The Intrinsic Beauty, Harmony and Interdisciplinarity in Einstein Velocity Addition Law: Gyrogroups and Gyrovector Spaces

Document Type : Special Issue: AIMC 51


North Dakota State University


The only justification for the Einstein velocity addition law ‎appeared to be its empirical adequacy‎, ‎so that the ‎intrinsic beauty and harmony in Einstein addition remained for a long time ‎a mystery to be conquered‎. ‎Accordingly‎, ‎the aim of this expository article is to present ‎(i) the Einstein relativistic vector addition‎, ‎(ii) the resulting Einstein scalar multiplication‎, ‎(iii) the Einstein relativistic mass‎, ‎and ‎(iv) the Einstein relativistic kinetic energy‎, ‎along with remarkable analogies with classical results in ‎groups and vector spaces that these ‎Einstein concepts capture in gyrogroups and gyrovector spaces‎. ‎Making the unfamiliar familiar‎, ‎these analogies uncover the ‎intrinsic beauty and harmony in the underlying ‎Einstein velocity addition law of relativistically admissible velocities‎, ‎as well as its interdisciplinarity‎.


Main Subjects

1. ‎‎T‎. ‎Abe‎, ‎Gyrometric preserving maps on Einstein gyrogroups‎, ‎Mobius‎ gyrogroups and proper velocity gyrogroups‎, ‎Nonlinear Funct‎. ‎Anal‎. ‎Appl. 19 (1) (2014) 1-17‎.

‎L‎. ‎V‎. ‎Ahlfors‎, Conformal Invariants‎: ‎Topics in Geometric Function Theory‎, ‎McGraw-Hill Series in Higher Mathematics‎, ‎McGraw-Hill Book Co.‎, ‎New York‎, ‎1973‎.

‎H‎. ‎Bacry‎, ‎Lectures on Group Theory and Particle Theory‎, ‎Documents on modern physics‎, ‎Gordon and Breach Science Publishers‎, ‎New York‎, ‎1977‎.

‎C‎. ‎Barbu‎, ‎Menelaus's theorem for hyperbolic quadrilaterals in the Einstein relativistic velocity model of hyperbolic geometry‎, ‎Sci‎. ‎Magna 6 (1) (2010) 19-24‎.

‎C‎. ‎Barbu‎, ‎Smarandache's pedal polygon theorem in the Poincare disc model of hyperbolic geometry‎, ‎Int‎. ‎J‎. Math‎. ‎Comb. 1 (2010) 99-102‎.
6. ‎‎‎C‎. ‎Barbu‎, ‎Trigonometric proof of {S}teiner-Lehmus theorem in hyperbolic geometry‎, ‎Acta Univ‎. ‎Apulensis Math‎. ‎Inform. (23) (2010) 63-67‎.

‎C‎. ‎Barbu‎, ‎The hyperbolic Stewart theorem in the Einstein relativistic velocity model of hyperbolic geometry‎, An‎. ‎Univ‎. ‎Oradea Fasc‎. ‎Mat. 18 (1) (2011) 133-138‎.

8. ‎W‎. ‎Barker‎, ‎R‎. ‎Howe‎, ‎Continuous Symmetry from Euclid to Klein, ‎American Mathematical Society‎, ‎Providence‎, RI‎, ‎2007‎.
9. A‎. ‎S‎. ‎Basarab‎, ‎K-loops‎, ‎Izv‎. ‎Akad‎. ‎Nauk Respub‎. ‎Moldova Mat. (1) (1992) 28-33‎, ‎90-91‎.

‎‎E‎. ‎Borel‎, ‎Introduction Geometrique a Quelques Theories Physiques‎, ‎Gauthier-Villars‎, ‎Paris‎, ‎1914‎.

‎F‎. ‎Chatelin‎, ‎Polymorphic information processing in weaving computation‎: ‎An approach through cloth geometry‎, ‎Cerfacs Tech. Rep. TR/PA/11/27 (2011) 67-79‎.

‎H‎. ‎S‎. ‎M‎. ‎Coxeter‎, Regular Polytopes‎, ‎Dover Publications Inc.‎, ‎New York‎, ‎third edition‎, ‎1973‎.

‎H‎. ‎S‎. ‎M‎. ‎Coxeter‎, ‎S‎. ‎L‎. ‎Greitzer‎, ‎Geometry Revisited, ‎New Mathematical Library‎, ‎19‎. ‎Random House‎, ‎Inc.‎, New York‎, ‎1967‎.

‎M‎. ‎J‎. ‎Crowe‎, ‎A History of Vector Analysis‎: ‎The Evolution of the Idea of a Vectorial System‎, ‎Corrected reprint of the 1985 edition‎, ‎Dover Publications‎, ‎Inc.‎, ‎New York‎, ‎1994‎.

