dorsal/arxiv
View SchemaThe Helmholtz Theorem and Superluminal Signals
| Authors | V. P. Oleinik |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0311124 |
| URL | https://arxiv.org/abs/quant-ph/0311124 |
Abstract
The conventional decomposition of a vector field into longitudinal (potential) and transverse (vortex) components (Helmholtz's theorem) is claimed in [1] to be inapplicable to the time-dependent vector fields and, in particular, to the retarded solutions of Maxwell's equations. Because of this, according to [1], a number of conclusions drawn in [2] on the basis of the Helmholtz theorem turns out to be erroneous. The Helmholtz theorem is proved in this letter to hold for arbitrary vector field, both static and time-dependent. Therefore, the conclusions of the paper [2] questioned in [1] are true. The validity of Helmholtz's theorem in the general case is due to the fact that the decomposition above of vector field does not influence the field time coordinate, which plays, thus, a passive role in the decomposition procedure. An analysis is given of the mistakes made in [1]. It is noted that for point particle the longitudinal and transverse components of electric field, taken separately, are characterized by the infinitely great velocity of propagation. However, superluminal contributions to the expression for the total electric field cancel each other.
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"abstract": "The conventional decomposition of a vector field into longitudinal\n(potential) and transverse (vortex) components (Helmholtz\u0027s theorem) is claimed\nin [1] to be inapplicable to the time-dependent vector fields and, in\nparticular, to the retarded solutions of Maxwell\u0027s equations. Because of this,\naccording to [1], a number of conclusions drawn in [2] on the basis of the\nHelmholtz theorem turns out to be erroneous. The Helmholtz theorem is proved in\nthis letter to hold for arbitrary vector field, both static and time-dependent.\nTherefore, the conclusions of the paper [2] questioned in [1] are true. The\nvalidity of Helmholtz\u0027s theorem in the general case is due to the fact that the\ndecomposition above of vector field does not influence the field time\ncoordinate, which plays, thus, a passive role in the decomposition procedure.\nAn analysis is given of the mistakes made in [1]. It is noted that for point\nparticle the longitudinal and transverse components of electric field, taken\nseparately, are characterized by the infinitely great velocity of propagation.\nHowever, superluminal contributions to the expression for the total electric\nfield cancel each other.",
"arxiv_id": "quant-ph/0311124",
"authors": [
"V. P. Oleinik"
],
"categories": [
"quant-ph"
],
"title": "The Helmholtz Theorem and Superluminal Signals",
"url": "https://arxiv.org/abs/quant-ph/0311124"
},
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