THE RESOLUTION OF FOUR-DIMENSIONAL VECTOR FIELD AND TENSOR FIELDS, AND ITS APPLICATION TO ELECTRODYNAMICS

1955 ◽  
Vol 33 (5) ◽  
pp. 235-240
Author(s):  
N. L. Balazs
Keyword(s):  
Electromagnetic Field ◽  
Potential Field ◽  
Vector Fields ◽  
Dimensional Vector ◽  
Field Tensor ◽  
Algebraic Condition ◽  
Tensor Fields ◽  
Lorentz Condition ◽  
Electromagnetic Field Tensor ◽  
Magnetic Poles

We generalize Helmholtz's theorem and apply it to four-dimensional vector fields and tensor fields. For vector fields the generalization is straightforward. Antisymmetric tensor fields of rank two exhibit a beautiful symmetry between the irrotational part of the tensor and the dual of the solenoidal component. The physical applications show that in Maxwell's theory the irrotational part of the four-potential field has no physical meaning and the Lorentz condition makes it identically zero. In Dirac's new electrodynamics an algebraic condition is imposed on the four-potential. Hence in this theory the irrotational part is not zero, and the algebraic condition establishes a relation between the sources and vortices of the four-potential field. If we apply the resolution to the electromagnetic field tensor we can see that the free charges are responsible for the sources, and the magnetic poles, if they exist, for the vortices, provided we use the customary association between the components of the electromagnetic field tensor and the components of the electric and magnetic fields.

Electromagnetism and special relativity

2018 ◽  
Author(s):  
J. Pierrus
Keyword(s):  
Electromagnetic Field ◽  
Special Relativity ◽  
Point Charge ◽  
Reference Frames ◽  
Classical Electrodynamics ◽  
Covariant Form ◽  
Field Tensor ◽  
Electromagnetic Field Tensor ◽  
Preferred Reference Frame ◽  
Physical Quantities

In 1905, when Einstein published his theory of special relativity, Maxwell’s work was already about forty years old. It is therefore both remarkable and ironic (recalling the old arguments about the aether being the ‘preferred’ reference frame for describing wave propagation) that classical electrodynamics turned out to be a relativistically correct theory. In this chapter, a range of questions in electromagnetism are considered as they relate to special relativity. In Questions 12.1–12.4 the behaviour of various physical quantities under Lorentz transformation is considered. This leads to the important concept of an invariant. Several of these are encountered, and used frequently throughout this chapter. Other topics considered include the transformationof E- and B-fields between inertial reference frames, the validity of Gauss’s law for an arbitrarily moving point charge (demonstrated numerically), the electromagnetic field tensor, Maxwell’s equations in covariant form and Larmor’s formula for a relativistic charge.

scholarly journals Toward Nonlocal Electrodynamics of Accelerated Systems

Universe ◽  
2020 ◽  
Vol 6 (12) ◽  
pp. 229
Author(s):  
Bahram Mashhoon
Keyword(s):  
Electromagnetic Field ◽  
Parity Violation ◽  
Spin Rotation ◽  
Field Tensor ◽  
Parity Conservation ◽  
Electromagnetic Field Tensor ◽  
Nonlocal Electrodynamics

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.

scholarly journals THE MAXWELL LAGRANGIAN IN PURELY AFFINE GRAVITY

2008 ◽  
Vol 23 (03n04) ◽  
pp. 567-579 ◽  
Author(s):  
NIKODEM J. POPŁAWSKI
Keyword(s):  
Electromagnetic Field ◽  
Ricci Tensor ◽  
Maxwell Equations ◽  
Legendre Transformation ◽  
Conformal Transformations ◽  
Field Tensor ◽  
Metric Tensor ◽  
Hamiltonian Density ◽  
Electromagnetic Field Tensor ◽  
Affine Gravity

The purely affine Lagrangian for linear electrodynamics, that has the form of the Maxwell Lagrangian in which the metric tensor is replaced by the symmetrized Ricci tensor and the electromagnetic field tensor by the tensor of homothetic curvature, is dynamically equivalent to the Einstein–Maxwell equations in the metric–affine and metric formulation. We show that this equivalence is related to the invariance of the Maxwell Lagrangian under conformal transformations of the metric tensor. We also apply to a purely affine Lagrangian the Legendre transformation with respect to the tensor of homothetic curvature to show that the corresponding Legendre term and the new Hamiltonian density are related to the Maxwell–Palatini Lagrangian for the electromagnetic field. Therefore the purely affine picture, in addition to generating the gravitational Lagrangian that is linear in the curvature, justifies why the electromagnetic Lagrangian is quadratic in the electromagnetic field.

