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This book aims to cover the material taught in an advanced undergraduate introductory course in general relativity and cosmology taught in many physics departments of Greek universities as an elective course. The recent discoveries in the subject area of gravity-cosmology, its fundamental role in physics, and the spectacular developments in observations and theoretical models have made it very attractive to many students. It stands out among similar books because it contains more extensive topics and the latest developments in these fields. The book consists of 17 chapters. The first 4 chapters describe special relativity, electromagnetism, the momentum-energy tensor, and mathematical concepts necessary for understanding theories. Chapters 5-10 cover general relativity, Einstein's equations, and observational confirmations of the theory. Chapter 11 describes the Schwarzschild solution and its properties. Chapters 12-17 are devoted to Cosmology. They start from the basic principles, the best-known cosmological models, the ways of measuring distances in the universe and reach the thermal history of the universe and the modern theories of dark matter and dark energy. At the same time, there is a description of the problems faced by the Big Bang model and the theories that exist to solve these problems are mentioned., Το παρόν βιβλίο στοχεύει να καλύψει την ύλη που διδάσκεται σε ένα προχωρημένο προπτυχιακό μάθημα εισαγωγής στη γενική θεωρία της σχετικότητας και την κοσμολογία που διδάσκεται σε πολλά τμήματα φυσικής ελληνικών Α.Ε.Ι. ως επιλεγόμενο μάθημα. Οι πρόσφατες ανακαλύψεις στη θεματική περιοχή της βαρύτητας-κοσμολογίας, ο θεμελιώδης ρόλος της στη φυσική και οι θεαματικές εξελίξεις σε παρατηρήσεις και θεωρητικά μοντέλα την έχουν καταστήσει πολύ ελκυστική σε μεγάλο πλήθος φοιτητών. Ξεχωρίζει ανάμεσα σε παρόμοια βιβλία γιατί περιέχει εκτενέστερη θεματολογία και τις πιο πρόσφατες εξελίξεις σε αυτούς τους τομείς. Το βιβλίο αποτελείται από 17 κεφάλαια. Στα 4 πρώτα κεφάλαια περιγράφεται η ειδική σχετικότητα, ο ηλεκτρομαγνητισμός, ο τανυστής ενέργειας ορμής και μαθηματικές έννοιες απαραίτητες για την κατανόηση των θεωριών. Τα κεφάλαια 5-10 αναφέρονται στην γενική σχετικότητα, στις εξισώσεις Einstein και στις παρατηρησιακές επιβεβαιώσεις της θεωρίας. Το κεφάλαια 11 περιγράφει την λύση Schwarzschild και τις ιδιότητές της. Τα κεφάλαια 12-17 είναι αφιερωμένα στην Κοσμολογία. Ξεκινούν από τις βασικές αρχές, τα πιο γνωστά κοσμολογικά μοντέλα, τους τρόπους μέτρησης αποστάσεων στο σύμπαν και φτάνουν μέχρι την θερμική ιστορία του σύμπαντος και τις σύγχρονες θεωρίες για την σκοτεινή ύλη και την σκοτεινή ενέργεια. Παράλληλα γίνεται περιγραφή των προβλημάτων που αντιμετωπίζει το μοντέλο της Μεγάλης Έκρηξης και αναφέρονται οι θεωρίες που υπάρχουν για την επίλυση αυτών των προβλημάτων.

We outline a regular way for solving Maxwell's equations. We take, as the starting point, the notion of vector potentials. The rationale for introducing this notion in electrodynamics is that the set of Maxwell's equations is seemingly overdetermined. We demonstrate the existence of two fundamental solutions to Maxwell's equations whose linear combinations comprise the whole variety of classical electromagnetic field configurations., 11 pages

In the frame of the general Gel’fand – Yaglom formalism, the Fradkin theory for a spin 3/2 particle in presence of external fields is investigated. Applying the standard requirements of relativistic invariance, P-symmetry, existence of a Lagrangian for the model, we derive a set of spinor equations, first in absence of external fields. The wave function consists of a bispinor and a vector-bispinor. It is shown that in absence of external fields the Fradkin model is reduced to the Pauli – Fierz theory. Taking into account the presence of external electromagnetic fields, the Fradkin theory can be turned to the minimal form of the equation for the main bispinor. This equation contains an additional interaction term governed by the electromagnetic tensor F αβ . Meanwhile, there appears a parameter in the Fradkin equation related to any characteristic of the particle additional to its charge. The theory is generalized for taking into account the pseudo-Riemannian space-time geometry. In this case, the Fradkin equation contains an additional interaction term, governed by the Ricci tensor R αβ . If the electric charge of the particle is zero, the Fradkin model remains correct and describes a neutral spin 3/2 particle of the Majorana type interacting nonminimally with the geometrical background through the Ricci tensor. To clarify the meaning of the additional physical characteristics underlying the Fradkin model in contrast to the Pauli – Fierz one we have considered nonrelativistic approximation for both theories in presence of an external uniform magnetic field, and found respective energy spectra. The structure of the ninrelativistic Fradkin equation permits to consider such an additional parameter as polarizability.

