volume | PIER Journals

Abstract

The Lorentz transformation group of the special theory of relativity is commonly represented in terms of observer′s, or coordinate, time and coordinate relative velocities. The aim of this article is to uncover the representation of the Lorentz transformation group in terms of traveler′s, or proper, time and proper relative velocities. Following a recent demonstration by M. Idemen, according to which the Lorentz transformation group is inherent in Maxwell equations, our proper velocity Lorentz transformation group may pave the way to uncover the proper time Maxwell equation

1. Hlavat′y, V., "Proper time, apparent time, and formal time in the twin paradox," J. Math. Mech., Vol. 9, 733-744, 1960.

2. Guven, J., "Classical and quantum mechanics of a relativistic system parametrized by proper time," Phys. Rev. D (3), Vol. 44, No. 10, 3360-3363, 1991.

3. Frisch, D. and J. Smith, "Measurement of the relativistic time dilation using μ-mesons," Amer. J. Phys., Vol. 31, No. 5, 342-355, 1963.
doi:10.1119/1.1969508

4. Idemen, M., "Derivation of the Lorentz transformation from the Maxwell equations," J. Electromagn. Waves Appl., Vol. 19, No. 4, 451-467, 2005.
doi:10.1163/1569393053303884

5. Sexl, R. U. and H. K. Urbantke, "Special relativity and relativistic symmetry in field and particle physics," Relativity, 2001.

6. Ungar, A. A., "Beyond the Einstein addition law and its gyroscopic Thomas precession: The theory of gyrogroups and gyrovector spaces," Fundamental Theories of Physics, Vol. 117, 2001.

7. Ungar, A. A., Analytic Hyperbolic Geometry: Mathematical Foundations and Applications, World Scientific, 2005.

8. Ungar, A. A., "Einstein′s special relativity: Unleashing the power of its hyperbolic geometry," Comput. Math. Appl., Vol. 49, 187-221, 2005.
doi:10.1016/j.camwa.2004.10.030

9. Woodhouse, N. M. J., Special Relativity, Springer Undergraduate Mathematics Series, 2003.

10. Sartori, L., Understanding Relativity: A Simplified Approach to Einstein′s Theories, University of California Press, 1996.

11. Schott, B. A., "On the motion of the Lorentz electron," Phil. Mag., Vol. 29, 49-62, 1915.

12. Chen, J.-L. and A. A. Ungar, "From the group sl(2, C)," Found. Phys., Vol. 31, No. 11, 1611-1639, 2001.
doi:10.1023/A:1012694816323

13. Ungar, A. A., "Gyrovector spaces and their differential geometry," Non-linear Funct. Anal. Appl., Vol. 10, No. 5, 791-834, 2005.

14. Vermeer, J., "A geometric interpretation of Ungar′s addition and of gyration in the hyperbolic plane," Topology Appl., Vol. 152, No. 3, 226-242, 2005.
doi:10.1016/j.topol.2004.10.012

15. Jackson, J. D., Classical Electrodynamics, second ed., 1975.

Dr. Abraham Ungar