Foundations of Physics

, Volume 27, Issue 6, pp 881–951 | Cite as

Thomas precession: Its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics

  • Abraham A. Ungar


Gyrogroup theory and its applications is introduced and explored, exposing the fascinating interplay between Thomas precession of special relativity theory and hyperbolic geometry. The abstract Thomas precession, called Thomas gyration, gives rise to grouplike objects called gyrogroups [A, A. Ungar, Am. J. Phys.59, 824 (1991)] the underlying axions of which are presented. The prefix gyro extensively used in terms like gyrogroups, gyroassociative and gyrocommutative laws, gyroautomorphisms, and gyrosemidirect products, stems from their underlying abstract Thomas gyration. Thomas gyration is tailor made for hyperbolic geometry. In a similar way that commutative groups underlie vector spaces, gyrocommutative gyrogroups underlie gyrovector spaces. Gyrovector spaces, in turn, provide a most natural setting for hyperbolic geometry in full analogy with vector spaces that provide the setting for Euclidean geometry. As such, their applicability to relativistic physics and its spacetime geometry is obvious.


Binary Operation Euclidean Geometry Hyperbolic Geometry Admissible Velocity Full Analogy 
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  1. 1.
    Abraham A. Ungar, “Thomas rotation and the parametrization of the Lorentz transformation group.”Found Phys. Lett. 1, 57–89 (1988).CrossRefMathSciNetGoogle Scholar
  2. 2.
    Abraham A. Ungar, “The Thomas rotation formalism underlying a nonassociative group structure for relativistic velocities.”Appl. Math. Lett. 1, 403–405 (1988).CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Arlan Ramsay and Robert D. Richtmyer,Introduction to Hyperbolic Geometry (Springer, New York, 1995), p. 251.zbMATHGoogle Scholar
  4. 4.
    Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler,Gravitation, Box 2.4. pp. 67–68 (W. H. Freeman, San Francisco, 1973). See also Jean-Marc Levy-Leblond, “Additivity, rapidity, relativity”,Am. J. Phys. 47, 1045–1049 (1979); Isaac Moiseevich Yaglom,A Simple Non-Euclidean Geometry and Its Physical Basis: an Elementary Account of Galilean Geometry and the Galilean Principle of relativity, translated from the Russian by Abe Shenitzer with the editorial assistance of Basil Gordon (Springer, New York, 1979, and Arlan Ramsay and Robert D. Richtmyer,Introduction to Hyperbolic Geometry (Springer, New York, 1995).Google Scholar
  5. 5.
    Cornelius Lanczos,Space through the Ages. The Evolution of Geometrical Ideas from Pythagoras to Hilbert and Einstein (Academic Press, New York, 1970), p. 66.zbMATHGoogle Scholar
  6. 6.
    Abraham A. Ungar, “Axiomatic approach to the nonassociative group of relativistic velocities.”Found. Phys. Lett. 2, 199–203 (1989).CrossRefMathSciNetGoogle Scholar
  7. 7.
    Abraham A. Ungar, “The relativistic noncommutative nonassociative group of velocities and the Thomas rotation.”Res. Math. 16, 168–179 (1989).MathSciNetzbMATHGoogle Scholar
  8. 8.
    Abraham A. Ungar, “The relativistic velocity composition paradox and the Thomas rotation,”Found. Phys. 19, 1385–1396 (1989).CrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Abraham A. Ungar, “Weakly associative groups.”Res. Math. 17, 149–168 (1990).MathSciNetzbMATHGoogle Scholar
  10. 10.
    Abraham A. Ungar, “The expanding Minkowski space.”Res. Math. 17, 342–354 (1990).MathSciNetzbMATHGoogle Scholar
  11. 11.
    Abraham A. Ungar, “Group-like structure underlying the unit ball in real inner product spaces.”Res. Math. 18, 355–364 (1990).MathSciNetzbMATHGoogle Scholar
  12. 12.
    Abraham A. Ungar, “Quasidirect product groups and the Lorentz transformation group,” in T. M. Rassias (ed.),Constantin Caratheodory: An International Tribute, Vol. II, (World Scientific Publ., New Jersey, 1991), pp. 1378–1392.Google Scholar
  13. 13.
