A link between the bounds on relativistic velocities and areas of hyperbolic triangles

American Journal of Physics

Volumn 69, Issue 3, 2001, Pages 306-310

  Criado, C ^a   Alamo, N ^a  

^a UNIVERSITY OF MÁLAGA   (Spain)

Author keywords

02; 03.30

Indexed keywords

EID: 23044523433     PISSN: 00029505     EISSN: 19432909     Source Type: Journal    
DOI: 10.1119/1.1323963     Document Type: Article

References (9)

1. Eugene P. Wigner, “The unreasonable effectiveness of mathematics in natural sciences,” Commun. Pure Appl. Math. CPAMAT 13, 1–14 (1960). CPAMAT A reprint of this famous essay can be found, for example, in The Word Treasury of Physics, Astronomy, and Mathematics, edited by Timothy Ferris (Little, Brown, Boston, 1991), pp. 526–540.
2. B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry-Methods and Applications, Part I (Springer-Verlag, New York, 1984), pp. 90–93.
3. The German physicist Arnold Sommerfeld (1868–1951) was the person who first connected Lobachevskian geometry with special relativity. In his paper “On the composition of velocities in relativity theory” [Arnold Sommerfeld, “Über die Zusammensetzung der Geschwindigkeiten in der Relativitätstheorie,” Phys. Z. PHZTAO 10, (22)826–829 (1909)] he established the relation between the formula for the addition of velocities in the theory of relativity and the trigonometric formulas for hyperbolic geometry.PHZTAO But it was the Yugoslav geometer Vladimir Varichak in the paper “On the non-Euclidean interpretation of the theory of relativity” [Vladimir Varichak, “Über die nichteuklidische Interpretation der Relativitätstheorie,” Jahrb. Deut. Math. Verein 21, 103–122 (1912)], who pointed out that those formulas were formulas of Lobachevskian geometry. These and other historical notes can be found in B. A. Rosenfeld, A History of Non-Euclidean Geometry (Springer-Verlag, New York, 1988), pp. 270–273.
4. Marvin J. Greenberg, Euclidean and Non-Euclidean Geometries. Development and History (Freeman, San Francisco, CA, 1980), 2nd ed., p. 187.
5. In spite of the name, it was Beltrami fourteen years before Poincaré who first discovered this model. See Tristan Needham, Visual Complex Analysis (Clarendon, Oxford, 1997), 315.
6. William P. Thurston, Three-dimensional Geometry and Topology (Princeton U.P., Princeton, NJ, 1997), Vol. 1, p. 66.
7. James W. Cannon, William J. Floyd, Richard Kenyon, and Walter R. Parry, “Hyperbolic Geometry” in Flavors of Geometry, edited by Silvio Levy (Cambridge U.P., Cambridge, 1997), pp. 59–116. Also in Ref. 7, p. 67.
8. H. S. M. Coxeter, Introduction to Geometry (Wiley, New York, 1969), 299.
9. W. Rindler, Essential Relativity (Springer-Verlag, New York, 1986), 2nd ed., p. 47.