Einstein’s special relativity: Unleashing the power of its hyperbolic geometry. (English) Zbl 1071.83505

Summary: We add physical appeal to Einstein velocity addition law of relativistically admissible velocities, thereby gaining new analogies with classical mechanics and invoking new insights into the special theory of relativity. We place Einstein velocity addition in the foundations of both special relativity and its underlying hyperbolic geometry, enabling us to present special relativity in full three space dimensions rather than the usual one-dimensional space, using three-geometry instead of four-geometry. Doing so we uncover unexpected analogies with classical results, enabling readers to understand the modern and unfamiliar in terms of the classical and familiar. In particular, we show that while the relativistic mass does not mesh up with the four-geometry, it meshes extraordinarily well with the three-geometry, providing unexpected insights that are not easy to come by, by other means.


83A05 Special relativity
83E05 Geometrodynamics and the holographic principle
Full Text: DOI


[1] Lenzen, V. F., Physical geometry, Amer. Math. Monthly, 46, 324-334 (1939) · Zbl 0022.07204
[2] Criado, C.; Alamo, N., A link between the bounds on relativistic velocities and areas of hyperbolic triangles, Amer. J. Phys., 69, 3, 306-310 (2001)
[3] Adler, C. G., Does mass really depend on velocity, dad?, Amer. J. Phys., 55, 8, 739-743 (1987)
[4] Ungar, A. A., The theory of gyrogroups and gyrovector spaces, (Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession, Volume 117 of Fundamental Theories of Physics (2001), Kluwer Academic Publishers Group) · Zbl 0972.83002
[5] Feynman, R. P.; Leighton, R. B.; Sands, M., The Feynman Lectures on Physics (1964), Addison-Wesley Publishing Co., Inc.: Addison-Wesley Publishing Co., Inc. Dordrecht · Zbl 0131.38703
[6] Einstein, A., Zur elektrodynamik bewegter körper, Ann. Physik, 17, 891-921 (1905)
[7] Einstein, A., Einstein’s Miraculous Years: Five Papers that Changed the Face of Physics, (Einstein’s dissertation on the determination of molecular dimensions-Einstein on Brownian motion — Einstein on the theory of relativity-Einstein’s early work on the quantum hypothesis. Einstein’s dissertation on the determination of molecular dimensions-Einstein on Brownian motion — Einstein on the theory of relativity-Einstein’s early work on the quantum hypothesis, A new English translation of Einstein’s 1905 paper on pp. 123-160 (1998), Princeton: Princeton Reading, MA), Edited and introduced by John Stachel. Includes bibliographical references
[8] Sexl, R. U.; Urbantke, H. K., Relativity, Groups, Particles, (Special relativity and relativistic symmetry in field and particle physics (2001), Springer-Verlag: Springer-Verlag Princeton, NJ), Revised and translated from the third German (1992) edition by Urbantke · Zbl 0966.83502
[9] Varičak, V., Beiträge zur nichteuklidischen geometrie, Jber. dtsch. Mat. Ver., 17, 70-83 (1908)
[10] Varičak, V., Anwendung der Lobatschefskjschen Geometrie in der Relativtheorie, Physikalische Zeitschrift, 11, 93-96 (1910)
[11] Varičak, V., Darstellung der Relativitätstheorie im Dreidimensionalen Lobatchefskijschen Raume (1924), Zaklada: Zaklada Vienna
[12] Lanczos, C., Space Through the Ages, (The Evolution of Geometrical Ideas from Pythagoras to Hilbert and Einstein (1970), Academic Press: Academic Press Zagreb) · Zbl 0196.23601
[13] Walter, S., The non-Euclidean style of Minkowskian relativity, (Gray, J. J., The Symbolic Universe (1999), Oxford Univ. Press: Oxford Univ. Press London), 91-127 · Zbl 1059.01513
