1.

Abraham A. Ungar, “Thomas rotation and the parametrization of the Lorentz transformation group.”

*Found Phys. Lett.*
**1**, 57–89 (1988).

CrossRefMathSciNet2.

Abraham A. Ungar, “The Thomas rotation formalism underlying a nonassociative group structure for relativistic velocities.”

*Appl. Math. Lett.*
**1**, 403–405 (1988).

CrossRefMathSciNetMATH3.

Arlan Ramsay and Robert D. Richtmyer,

*Introduction to Hyperbolic Geometry* (Springer, New York, 1995), p. 251.

MATH4.

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).

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.

MATH6.

Abraham A. Ungar, “Axiomatic approach to the nonassociative group of relativistic velocities.”

*Found. Phys. Lett.*
**2**, 199–203 (1989).

CrossRefMathSciNet7.

Abraham A. Ungar, “The relativistic noncommutative nonassociative group of velocities and the Thomas rotation.”

*Res. Math.*
**16**, 168–179 (1989).

MathSciNetMATH8.

Abraham A. Ungar, “The relativistic velocity composition paradox and the Thomas rotation,”

*Found. Phys.*
**19**, 1385–1396 (1989).

CrossRefMathSciNetADS9.

Abraham A. Ungar, “Weakly associative groups.”

*Res. Math.*
**17**, 149–168 (1990).

MathSciNetMATH10.

Abraham A. Ungar, “The expanding Minkowski space.”

*Res. Math.*
**17**, 342–354 (1990).

MathSciNetMATH11.

Abraham A. Ungar, “Group-like structure underlying the unit ball in real inner product spaces.”

*Res. Math.*
**18**, 355–364 (1990).

MathSciNetMATH12.

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.

13.

Abraham A. Ungar, “Successive Lorentz transformations of the electromagnetic field,”

*Found. Phys.*
**21**, 569–589 (1991).

CrossRefMathSciNetADS14.

Abraham A. Ungar, “Thomas precession and its associated grouplike structure.”

*Amer. J. Phys.*
**59**, 824–834 (1991).

CrossRefADSMathSciNet15.

Abrhama A. Ungar, “A note on the Lorentz transformations linking initial and final 4-vectors.”

*J. Math. Phys.*
**33**, 84–85 (1992).

CrossRefADSMathSciNet16.

Abraham A. Ungar, “The abstract Lorentz transformation group.”

*Amer. J. Phys.*
**60**, 815–828 (1992).

CrossRefADSMathSciNet17.

Abraham A. Ungar, “The holomorphic automorphism group of the complex disk.”

*Aequat. Math.*
**47**, 240–254 (1994).

CrossRefMathSciNetMATH18.

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).

CrossRefADSMathSciNetMATH19.

Abraham A. Ungar, “The abstract complex Lorentz transformation group with real metric II: The invariance group of the form |

*t*|

^{2}-∥

**x**∥

^{2},”

*J. Math. Phys.*
**35**, 1881–1913 (1994).

CrossRefADSMathSciNetMATH20.

Yaakov Friedman and Abraham A. Ungar, “Gyrosemidirect product structure of bounded symmetric domains.”

*Res. Math.*
**26**, 28–38 (1994).

MathSciNetMATH21.

Yuching You and Abraham A. Ungar, “Equivalence of two gyrogroup structures on unit balls.”

*Res. Math.*
**28**, 359–371 (1995).

MathSciNetMATH22.

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).

CrossRefMathSciNetMATH23.

W. Benz,*Geometrische Transformationen Unter Besonderer Berücksichtigung der Lorentztranformationen* (Wissenschaftsverlag, Wien, 1992), Chap. 6.

24.

R. Sexl and H. K. Urbantke,

*Relativität, Gruppen Teilchen* Springer, New York, 1992), pp. 40, 138.

MATH25.

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. Smith*Post-modern Algebra*, Vol. 2 (Wiley, New York, in preparation).

26.

J. Dwayne Hamilton, “Relativistic precession.”

*Am. J. Phys.*
**64**, 1197–1201 (1996).

CrossRefADS27.

E. G. Peter Rowe, “Rest frames for a point particle in special relativity,”

*Am. J. Phys.*
**64**, 1184–1196 (1996).

CrossRefADSMathSciNet28.

R. J. Philpott, “Thomas precession and the Lienard-Wiechert field.”

*Am. J. Phys.*
**64**, 552–556 (1996).

CrossRefADS29.

