Skip to main content
SpringerLink
Log in

On the Appeal to a Pre-Established Harmony Between Pure Mathematics and Relativity Physics

  • Published:
Foundations of Physics Letters

Abstract

Soon after its appearance in 1905, the Einsteinian relativity with its relativistically admissible 3-velocities was recognized by Vladimir Varičak in 1908 as the realization in physics of the hyperbolic geometry of Bolyai and Lobachevski. At the same time, however, during the years 1907–1909 Minkowski reformulated the Einsteinian relativity in terms of a space of 4-velocities that now bears his name. As a result, the special theory of relativity that we find in the mainstream literature is not the one originally formulated by Einstein but, rather, the one reformulated by Minkowski. Thus, in particular, one of the most powerful ideas of Einstein in 1905, the Einstein addition of relativistically admissible 3-velocities that need not be parallel, is unheard of in most texts on relativity physics. Following our recently published book, Beyond the Einstein Addition, Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces [1], the aim of this article is to employ the principle of pre-established harmony between mathematics and physics to demonstrate that the original Einsteinian relativity, as opposed to the Minkowskian relativity, is the legitimate formulation of special relativity whose time has returned.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. A. A. Ungar, Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces (Kluwer Academic, Dordrecht, 2001).

    Book  MATH  Google Scholar 

  2. L. Pyenson, “Relativity in late Wilhelmian Germany: The appeal to a preestablished harmony between mathematics and physics,” Arch. Hist. Exact Sci., 27(2), 137-159 (1982).

    MathSciNet  MATH  Google Scholar 

  3. V. Varičak, “Beiträge zur nichteuklidischen geometrie [Contributions to non-Euclidean geometry],” Jber. dtsch. Mat. Ver., 17, 70-83 (1908).

    MATH  Google Scholar 

  4. V. Varičak, “Anwendung der Lobatschefskjschen Geometrie in der Relativtheorie,” Phys. Z. 11, 93-96 (1910).

    MATH  Google Scholar 

  5. V. Varičak, Darstellung der Relativitätstheorie im dreidimensionalen Lobatchefskijschen Raume [Presentation of the Theory of Relativity in the Three-Dimensional Lobachevskian Space]. (Zaklada, Zagreb, 1924).

    Google Scholar 

  6. J.-L. Chen and A. Ungar, “The Bloch gyrovector,” Found. Phys. 32(4) 531-565 (2002).

    Article  MathSciNet  Google Scholar 

  7. J.-L. Chen and A. A. Ungar, “Introducing the Einstein metric to quantum computation and quantum information geometry,” Found. Phys. Lett. 15(2) 189-197 (2002).

    Article  MathSciNet  Google Scholar 

  8. A. A. Ungar, “The hyperbolic geometric structure of the density matrix for mixed-state qubits,” Found. Phys. 32(11), 1-29 (2002).

    Article  MathSciNet  Google Scholar 

  9. C. Lanczos, Space Through the Ages. The Evolution of Geometrical ideas from Pythagoras to Hilbert and Einstein (Academic, London, 1970).

    MATH  Google Scholar 

  10. A. Einstein, “Zur Elektrodynamik Bewegter Körper [On the electrodynamics of moving bodies],” Ann. Phys. (Leipzig) 17, 891-921 (1905).

    Article  ADS  Google Scholar 

  11. A. Einstein, Einstein's Miraculous Years: Five Papers that Changed the Face of Physics (Princeton University Press, Princeton, 1998), edited and introduced by John Stachel. Includes bibliographical references, Einstein's dissertation on the determination of molecular dimensions, Einstein on Brownian motion, Einstein on the theory of relativity, Einstein's early work on the quantum hypothesis: a new English translation of Einstein's 1905 paper appears on pp. 123–160.

    Google Scholar 

  12. S. Walter, “The non-Euclidean style of Minkowskian relativity,” in The Symbolic Universe, Jeremy J. Gray, ed. (Oxford University Press, Oxford, 1999), pp. 91-127.

    Google Scholar 

  13. J. F. Barrett, “Special relativity and hyperbolic geometry,” in Proceedings, Physical Interpretations of Relativity Theory, London, UK 11–14 September 1998 (University of Sunderland, Sunderland, 1998).

    Google Scholar 

  14. L. Corry, “The influence of David Hilbert and Hermann Minkowski on Einstein's views over the interrelation between physics and mathematics,” Endeavor 22(3), 95-97 (1998).

