Algebraic and Geometric Structures in Special Relativity

  • D. Giulini
Part of the Lecture Notes in Physics book series (LNP, volume 702)


I review, on an advanced level, some of the algebraic and geometric structures that underlie the theory of Special Relativity. This includes a discussion of relativity as a symmetry principle, derivations of the Lorentz group, its composition law, its Lie algebra, comparison with the Galilei group, Einstein synchronization, the lattice of causally and chronologically complete regions in Minkowski space, rigid motion, and the geometry of rotating reference frames. Representation-theoretic aspects of the Lorentz group are not included. A series of appendices present some related mathematical material.


Special Relativity Minkowski Space Lorentz Transformation Lorentz Group Rigid Motion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alexander Danilovich Alexandrov. Mappings of spaces with families of cones and space-time transformations. Annali di Matematica (Bologna), 103(8):229–257, 1975.MathSciNetCrossRefGoogle Scholar
  2. 2.
    James Anderson. Principles of Relativity Physics. Academic Press, New York, 1967.Google Scholar
  3. 3.
    Henri Bacry and Jean-Marc Lévy-Leblond. Possible kinematics. Journal of Mathematical Physics, 9(10):1605–1614, 1968.CrossRefzbMATHMathSciNetADSGoogle Scholar
  4. 4.
    Frank Beckman and Donald Quarles. On isometries of euclidean spaces. Proceedings of the American Mathematical Society, 4:810–815, 1953.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Enrico Beltrametti and Gianni Cassinelli. The Logic of Quantum Mechanics. Encyclopedia of Mathematics and its Application Vol. 15. Addison-Wesley, Reading, Massachusetts, 1981.Google Scholar
  6. 6.
    Marcel Berger. Geometry, volume I. Springer Verlag, Berlin, first edition, 1987. Corrected second printing 1994.CrossRefGoogle Scholar
  7. 7.
    Marcel Berger. Geometry, volume II. Springer Verlag, Berlin, first edition, 1987. Corrected second printing 1996.Google Scholar
  8. 8.
    Vittorio Berzi and Vittorio Gorini. Reciprocity principle and the Lorentz transformations. Journal of Mathematical Physics, 10(8):1518–1524, 1969.MathSciNetCrossRefzbMATHADSGoogle Scholar
  9. 9.
    David Bleecker. Gauge Theory and Variational Principles. Number 1 in Global Analysis, Pure and Applied. Addison-Wesley, Reading, Massachusetts, 1981.zbMATHGoogle Scholar
  10. 10.
    Hans-Jürgen Borchers and Gerhard Hegerfeld. The structure of space-time transformations. Communications in Mathematical Physics, 28:259–266, 1972.MathSciNetCrossRefADSzbMATHGoogle Scholar
  11. 11.
    Émile Borel. La théorie de la relativité et la cinématique. In Œuvres de Émile Borel, volume 3, pages 1809–1811. Editions du Centre National de la Recherche Scientifique, Paris, 1972. First appeared in Comptes Rendus des séances de l’Académie des Sciences 156 (1913): 215–217.Google Scholar
  12. 12.
    Max Born. Die Theorie des starren Elektrons in der Kinematik des Relativitätsprinzips. Annalen der Physik (Leipzig), 30:1–56, 1909.CrossRefADSGoogle Scholar
  13. 13.
    Horacio Casini. The logic of causally closed spacetime subsets. Classical and Quantum Gravity, 19:6389–6404, 2002.MathSciNetCrossRefADSzbMATHGoogle Scholar
  14. 14.
    Wojciech Cegła and Arkadiusz Jadczyk. Logics generated by causality structures. covariant representations of the galilei group. Reports on Mathematical Physics, 9(3):377–385, 1976.MathSciNetCrossRefzbMATHADSGoogle Scholar
  15. 15.
    Wojciech Cegła and Arkadiusz Jadczyk. Causal logic of Minkowski space. Communications in Mathematical Physics, 57:213–217, 1977.MathSciNetCrossRefADSzbMATHGoogle Scholar
  16. 16.