Heinrich Wefelscheid, “On K-loops,”*J. Geom.*
**44**, 22–23 (1992); and H. Wefelscheid, “On K-loops,”*J. Geom.*
**53**, 26 (1995).

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).

MathSciNetCrossRefMATH31.

Alexander Kreuzer, “Inner mappings of Bruck loops,” preprint.

32.

Hala O. Pflugfelder,*Quasigroups and Loops: Introduction* (Heldermann Verlag, Berlin, 1990).

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).

34.

In group theory a

*loop* is a groupoid (

*S*,+) with an identity element in which each of the two equations

*a+x=b* and

*y+a=b* for the unknowns

*x* and

*y* 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).

MathSciNetMATH35.

Abraham A. Ungar, “Midpoints in gyrogroups.”

*Found Phys.*
**26**, 1277–1328 (1996).

CrossRefMathSciNetADS36.

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.

CrossRefADS37.

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.

38.

Michael K. Kinyon and Abraham A. Ungar, “The complex unit disk,” Preprint.

39.

See, for instance, L. Silberstein,

*The Theory of Relativity* (Macmillan, London, 1914), p. 169.

MATH40.

I. C. Mocanu, “On the relativistic velocity composition paradox and the Thomas rotation,”

*Found. Phys. Lett.*
**5**, 443–456 (1992).

CrossRefMathSciNet41.

Jonathan D. H. Smith and Abraham A. Ungar, “Abstract space-times and their Lorentz groups,”

*J. Math. Phys.*
**37**, 3073–3098 (1996).

CrossRefADSMathSciNetMATH42.

John D. Jackson

*Classical Electrodynamics*, 2nd ed. (Wiley, New York, 1975), p. 524.

MATH43.

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).

MathSciNetMATH44.

Marvin J. Greenberg,

*Euclidean and Non-Euclidean Geometries* (W. H. Freeman, San Francisco, 1980), pp. 208–209.

MATH45.

P. Fraundorf, “Proper velocity and frame-invariant acceleration in special relativity,” preprint, available from physics/9611011 (xxx.lanl.gov archive, Los Alamos, NM, 1996).

46.

Abraham A. Ungar, “Formalism to deal with Reichenbach's special theory of relativity,”

*Found. Phys.*
**21**, 691–726 (1991).

CrossRefMathSciNetADS47.

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).

CrossRefADS48.

Robert W. Brehme, “The advantage of teaching relativity with four-vectors.”

*Am. J. Phys.*
**36**, 896–901 (1968).

CrossRefADS49.

J. A. Winne, “Special Relativity without One-Way Velocity Assumptions: Parts I and II,”

*Philos. Sci.*,

**37**, 81–99, 223–238 (1970).

CrossRef50.

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).

MATH51.

Thomas A. Moore,*A Traveler's Guide to Spacetime* (McGraw Hill, New York, 1995), fn. 1, p. 54.

52.

Walter Rudin,*Function Theory in the Unit Ball of ℂ*
^{
n
} Springer-Verlag, New York, 1980).

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.

ADS54.

Helgason Sigurdur,

*Differential Geometry, Lie Groups, and Symmetric Spaces* (Academic Press, New York, 1978).

MATH55.

Oliver Jones, “On the equivalence of the categories of Riemannian globally symmetric spaces of non-compact type and gyrovector spaces,” in preparation.

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.

57.

Helmuth K. Urbantke, “Physical holonomy, Thomas precession, and Clifford algebra,” Sect. III,

*Amer. J. Phys.*
**58**, 747–750 (1990).

CrossRefADSMathSciNet58.

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).

CrossRefADSMathSciNet59.

Richard S. Millman and George D. Parker,

*Geometry A Metric Approach with Models*, 2nd ed. (Springer-Verlag, (New York, 1991).

MATH60.

Edward C. Wallace and Stephen F. West,

*Roads To Geometry* (Prentice Hall, Englewood Cliffs, NJ, 1992).

MATH61.

62.

An observation made by Michael K. Kinyon.

63.

An observation made by Oliver Jones.

64.

John D. Jackson,

*Classical Electrodynamics*, 2nd edn. (Wiley, New York, 1975), p. 517.

MATH65.

C. B. van Wyk, “Lorentz transformations in terms of initial and final vectors.”

*J. Math. Phys.*
**27**, 1311–1314 (1986).

CrossRefADSMathSciNet66.

“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).

67.

Helmuth Urbantke, “Comment on “The expanding Minkowski space’ by A. A. Ungar,”

*Res. Math.*,

**19**, 189–191 (1991).

MathSciNetMATH68.

“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,” in*The 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).