    Abraham A. Ungar, “Successive Lorentz transformations of the electromagnetic field,”Found. Phys. 21, 569–589 (1991).CrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Abraham A. Ungar, “Thomas precession and its associated grouplike structure.”Amer. J. Phys. 59, 824–834 (1991).CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Abrhama A. Ungar, “A note on the Lorentz transformations linking initial and final 4-vectors.”J. Math. Phys. 33, 84–85 (1992).CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Abraham A. Ungar, “The abstract Lorentz transformation group.”Amer. J. Phys. 60, 815–828 (1992).CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Abraham A. Ungar, “The holomorphic automorphism group of the complex disk.”Aequat. Math. 47, 240–254 (1994).CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Abraham A. Ungar, “The abstract complex Lorentz transformation group with real metric I: Special relativity formalism to deal with the holomorphic automorphism group of the Unit ball in any complex Hilbert space.”J. Math. Phys. 35, 1408–1425 (1994); and Erratum: “The abstract complex Lorentz transformation group with real metric I: Special relativity formalism to deal with the holomorphic automorphism group of the unit ball in any complex Hilbert space”,J. Math. Phys. 35, 3770 (1994).CrossRefADSMathSciNetzbMATHGoogle Scholar
  19. 19.
    Abraham A. Ungar, “The abstract complex Lorentz transformation group with real metric II: The invariance group of the form |t|2-∥x2,”J. Math. Phys. 35, 1881–1913 (1994).CrossRefADSMathSciNetzbMATHGoogle Scholar
  20. 20.
    Yaakov Friedman and Abraham A. Ungar, “Gyrosemidirect product structure of bounded symmetric domains.”Res. Math. 26, 28–38 (1994).MathSciNetzbMATHGoogle Scholar
  21. 21.
    Yuching You and Abraham A. Ungar, “Equivalence of two gyrogroup structures on unit balls.”Res. Math. 28, 359–371 (1995).MathSciNetzbMATHGoogle Scholar
  22. 22.
    Abrham A. Ungar, “Extension of the unit disk gyrogroup into the unit ball of any real inner product space,”J. Math. Anal. Appl. 202, 1040–1057 (1996).CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    W. Benz,Geometrische Transformationen Unter Besonderer Berücksichtigung der Lorentztranformationen (Wissenschaftsverlag, Wien, 1992), Chap. 6.Google Scholar
  24. 24.
    R. Sexl and H. K. Urbantke,Relativität, Gruppen Teilchen Springer, New York, 1992), pp. 40, 138.zbMATHGoogle Scholar
  25. 25.
    Abraham A. Ungar and Michael K. Kinyon,Gyrogroups: The Symmetries of Thomas Precession (Kluwer Academic Publishers, Doydrecht, in preparation); and A. B. Romanowska and J. D. H. SmithPost-modern Algebra, Vol. 2 (Wiley, New York, in preparation).Google Scholar
  26. 26.
    J. Dwayne Hamilton, “Relativistic precession.”Am. J. Phys. 64, 1197–1201 (1996).CrossRefADSGoogle Scholar
  27. 27.
    E. G. Peter Rowe, “Rest frames for a point particle in special relativity,”Am. J. Phys. 64, 1184–1196 (1996).CrossRefADSMathSciNetGoogle Scholar
  28. 28.
    R. J. Philpott, “Thomas precession and the Lienard-Wiechert field.”Am. J. Phys. 64, 552–556 (1996).CrossRefADSGoogle Scholar
  29. 29.
    Heinrich Wefelscheid, “On K-loops,”J. Geom. 44, 22–23 (1992); and H. Wefelscheid, “On K-loops,”J. Geom. 53, 26 (1995).Google Scholar
  30. 30.
    Helmut Karzel, “Zusammenhange zwischen Fastbereichen scharf 2-fach transitiven Permutationsgruppen und 2-Strukturen mit Rechteksaxiom.”Abh. Math. Sem. Univ. Hamburg 32, 191–206 (1968).MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Alexander Kreuzer, “Inner mappings of Bruck loops,” preprint.Google Scholar
  32. 32.
