Abstract
The Thomas precession of relativity physics gives rise to important isometries in hyperbolic geometry that expose analogies with Euclidean geometry. These, in turn, suggest our bifurcation approach to hyperbolic geometry, according to which Euclidean geometry bifurcates into two mutually dual branches of hyperbolic geometry in its transition to non-Euclidean geometry. One of the two resulting branches turns out to be the standard hyperbolic geometry of Bolyai and Lobachevsky. The corresponding bifurcation of Newtonian mechanics in the transition to Einsteinian mechanics indicates that there are two, mutually dual, kinds of uniform accelerations. Furthermore, while current hyperbolic geometry “does not use the notion of vector at all” (I. M. Yaglom, Geometric Transformations III, p. 135, trans. by Abe Shenitzer, Random House, New York, 1973), our bifurcation approach exposes the elusive hyperbolic vectors, that we call gyrovectors.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
W. Cannon, W. J. Floyd, R. Kenyon, and W. R. Parry, “Hyperbolic geometry,” in Flavors of Geometry, S. Levy, ed. (University Press, Cambridge, 1997), pp. 59–115.
A. A. Ungar, “Thomas precession: Its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics,” Found Phys. 27 (6), 881–951 (1997).
A. A. Ungar, “Thomas rotation and the parametrization of the Lorentz transformation group,” Found. Phys. Lett. 1 (1), 57–89 (1988).
A. A. Ungar, “Thomas precession and its associated grouplike structure,” Am. J. Phys. 59 (9), 824–834 (1991).
A. A. Ungar, “From Pythagoras to Einstein: The hyperbolic Pythagorean theorem,” Found. Phys. 28 (8), 1283–1321 (1998).
L. V. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory (McGraw-Hill, New York, 1973).
S. G. Krantz, Complex Analysis: The Geometric Viewpoint (Mathematical Association of America, Washington, DC, 1990).
H. O. Pflugfelder, Quasigroups and Loops: Introduction (Heldermann, Berlin, 1990).
J. D. H. Smith, and A. B. Romanowska, Post-modern Algebra (Wiley, New York, 1999).
A. A. Ungar, “Extension of the unit disk gyrogroup into the unit ball of any real inner product space,” J. Math. Anal. Appl. 202 (3), 1040–1057 (1996).
J. D. H. Smith, and A. A. Ungar, “Abstract space-times and their Lorentz group,” J. Math. Phys. 37(6), 3073–3098 (1996).
V. Fock, The Theory of Space, Time and Gravitation, 2nd revised edn. (Macmillan, New York, 1964); translated from the Russian by N. Kemmer.
H. Bacry, Lectures on Group Theory and Particle Theory (Gordon 6 Breach, New York, 1977).
R. U. Sexl, and H. K. Urbantke, Relativität, Gruppen, Teilchen, 2nd edn. (Springer, Vienna, 1982).
A. A. Ungar, “Gyrovector spaces in the service of hyperbolic geometry,” in Mathematical Analysis and Applications, Th. M. Rassias, ed. (Hadronic, Florida, 2000).
J. McCleary, Geometry from a Differentiable Viewpoint (University Press, Cambridge, 1994).
L. Marder, “On uniform acceleration in special and general relativity,” Proc. Cambridge Philos. Soc. 53, 194–198 (1957).
H. Urbantke, “Physical holonomy, Thomas precession, and Clifford algebra,” Am. J. Phys. 58(8), 747–750 (1990).
A. A. Ungar, Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces (Kluwer Academic, Dordrecht, forthcoming).
A. A. Ungar, “Midpoints in gyrogroups,” Found. Phys. 26 (10), 1277–1328 (1996).
M. J. Greenberg, Euclidean and Non-Euclidean Geometries, 3rd ed. (Freeman, New York, 1993).
S. Walter, “The non-Euclidean style of minkowskian relativity,” in The Symbolic Universe: Geometry and Physics 1890-1930 (Birkhäuser, Boston, 1999), pp. 91–127.
V. Varicak, Darstellung der Relativitätstheorie im dreidimensionalen Lobatchefskijschen Raume [Presentation of the Theory of Relativity in the Threedimensional Lobachevskian Space] (Zaklada, Zagreb, 1924).
H. S. M. Coxeter, Non-Euclidean Geometry, 6th ed. (Mathematical Association of America, Washington, DC, 1998).
J. Stillwell, Sources of Hyperbolic Geometry (American Mathematical Society, Providence, RI, 1996).
A. Tarski, and S. Givant, “Taraki's system of geometry,” Bull. Symb. Logic 5 (2), 175–214 (1999).
V. Pambuccian, “On the simplicity of an axiom system for plane Euclidean geometry,” Demonstratio Math. 30(3), 509–512 (1997).
B. J. Fei, and Z. G. Li, “Relativistic velocity and hyperbolic geometry,” Phys. Essays 10 (2), 248–255 (1997).
F. G. Kard, “Hyperbolic trigonometry in a relativistic velocity space,” Tartu Riikl. Ñl. Toimetised 117 (417) Metodolog. Voprosy Fiz. 3, 57-69 (1977).
D. K. Sen, “3-dimensional hyperbolic geometry and relativity,” in Proceedings of the 2nd Canadian Conference on General Relativity and Relativistic Astrophysics (Toronto, Ontario, 1987), (World Scientific, Singapore, 1988), pp. 264–266.
A. Ramsay, and R. D. Richtmyer, Introduction to Hyperbolic Geometry (Springer, New York, 1995).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ungar, A.A. The Bifurcation Approach to Hyperbolic Geometry. Foundations of Physics 30, 1257–1282 (2000). https://doi.org/10.1023/A:1003636505621
Issue Date:
DOI: https://doi.org/10.1023/A:1003636505621