As the laws of physics are expressed in a manner that makes their invariance under coordinate transformations manifest, they should be written in terms of tensors. Furthermore, tensors make manifest the characteristics and behaviors of electromagnetic fields through inhomogeneous, anisotropic, and compressible media. Electromagnetic fields are expressed completely in tensor form, Fαβ, which implies both electric field E → and magnetic field B → rather than separately in the vector fields. This study presents the Mathematica platform that generates and transforms a second-rank antisymmetric field-strength tensor Fαβ and whiskbroom pattern in Minkowski space. The platforms enhance the capabilities of students and researchers in tensor analysis and improves comprehension of the elegant features of complete structure in physics.

The aim of this research is a better understanding of the quantization in physics. The true origin of the quantization is the existence of the quantized kinetic momentum of electrons, neutrinos, protons and neutrons with the value. It is a consequence of the extended relativistic invariance of the wave of fundamental particles with spin 1/2. This logical link is due to properties of the quantum waves of fermions, which are functions of space-time with value into the and End(Cl3) Lie groups. Space-time is a manifold forming the auto-adjoint part of . The Lagrangian densities are the real parts of the waves. The equivalence between the invariant form and the Dirac form of the wave equation takes the form of Lagrange's equations. The momentum-energy tensor linked by Noether's theorem to the invariance under space-time translations has components which are directly linked to the electromagnetic tensor. The invariance under of the kinetic momentum tensor gives eight vectors. One of these vectors has a time component with value . Resulting aspects of the standard model of quantum physics and of the relativistic theory of gravitation are discussed.

We employ geometric algebra as the mathematical tool to provide a neat proof for a rarely seen formula in special relativity.

Gauss integral theorems for electric and magnetic fields, Faradays law of electromagnetic induction, magnetic field circulation theorem, theorems on the flux and circulation of vector potential, which are valid in curved spacetime, are presented in a covariant form. Covariant formulas for magnetic and electric fluxes, for electromotive force and circulation of the vector potential are provided. In particular, the electromotive force is expressed by a line integral over a closed curve, while in the integral, in addition to the vortex electric field strength, a determinant of the metric tensor also appears. Similarly, the magnetic flux is expressed by a surface integral from the product of magnetic field induction by the determinant of the metric tensor. A new physical quantity is introduced - the integral scalar potential, the rate of change of which over time determines the flux of vector potential through a closed surface. It is shown that the commonly used four-dimensional Kelvin-Stokes theorem does not allow one to deduce fully the integral laws of the electromagnetic field and in the covariant notation requires the addition of determinant of the metric tensor, besides the validity of the Kelvin-Stokes theorem is limited to the cases when determinant of metric tensor and the contour area are independent from time. This disadvantage is not present in the approach that uses the divergence theorem and equation for the dual electromagnetic field tensor. A new effect is predicted, according to which the circulation of magnetic field can appear even in the absence of electric current and with a constant electric field through the contour, if the area of this contour would change. By analogy with electromagnetic induction, for the magnetic field circulation to appear it is important that electric field flux that passes through the area of the contour would change over time., 25 pages

For electromagnetic field theories, canonical energy-momentum conservation laws can be derived from the underpinning spacetime translation symmetry according to the Noether procedure. However, the canonical energy-momentum tensors (EMTs) are neither symmetric nor gauge-symmetric (gauge invariant). The Belinfante-Rosenfeld (BR) method is a well-known procedure to symmetrize the EMTs, which also renders them gauge symmetric for first-order field theories. High-order electromagnetic field theories appear in the study of gyrokinetic systems for magnetized plasmas and the Podolsky system for the radiation reaction of classical charged particles. For these high-order field theories, gauge-symmetric EMTs are not necessarily symmetric and vice versa. In the present study, we develop a new gauge-symmetrization method for EMTs in high-order electromagnetic field theories. The Noether procedure is carried out using the Faraday tensor ${F}_{\ensuremath{\mu}\ensuremath{\nu}}$, instead of the 4-potential ${A}_{\ensuremath{\mu}}$, to derive a canonical EMT ${T}_{\mathrm{N}}^{\ensuremath{\mu}\ensuremath{\nu}}$. We show that the gauge-dependent part of ${T}_{\mathrm{N}}^{\ensuremath{\mu}\ensuremath{\nu}}$ can be removed using the displacement-potential tensor ${\mathcal{F}}^{\ensuremath{\sigma}\ensuremath{\mu}\ensuremath{\nu}}\ensuremath{\equiv}{\mathcal{D}}^{\ensuremath{\sigma}\ensuremath{\mu}}{A}^{\ensuremath{\nu}}/4\ensuremath{\pi}$, where ${\mathcal{D}}^{\ensuremath{\sigma}\ensuremath{\mu}}$ is the antisymmetric electric displacement tensor. This method gauge-symmetrizes the EMT without necessarily making it symmetric, which is adequate for applications not involving general relativity. For first-order electromagnetic field theories, such as the standard Maxwell system, ${\mathcal{F}}^{\ensuremath{\sigma}\ensuremath{\mu}\ensuremath{\nu}}$ reduces to the familiar BR superpotential ${\mathcal{S}}^{\ensuremath{\sigma}\ensuremath{\mu}\ensuremath{\nu}}$, and the method developed can be used as a simpler procedure to calculate ${\mathcal{S}}^{\ensuremath{\sigma}\ensuremath{\mu}\ensuremath{\nu}}$ without employing the angular momentum tensor in 4D spacetime. When the electromagnetic system is coupled to classical charged particles, the gauge-symmetrization method for EMTs is shown to be effective as well.