On a duality invariant action principle in vacuum electrodynamics

1970 ◽  
Vol 48 (20) ◽  
pp. 2423-2426 ◽  
Author(s):  
G. M. Levman
Keyword(s):  
Electromagnetic Field ◽  
Electromagnetic Fields ◽  
Maxwell Equations ◽  
Lagrangian Density ◽  
Vacuum Field ◽  
Field Tensor ◽  
Field Equations ◽  
Invariant Action ◽  
Electromagnetic Field Tensor ◽  
Derivatives Of

Although Maxwell's vacuum field equations are invariant under the so-called duality rotation, the usual Lagrangian density for the electromagnetic field, which is bilinear in the first derivatives of the electromagnetic potentials, does not exhibit that invariance. It is shown that if one takes the components of the electromagnetic field tensor as field variables then the most general Lorentz invariant Lagrangian density bilinear in the electromagnetic fields and their first derivatives is determined uniquely by the requirement of duality invariance. The ensuing field equations are identical with the iterated Maxwell equations.

THE DYNAMICAL THEORY OF MAGNETIC MONOPOLES

1952 ◽  
Vol 30 (3) ◽  
pp. 218-225 ◽  
Author(s):  
S. Shanmugadhasan
Keyword(s):  
Electromagnetic Field ◽  
Equations Of Motion ◽  
Natural Generalization ◽  
Magnetic Monopoles ◽  
Dynamical Theory ◽  
The Other ◽  
Field Tensor ◽  
Physical Content ◽  
Electromagnetic Field Tensor ◽  
Set Up

The theory of electric charges and magnetic monopoles has been set up by Dirac by expressing the electromagnetic field tensor in terms of one four-potential and of the variables describing the strings attached to each magnetic mono-pole. In this reformulation of Dirac's theory the field tensor is expressed in terms of two four-potentials, one corresponding to charges and the other to monopoles, and the action principle for the equations of motion is set up in terms of the two four-potentials and of the tensors dual to them. Thus there is formal symmetry as far as is possible in the treatment of the charges and the monopoles. Also the mathematics is direct and neat. Though the physical content is the same as that of Dirac, a natural generalization of the Fermi form of electrodynamics subject to the restriction that the same particle cannot have both charge and monopole is obtained here.

On Killing vectors and invariance transformations of the Einstein–Maxwell equations

1976 ◽  
Vol 80 (2) ◽  
pp. 357-364 ◽  
Author(s):  
M. L. Woolley
Keyword(s):  
Electromagnetic Field ◽  
Maxwell Equations ◽  
Invariance Group ◽  
Field Tensor ◽  
Metric Tensor ◽  
Divergence Free Vector ◽  
Point Transformations ◽  
Maxwell Space ◽  
Electromagnetic Field Tensor ◽  
Simply Connected

AbstractIt is shown that, in a simply connected four dimensional Riemannian space, an arbitrary divergence-free vector generates a one-parameter group of point transformations which leaves Maxwell's equations unchanged. This result is used to show that, if the metric tensor of a simply connected vacuum Einstein–Maxwell space-time admits a group of motions which is also an invariance group of the electromagnetic field tensor, then there exists a one-parameter family of metric tensors all of which satisfy the Einstein–Maxwell equations with the invariant electromagnetic field as source.

Canonical relativistic formulation to calculate the Casimir force four-vector on anisotropic conductor polarizable and magnetizable media

2019 ◽  
Vol 34 (30) ◽  
pp. 1950247
Author(s):  
Majid Amooshahi
Keyword(s):  
Electromagnetic Field ◽  
Vector Fields ◽  
Casimir Force ◽  
Tangential Component ◽  
Antisymmetric Tensor ◽  
Relativistic Wave Equation ◽  
Total System ◽  
Tensor Fields ◽  
Relativistic Formulation ◽  
Multilayer Medium

A canonical relativistic formulation to calculate the Casimir force four-vector exerted on anisotropic conductor polarizable and magnetizable media is provided. The anisotropic conductor polarizable and magnetizable medium is modeled by a continuum collection of the antisymmetric tensor fields of the second rank and a continuum collection of the vector fields in Minkowski spacetime. The collection of the antisymmetric tensor fields describes the polarization and the magnetization properties of the medium. The collection of the vector fields describes the conductivity property of the medium. The quantum relativistic wave equation of the four-vector potential of the electromagnetic field is solved using an iteration method. According to the conservation principle of the energy–momentum four-vector and using of the energy–momentum tensors of the relativistic dynamical fields, contained in the theory, the Casimir force four-vector on the anisotropic conductor polarizable and magnetizable medium is obtained. The Casimir force four-vector is obtained in terms of the relativistic dynamical fields, contained in the theory and the coupling tensors that couple the electromagnetic field to the anisotropic conductor polarizable and magnetizable medium. The Casimir force four-vector exerted on the medium is calculated in the vacuum state of the total system. As a special case, the formulation is applied to a multilayer medium. The tangential component of the Casimir force exerted on a multilayer medium vanish when the anisotropic conductor polarizable and magnetizable medium is converted to an isotropic one.

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