Helmholtz Theorems, Gauge Transformations, General Covariance and the Empirical Meaning of Gauge Conditions

It is well known that the use of Helmholtz decomposition theorem for static vector fields , when applied to the time dependent vector fields , which represent the electromagnetic field, allows us to obtain instantaneous-like solutions all along . For this reason, some people thought (see e.g. [1] and references therein) that the Helmholtz theorem cannot be applied to time dependent vector fields and some modification is wanted in order to get the retarded solutions. However, the use of the Helmholtz theorem for static vector fields is correct even for time dependent vector fields (see, e.g. [2]), so a relation between the solutions was required, in such a way that a retarded solution can be transformed in an instantaneous one, and conversely. On this paper we want to suggest, following most of the time the mathematical formalism of Woodside in [3], that: 1) there are many Helmholtz decompositions, all equally consistent, 2) each one is naturally related to a space-time structure, 3) when we use the Helmholtz decomposition for the electromagnetic potentials it is equivalent to a gauge transformation, 4) there is a natural methodological criterion for choosing the gauge according to the structure postulated for a global space-time, 5) the Helmholtz decomposition is the manifestation at the level of the fields that a gauge is involved. So, when we relate the retarded solution to the instantaneous one what we do is to change the gauge and the space-time. And, if the Helmholtz decompositions are related to a space-time structure, and are equivalent to gauge transformations, each gauge transformation is natural for a specific space-time. In this way, a Helmholtz decomposition for Euclidean space is equivalent to the Coulomb gauge and a Helmholtz decomposition for the Minkowski space is equivalent to the Lorenz gauge. This leads us to consider that the theories defined by different gauges may be mathematically equivalent, because they can be related by means of a gauge transformation, but they are not empirically equivalent, because they have quite different observational consequences due to the different space-time structure involved.

Gauge Invariance of Gravitodynamical Potentials in the Jefimenko’s Generalized Theory of Gravitation

In the Jefimenko’s generalized theory of gravitation, it is proposed the existence of certain potentials to help us to calculate the gravitational and cogravitational fields, such potentials are also presumed non-invariant under certain gauge transformations. In return, we propose that there is a way to perform the calculation of certain potentials that can be derived without using some kind of gauge transformation, and to achieve this we apply the Helmholtz’s theorem. This procedure leads to the conclusion that both gravitational and cogravitational fields propagate simultaneously in a delayed and in an instant manner. On the other hand, it is also concluded that these potentials thus obtained can be real physical quantities, unlike potentials obtained by Jefimenko, which are only used as a mathematical tool for calculating gravitational and cogravitational fields.

On the Question of the Invariance of the Light Speed

This article is devoted to the key concept of modern electrodynamics—the invariance of the speed of light. The general principle of relativity is considered in detail. Some critical remarks to the relativistic invariance and to the Lorentz transformations are presented. The general invariance of Maxwell equations is discussed. Different theoretical expectations for possible results of Michelson-Morley experiment and some physical consequences are considered. Some critical remarks to the notion of the light speed and its constancy are given. The relativistic law for velocity addition, including strangeness of a noncollinear addition and a superluminal motion, is discussed. Critical analysis of two works which proof the need for existence of an invariant velocity is consequentially made.

On the Jefimenko’s Non-Einsteinian Clocks and Synchronicity of Moving Clocks

In this work we analyze the concept of time dilation in its application to the rate of moving clocks. The rates of two equiform elementary electromagnetic clocks of different orientations relative to their direction of motion are computed on the basis of relativistic transformations of force and coordinates for the case when the clocks are at rest in a stationary reference frame and for the case when they are moving at constant speed relative to the stationary reference frame. It is shown that, although both clocks run slower when they are moving than when they are at rest, the rate of the moving clocks is affected by their orientation relative to their direction of motion, rather than by the kinematic (relativistic) time dilation as it is now generally assumed. The implication of this result for the experimental proofs of the existence of the kinematic the dilation is discussed.

On the Existence of Non-Intersecting Straight Lines on the Plane

In this brief note, we adduce the logical rationale that if at least one infinite straight line non-intersecting with the given straight line passes through a given point not lying on a given straight line, then it must be unique.

Conceptual Content of the Generalized Theory of Gravitation of Jefimenko

In this work, we make a brief exposition of the Jefimenko’s generalized theory of gravitation, describe its conceptual content, explain the mathematical apparatus used for the formulations of the theory and present the fundamental equations of the theory. We elucidate the main difference between Newton’s original theory of gravitation and the generalized theory of gravitation.