Maxwell tensor

Note that Bell et al. [7] derived Eq. (11.73) by the usual method, that is, by integrating the osmotic pressure and the Maxwell tensor over an arbitrary closed surface enclosing either sphere. Equation (11.73) agrees with the leading term of the exact... 

Patchwise treatment of surfactant aggregates and surfactant-laden curved interfaces, as implicitly assumed here, has been advanced in recent years and is, in principle, feasible in spite of the electrostatic interactions being long range as they can be handled in a localized manner by means of the Maxwell tensor [61], whereas other (dispersion) interactions that are of long range per se for the most part are fairly weak in surfactant systems. 

Pj is the body-force tensor which accounts for the action of body forces such as gravity, electrostatic forces (the Maxwell tensor), etc. 

A frequently used example of Oldroyd-type constitutive equations is the Oldroyd-B model. The Oldroyd-B model can be thought of as a description of the constitutive behaviour of a fluid made by the dissolution of a (UCM) fluid in a Newtonian solvent . Here, the parameter A, called the retardation time is de.fined as A = A (r s/(ri + s), where 7]s is the viscosity of the solvent. Hence the extra stress tensor in the Oldroyd-B model is made up of Maxwell and solvent contributions. The Oldroyd-B constitutive equation is written as... 

The relativistic invariance of the electromagnetic field is conveniently expressed in tensor notation. Factorized in Minkowski space the Maxwell equa-... 

Tensor formalism can be used to generate all aspects of the electromagnetic field. The inhomogeneous Maxwell equation (2) in tensor form... 

We shall be concerned with the doubled operators describing the energy-momentum tensor of the free Maxwell and Dirac fields according to the tilde conjugation rules, we have, respectively ... 

Considering such choices for the parameters ay and using the explicit form of Gq1(x — x +y) in the 4-dimensional space-time (corresponding to N = 3), we obtain the renormalized o-dependenl, energy-momentum tensor in the general case, for both Maxwell and Dirac fields ... 

We can now identify the first term in (A.4) with Maxwell s stress tensor, which acts on any dielectric in an electric field. The magnitude of this force Pe is given by... 

These concepts of 0(3) electrodynamics also completely resolve the problem that, in Maxwell-Heaviside electrodynamics, the energy momentum of radiation is defined through an integral over the conventional tensor and for this reason is not manifestly covariant. To make it so requires the use of special hypersurfaces as attempted, for example, by Fermi and Rohrlich [40]. The 0(3) energy momentum (78), in contrast, is generally covariant in 0(3) electrodynamics [11-20]. 

In this second technical appendix, it is shown that the Maxwell-Heaviside equations can be written in terms of a field 4-vector = (0, cB + iE) rather than as a tensor. Under Lorentz transformation, GM transforms as a 4-vector. This shows that the field in electromagnetic theory is not uniquely defined as a... 

The only common factor is that the charge-current 4-tensor transforms in the same way. The vector representation develops a time-like component under Lorentz transformation, while the tensor representation does not. However, the underlying equations in both cases are the Maxwell-Heaviside equations, which transform covariantly in both cases and obviously in the same way for both vector and tensor representations. 

To consider magnetic flux density components of IAIV, Q must have the units of weber and R, the scalar curvature, must have units of inverse square meters. In the flat spacetime limit, R 0, so it is clear that the non-Abelian part of the field tensor, Eq. (6), vanishes in special relativity. The complete field tensor F vanishes [1] in flat spacetime because the curvature tensor vanishes. These considerations refute the Maxwell-Heaviside theory, which is developed in flat spacetime, and show that 0(3) electrodynamics is a theory of conformally curved spacetime. Most generally, the Sachs theory is a closed field theory that, in principle, unifies all four fields gravitational, electromagnetic, weak, and strong. 

Similarly, the 4-current J depends directly on the curvature tensor / [1], and there can exist no 4-current in the Heaviside-Maxwell theory, so the... 

But the Minkowski spacetime R4 has trivial cohomology. This means that the Maxwell equation implies that. is a closed 2-form, so it is also an exact form and we can write. = d d, where ( is another potential 1-form in the Minkowski space. Now the dynamical equation becomes another Bianchi identity. This simple idea is a consequence of the electromagnetic duality, which is an exact symmetry in vacuum. In tensor components, with sJ = A dx and ((i = C(1dxt we have b iV = c, /tv — and b iV = SMCV - SvC or, in vector components... 

It follows immediately that both Fap (<[>) and Fap (0) obey the Maxwell equations in empty space. In fact, the first pair for both tensors... 

Property 1. In a theory based on the pair of fields (, 0) with action integral equal to (118), submitted to the duality constraint (119), both tensors Fap and Fap obey the Maxwell equations in empty space. As the duality constraint is naturally conserved in time, the same result is obtained if it is imposed just at t = 0. 

Property 2. If two scalar fields <[), 0 form an arbitrary pair of dual fields, in the sense of Eq. (15) [or, equivalently, if they verify (119)], the tensors Fap and F.jfi satisfy the Maxwell equations in empty space at any time. 

In utilizing a complex three-vector (self-dual tensor) rather than a real antisymmetric tensor to describe the electromagnetic field, Hillion and Quinnez discussed the equivalence between the 2-spinor field and the complex electromagnetic field [63]. Using a Hertz potential [64] instead of the standard 4-vector potential in this model, they derived an energy momentum tensor out of which Beltrami-type field relations emerged. This development proceeded from the Maxwell equations in free homogeneous isotropic space... 

In classical electrodynamics, the field equations for the Maxwell field A/( depend only on the antisymmetric tensor which is invariant under a gauge transformation A/l A/l + ticduxix), where x is an arbitrary scalar field in space-time. Thus the vector field A/( is not completely determined by the theory. It is customary to impose an auxiliary gauge condition, such as 9/x/Fx = 0, in order to simplify the field equations. In the presence of an externally determined electric current density 4-vector j11, the Maxwell Lagrangian density is... 

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