Título
Helmholtz Theorems, Gauge Transformations, General Covariance and the Empirical Meaning of Gauge Conditions
Autor
Andrew Chubykalo
Nivel de Acceso
Acceso Abierto
Materias
Resumen o descripción
It is well known that the use of Helmholtz decomposition theorem for static vector fields C : R3 → R3 ,
when applied to the time dependent vector fields E : R4 → R3 , B : R4 → R3 which represent the
electromagnetic field, allows us to obtain instantaneous-like solutions all along R3 . 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 spacetime,
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.
Producción Científica de la Universidad Autónoma de Zacatecas UAZ
Fecha de publicación
mayo de 2016
Tipo de publicación
Artículo
Recurso de información
Formato
application/pdf
Idioma
Inglés
Audiencia
Público en general
Repositorio Orígen
Repositorio Institucional Caxcán
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