Abstract
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.
Highlights
Starting directly from Maxwell equations for the pair of time dependent vector fields that represent the electromagnetic field, E and B, it is possible to deduce a pair of second order, inhomogeneous, partial differential equations of apparently hyperbolic type given by: ∆ − 1 c2 ∂2 ∂t 2 E ( x, t )= 4π∇ + 4π c2
We have find little role for the gauge transformations, and specially for the doctrine that all gauges are equivalent
We say that the gauge condition presents “local empirical irrelevance” because at the level of direct human experience it is irrelevant the global space-time form
Summary
Starting directly from Maxwell equations for the pair of time dependent vector fields that represent the electromagnetic field, E and B , it is possible to deduce a pair of second order, inhomogeneous, partial differential equations of apparently hyperbolic type given by:. We shall prove that each the Helmholtz decomposition of the electromagnetic potentials is equivalent to a gauge transformation, and because the solutions of the Maxwell equations for the potentials at each different gauge give rise to the corresponding solutions for the field strengths, we can understand that the Helmholtz decomposition that relates retarded and instantaneous solutions is a gauge transformation. Because the Helmholtz decomposition is related to a gauge in a unique way the space-time form is related to the choice of gauge If this is so, we conclude that the Maxwell electromagnetic field equations, its theoretical and operational concepts and the whole of its experimental confirmations are not enough to make the theory complete, and for this reason, and this reason alone, the instantaneous and the retarded solutions are possible solutions of the field equations.
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