[14] Pyenson, L., Relativity in late Wilhelmian Germany: The appeal to a preestablished harmony between mathematics and physics, Arch. Hist. Exact Sci., 27, 2, 137-155 (1982) · Zbl 0493.01023
[15] Barrett, J. F., Special Relativity and Hyperbolic Geometry, (Physical Interpretations of Relativity Theory Proceedings. Physical Interpretations of Relativity Theory Proceedings, London, U.K., Sept. 11-14 (1998), Univ. Sunderland: Univ. Sunderland New York) · Zbl 0197.42703
[16] Walter, S., Minkowski, mathematicians, and the mathematical theory of relativity, (The Expanding Worlds of General Relativity (Berlin, 1995) (1999), Birkhäuser Boston: Birkhäuser Boston Sunderland, U.K.), 45-86 · Zbl 0943.83005
[17] Corry, L., The influence of David Hilbert and Hermann Minkowski on Einstein’s views over the interrelation between physics and mathematics, Endeavor, 22, 3, 95-97 (1998)
[18] (Schilpp, P. A., Albert Einstein: Philosopher-Scientist (1949), The Library of Living Philosophers, Inc.: The Library of Living Philosophers, Inc. Boston, MA) · Zbl 0038.14804
[19] Penrose, R., The rediscovery of gravity: The Einstein equation of general relativity, (Farmelo, G., It Must be Beautiful: Great Equations of modern Sciences (2002), Granta: Granta Evanston, IL)
[20] Misner, C. W.; Thorne, K. S.; Wheeler, J. A., Gravitation (1973), W. H. Freeman and Co.: W. H. Freeman and Co. London
[21] Fock, V., The theory of space, time and gravitation (1964), The Macmillan Co.: The Macmillan Co. San Francisco, CA, Translated from the Russian by N. Kemmer. A Pergamon Press Book · Zbl 0085.42301
[22] Bacry, H., Documents on Modern Physics, (Lectures on Group Theory and Particle Theory (1977), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers New York) · Zbl 0363.20001
[23] Brehme, R. W., The advantage of teaching relativity with four-vectors, Amer. J. Phys., 36, 10, 896-901 (1968)
[24] Okun, L,B., The concept of mass, Phys. Today, 31-36 (1989), June
[25] Tsai, L., The relation between gravitational mass inertial mass and velocity, Amer. J. Phys., 54, 340-342 (1986)
[26] Gabrielse, G., Relativistic mass increase at slow speeds, Amer. J. Phys., 63, 6, 568-569 (1995)
[27] Sommerfeld, A., Über die Zusammensetzung der Geschwindigkeiten in der Relativtheorie, Physikalische Zeitschrift, 10, 826-829 (1909)
[28] Levy-Leblond, J.-M., Additivity, rapidity, relativity, Amer. J. Phys., 47, 1045-1049 (1979)
[29] Rosenfeld, B. A., A History of Non-Euclidean Geometry (1988), Springer-Verlag: Springer-Verlag New York · Zbl 0657.01002
[30] Vigoureux, J. M., Calculations of the wigner angle, European Journal of Physics, 22, 2, 149-155 (2001) · Zbl 0980.83005
[31] Barrett, J. F., On Carathéodory’s approach to relativity and its relation to hyperbolic geometry, (Constantin Carathéodory in His … Origins, (Vissa-Orestiada, 2000) (2001), Hadronic Press: Hadronic Press New York), 81-90
[32] Ungar, A. A., Thomas rotation and the parametrization of the Lorentz transformation group, Found. Phys. Lett., 1, 1, 57-89 (1988)
[33] Mocanu, C. I., On the relativistic velocity composition paradox and the Thomas rotation, Found. Phys. Lett., 5, 5, 443-456 (1992)
[34] Silberstein, L., The Theory of Relativity (1914), MacMillan: MacMillan Palm Harbor, FL
[35] Ungar, A. A., Thomas precession and its associated grouplike structure, Amer. J. Phys., 59, 9, 824-834 (1991)
[36] Jackson, J. D., Classical Electrodynamics (1975), John Wiley & Sons Inc.: John Wiley & Sons Inc. London · Zbl 0114.42903
[37] Ungar, A. A., Thomas precession: Its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics, Found. Phys., 27, 6, 881-951 (1997)
[38] Ungar, A. A., The hyperbolic geometric structure of the density matrix for mixed state qubits, Found. Phys., 32, 11, 1671-1699 (2002)
[39] Thomas, L. H., The motion of the spinning electron, Nature, 117, 514 (1926)
