In this paper the Lorentz transformations (LT) and the standard transformations (ST) of the usual Maxwell equations (ME) with the three-dimensional (3D) vectors of the electric and magnetic fields, E and B, respectively, are examined using both the geometric algebra and tensor formalisms. Different 4D algebraic objects are used to represent the usual observer dependent and the new observer independent electric and magnetic fields. It is found that the ST of the ME differ from their LT and consequently that the ME with the 3D E and B are not covariant upon the LT but upon the ST. The obtained results do not depend on the character of the 4D algebraic objects used to represent the electric and magnetic fields. The Lorentz invariant field equations are presented with 1-vectors E and B, bivectors E Hv and B Hv and the abstract tensors, the 4-vectors Ea and Ba. All these quantities are defined without reference frames, i.e., as absolute quantities. When some basis has been introduced, they are represented as coordinate-based geometric quantities comprising both components and a basis. It is explicitly shown that this geometric approach agrees with experiments, e.g., the Faraday disk, in all relatively moving inertial frames of reference, which is not the case with the usual approach with the 3D bf E and B and their ST.
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References
T. Ivezić (2003) Found. Phys. 33 1339 Occurrence Handle2004k:78003
T. Ivezić, physics/0304085.
H. A. Lorentz, Proceedings of the Academy of Sciences of Amsterdam 6 (1904), W. Perrett and G. B. Jeffery, eds., in The Principle of Relativity (Dover, New York, 1952)
A. Einstein, Ann. Physik. 17, 891 (1905), tr. W. Perrett and G. B. Jeffery eds. in The Principle of Relativity (Dover, New York, 1952)
J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1977), 2nd edn.; L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, Oxford, 1979), 4th edn.;
C.W. Misner K.S. Thorne J.A. Wheeler (1970) Gravitation Freeman San Francisco
D. Hestenes, Space–Time Algebra (Gordon & Breach, New York, 1966); Space–Time Calculus; available at: http://modelingnts.la. asu.edu/evolution. html; New Foundations for Classical Mechanics (Kluwer Academic, Dordrecht, 1999), 2nd. edn.; Am. J Phys. 71, 691 (2003)
C. Doran A. Lasenby (2003) Geometric Algebra for Physicists Cambridge University Press Cambridge
B. Jancewicz (1989) Multivectors and Clifford Algebra in Electrodynamics World Scientific Singapore
T. Ivezić, physics/0305092.
T. Ivezić (2001) Found. Phys. 31 1139 Occurrence Handle2002m:83004
T. Ivezić (2002) Found. Phys. Lett. 15 27 Occurrence Handle2002m:83005
T. Ivezić, physics/0311043
M. Riesz, Clifford Numbers and Spinors, Lecture Series No. 38, The Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, MD (1958)
T. Ivezić, hep-th/0207250; hep-ph/0205277
A. Einstein, Ann. Physik 49, 769 (1916), translated by W. Perrett and G. B. Jeffery, in The Principle of Relativity (Dover, New York, 1952)
H.N. Núñez Yépez A.L. Salas Brito C.A. Vargas (1988) Revista Mexicana de Física 34 636
T. Matolcsi (1993) Spacetime without Reference Frames Akadémiai Kiadó Budapest
D. Hestenes G. Sobczyk (1984) Clifford Algebra to Geometric Calculus Reidel Dordrecht
T. Ivezić (2002) Ann. Fond. Louis de Broglie 27 287 Occurrence Handle2004b:83004
S. Esposito (1998) Found. Phys. 28 231 Occurrence Handle10.1023/A:1018752803368 Occurrence Handle99a:78013
T. Ivezić, Found. Phys. Lett. 12, 105 (1999); Found. Phys. Lett. 12, 507 (1999)
R.M. Wald (1984) General Relativity Chicago University Chicago
M. Ludvigsen, General Relativity, A Geometric Approach (Cambridge University, Cambridge, 1999). S. Sonego and M. A. Abramowicz, J. Math. Phys. 39, 3158 (1998). D. A. T. Vanzella, G. E. A. Matsas, H. W. Crater, Am. J. Phys. 64, 1075 (1996)
F. Rohrlich (1966) Nuovo Cimento B 45 76 Occurrence Handle0139.45503
A. Gamba (1967) Am. J. Phys. 35 83 Occurrence Handle10.1119/1.1973974
G. Saathoff et al., Phys. Rev. Lett. 91, 190403 (2003). H. Müller et al. Phys. Rev. Lett. 91, 020401 (2003). P. Wolf et al. Phys. Rev. Lett. 90, 060402 (2003)
W.K.H. Panofsky M. Phillips (1962) Classical electricity and magnetism EditionNumber2 Addison-Wesley Reading, MA
L. Nieves M. Rodriguez G. Spavieri E. Tonni (2001) Nuovo Cimento B 116 585 Occurrence Handle2001NCimB.116..585N
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Ivezić, T. 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. Found Phys 35, 1585–1615 (2005). https://doi.org/10.1007/s10701-005-6484-y
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DOI: https://doi.org/10.1007/s10701-005-6484-y