Killing vector fields can be used as gauge vector potentials since the associated electromagnetic field tensor automatically satisfies the source-free Maxwell’s equations in vacuum spacetimes. This fact enabled Wald to find the form of the electromagnetic tensor corresponding to the Kerr black hole immersed in a uniform test magnetic field. We present the generalisation of this result, which is valid for static black holes surrounded by nonlinear electromagnetic fields. The first obstacle we encountered when dealing with the nonlinear electrodynamics was that the above- described ansatz no longer works. Secondly, finding the exact solution in a closed form proved to be a rather challenging task because it would require solving a highly nonlinear differential equation. The alternative approach is via perturbative expansion around the original Wald’s solution. We obtain the equation which determines the lowest order correction to the gauge vector field 1- form and magnetic scalar potential. With the main focus on the Born–Infeld and Euler–Heisenberg theories on the Schwarzschild background, we calculate the aforementioned correction. Additionally, we show that this perturbative correction does not change electric and magnetic Komar charges or the asymptotic behaviour of the field. Finally, stating physical arguments, we justify the usage of the perturbative approach.

The Alcubierre warp drive metric is a spacetime construction where a massive particle located inside a spacetime distortion, called warp bubble, travels at velocities arbitrarily higher than the velocity of light. This theoretically constructed spacetime geometry is a consequence of general relativity where global superluminal velocities, also known as warp speeds, are possible, whereas local speeds are limited to subluminal ones as required by special relativity. In this work we analyze the solutions of the Einstein equations having charged dust energy-momentum tensor as source for warp velocities. The Einstein equations with the cosmological constant are written and all solutions having energy-momentum tensor components for electromagnetic fields generated by charged dust are presented, as well as the respective energy conditions. The results show an interplay between the energy conditions and the electromagnetic field such that in some cases the former can be satisfied by both positive and negative matter density. In other cases the dominant and null energy conditions are violated. A result connecting the electric energy density with the cosmological constant is also presented, as well as the effects of the electromagnetic field on the bubble dynamics., Comment: 21 pages, 1 table. LaTeX. Accepted for publication in General Relativity and Gravitation

Most students in engineering and science have heard the term “Maxwell’s equations,” and some may also have heard that Maxwell’s equations “describe all electromagnetic phenomena.” However, it is not always clear what exactly do we mean by these equations. How are they any different from what we have studied in the previous chapters? We recall discussing static and time-dependent fields and, in the process, discussed many applications. To do so, we used the definitions of the curl and divergence of the electric and magnetic fields: what we called “the postulates.” Are Maxwell’s equations different? Do they add anything to the previously described phenomena? Perhaps the best question to ask is the following: Is there any other electromagnetic phenomenon that was not discussed in the previous chapters because the definitions we used were not sufficient to do so? If so, do Maxwell’s equations define these yet unknown properties of the electromagnetic field? The answer to the latter is emphatically yes.

It has recently been revealed that massless scalar fields which are non-minimally coupled to the Maxwell electromagnetic tensor can be supported in the exterior spacetime regions of spherically symmetric charged black holes. The boundary between scalarized charged black-hole spacetimes and bald (scalarless) Reissner–Nordström black holes is determined by the presence of a critical existence-line which describes spatially regular linearized scalar ‘clouds’ that are supported in the black-hole spacetime. In the present paper we use analytical techniques in order to solve the Klein–Gordon wave equation for the non-minimally coupled linearized scalar fields in the spacetimes of near-extremal supporting black holes. In particular, we derive a remarkably compact analytical formula for the discrete resonant spectrum $$\{\alpha (l,Q/M;n)\}^{n=\infty }_{n=1}$$ { α ( l , Q / M ; n ) } n = 1 n = ∞ which characterizes the dimensionless coupling parameter of the composed Reissner–Nordström-black-hole-nonminimally-coupled-massless-scalar-field configurations along the critical existence-line of the Einstein–Maxwell-scalar theory (here Q/M is the dimensionless charge-to-mass ratio of the central supporting black hole and l is the angular harmonic index of the supported scalar configurations).

We revisit acceleration-induced nonlocal electrodynamics and the phenomenon of photon spin-rotation coupling. The kernel of the theory for the electromagnetic field tensor involves parity violation under the assumption of linearity of the field kernel in the acceleration tensor. However, we show that parity conservation can be maintained by extending the field kernel to include quadratic terms in the acceleration tensor. The field kernel must vanish in the absence of acceleration; otherwise, a general dependence of the kernel on the acceleration tensor cannot be theoretically excluded. The physical implications of the quadratic kernel are briefly discussed., 29 pages; v2: minor improvements, references added, invited contribution --- Universe: 5th Anniversary

We evaluate the correlators for the vector potential and for the field strength tensor of the electromagnetic field in the geometry of two parallel planar plates in AdS spacetime. Two types of boundary conditions are considered on the plates. The first one is a generalization of perfect conductor boundary condition and the second one corresponds to the confining boundary conditions. By using the expressions for the correlators, the vacuum expectation values (VEVs) of the photon condensate and of the electric and magnetic fields squared are investigated. As another important local characteristic of the vacuum state we consider the VEV of the energy-momentum tensor. The Casimir forces acting on the plates are decomposed into the self-action and interaction parts. It is shown that the interaction forces are attractive for both types of boundary conditions. At separations between the plates larger than the curvature radius of the background geometry they decay exponentially as functions of the proper distance. The self-action force per unit surface of a single plate does not depend on its location and depending on the boundary condition and on the number of spatial dimensions can be either attractive or repulsive with respect to the AdS boundary. By using the generalized zeta function technique we also evaluate the total Casimir energy. Applications are given in $Z_{2}$-symmetric braneworld models of the Randall-Sundrum type for vector fields with even and odd parities., Comment: 33 pages, 5 figures

