Advertisement

Foundations of Physics

, Volume 36, Issue 10, pp 1511–1534 | Cite as

Four-Dimensional Geometric Quantities versus the Usual Three-Dimensional Quantities: The Resolution of Jackson’s Paradox

  • Tomislav IvezićEmail author
Article

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 Ns, Nt and Ms, Mt, respectively, are given. The torques Ns, Nt (the angular momentums Ms, Mt), taken together, contain the same physical information as the bivector N (the bivector M). The usual approaches that deal with the 3D quantities \(\varvec{E,\,B,\,F,\,L,\,N}\) 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 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.

Keywords

4D geometric quantities 3D quantities Jackson’s paradox 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Jackson J.D. (1977). Classical Electrodynamics, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  2. 2.
    Jackson J.D. (2004). Am. J. Phys. 72: 1484CrossRefADSGoogle Scholar
  3. 3.
    Jefimenko O.D. (1997). Retardation and Relativity. Electret Scientific, Star City, WVGoogle Scholar
  4. 4.
    Jefimenko O.D. (1999). J. Phys. A: Math. Gen 32: 3755zbMATHMathSciNetCrossRefADSGoogle Scholar
  5. 5.
    Ivezić T. (1999). Found Phys. Lett. 12: 507MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ivezić T. (2001). Found. Phys. 31:1139MathSciNetCrossRefGoogle Scholar
  7. 7.
    T. Ivezić, Found. Phys. Lett. 15, 27 (2002); physics/0103026; physics/0101091.Google Scholar
  8. 8.
    Ivezić T. (2003). Found. Phys. 33:1339MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ivezić T., hep-th/0207250; hep-ph/0205277.Google Scholar
  10. 10.
    Ivezić T. (2005). Found. Phys. Lett. 18:301CrossRefGoogle Scholar
  11. 11.
    Ivezić T. (2005). Found. Phys. 35:1585MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ivezić T. (2005). Found. Phys. Lett. 18: 401MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ivezić T., physics/0505013.Google Scholar
  14. 14.
    Ivezić T. (1999). Found. Phys. Lett. 12:105MathSciNetCrossRefGoogle Scholar
  15. 15.
    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, 2nd edn. (Kluwer Academic, Dordrecht, 1999); Am. J Phys. 71, 691 (2003).Google Scholar
  16. 16.
    Doran C., Lasenby A. (2003). Geometric algebra for physicists. Cambridge University, CambridgezbMATHGoogle Scholar
  17. 17.
    A. Einstein, Ann. Physik 17, 891 (1905), translated by W. Perrett and G. B. Jeffery, The Principle of Relativity (Dover, New York, 1952).Google Scholar
  18. 18.
    Rohrlich F. (1966). Nuovo Cimento B 45: 76zbMATHGoogle Scholar
  19. 19.
    Gamba A. (1967). Am. J. Phys. 35: 83CrossRefGoogle Scholar
  20. 20.
    R. M. Wald, General Relativity (Chicago University, Chicago, 1984); 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).Google Scholar
  21. 21.
    Hestenes D., Sobczyk G. (1984). Clifford Algebra to Geometric Calculus. Reidel, DordrechtzbMATHGoogle Scholar
  22. 22.
    A. Einstein, Ann. Physik 49, 769 (1916), translatted by W. Perrett and G. B. Jeffery, The Principle of Relativity (Dover, New York, 1952).Google Scholar
  23. 23.
    Rosser W.G.W. (1968). Classical Electromagnetism via Relativity. Plenum, New YorkGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Ruđer Bošković InstituteZagrebCroatia

Personalised recommendations