[40] Borel, E., Introduction Géométrique a Quelques Théories Physiques (1914), Gauthier-Villars: Gauthier-Villars New York
[41] Stachel, J. J., History of relativity, (Brown, L. M.; Pais, A.; Pippard, B., Twentieth century physics, Volume I (1995), Published jointly by the Institute of Physics Publishing: Published jointly by the Institute of Physics Publishing Paris), 249-356 · Zbl 0846.00003
[42] Belloni, L.; Reina, C., Sommerfeld’s way to the Thomas precession, European J. Phys., 7, 55-61 (1986)
[43] Eddington, A. S., The Mathematical Theory of Relativity (1924), Cambridge
[44] Chen, J.-L.; Ungar, A. A., The Bloch gyrovector, Found. Phys., 32, 531-565 (2002)
[45] MacKeown, P. K., Question 57: Thomas precession, Amer. J. Phys., 65, 2, 105 (1997)
[46] Perdigao do. Carmo, M., Differential Geometry of Curves and Surfaces (1976), Prentice-Hall: Prentice-Hall Bristol · Zbl 0326.53001
[47] Kreyszig, E., Differential Geometry (1991), Dover Publications Inc., Reprint of the 1963 edition
[48] McCleary, J., Geometry from a Differentiable Viewpoint (1994), Cambridge University Press: Cambridge University Press New York
[49] Hausner, M., A Vector Space Approach to Geometry (1998), Dover Publications Inc.: Dover Publications Inc. Cambridge, Reprint of the 1965 original · Zbl 0124.13202
[50] Greenberg, M. J., Euclidean and Non-Euclidean Geometries, (Development and History (1993), W.H. Freeman and Company: W.H. Freeman and Company Mineola, NY) · Zbl 0442.51008
[51] Krantz, S. G., A matter of gravity, Amer. Math. Monthly, 110, 6, 465-481 (2003) · Zbl 1045.51007
[52] Dubrovskii, V. N.; Smorodinskii, Ya. A.; Surkov, E. L., The World of Relativity (1984), Nauka: Nauka New York, (Russian)
[53] Ungar, A. A., Quasidirect product groups and the Lorentz transformation group, (Rassias, T. M., Constantin Carathéodory: An International Tribute, Volumes I and II (1991), World Sci. Publishing: World Sci. Publishing Moscow), 1378-1392 · Zbl 0746.22009
[54] Møller, C., The Theory of Relativity (1952), at the Clarendon Press: at the Clarendon Press Oxford · Zbl 0047.20602
[55] Rivas, M.; Valle, M. A.; Aguirregabiria, J. M., Composition law and contractions of the Poincaré group, European J. Phys., 7, 1, 1-5 (1986)
[56] Sard, R. D., Relativistic Mechanics: Special Relativity and Classical Particle Dynamics (1970), W.A. Benjamin
[57] Halpern, F. R., Special Relativity and Quantum Mechanics (1968), Prentice-Hall Inc.: Prentice-Hall Inc. New York · Zbl 0162.29101
[58] Coll, B.; San José Martínez, F., Composition of Lorentz transformations in terms of their generators, Gen. Relativity Gravitation, 34, 9, 1345-1356 (2002) · Zbl 1016.83006
[59] Ungar, A. A., A note on the Lorentz transformations linking initial and final four-vectors, J. Math. Phys., 33, 1, 84-85 (1992)
[60] Ungar, A. A., The abstract Lorentz transformation group, Amer. J. Phys., 60, 9, 815-828 (1992) · Zbl 1219.83029
[61] van Wyk, C. B., Lorentz transformations in terms of initial and final vectors, J. Math. Phys., 27, 5, 1311-1314 (1986)
[62] Urbantke, H., Lorentz transformations from reflections: some applications, Found. Phys. Lett., 16, 2, 111-117 (2003)
[63] Yiu, P., The uses of homogeneous barycentric coordinates in plane Euclidean geometry, Internat. J. Math. Ed. Sci. Tech., 31, 4, 569-578 (2000) · Zbl 1019.51021
[64] Weisstein, E. W., CRC Concise Encyclopedia of Mathematics (2003), Chapman & Hall/CRC: Chapman & Hall/CRC Englewood Cliffs, NJ · Zbl 1079.00009
[65] Mumford, D.; Series, C.; Wright, D., Indra’s Pearls, (The Vision of Felix Klein (2002), Cambridge University Press: Cambridge University Press Boca Raton, FL) · Zbl 1314.00007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.