A model of dark matter (DM) that communicates with the Standard Model (SM) exclusively through suppressed dimension five operator is discussed. The SM is augmented with a symmetry $U(1)_X \otimes Z_2$, where $U(1)_X$ is gauged and broken spontaneously by a very heavy decoupled scalar. The massive $U(1)_X$ vector boson ($X^\mu$) is stabilized being odd under unbroken $Z_2$ and therefore may contribute as the DM component of the universe. Dark sector field strength tensor $X^{\mu\nu}$ couples to the SM hypercharge tensor $B^{\mu\nu}$ via the presence of a heavier $Z_2$ odd real scalar $\Phi$, i.e. $1/\Lambda \; X^{\mu\nu}B_{\mu\nu}\Phi$, with $\Lambda$ being a scale of new physics. The freeze-in production of the vector boson dark matter feebly coupled to the SM is advocated in this analysis. Limitations of the so-called UV freeze-in mechanism that emerge when the maximum reheat temperature $T_\text{RH}$ drops down close to the scale of DM mass are discussed. The parameter space of the model consistent with the observed DM abundance is determined. The model easily and naturally avoids both direct and indirect DM searches. Possibility for detection at the Large Hadron Collider (LHC) is also considered. A Stueckelberg formulation of the model is derived., Comment: 40 pages, 14 figures, 3 tables, new references and figures added. Version accepted in JHEP

We consider the dynamics of a classical charge in flat spacetime of six dimensions. The mass shell relation of a free charge admits nonlinear oscillations. Having analyzed the problem of on eigenvalues and eigenvectors of Faraday tensor, we establish the algebraic structure of electromagnetic field in 6D. We elaborate the classification scheme based on three field's invariants. Using the basic algebraic properties of the electromagnetic field tensor we analyze the motion of a charge in constant electromagnetic field. Its world line is a combination of hyperbolic and circular orbits which lie in three mutually orthogonal sheets of two dimensions. Within the braneworld scenario, we project the theory on the de Sitter space of four dimensions. Actually, as it turns out, spins of elementary particles themselves are manifestations of extra dimensions., 16 pages, 5 figures

We extend a previous numerical study of SU(3) Yang-Mills theory in which we measured the spatial distribution of all components of the color fields surrounding a static quark-antiquark pair for a wide range of quark-antiquark separations, and provided evidence that the simulated gauge invariant chromoelectric field can be separated into a Coulomb-like 'perturbative' field and a 'non-perturbative' field, identified as the confining part of the SU(3) flux tube field. In this paper we hypothesize that the fluctuating color fields not measured in our simulations do not contribute to the string tension. Under this assumption the string tension is determined by the color fields we measure, which form a tensor $F_{\mu \nu}$ pointing in a single direction in color space. We call this the Maxwell mechanism of confinement. We provide an additional procedure to isolate the non-perturbative (confining) field. We then extract the string tension from a stress energy-momentum tensor $T_{\mu \nu}$ having the Maxwell form, constructed from the non-perturbative part of the tensor $F_{\mu \nu}$ obtained from our simulations. To test our hypothesis we calculate the string tension from our simulations of the color fields for ten values of the quark-antiquark separation ranging from 0.37 fm to 1.2 fm. We also calculate the spatial distributions of the energy-momentum tensor $T_{\mu \nu}$ surrounding static quarks for this range of separations, and we compare these distributions with those obtained from direct simulations of the energy-momentum tensor in SU(3) Yang-Mills theory., Comment: 16 pages, 12 figures. arXiv admin note: text overlap with arXiv:1810.07133

Fermi-Walker derivative and biharmonic particle play an important role in skillful applications. We obtain a new characterization on binormal spherical indicatrix by using the Fermi-Walker derivative and parallelism in space. We suggest that an inextensible flow is the necessary and sufficient condition for this particle. Finally, we give some characterizations for a non-rotating frame of this binormal spherical indicatrix.

We consider the conformal transformation of electrodynamics in curved space. The invariance of the Lorenz gauge condition under the conformal transformation yields the massive Lorenz gauge condition. The conformal massless electrodynamics in curved space yields massive electrodynamics in Minkowski space. The conformal Klein–Gordon equation in the conformal frame is found to yield the quantum Telegraph equation. The conformal transformation of a given field yields the matter (particle) of the field confirming that the field, like a particle, has a dual nature. A special type of conformal transformation is shown to give massless fields a mass that is tantamount to Higgs mechanics. The parameter (weight) ξ = ± 1 in the conformal transformation of the electromagnetic tensor, F ˜ μ ν = Ω ± 1 F μ ν , defines the photon mass in the Minkowski frame.