    Hala O. Pflugfelder,Quasigroups and Loops: Introduction (Heldermann Verlag, Berlin, 1990).Google Scholar
  33. 33.
    O. Chein, Hala O. Pflugfelder, and Jonathan D. H. Smith (eds.)Quasigroups and Loops Theory and Applications, Sigma Series in Pure Mathematics, Vol. 8 (Heldermann Verlag, Berlin, 1990).Google Scholar
  34. 34.
    In group theory aloop is a groupoid (S,+) with an identity element in which each of the two equationsa+x=b andy+a=b for the unknownsx andy possesses a unique solution. Several of our identities can be found in the literature on loop theory; see, e.g., Alexander Kreuzer and Heinrich Wefelscheid, “On K-loops of finite order, To the memory of Hans Zassenhaus,”Res. Math. 25, 79–102 (1994).MathSciNetzbMATHGoogle Scholar
  35. 35.
    Abraham A. Ungar, “Midpoints in gyrogroups.”Found Phys. 26, 1277–1328 (1996).CrossRefMathSciNetADSGoogle Scholar
  36. 36.
    See, for instance, A. Aurilia, “Invariant relative velocity,”Am. J. Phys. 43, 261–264 (1975). It is difficult to find in the literature Einstein's relativistic velocity addition law for not necessarily parallel velocities in a vector form. It can, however, readity be derived from the vector Lorenz transformation (8.14) which, in turm, can be found in the literature; see, e.g., Ref. 64.CrossRefADSGoogle Scholar
  37. 37.
    During a seminar on spacetime geometry that the author delivered at The University of Sydney, School of Mathematics and Statistics, April 18, 1996, Dr. Hugh Luckock stated that the splitting of spacetime into time and space, offered in gyrogroup theory by means of Thomas precession, may provide an answer to the desire to split the general relativistic notion of spacetime into space and time, expressed in: Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler,Gravitation (W. H. Freeman, San Francisco, 1973), Section 24.1, p. 505.Google Scholar
  38. 38.
    Michael K. Kinyon and Abraham A. Ungar, “The complex unit disk,” Preprint.Google Scholar
  39. 39.
    See, for instance, L. Silberstein,The Theory of Relativity (Macmillan, London, 1914), p. 169.zbMATHGoogle Scholar
  40. 40.
    I. C. Mocanu, “On the relativistic velocity composition paradox and the Thomas rotation,”Found. Phys. Lett. 5, 443–456 (1992).CrossRefMathSciNetGoogle Scholar
  41. 41.
    Jonathan D. H. Smith and Abraham A. Ungar, “Abstract space-times and their Lorentz groups,”J. Math. Phys. 37, 3073–3098 (1996).CrossRefADSMathSciNetzbMATHGoogle Scholar
  42. 42.
    John D. JacksonClassical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 524.zbMATHGoogle Scholar
  43. 43.
    See Eq. (3.3) in Lars V. Ahlfors “Old and new in Möbius groups,”Ann. Acad. Sci. Fenn. Ser. A. I Math. 9, 93–105 (1984); and p. 25 in Lars V. Ahlfors,Möbius Transformations in Several Dimensions, Lecture Notes (University of Minnesota, Minneapolis, 1981).MathSciNetzbMATHGoogle Scholar
  44. 44.
    Marvin J. Greenberg,Euclidean and Non-Euclidean Geometries (W. H. Freeman, San Francisco, 1980), pp. 208–209.zbMATHGoogle Scholar
  45. 45.
    P. Fraundorf, “Proper velocity and frame-invariant acceleration in special relativity,” preprint, available from physics/9611011 ( archive, Los Alamos, NM, 1996).Google Scholar
  46. 46.
    Abraham A. Ungar, “Formalism to deal with Reichenbach's special theory of relativity,”Found. Phys. 21, 691–726 (1991).CrossRefMathSciNetADSGoogle Scholar
  47. 47.
    Carl G. Adler, “Does mass really depend on velocity, dad?.”Am. J. Phys. 55, 739–743 (1987); but see T. R. Sandin, “In defense of relativistic mass,”Am. J. Phys. 59, 1032–1036 (1991).CrossRefADSGoogle Scholar
  48. 48.