‎O‎. ‎Demirel‎, ‎The theorems of Stewart and Steiner in the Poincare disc model of hyperbolic geometry‎,
Comment‎. ‎Math‎. ‎Univ‎. ‎Carolin. 50 (3) (2009) 359-371‎.

‎O‎. ‎Demirel‎, ‎E‎. ‎Soyturk‎, ‎The hyperbolic Carnot theorem in the poincare disc model of hyperbolic geometry‎,
Novi Sad J‎. ‎Math. 38(2) (2008) 33-39‎.

‎A‎. ‎Einstein‎, ‎Zur Elektrodynamik Bewegter Korper‎, ‎
Ann‎. ‎Physik (Leipzig) 17 (1905) 891-921‎.

‎A. Einstein‎,
Einstein's Miraculous Years‎: ‎Five Papers that Changed the Face‎ of Physics, Princeton‎, ‎Princeton‎, ‎NJ‎, ‎1998‎.

‎T. Feder‎,
Strong near subgroups and left gyrogroups‎, J‎. ‎Algebra 259 (1) (2003) 177-190‎.

‎M. Ferreira‎,
Factorizations of Mobius gyrogroups‎, Adv‎. ‎Appl‎. ‎Clifford Algebr. 19 (2) (2009) 303-323‎.
21. ‎M. Ferreira‎, Gyrogroups in projective hyperbolic Clifford analysis‎, in‎: ‎ Hypercomplex Analysis and Applications‎, ‎Trends Math.‎, ‎pages‎ 61-80‎. ‎Birkhauser/Springer Basel AG‎, ‎Basel‎, ‎2011‎.

22. ‎M. Ferreira‎, ‎G. Ren‎,
Mobius gyrogroups‎: ‎a Clifford algebra approach‎, J‎. ‎Algebra 328 (1) (2011) 230-253‎.

23. ‎M. Ferreira‎, ‎F. Sommen‎,
Complex boosts‎: ‎a Hermitian Clifford algebra approach‎, Adv‎. ‎Appl‎. ‎Clifford Algebr. 23 (2) (2013) 339-362‎.

24. ‎V. Fock‎,
The Theory of Space‎, ‎Time and Gravitation, Second revised edition‎. ‎Translated from the Russian by N‎. ‎Kemmer‎. ‎A‎ Pergamon Press Book‎, The Macmillan Co.‎, ‎New York‎, ‎1964‎.

T. G‎. ‎Jaiyeola, ‎A. R. T‎. ‎Solarin‎, ‎J. O‎. Adeniran‎, Some Bol-Moufang characterization of the Thomas precession of a‎ gyrogroup‎, Algebras Groups Geom. 31 (3) (2014) 341-362‎.

26. H. Kiechle,
Theory of K-Loops, Springer-Verlag, Berlin, 2002.

27. S. Kim, Distributivity on the Gyrovector spaces,
Kyungpook Math. J. 55 (1) (2015) 13–20.

28. S. G. Krantz,
Complex Analysis: the Geometric Viewpoint, Mathematical Association of America, Washington, DC, 1990.

29. R. Lal, A. C. Yadav, Topological right Gyrogroups and Gyrotransversals,
Comm. Algebra 41 (9) (2013) 3559–3575.

30. P. Lévay, Thomas rotation and the mixed state geometric phase,
J. Phys. A 37 (16) (2004) 4593–4605.

31. ‎H. A‎. ‎Lorentz‎, ‎A. Einstein‎, ‎H. Minkowski‎, ‎H. Weyl‎,
The Principle of Relativity‎, With notes by A‎. ‎Sommerfeld‎, ‎ Translated by W‎. ‎Perrett and G‎. ‎B‎. Jeffery‎, ‎A collection of original memoirs on the special and general theory‎of relativity‎, Dover Publications Inc.‎, ‎New York‎, ‎N‎. ‎Y.‎, ‎undated‎.

32. J. E. Marsden,
Elementary Classical Analysis, With the assistance of Michael Buchner, Amy Erickson, Adam Hausknecht, Dennis Heifetz, Janet Macrae and William Wilson, and with contributions by Paul Chernoff, István Fáry and Robert Gulliver. W. H. Freeman and Co., San Francisco, 1974.

33. A. I. Miller,
Albert Einstein’s Special Theory of Relativity, Emergence (1905) and early interpretation (1905–11), Includes a translation by the author of Einstein’s “On the electrodynamics of moving bodies”, Reprint of the 1981 edition. Springer-Verlag, New York, 1998.

34. C. Møller,
The Theory of Relativity, Oxford, at the Clarendon Press, 1952.
35. D. Mumford, C. Series, D. Wright, Indra’s Pearls: The Vision of Felix Klein, Cambridge University Press, New York, 2002.

‎T. Needham‎,
Visual Complex Analysis, The Clarendon Press Oxford University Press‎, ‎New York‎, ‎1997‎.