We implement the KORAL code to perform two sets of very long general relativistic radiation magnetohydrodynamic simulations of an axisymmetric optically thin magnetized flow around a non-rotating black hole: one with a new term in the electromagnetic field tensor due to the radiation pressure felt by the plasma electrons on the comoving frame of the electron-proton plasma, and one without. The source of the radiation is the accretion flow itself. Without the new term, the system evolves to a standard accretion flow due to the development of the magneto-rotational instability (MRI). With the new term, however, the system eventually evolves to a magnetically arrested state (MAD) in which a large scale jet-like magnetic field threads the black hole horizon. Our results confirm the secular action of the Cosmic Battery in accretion flows around astrophysical black holes., Comment: 7 pages, 4 figures, submitted to Monthly Notices

Einstein’s theory of relativity is based on the Principle of Equivalence, Hilbert’s on invariant theory and the calculus of variations. The two paradigms are not equivalent. Using the universality of Maxwell’s equations, Hilbert’s variational method is used to determine the energy–momentum tensor uniquely, and to show that general relativity can be formulated on the basis of Maxwellian, rather than specific physical force fields. A unified field theory is proved in which the Maxwellian force fields are all on an equal footing, distinct from the geometric field.

In the numerical solution of Maxwell's equations for dynamic electromagnetic fields, the two Gauss's laws are usually not considered since they are a natural consequence of Faraday's and MaxwellAmpre's laws if the charge conservation law is satisfied. However, when the charge conservation law is not satisfied, the numerical errors of Gauss's laws will increase unbounded, leading to numerical instability or breakdown. Unfortunately, the charge conservation law can be easily violated in self-consistent waveparticle simulations. In the meantime, the violation of Gauss's laws will also result in an increased error in the normal flux continuity. In the simulations of pure electromagnetic problems, the satisfaction of tangential field continuity across a material interface is sufficient to yield accurate numerical results. However, in a self-consistent waveparticle simulation, the normal components are as important as the tangential components, since they are critical in predicting the particle kinetics. In this paper, a divergence-cleaning method is presented to enforce Gauss's laws and normal flux continuity by introducing auxiliary variables and damping terms into Maxwell's equations in inhomogeneous media, which yield hyperbolic and mixed hyperbolicparabolic Maxwell equations. The numerical solution schemes of the resulting purely and damped hyperbolic Maxwell equations are introduced under the framework of the discontinuous Galerkin time-domain method. It is shown through several numerical examples that the proposed method is able to preserve the continuity conditions for both tangential field components and normal flux components, and is effective in cleaning the numerical errors in both Gauss's laws.

A new unification of the Maxwell equations is given in the domain of Clifford algebras with in a fashion similar to those obtained with Pauli and Dirac algebras. It is shown that the new electromagnetic field multivector can be obtained from a potential function that is closely related to the scalar and the vector potentials of classical electromagnetics. Additionally it is shown that the gauge transformations of the new multivector and its potential function and the Lagrangian density of the electromagnetic field are in agreement with the transformation rules of the second-rank antisymmetric electromagnetic field tensor, in contrast to the results obtained by applying other versions of Clifford algebras.

Made available in DSpace on 2020-12-10T17:30:50Z (GMT). No. of bitstreams: 0 Previous issue date: 2019-12-01 National Natural Science Foundation of China (NNSFC) Post-doctoral Science Foundation of China Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) In this paper, the scalar quasinormal modes of nonlinear charged black hole metrics in Rastall gravity is investigated. The electromagnetic tensor presented in the background spacetime possesses a power-law form, and the system is assumed to be surrounded by a quintessence field. By utilizing the recently obtained analytic form of the metric, the quasinormal frequencies are obtained via the matrix method. The numerical values have been further compared against those evaluated by using the WKB approximation up to thirteenth order. Also, the finite difference method is utilized to study the temporal evolution of the scalar perturbations. In terms of calculations carried out for both massless and massive scalar fields, we discuss the properties of the resultant quasinormal frequencies, as well as their dependences on the model parameters describing the background black hole. To be specific, the effect of the electric charge, mass of the scalar field, and equation of state of the quintessence are examined. Besides, the quasinormal frequencies associated with an extremal black hole regarding the Nariai limit is explored, where the obtained results are found to be consistent with theoretical arguments. The black hole metric is found to be stable against scalar perturbations, in the presence of both linear and nonlinear electromagnetic fields. Copyright (C) EPLA, 2020 Huazhong Univ Sci & Technol, MOE Key Lab Fundamental Phys Quant Measurement, Hubei Key Lab Gravitat & Quantum Phys, PGMF, Wuhan 430074, Hubei, Peoples R China Huazhong Univ Sci & Technol, Sch Phys, Wuhan 430074, Hubei, Peoples R China China Univ Geosci, Hubei Subsurface Multiscale Imaging Key Lab, Inst Geophys & Geomat, Wuhan 430074, Hubei, Peoples R China Univ Sao Paulo, Escola Engn Lorena, BR-12602810 Lorena, SP, Brazil Univ Estadual Paulista, Fac Engn Guaratingueta, BR-12516410 Guaratingueta, SP, Brazil Yangzhou Univ, Ctr Gravitat & Cosmol, Sch Phys Sci & Technol, Yangzhou 225002, Jiangsu, Peoples R China Univ Estadual Paulista, Fac Engn Guaratingueta, BR-12516410 Guaratingueta, SP, Brazil National Natural Science Foundation of China (NNSFC): 11805166 National Natural Science Foundation of China (NNSFC): 11805074 National Natural Science Foundation of China (NNSFC): 91636221 Post-doctoral Science Foundation of China: 2017M620308 Post-doctoral Science Foundation of China: 2018T110750

