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The Einstein Relativistic Velocity Model of Hyperbolic Geometry and Its Plane Separation Axiom

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Abstract

The relativistically admissible velocities of Einstein’s special theory of relativity are regulated by the Beltrami–Klein ball model of the hyperbolic geometry of Bolyai and Lobachevsky. It is shown in this expository article that the Einstein velocity addition law of relativistically admissible velocities enables Cartesian coordinates to be introduced into hyperbolic geometry, resulting in the Cartesian–Beltrami-Klein ball model of hyperbolic geometry. Suggestively, the latter is increasingly becoming known as the Einstein Relativistic Velocity Model of hyperbolic geometry. Möbius addition is a transformation of the ball linked to Clifford algebra. Einstein addition and Möbius addition in the ball of the Euclidean n-space are isomorphic to each other, and they share remarkable analogies with vector addition. Thus, in particular, Einstein (Möbius) addition admits scalar multiplication, giving rise to gyrovector spaces, just as vector addition admits scalar multiplication, giving rise to vector spaces. Moreover, the resulting Einstein (Möbius) gyrovector spaces form the algebraic setting for the Beltrami-Klein (Poincaré) ball model of n-dimensional hyperbolic geometry, just as vector spaces form the algebraic setting for the standard Cartesian model of n-dimensional Euclidean geometry. As an illustrative novel example special attention is paid to the study of the plane separation axiom (PSA) in Euclidean and hyperbolic geometry.

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References

  1. Lars V.: Ahlfors, Old and new in Möbius groups. Ann. Acad. Sci. Fenn. Ser. A I Math. 9, 93–105 (1984)

    MathSciNet  MATH  Google Scholar 

  2. Cătăalin Barbu, Menelaus’s theorem for hyperbolic quadrilaterals in the Einstein relativistic velocity model of hyperbolic geometry. Sci. Magna 6 (1) (2010), 19–24.

  3. Cătăalin Barbu, Smarandache's pedal polygon theorem in the Poincar’e disc model of hyperbolic geometry. Int. J. Math. Comb. 1 (2010), 99–102.

  4. Cătăalin Barbu: Trigonometric proof of Steiner-Lehmus theorem in hyperbolic geometry. Acta Univ. Apulensis Math. Inform. 23, 63–67 (2010)

    Google Scholar 

  5. Karol Borsuk, Multidimensional analytic geometry. Translated from the Polish by Halina Spali´nska. Monografie Matematyczne, Tom 50. Polish Scientific Publishers, Warsaw, 1969.

  6. Michael A.: Carchidi, Generating exotic-looking vector spaces. College Math. J. 29(4), 304–308 (1998)

    Article  Google Scholar 

  7. Oğguzhan Demirel, The theorems of Stewart and Steiner in the Poincaré disc model of hyperbolic geometry. Comment. Math. Univ. Carolin. 50 (3) (2009), 359–371.

  8. Oğguzhan Demirel, Emine Soytürk: The hyperbolic Carnot theorem in the poincaré disc model of hyperbolic geometry. Novi Sad J.Math. 38(2), 33–39 (2008)

    MathSciNet  Google Scholar 

  9. Albert Einstein, The collected papers of Albert Einstein. Vol. 2. Princeton University Press, Princeton, NJ, 1989. The Swiss years: writings, 1900–1909, Edited by John Stachel, Translations from the German by Anna Beck.

  10. Tomás Feder: Strong near subgroups and left gyrogroups. J. Algebra 259(1), 177–190 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Milton Ferreira: Factorizations of Möbius gyrogroups. Adv. Appl. Clifford Algebra 19(2), 303–323 (2009)

    Article  Google Scholar 

  12. Milton Ferreira: Spherical continuous wavelet transforms arising from sections of the Lorentz group. Appl. Comput. Harmon. Anal. 26(2), 212–229 (2009)

    Article  MathSciNet  Google Scholar 

  13. Milton Ferreira, Ren G.: Möbius gyrogroups: a Clifford algebra approach. J. Algebra 328(1), 230–253 (2011)

    Article  MathSciNet  Google Scholar 

  14. V. Fock, The theory of space, time and gravitation. The Macmillan Co., New York, 1964. Second revised edition. Translated from the Russian by N. Kemmer. A Pergamon Press Book.

  15. Tuval Foguel and Abraham A. Ungar, Involutory decomposition of groups into twisted subgroups and subgroups. J. Group Theory 3 (1) (2000), 27–46.

  16. Tuval Foguel and Abraham A. Ungar, Gyrogroups and the decomposition of groups into twisted subgroups and subgroups. Pac. J. Math. 197 (1) 2001, 1–11.

