutions of the field equations are presented.

Perturbation Theory for Electromagnetic Coupling to Elastic Surface Waves on Piezoelectric Substrates

Starting from the fundamental acoustic and electromagnetic field equations an approximate expression is obtained for the perturbation in propagation wavenumber of a surface acoustic wave in a piezoelectric crystal. The source of perturbation is taken to be a surface spaced an air-gap distance h above the piezoelectric and is described in terms of an electrical impedance. The resulting perturbation is found in terms of the perturbing electrical impedance, an effective dielectric constant for the piezoelectric, the air-gap spacing h, and a perturbation coupling constant defined in terms of the unperturbed electric potential at the piezoelectric surface and the average acoustic power flow per unit frequency. The theory is applied to the case of a short-circuit perturbing surface and found to be in excellent agreement with certain numerical results for Y-cut Z-propagating LiNbO3 and several cuts of Bi12 GeO20. In the general case of a complex perturbing impedance, such as that exhibited by a semiconductor, the theory indicates that attenuation or gain and dispersion may be introduced by the perturbation, in close agreement with experimental observations.

Matrix theory of skin effect in laminations

This paper presents a mathematical technique for determining the magnetic and electric field intensities and the current density in plane conducting metal plates of constant permeability produced by an external impressed alternating magnetic field. The fundamental Maxwell field equations are used as the basis of the analysis and expressions for the amplitudes of the fields and currents are written in matrix form. It is demonstrated how the field and current distribution may be simply and systematically determined in composite laminated plates such as those used in transformers by a simple matrix multiplication procedure. The technique simplifies the calculations involving skin effect and magnetic shielding.

The Meson Fields and the Equation of Motion of a Spinning Particle

Following Riesz, the meson fields generated by a point dipole at a point outside the world line have been defined in terms of the derivatives of an integral which is an analytical function of a parameter, $\ensuremath{\alpha}$, and converges for $\ensuremath{\alpha}$ lying between $2<\ensuremath{\alpha}<3$. It is then shown that by the process of analytical continuation of the parameter to the value $\ensuremath{\alpha}=2$ for which the Riesz fields satisfy the fundamental meson field equations, we not only obtain the classical Li\'enard-Wiechert-Bhabha fields at a point outside the world line but also a field at a point on the world line itself which is finite and represents the reaction of the field correctly. The equations for rotational and translational motions follow immediately from the usual electrodynamical equations of a dipole given by Frenkel. The equations thus obtained are free from singularities and do not contain any arbitrary constant except the mass and spin angular momentum of the dipole.