Starting from the suitable action of four-dimensional Einstein-dilaton gravity with the logarithmic nonlinear electrodynamics, the explicit form of the coupled scalar, electromagnetic and gravitational field equations have been obtained. Through the exact solution of the field equations, the nonvanishing component of the electromagnetic tensor have been obtained and it has been shown that dilaton potential can be written as the suitable combination of two terms similar to the Liouville potentials. Three classes of novel electrically charged hairy black holes have been introduced with the regular and well-defined metric functions for all values of the parameters in the theory. The solutions reduce to their corresponding values in the Einstein-Maxwell-dilaton theory when the nonlinearity parameter of the electrodynamics goes to infinity. The thermodynamic quantities, such as temperature, entropy, mass, electric charge and electric potential, have been calculated and it has been demonstrated that the first law of black hole thermodynamics is valid for all of the new solutions. Thermodynamic stability or phase transition of the black holes have been investigated based on the canonical ensemble method. By calculating the black hole heat capacity type-1 and type-2 phase transition points have been determined. Also, it has been found that in order for the new nonlinearly charged dilaton black holes to remain locally stable some simple conditions must be fulfilled.

The electrodynamics of Weyl semimetals is an extension of Maxwell's theory where in addition to field strength tensor ${F}_{\ensuremath{\mu}\ensuremath{\nu}}$, an axion field enters the theory which is parametrized by a four-vector ${b}^{\ensuremath{\mu}}=({b}_{0},\mathbit{b})$. In tilted Weyl materials (TWMs) an additional set of parameters $\mathbit{\ensuremath{\zeta}}=({\ensuremath{\zeta}}_{x},{\ensuremath{\zeta}}_{y},{\ensuremath{\zeta}}_{z})$ enters the theory that can be encoded into the metric of the spacetime felt by electrons in TWMs. This allows an extension of Maxwell's electrodynamics that describes electric and magnetic fields in TWMs, and tilted Dirac materials (TDMs) that correspond to ${b}^{\ensuremath{\mu}}=0$. The tilt parameter $\mathbit{\ensuremath{\zeta}}$ appearing as an off-diagonal metric matrix element, mixing time and space components, which mingles $\mathbit{E}$ and $\mathbit{B}$ fields, whereby the inhomogeneous Maxwell's equations are modified. As an example of the application of the electrodynamics of TWMs, we study the surface plasmon polariton (SPP) in these systems. The peculiarity of the SPP on the surface of TWMs or TDMs is that it is not merely the propagation of electromagnetic modes on the surface of a conductor. It also describes the propagation of electromagnetic waves at the interface of two different spacetime geometries. In the case of TDMs, we find a characteristic dependence of the SPP spectrum on the tilt parameter $\ensuremath{\zeta}$ which can be used to map $\ensuremath{\zeta}$ from SPP measurements. In the case of TWMs, depending on whether the interface with vacuum supports a Fermi arc or not, and whether the propagation direction is along the Fermi arc or transverse to it, we find many unusual spectral features for SPP modes. These include (1) surface plasmons with much higher frequency than bulk plasmon frequency, (2) soft SPP modes at short length scales, (3) a tilt-controlled SPP window beyond which SPP modes are unstable, (4) a kink in the SPP dispersion, (5) uniform group velocity near the ``horizon'' ($\ensuremath{\zeta}=1$), and (6) possible negative group velocity. Our detailed study of the dependence of SPP spectra on the arrangements of three vectors $(\mathbit{b},\mathbit{q},\mathbit{\ensuremath{\zeta}})$, the first two of which are at our control, can be utilized to map the tilt characteristics and Fermi arc characteristics from SPP measurements.

It would be very useful to find a way of reducing excited-state effects in lattice QCD calculations of nucleon structure that has a low computational cost. We explore the use of hybrid interpolators, which contain a nontrivial gluonic excitation, in a variational basis together with the standard interpolator with tuned smearing width. Using the clover discretization of the field strength tensor, a calculation using a fixed linear combination of standard and hybrid interpolators can be done using the same number of quark propagators as a standard calculation, making this a cost-effective option. We find that such an interpolator, optimized by solving a generalized eigenvalue problem, reduces excited-state contributions in two-point correlators. However, the effect in three-point correlators, which are needed for computing nucleon matrix elements, is mixed: for some matrix elements such as the tensor charge, excited-state effects are suppressed, whereas for others such as the axial charge, they are enhanced. The results illustrate that the variational method is not guaranteed to reduce the net contribution from excited states except in its asymptotic regime, and suggest that it may be important to use a large basis of interpolators capable of isolating all of the relevant low-lying states., 12 pages, 8 figures

In this paper, we prove the linear stability to gravitational and electromagnetic perturbations of the Reissner-Nordstr\"om family of charged black holes with small charge. Solutions to the linearized Einstein-Maxwell equations around a Reissner-Nordstr\"om solution arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to a linearized Kerr-Newman metric. We express the perturbations in geodesic outgoing null foliations, also known as Bondi gauge. To obtain decay of the solution, one must add a residual pure gauge solution which is proved to be itself controlled from initial data. Our results rely on decay statements for the Teukolsky system of spin $\pm2$ and spin $\pm1$ satisfied by gauge-invariant null-decomposed curvature components, obtained in earlier works. These decays are then exploited to obtain polynomial decay for all the remaining components of curvature, electromagnetic tensor and Ricci coefficients. In particular, the obtained decay is optimal in the sense that it is the one which is expected to hold in the non-linear stability problem., Comment: 121 pages, 2 figures, version accepted for publication

