Abstract
In this paper we give a brief review of the pseudo-Riemannian geometry of the five-dimensional homogeneous space for the conformal group O(4, 2). Its topology is described and its relation to the conformally compactified Minkowski space is discussed. Its metric and geodesics are calculated using a generalized half-space representation. Compactification via Lie-sphere geometry is outlined. Possible applications to Jaime Keller’s START theory may follow by using its predecessor - the 5-optics of Yu. B. Rumer. The point of view of Rumer is given extensively in the last section of the paper.
Similar content being viewed by others
References
Keller J.: Spinors, Twistors, Mexors and the Massive Spinning Electron. Advances in Applied Clifford Algebras. 7, 439–455 (1997)
R. da Rocha, J. Vaz, Conformal Structures and Twistors in the Paravector Model of Spacetime. International Journal of Geometric Methods in Modern Physics, 4 (2007), 547–576, http://arxiv.org/abs/math-ph/0412074 .
Yu. B. Rumer, Studies on 5-Optics. Gosizdat tech-teor, Moskva, 1956 (in Russian).
Yu B.: Rumer Action as Space Coordinate. X. Soviet Physics JETP. 36((9), 1348–1353 (1959)
J. Keller, The Theory of the Electron. A Theory of Matter from START. Fundamental Theories of Physics, Vol. 115, Kluwer Acad Publ. (2001).
Keller J.: General Relativity as a Symmetry of Unified Space-Time-Action Geometrical Space. Proceedings of the Institute of Mathematics of Ukraine. 43, 557–568 (2002)
Keller J.: START: Inventio Principia Geometrica Physicae. Advances in Applied Clifford Algebras. 18, 772–805 (2008)
P. Anglés, Conformal Groups in Geometry and Spin Structures. Birkhäuser, Progress in Mathematical Physics, Vol. 50, 2008.
A. Jadczyk, On Conformal Infinity and Compactifications of the Minkowski Space. Advances in Applied Clifford Algebras, 21 (2011), 721–756, http://arxiv.org/abs/math-ph/0412074 .
A. Jadczyk, Geometry and Shape of Minkowski’s Space Conformal Infinity. Reports on Mathematical Physics (to appear), http://arxiv.org/abs/1107.0933 .
Th. Müller and D. Weiskopf, Detailed Study of Null and Time-Like Geodesics in the Alcubierre Warp Spacetime. Gen. Rel. Grav. (to appear), DOI:10.1007/s10714-011-1289-0, http://arxiv.org/abs/1107.5650 .
J. A. Wolf, Spaces of Constant Curvature. Fifth Edition, Publish or Perish, Inc., Wilmington, 1984.
J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry, Hyperbolic Geometry. In Flavors in Geometry, Silvio Levy, Ed., Cambridge University Press, MSRI Publications, Vol. 31 (1997), 59–115.
Ingraham R.L.: Conformal Relativity. Proc. Natl. Acad. Sci. U.S.A. 38(10), 921–925 (1952)
R. L. Ingraham, The Angle-Geometry of Spacetime and Classical Charged Particle Motion. Int. J. Modern Physics D, Vol. 7 (1998), 603–621.
R. L. Ingraham, Particle Masses and the Fifth Dimension. Annales de la Fondation Louis de Broglie, Vol. 29 Hors série 2 (2004), 989–1004.
Elstrodt J., Grunewald F., Mennicke J.: Groups Acting on Hyperbolic Space. Springer, Berlin (1998)
J. B. Hartle, Gravity: An Introduction to Einstein’s General Relativity. Addison- Wesley, 2002, http://web.physics.ucsb.edu/~gravitybook/math/ .
Cecil Th.E.: Lie Sphere Geometry. Springer-Verlag, New York (2000)
S. Lie, Über Komplexe, inbesondere Linien- und Kugelkomplexe, mit Anwendung auf der Theorie der partieller Differentialgleichungen. Math. Ann., 5 (1872), 145–208, 209–256 (Ges. Abh. 2, 121).
W. Blaschke, Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Vol. 3, Springer-Verlag, Berlin, 1929.
Wyler A.: On the Conformal Groups in the Theory of Relativity and their Unitary Representations. Arch. Rat. Mech. and Anal. 31, 35–50 (1968)
A. Wyler, The Complex Light Cone, Symmetric Space of the Conformal Group. Preprint IAS, Princeton, 1972.
Th. Kaluza, Zum Unitätsproblem der Physik. Sitzungsberichte d. Preuss. Akad. (1921), 966–972.
Klein O.: Quantentheorie und fünfdimensionale Relativitätstheorie. Zeits. f. Phys. 37, 895–906 (1926)
Fock V.A.: Über die invariante Form der Wellen-und Bewegungsgleichungen für einen geladenen Massenpunkt. Zeits. f. Phys. 39, 226–232 (1926)
Einstein A., Bergmann P.: On a Generalization of Kaluza’s Theory of Electricity. Ann. Math. 39, 683–701 (1938)
I. F. Ginzburg, M. Yu. Mikhailov (Rumer), V. L. Pokrovsky, “Pictures of Yuri Borisovich Rumer”. 2011, http://ru.dleex.com/read/7014 (in Russian).
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is dedicated to the memory of Jaime Keller.
Rights and permissions
About this article
Cite this article
Jadczyk, A. START in a Five-Dimensional Conformal Domain. Adv. Appl. Clifford Algebras 22, 689–701 (2012). https://doi.org/10.1007/s00006-012-0355-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00006-012-0355-3