    Robert W. Brehme, “The advantage of teaching relativity with four-vectors.”Am. J. Phys. 36, 896–901 (1968).CrossRefADSGoogle Scholar
  49. 49.
    J. A. Winne, “Special Relativity without One-Way Velocity Assumptions: Parts I and II,”Philos. Sci.,37, 81–99, 223–238 (1970).CrossRefGoogle Scholar
  50. 50.
    See Wolfgang Pauli,Theory of Relativity, translated by G. Field (Pergamon, New York, 1958) p. 74 A. Sommerfeld, “Ueber die Zusammensetzung der Geschwindigkeiten in der Relativitatstheorie,”Phys. Z. 10, 826–829 (1909); Vladimir Varićak, “Anwendung der Lobatschefkijschen Geometrie in der Relativtheorie,”Phys. Z. 11, 93–96 and 287–293 (1910); and Vladimir Varićak, “Ueber die nichteuklidische Interpretation der Relativitatstheorie,”Jahresber. Dtsch. Math. Ver. 21, 103–127 (1912). An extension of the study of the hyperbolic structure of relativity velocity spaces from one to three dimensions is available in the literature; see D. K. Sen, “3-dimensional hyperbolic geometry and relativity,” in A. Coley, C. Dyer, and T. Tupper (eds),Proceedings of the 2nd Canadian Conference on General Relativity and Relativistic Astrophysics, pp. 264–266 (World Scientific, 1988); and Lars-Erik Lundberg, “Quantum theory, hyperbolic geometry and relativity,”Rev. Math. Phys. 6, 39–49 (1994).zbMATHGoogle Scholar
  51. 51.
    Thomas A. Moore,A Traveler's Guide to Spacetime (McGraw Hill, New York, 1995), fn. 1, p. 54.Google Scholar
  52. 52.
    Walter Rudin,Function Theory in the Unit Ball of ℂ n Springer-Verlag, New York, 1980).Google Scholar
  53. 53.
    A. Einstein, Zur Elektrodynamik Bewegter Körper (On the Electrodynamics of Moving Bodies),Ann. Phys. (Leipzig) 17 891–921, (1905). For English translation see H. M. Schwartz, “Einstein's first paper on relativity,” (covers the first of the two parts of Einstein's paper),Am. J. Phys. 45, 18–25, (1977); and H. A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl,The Principle of Relativity (Dover, New York, translated by W. Perrett and G. B. Jeffrey, 1952, first published in 1923), pp. 37–65.ADSGoogle Scholar
  54. 54.
    Helgason Sigurdur,Differential Geometry, Lie Groups, and Symmetric Spaces (Academic Press, New York, 1978).zbMATHGoogle Scholar
  55. 55.
    Oliver Jones, “On the equivalence of the categories of Riemannian globally symmetric spaces of non-compact type and gyrovector spaces,” in preparation.Google Scholar
  56. 56.
    P. O. Miheev and L. V. Sabinin,Quasigroups and Differential Geometry, Chap. XII in O. Chein, Hala, O. Pflugfelder, and J. D. H. Smith (eds.),Quasigroups and Loops: Theory and Applications (Sigma Series in Pure Mathematics, Vol. 8, Heldermann Verlag, Berlin, 1990) pp. 357–430; and H. Karzel and M. J. Thomsen, “Near-rings, generalizations, near-rings with regular elements and applications, a report,”Contributions to General Algebra 8,Proc. Conf. on Near-Rings and Near-Fields, Linz, Austria, July 14–20, 1991.Google Scholar
  57. 57.
    Helmuth K. Urbantke, “Physical holonomy, Thomas precession, and Clifford algebra,” Sect. III,Amer. J. Phys. 58, 747–750 (1990).CrossRefADSMathSciNetGoogle Scholar
  58. 58.
    W. Rindler and L. Mishra, “The nonreciprocity of relative acceleration in relativity.”Phys. Lett A 173, 105–108 (1993). See also Gonzalez-Dîaz, “Relativistic negative acceleration components,”Am. J. Phys. 46, 932–934 (1977).CrossRefADSMathSciNetGoogle Scholar
  59. 59.
    Richard S. Millman and George D. Parker,Geometry A Metric Approach with Models, 2nd ed. (Springer-Verlag, (New York, 1991).zbMATHGoogle Scholar
  60. 60.