‎R. Oláh-Gál‎, ‎J. Sandor‎,
On trigonometric proofs of the Steiner-Lehmus theorem‎, Forum Geom. 9 (2009) 155-160‎.

‎L.-I‎. ‎Piscoran‎, ‎C. Barbu‎,
Pappus's harmonic theorem in the Einstein relativistic velocity‎ model of hyperbolic geometry‎, Stud‎. ‎Univ‎. ‎Babec s-Bolyai Math. 56 (1) (2011) 101-107‎.

39. ‎T. M‎. ‎Rassias‎,
book review‎: ‎ Analytic Hyperbolic Geometry and Albert‎ Einstein's Special Theory of Relativity‎, ‎by Abraham A. Ungar‎, Nonlinear Funct‎. ‎Anal‎. ‎Appl. 13 (1) (2008) 167-177‎.

40. T. M. Rassias, book review:
A Gyrovector Space Approach to Hyperbolic Geometry, by Abraham A. Ungar, J. Geom. Symmetry Phys. 18 (2010) 93–106.

‎R. U‎. ‎Sexl‎, ‎H. K‎. ‎Urbantke‎,
Relativitat‎, ‎ Gruppen‎, ‎Teilchen, Spezielle Relativitatstheorie als Grundlage der Feld‎-und‎ Teilchenphysik‎,‎ Springer-Verlag‎, ‎Vienna‎, ‎New York‎, ‎third edition‎, ‎1992‎.

‎R. U‎. ‎Sexl‎, ‎H. K‎. ‎Urbantke‎,
Relativity‎, ‎Groups‎, ‎Particles‎, Special relativity and relativistic symmetry in field and particle‎ physics‎, ‎Revised and translated from the third German (1992) edition by‎ Urbantke‎, Springer Physics‎. ‎Springer-Verlag‎, ‎Vienna‎, ‎2001‎.

43. L. Silberstein,
The Theory of Relativity, MacMillan, London, 1914.

44. F. Smarandache, C. Barbu, The hyperbolic Menelaus theorem in the Poincaré disc model of hyperbolic geometry,
Ital. J. Pure Appl. Math. (30) (2013) 67– 72.

45. L. R. So˘ıkis, The special loops, in:
Questions of the Theory of Quasigroups and Loops, pages 122–131. Redakc.-Izdat. Otdel Akad. Nauk Moldav. SSR, Kishinev, 1970.

46. N. Sönmez, A. A. Ungar, The Einstein relativistic velocity model of hyperbolic geometry and its plane separation axiom,
Adv. Appl. Clifford Alg. 23 (2013) 209–236.

47. J. Stachel, History of relativity, in:
Twentieth Century Physics, Vol. I, L. M. Brown, A. Pais, B. Pippard (Eds.), pages 249–356, Published jointly by the Institute of Physics Publishing, Bristol, 1995.
48. ‎T. Suksumran‎, The algebra of gyrogroups‎: ‎Cayley's theorem‎, Lagrange's theorem‎ and isomorphism theorems‎, Essays in Mathematics and its Applications‎: ‎In Honor of‎ ‎Vladimir Arnold, ‎T‎. ‎M‎. ‎Rassias‎, ‎P‎. ‎M‎. ‎Pardalos (Eds.)‎, ‎Springer‎, ‎New York‎, ‎2016‎.
49. T. Suksumran, K. Wiboonton, Isomorphism theorems for gyrogroups and L-subgyrogroups,
J. Geom. Symmetry Phys. 37 (2015) 67–83.

50. T. Suksumran, K. Wiboonton, Lagrange’s theorem for gyrogroups and the cauchy property,
Quasigroups Related Systems 22 (2015) 283–294.

51. T. Suksumran, Gyrogroup actions: a generalization of group actions,
J. Algebra 454 (2016) 70–91.

52. ‎T. Suksumran‎, ‎A. A‎. ‎Ungar‎,
The Cauchy property‎: ‎from groups to gyrogroups‎, (2016)‎ preprint‎.

53. L. H. Thomas, The motion of the spinning electron,
Nature 117 (1926) 514.

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

55. A. A. Ungar, The Thomas rotation formalism underlying a nonassociative group structure for relativistic velocities,
Appl. Math. Lett. 1 (4) (1988) 403–405.

‎A. A‎. ‎Ungar‎,
The relativistic noncommutative nonassociative group of velocities‎ and the Thomas rotation‎, Resultate Math. 16 (1-2) (1989) 168-179‎.

57. A. A. Ungar, Weakly associative groups,
Resultate Math. 17 (1-2) (1990) 149–168.

58. A. A. Ungar, Thomas precession and its associated grouplike structure,
Amer. J. Phys. 59 (9) (1991) 824–834.

59. A. A. Ungar, Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics,
Found. Phys. 27 (6) (1997) 881–951.

60. A. 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.

61. A. A. Ungar, The density matrix for mixed state qubits and hyperbolic geometry,
Quantum Inf. Comput. 2 (6) (2002) 513–514.