    Article  MathSciNet  Google Scholar 

  15. H. Schmidt, Review of A. A. Ungar, Beyond the Einstein addition law and its gyroscopic Thomas precession, Zbl. Math. 0972.83002 (2001).

  16. G. Gibbons, Review of A. A. Ungar, Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession, The Observatory 121, 394 (2001).

    Google Scholar 

  17. V. Fock, The Theory of Space, Time and Gravitation (Macmillan, New York, 1964), 2nd revised edition, translated from the Russian by N. Kemmer.

    MATH  Google Scholar 

  18. H. Bacry, Lectures on Group Theory and Particle Theory (Gordon & Breach, New York, 1977), documents on modern physics.

    MATH  Google Scholar 

  19. R. U. Sexl and H. K. Urbantke, Relativity, Groups, Particles (Springer, Vienna, 2001) (special relativity and relativistic symmetry in field and particle physics), revised and translated from the 3rd German (1992) edn. by H. K. Urbantke.

    Book  MATH  Google Scholar 

  20. S. Walter, Review of A. A. Ungar, Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession, Found. Phys. Lett. 32(2), 327-330 (2002).

    Google Scholar 

  21. C. B. van Wyk, “Lorentz transformations in terms of initial and final vectors,” J. Math. Phys. 27(5), 1311-1314 (1986).

    Article  ADS  MathSciNet  Google Scholar 

  22. F. R. Halpern, Special Relativity and Quantum Mechanics (Prentice-Hall, Englewood Cliffs, 1968).

    MATH  Google Scholar 

  23. A. A. Ungar, “Applications of hyperbolic geometry in relativity physics,” in A. Prekopa, E. Kiss, Gy. Staar and J. Szenthe, eds., Janos Bolyai Memorial Volume (Vince, Budapest, 2002).

    Google Scholar 

  24. H. Goldstein, Classical Mechanics. 2nd edn. (Addison-Wesley, Reading, MA, 1980).

    MATH  Google Scholar 

  25. A. A. Ungar, “Thomas rotation and the parametrization of the Lorentz transformation group,” Found. Phys. Lett. 1(1), 57-89 (1988).

    Article  MathSciNet  Google Scholar 

  26. L. Silberstein, The Theory of Relativity (MacMillan, London, 1914).

    MATH  Google Scholar 

  27. E. Cunningham, The Principle of Relativity (University Press, Cambridge, 1914).

    MATH  Google Scholar 

  28. M. v. Laue, Das Relativitatsprinzip (Vieweg, Braunschweig, 1911).

    MATH  Google Scholar 

  29. M. B. Weinstein, Die Physik der bewegten Materie und die Relativitatstheorie (Barth, Leipzig, 1913).

    MATH  Google Scholar 

  30. M. Rivas, “Classical relativistic spinning particles,” J. Math. Phys. 30(2), 318-329 (1989).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. E. Wigner, “On unitary representations of the inhomogeneous lorentz group,” Ann. Math. 40, 149-204 (1939).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. D. Vassiliadis, “Wigner's little group and decomposition of Lorentz transformations,” J. Math. Phys. 30(9), 2177-2180 (1989).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Y. S. Kim and M. E. Noz, Theory and Applications of the Poincaré Group (Reidel, Dordrecht, 1986).

    Book  Google Scholar 

  34. M. H. L. Pryce, “The mass-centre in the restricted theory of relativity and its connexion with the quantum theory of elementary particles,” Proc. Roy. Soc. London. Ser. A. 195, 62-81 (1948).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. J. L. Synge, Relativity: The Special Theory (North-Holland, Amsterdam, 1956).

    MATH  Google Scholar 

  36. W. Rindler, Essential Relativity: Special, General, and Cosmological (Springer, New York, 1977).

    Book  MATH  Google Scholar 

  37. W. Rindler, Introduction to Special Relativity (Clarendon Oxford, 1982).

  38. D. Hestenes, New Foundations for Classical Mechanics, 2nd edn. (Kluwer Academic, Dordrecht, 1999).

    MATH  Google Scholar 

  39. M. Hausner, A Vector Space Approach to Geometry (Dover, New York, 1998), reprint of the 1965 original.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, North Dakota State University, Fargo, North Dakota, 58105

    Abraham A. Ungar

Authors
  1. Abraham A. Ungar
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ungar, A.A. On the Appeal to a Pre-Established Harmony Between Pure Mathematics and Relativity Physics. Found Phys Lett 16, 1–23 (2003). https://doi.org/10.1023/A:1024136905975

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1024136905975

Search

Navigation