A relativistic particle undergoing successive boosts which are non collinear will experience a rotation of its coordinate axes with respect to the boosted frame. This rotation of coordinate axes is caused by a relativistic phenomenon called Thomas Rotation. We assess the importance of Thomas rotation in the calculation of physical quantities like electromagnetic fields in the relativistic regime. We calculate the electromagnetic field tensor for general three dimensional successive boosts in the particle's rest frame as well as the laboratory frame. We then compare the electromagnetic field tensors obtained by a direct boost $\vec{\beta} + \delta \vec{\beta}$ and successive boosts $\vec{\beta}$ and $\Delta \vec{\beta}$ and check their consistency with Thomas rotation. This framework might be important to situations such as the calculation of frequency shifts for relativistic spin-1/2 particles undergoing Larmor precession in electromagnetic fields with small field non-uniformities., Comment: 18 pages, 4 figures

Detection of axion field can unfold intriguing facets of our universe in several astrophysical and cosmological scenarios. In four dimensions, such a field owes its origin to the completely anti-symmetric Kalb-Ramond field strength tensor. Its invisibility in the solar system based tests compels one to look for its signatures in the strong field regime. The recent observation of the shadow of the supermassive black hole in the galaxy M87 ushers in a new opportunity to test for the footprints of axion in the near horizon region of black holes, where the gravity is expected to be strong. In this paper, we explore the impact of axion on the black hole shadow and compare the result with the available image of M87*. Our analysis indicates that axion which violates the energy condition seems to be favored by observations. The implications are discussed., Comment: 20 pages, 6 figures

We study metric transformations including not just the field strength tensor of a $U(1)$ gauge field, but also its dual tensor. We first consider an arbitrary symmetric matrix built up with these two tensors in the metric transformation. It turns out the form of transformation reduces to a quite simple form on imposing the parity evenness of the transformed metric and by utilising the Cayley-Hamilton theorem as well as other useful identities. Interestingly, the same form for the transformation was recently argued in the process of seeking for a generic metric transformation but without the inclusion of the dual tensor., Comment: 6 pages

Considering a complex Lagrange space ([24]), in this paper the complex electromagnetic tensor fields are defined as the sum between the differential of the complex Liouville 1-form and the symplectic 2-form of the space relative to the adapted frames of the Chern–Lagrange complex nonlinear connection. In particular, an electrodynamics theory on a complex Finsler space is obtained. We show that our definition of the complex electrodynamics tensors has physical meaning and these tensors generate an adequate field theory which offers the opportunity of coupling with the gravitation. The generalized complex Maxwell equations are written. A gauge field theory of electrodynamics on the holomorphic tangent bundle is put over T′M and the gauge invariance to phase transformations is studied. An extension of the Dirac Lagrangian on T′M coupled with the electrodynamics Lagrangian is studied and it offers the framework for a unified gauge theory of fields.

We investigate the Plebanski class of electrodynamical theories, i.e., theories of nonlinear vacuum electrodynamics that derive from a Lorentz-invariant Lagrangian (or Hamiltonian). In any such theory the light rays are the lightlike geodesics of two optical metrics that depend on the electromagnetic background field. A set of necessary and sufficient conditions is found whose fulfillment secures that the optical metrics are causal in the sense that the light rays are lightlike or timelike with respect to the underlying space-time metric. Thereupon we derive conditions on the Lagrangian, or the Hamiltonian, of the theory such that the causality conditions are satisfied for all allowed background fields. (The allowed values of the field strength tensor are those for which the excitation tensor is finite and real.) The general results are illustrated with several examples.

We investigate the compressible Navier–Stokes equations where the constitutive law for the stress tensor given by Maxwell’s law is revised to a system of relaxation equations for two parts of the tensor. The global well-posedness is proved as well as the compatibility with the classical compressible Navier–Stokes system in the sense that, for vanishing relaxation parameters, the solutions to the Maxwell system are shown to converge to solutions of the classical system.

A theory where the gravitational, Maxwell and Dirac fields (mathematically represented as particular sections of a convenient Clifford bundle) are treated as fields in Faraday’s sense living in Minkowski spacetime is presented. In our theory we obtain a genuine energy-momentum tensor for the gravitational field and a genuine energy-momentum conservation law for the system of the interacting gravitational, Maxwell and Dirac fields. Moreover, the energy-momentum tensors of the Maxwell and Dirac fields are symmetric, and it is shown that the equations of motion for the gravitational potentials are equivalent to Einstein equation of General Relativity. Precisely, the Einstein equation in which the second member is the sum of the energy-momentum tensors of the Maxwell, Dirac and the interaction Maxwell–Dirac fields all defined in an effective Lorentzian spacetime whose use is eventually no more than a question of mathematical convenience.

We construct explicit solutions of the Einstein–Maxwell equations in the case when distributions defined by the Faraday tensor are completely integrable and totally geodesic.

