The 4D geometric quantities versus the usual 3D quantities. The resolution of Jackson's paradox

Abstract

In this paper we present definitions of different four-dimensional (4D) geometric quantities (Clifford multivectors). New decompositions of the torque N and the angular momentum M (bivectors) into 1-vectors N_{s}, N_{t} and M_{s}, M_{t} respectively are given. The torques N_{s}, N_{t} (the angular momentums M_{s}, M_{t}), taken together, contain the same physical information as the bivector N (the bivector M). The usual approaches that deal with the 3D quantities $\mathbf{E}$, $\mathbf{B}$, $\mathbf{F}$, $\mathbf{L}$, $\mathbf{N}$, etc. and their transformations are objected from the viewpoint of the invariant special relativity (ISR). In the ISR it is considered that 4D geometric quantities are well-defined both theoretically and \emph{experimentally} in the 4D spacetime. This is not the case with the usual 3D quantities. It is shown that there is no apparent electrodynamic paradox with the torque, and that the principle of relativity is naturally satisfied, when the 4D geometric quantities are used instead of the 3D quantities.

13 pages, revtex

References in corpus (3)

• The Proof that the Standard Transformations of E and B are not the Lorentz Transformations. Clifford Algebra Formalism
• The Proof that Maxwell Equations with the 3D E and B are not Covariant upon the Lorentz Transformations but upon the Standard Transformations. The New Lorentz Invariant Field Equations
• Axiomatic geometric formulation of electromagnetism with only one axiom: the field equation for the bivector field F with an explanation of the Trouton-Noble experiment

Cited by in corpus (7)

• Lorentz Transformations of the Electric and Magnetic Fields According to Minkowski
• Trouton-Noble paradox revisited
• Generalized Uhlenbeck-Goudsmit hypothesis 'Magnetic' S^{a} and 'Electric' Z^{a} Spins
• The Lorentz transformations of the vectors E, B, P, M and the external electric fields from a stationary superconducting wire with a steady current and from a stationary permanent magnet
• Electric and magnetic fields as explicitly observer dependent four-dimensional vectors and their Lorentz transformations according to Minkowski-Ivezić
• The manifestly covariant Aharonov-Bohm effect in terms of the 4D fields
• Jackson's paradox and its resolution by the four-dimensional geometric quantities