  17. Nourou Issa A.: Gyrogroups and homogeneous loops. Rep. Math. Phys. 44(3), 345–358 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nourou Issa A.: Left distributive quasigroups and gyrogroups. J. Math. Sci. Univ. Tokyo 8(1), 1–16 (2001)

    MathSciNet  MATH  Google Scholar 

  19. Azniv K. Kasparian and Abraham A. Ungar, Lie gyrovector spaces. J. Geom. Symm. Phys. 1 (1) (2004), 3–53.

  20. Michihiko Kikkawa: Geometry of homogeneous Lie loops. Hiroshima Math. J. 5(2), 141–179 (1975)

    MathSciNet  MATH  Google Scholar 

  21. Michihiko Kikkawa, Geometry of homogeneous left Lie loops and tangent Lie triple algebras. Mem. Fac. Sci. Eng. Shimane Univ. Ser. B Math. Sci. 32 (1999), 57–68.

  22. Eugene Kuznetsov: Gyrogroups and left gyrogroups as transversals of a special kind. Algebra Discrete Math. 3, 54–81 (2003)

    Google Scholar 

  23. Richard S. Millman and George D. Parker, Geometry. Springer-Verlag, New York, second edition, 1991. A metric approach with models.

  24. C. Møller, The theory of relativity. Oxford, at the Clarendon Press, 1952.

  25. Róbert Oláh-Gál and József Sándor, On trigonometric proofs of the Steiner- Lehmus theorem. Forum Geom. 9 (2009), 155–160.

  26. Laurian-Ioan Pişcoran and Căatăalin Barbu, Pappus’s harmonic theorem in the Einstein relativistic velocity model of hyperbolic geometry. Stud. Univ. Babeş- Bolyai Math. 56 (1) (2011), 101–107.

  27. Roman U. Sexl and Helmuth K. Urbantke, Relativity, groups, particles. Springer Physics. Springer-Verlag, Vienna, 2001. Special relativity and relativistic symmetry in field and particle physics, Revised and translated from the third German (1992) edition by Urbantke.

  28. Abraham A.: Ungar, Thomas rotation and the parametrization of the Lorentz transformation group. Found. Phys. Lett. 1(1), 57–89 (1988)

    Article  MathSciNet  Google Scholar 

  29. Abraham A. Ungar, Beyond the Einstein addition law and its gyroscopic Thomas precession: The theory of gyrogroups and gyrovector spaces. Volume 117 of Fundamental Theories of Physics. Kluwer Academic Publishers Group, Dordrecht, 2001.

  30. Abraham A. Ungar, Analytic hyperbolic geometry: Mathematical foundations and applications. World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ, 2005.

  31. Abraham A.: Ungar, Gyrovector spaces and their differential geometry. Nonlinear Funct. Anal. Appl. 10(5), 791–834 (2005)

    MathSciNet  MATH  Google Scholar 

  32. Abraham A. Ungar, Analytic hyperbolic geometry and Albert Einstein's special theory of relativity. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.

  33. Abraham A.: Ungar, From Möbius to gyrogroups. Amer.Math. Monthly 115(2), 138–144 (2008)

    MathSciNet  MATH  Google Scholar 

  34. Abraham A. Ungar, A gyrovector space approach to hyperbolic geometry. Morgan & Claypool Pub., San Rafael, California, 2009.

  35. Abraham A.: Ungar, Hyperbolic barycentric coordinates. Aust. J. Math. Anal. Appl. 6(1), 1–35 (2009)

    MathSciNet  Google Scholar 

  36. Abraham A. Ungar, The hyperbolic triangle incenter. Dedicated to the 30th anniversary of Th.M. Rassias' stability theorem. Nonlinear Funct. Anal. Appl. 14 (5) (2009), 817–841.

  37. Abraham A. Ungar, Barycentric calculus in Euclidean and hyperbolic geometry: A comparative introduction. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010.

  38. Abraham A. Ungar, Hyperbolic triangle centers: The special relativistic approach. Springer-Verlag, New York, 2010.

  39. Abraham A.: Ungar, When relativistic mass meets hyperbolic geometry. Commun. Math. Anal. 10(1), 30–56 (2011)

    MathSciNet  MATH  Google Scholar 

  40. Kehe Zhu, Spaces of holomorphic functions in the unit ball. Volume 226 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2005.

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Authors and Affiliations

  1. Department of Mathematics, ANS Campus, Faculty of Science and Arts, Afyon Kocatepe University, 03200, Afyonkarahisar, Turkey

    Nilgün Sönmez

  2. Department of Mathematics, North Dakota State University, Fargo, ND, 58105, USA

    A.A. Ungar

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  1. Nilgün Sönmez
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  2. A.A. Ungar
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Correspondence to Nilgün Sönmez.

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Sönmez, N., Ungar, A. The Einstein Relativistic Velocity Model of Hyperbolic Geometry and Its Plane Separation Axiom. Adv. Appl. Clifford Algebras 23, 209–236 (2013). https://doi.org/10.1007/s00006-012-0367-z

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  • DOI: https://doi.org/10.1007/s00006-012-0367-z

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