Maxwell's theory of multicomponent diffusion and subsequent extensions are based on systems of mass and momentum conservation equations. The partial stress tensor, which is involved in these equations, is expressed in terms of the gradients of velocity fields by statistical and continuum mechanical methods. We propose a method for the solution of Maxwell's equations of diffusion coupled with Muller's expression for the partial stress tensor. The proposed method consists in a singular perturbation process, followed by a weak (finite element) analysis of the resulting PDE systems. The singularity involved in the obtained equations was treated by a special technique, by which lower-order systems were supplemented by proper combinations of higher-order equations. The method proved particularly efficient for the solution of the Maxwell-Muller system, eventually reducing the number of unknown fields to that of the classical Navier-Stokes/Fick system. It was applied to the classical Stefan tube problem and the Hagen-Poiseuille flow in a hollow-fiber membrane tube. Numerical results for these problems are presented, and compared with the Navier-Stokes/Fick approximation. It is shown that the 0-th order term of the Maxwell-Muller equations differs from a properly formulated Navier-Stokes/Fick system, by a numerically insignificant amount. Numerical results for 1st-order terms indicate a good agreement of the classical approximation (with properly formulated Navier-Stokes and Fick's equations) with the Maxwell-Muller system, in the studied cases. We propose a method for the solution of Maxwell's equations of diffusion with Muller's expression for the stress tensor.The method is particularly efficient for the solution of the Maxwell-Muller system for low Mach numbers.It reduces the number of unknown fields to that of the classical Navier-Stokes/Fick system.It is applied to Stefan tube and Hagen-Poiseuille flow in a hollow-fiber membrane tube.Numerical results indicate good agreement with classical approximation.

We study the relation between the energy-momentum tensor of the electromagnetic field and the group velocity of quasi-monochromatic waves in a nonabsorptive, isotropic, spatially and temporally dispersive dielectric. It is shown that the Abraham force acting on a dielectric is not needed for the momentum conservation law to hold if the dielectric is free of external charges and currents and if the Abraham momentum density is used. The energy-momentum tensor turns out to be symmetric, and the Maxwell stress tensor is expressed either in terms of the momentum density vector and the group velocity or in terms of the energy density and the group velocity. The stress tensor and the energy density are essentially dependent on the frequency and wave vector derivatives of the functions that describe the electromagnetic properties of the medium (i.e., the dielectric permittivity and the magnetic permeability). The obtained results are applicable to both ordinary and left-handed media. The results are compared with those of other authors. The pressure a wave exerts on the interface between two media is calculated. For both ordinary and left-handed media, either 'radiation pressure' or 'radiation attraction' can occur at the interface, depending on the material parameters of the two media. For liquid dielectrics, the striction effect is considered.

We find an explicit solution of the source free Maxwell equations in a generalized Kerr-NUT-(A)dS spacetime in all dimensions. This solution is obtained as a linear combination of the closed conformal Killing-Yano tensor $h_{ab}$, which is present in such a spacetime, and a derivative of the primary Killing vector, associated with $h_{ab}$. For the vanishing cosmological constant the obtained solution reduces to the Wald's electromagnetic field generated from the primary Killing vector., Comment: 4 pages, no figures v2: added references

In this paper we present two results in ($2+1$) gravity coupled to nonlinear electrodynamics. First we determine the general form of the electromagnetic field tensor in ($2+1$) gravity coupled to nonlinear electrodynamics in stationary cyclic spacetimes. Secondly, we determine a family of exact solutions in ($2+1$) gravity sourced by a nonlinear electromagnetic field. The solutions are characterized by five parameters: mass $M$, angular momentum $J$, cosmological constant $\mathrm{\ensuremath{\Lambda}}$, and two electromagnetic charges ${q}_{\ensuremath{\alpha}}$ and ${q}_{\ensuremath{\beta}}$. Remarkably, the solution can be interpreted as a traversable wormhole, provided the fulfillment of certain inequalities by the characteristic parameters; fine-tuning of the cosmological constant leads to an extreme black hole, whereas by switching off one of the electromagnetic charges, we obtain the Ba\~nados-Teitelboim-Zanelli (BTZ) black hole.

In this chapter, I present a parallel development of electromagnetic theory based on the traditional Gibbsian vector formalism (using grad, div, and curl), and using the methods of differential forms. In the framework of this new approach, Maxwell's equations appear naturally from the basic equations of the field theory for the differential forms. I provide the physical interpretation of Maxwell's equations as well. The boundary conditions for the vector fields are introduced based on Maxwell's equations. This chapter also develops the basic aspects of electromagnetic theory which are common to the many techniques of electrical exploration that will be discussed later in the book. We introduce the basic theorems characterizing the distribution of the energy of electromagnetic field (Poynting's theorems) and the concept of electromagnetic Green's tensors. The chapter concludes with a description of the reciprocity relations and Lorentz lemma.

In the present work, the Minkowski equations obtained on the basis of theory of relativity are used to describe electromagnetic fields in moving media. But important electromagnetic processes run under non-relativistic conditions of slow-moving media. Therefore, one should carry out its description in terms of classical mechanics. Hertz derived electrodynamic equations for moving media within the frame of classical mechanics on the basis of the Maxwell theory. His equations disagree with the experimental data concerned with the moving dielectrics. In the paper, a way of description of electromagnetic fields in slow-moving media on the basis of the Maxwell theory within the frame of classical mechanics is offered by combining the Hertz approach and the experimental data concerned with the movement of dielectrics in electromagnetic fields. Received Maxwell equations lack asymmetry in the description of the reciprocal electrodynamic action of a magnet and a conductor and conform to known experimental data. Comparative analysis of the Minkowski and Maxwell models is carried out.