62. A. A. Ungar, The hyperbolic geometric structure of the density matrix for mixed state qubits,
Found. Phys. 32 (11) (2002) 1671–1699.
63. A. A. Ungar, On the appeal to a pre-established harmony between pure mathematics and relativity physics, Found. Phys. Lett. 16 (1) (2003) 1–23.

64. A. A. Ungar,
Analytic Hyperbolic Geometry: Mathematical Foundations and Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.

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

‎A. A‎. ‎Ungar‎,
Newtonian and relativistic kinetic energy‎: ‎analogous consequences of‎ their conservation during elastic collisions‎, European J‎. ‎Phys. 27 (5) (2006) 1205-1212‎.

67. A. A. Ungar,
Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativity, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.

68. A. A. Ungar, On the origin of the dark matter/energy in the universe and the Pioneer anomaly,
Prog. Phys. 3 (2008) 24–29.

69. A. A. Ungar,
A Gyrovector Space Approach to Hyperbolic Geometry, Morgan & Claypool Pub., San Rafael, California, 2009.

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

71. A. A. Ungar,
Barycentric Calculus in Euclidean and Hyperbolic Geometry: A Comparative Introduction, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010.

72. A. A. Ungar,
Hyperbolic Triangle Centers: The Special Relativistic Approach, Springer-Verlag, New York, 2010.

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

74. A. A. Ungar, Gyrations: the missing link between classical mechanics with its underlying Euclidean geometry and relativistic mechanics with its underlying hyperbolic geometry, in:
Essays in Mathematics and its Applications in Honor of Stephen Smale’s 80th Birthday, P. M. Pardalos, T. M. Rassias (Eds.), pages 463–504, Springer, Heidelberg, 2012.

75. A. A. Ungar, An introduduction to hyperbolic barycentric coordinates and their applications, (2013) arXiv 1304.0205 (math-ph).

76. ‎A. A‎. ‎Ungar‎,
Hyperbolic geometry‎, in‎:  Geometry‎, ‎Integrability and Quantization XV, ‎pages‎ 259-282‎, Avangard Prima‎, ‎Sofia‎, ‎2014‎. 77. ‎A. A‎. ‎Ungar‎, An introduduction to hyperbolic barycentric coordinates and their‎ applications‎, in‎: ‎ Mathematics without Boundaries‎: ‎Surveys in‎ Interdisciplinary Research‎, ‎P‎. ‎M‎. ‎Pardalos‎, ‎T‎. ‎M‎. Rassias Eds.‎, ‎ Springer Optim‎. ‎Appl.‎, ‎pages 577-648‎, Springer‎, ‎New York‎, ‎2014‎.

‎A. A‎. ‎Ungar‎,
On the study of hyperbolic triangles and circles by hyperbolic‎ barycentric coordinates in relativistic hyperbolic geometry‎, in‎: ‎Mathematics without Boundaries‎: ‎ Surveys in Pure and‎ ‎Applied Mathematics‎, ‎T‎. ‎M‎. ‎Rassias‎, ‎P‎. ‎M‎. ‎Pardalos Eds.,‎, ‎pages 569-649‎, ‎Springer‎, ‎New York‎, ‎2014‎.

‎A. A‎. ‎Ungar‎,
Analytic Hyperbolic Geometry in n Dimensions‎: ‎An Introduction‎, CRC Press‎, ‎Boca Raton‎, ‎FL‎, ‎2015‎.

80. A. A. Ungar, Parametric realization of the Lorentz transformation group in pseudo-euclidean spaces,
J. Geom. Symmetry Phys. 38 (2015) 39–108.

‎D. K‎. ‎Urribarri‎, ‎S. M‎. ‎Castro‎, ‎S. R‎. ‎Martig‎,
Gyrolayout‎: ‎A hyperbolic level-of-detail tree layout‎, J‎. ‎Universal Comput. Sci. 19 (2013) 132-156‎.

‎S. Walter‎,
The non-Euclidean style of Minkowskian relativity‎, in‎: ‎ The Symbolic Universe‎: ‎Geometry and‎ Physics 1890-1930‎, ‎J. J‎. ‎Gray Ed., ‎pages 91-127‎, ‎Oxford Univ‎. ‎Press‎, ‎New York‎, ‎1999‎.

83. S. Walter, book review:
Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces, by Abraham A. Ungar, Found. Phys. 32(2) (2002) 327–330.

84. P. Yiu, The uses of homogeneous barycentric coordinates in plane Euclidean geometry,
Internat. J. Math. Ed. Sci. Tech. 31 (4) (2000) 569–578.