    Edward C. Wallace and Stephen F. West,Roads To Geometry (Prentice Hall, Englewood Cliffs, NJ, 1992).zbMATHGoogle Scholar
  61. 61.
    See Ref. 59. p. 304.zbMATHGoogle Scholar
  62. 62.
    An observation made by Michael K. Kinyon.Google Scholar
  63. 63.
    An observation made by Oliver Jones.Google Scholar
  64. 64.
    John D. Jackson,Classical Electrodynamics, 2nd edn. (Wiley, New York, 1975), p. 517.zbMATHGoogle Scholar
  65. 65.
    C. B. van Wyk, “Lorentz transformations in terms of initial and final vectors.”J. Math. Phys. 27, 1311–1314 (1986).CrossRefADSMathSciNetGoogle Scholar
  66. 66.
    “The possible existence of extra dimensions to spacetime can be tested astrophysically” perhaps by Gravity Probe B(68); see D. Kalligas, P. S. Wesson, and C. F. W. Everitt, “The Classical tests in Kaluza-Klein gravity,” preprint. A few references from the literature on six-dimensional relativity are: H. C. Chandola and B. S. Rajput, “Maxwell's equations in six-dimensional space-time,”Indian J. Pure Appl. Phys. 24, 58–64 (1986); E. A. B. Cole, “Centre-of-mass frame in six-dimensional special relativity,”Lett. Nuovo Cimento,28, 171–174 (1980); E. A. B. Cole, “New electromagnetic fields in six-dimensional special relativity,”Nuovo Cimento,60A, 1–11 (1980); E. A. B. Cole and S. A. Buchanan, “Spacetime transformations in six-dimensional special relativity,”J. Phys. A. Math. Gen. 15, L255–L257 (1982); I. Merches, and C. Dariescu, “The use of six-vectors in the theory of special relativity,”Acta Phys. Hung. 70, 63–70 (1991); P. T. Papas, “The three-dimensional time equation,”Lett. Nuovo Cimento 25, 429–434 J. Strnad, “Six-dimensional spacetime and the Thomas precession,”Lett Nuovo Cimento 26, 535–536 (1970); and M. T. Teli, “General Lorentz transformations in six-dimensional space-time,”Phys. Lett. A 128, 447–450 (1987).Google Scholar
  67. 67.
    Helmuth Urbantke, “Comment on “The expanding Minkowski space’ by A. A. Ungar,”Res. Math.,19, 189–191 (1991).MathSciNetzbMATHGoogle Scholar
  68. 68.
    “Thumbs partly up for Gravity Probe B.”Science News 147, No. 23, p. 367 (1995). Gravity Probe B is a drag-free satellite carrying gyroscopes around Earth. For details see C. W. Francis Everitt, William M. Fairbank, and L. I. Schiff, “Theoretical background and present status of the Stanford relativity-gyroscope experiment,” inThe Significance of Space Research for Fundamental Physics, Proc. Colloq. of the European Space Research Org. at Interlaken, Swizerland, 4 Sept. 1969; R. Vassar, J. V. Breakwell, C. W. F. Everitt, and R. A. VanPatten, “Orbit selection for the Stanford relativity gyroscope experiment,”J. Spacecraft Rockets 19, 66–71 (1986). The general-relativistic Thomas precession involves several terms one of which is the special-relativistic Thomas precession studied in this article. The NASA program to perform a Thomas precession test of Einstein's theory of general relativity by measuring the precession of gyroscopes in Earth orbit was initiated by William M. Fairbank; see C. W. F. Everitt “Gravity Probe B: I. The scientific implications,” The Sixth Marcel Grossmann Meeting on Relativity, Kyoto, Japan, June 23–29, 1991 (World Scientific Publ.); J. D. Fairbank, B. S. Deaver, Jr., C. W. F. Everitt, and P. F. Michelson,Near Zero: New Frontiers of Physics (Freeman, New York, 1988); “William Martin Fairbank (1917–1989)”,Nature 342, 125 (1989).Google Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • Abraham A. Ungar
    • 1
  1. 1.Department of MathematicsNorth Dakota State UniversityFargo

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