    Alexander Chubarev and Iosif Pinelis. Linearity of space-time transformations without the one-to-one, line-onto-line, or constancy-of-speed-of-light assumption. Communications in Mathematical Physics, 215:433–441, 2000.MathSciNetCrossRefADSzbMATHGoogle Scholar
  17. 17.
    Paul Ehrenfest. Gleichförmige Rotation starrer Körper und Relativitätstheorie. Physikalische Zeitschrift, 10(23):918, 1909.Google Scholar
  18. 18.
    Vladimir Fock. The Theory of Space Time and Gravitation. Pergamon Press, London, first english edition, 1959.zbMATHGoogle Scholar
  19. 19.
    Philipp Frank and Hermann Rothe. Über die Transformation der Raumzeitkoordinaten von ruhenden auf bewegte Systeme. Annalen der Physik (Leipzig), 34(5):825–855, 1911.CrossRefADSGoogle Scholar
  20. 20.
    Philipp Frank and Hermann Rothe. Zur Herleitung der Lorentztransformation. Physikalische Zeitschrift, 13:750–753, 1912. Erratum: ibid, p. 839.Google Scholar
  21. 21.
    Wilhelm I. Fushchich, Vladimir M. Shtelen, and N.I. Serov. Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics. Kluwer Academic Publishers, Dordrecht, 1993.zbMATHGoogle Scholar
  22. 22.
    Domenico Giulini. Advanced Special Relativity. Oxford University Press, Oxford. To appear.Google Scholar
  23. 23.
    Domenico Giulini. On Galilei invariance in quantum mechanics and the Bargmann superselection rule. Annals of Physics (New York), 249(1):222–235, 1996.MathSciNetCrossRefADSzbMATHGoogle Scholar
  24. 24.
    Domenico Giulini. Uniqueness of simultaneity. Britisch Journal for the Philosophy of Science, 52:651–670, 2001. 110 D. GiuliniMathSciNetGoogle Scholar
  25. 25.
    Domenico Giulini. Das Problem der Trägheit. Philosophia Naturalis, 39(2):843–374, 2002.MathSciNetGoogle Scholar
  26. 26.
    Domenico Giulini and Norbert Straumann. Einstein’s impact on the physics of the twentieth century. Studies in the History and Philisophy of Modern Physics, to appear, 2006. ArXiv physics/0507107.Google Scholar
  27. 27.
    Rudolf Haag. Local Quantum Physics. Springer Verlag, Berlin, first 1991 second revised and enlarged 1996 edition, 1996.zbMATHGoogle Scholar
  28. 28.
    Gerhard Hegerfeld. The Lorentz transformations: Derivation of linearity and scale factor. Il Nuovo Cimento, 10 A(2):257–267, 1972.ADSGoogle Scholar
  29. 29.
    Gustav Herglotz. Über den vom Standpunkt des Relativitätsprinzips aus als “starr” zu bezeichnenden Körper. Annalen der Physik (Leipzig), 31:393–415, 1910.CrossRefADSGoogle Scholar
  30. 30.
    Wladimir von Ignatowsky. Einige allgemeine Bemerkungen zum Relativitätsprinzip. Verhandlungen der Deutschen Physikalischen Gesellschaft, 12:788–796, 1910.Google Scholar
  31. 31.
    Erdal Inönü and Eugen Wigner. On the cotraction of groups and their representations. Proceedings of the National Academy of Sciences, 39(6):510–524, 1953.CrossRefADSzbMATHMathSciNetGoogle Scholar
  32. 32.
    Nathan Jacobson. Basic Algebra I. W.H. Freeman and Co., New York, second edition, 1985.zbMATHGoogle Scholar
  33. 33.
    Josef M. Jauch. Foundations of Quantum Mechanics. Addison-Wesley, Reading, Massachusetts, 1968.zbMATHGoogle Scholar
  34. 34.
    Theodor Kaluza. Zur Relativitätstheorie. Physikalische Zeitschrift, 11:977–978, 1910.Google Scholar
  35. 35.
    Felix Klein. Vergleichende Betrachtungen über neuere geometrische Forschungen. Verlag von Andreas Deichert, Erlangen, first edition, 1872. Reprinted in Mathematische Annalen (Leipzig) 43 (1892) 43–100.Google Scholar
  36. 36.
    Max von Laue. Zur Diskussion über den starren Körper in der Relativitätstheorie. Physikalische Zeitschrift, 12:85–87, 1911.Google Scholar
  37. 37.
    Fritz Noether. Zur Kinematik des starren Körpers in der Relativitätstheorie. Annalen der Physik (Leipzig), 31:919–944, 1910.CrossRefADSGoogle Scholar
  38. 38.
    Felix Pirani and Gareth Williams. Rigid motion in a gravitational field. Séminaire JANET (Mécanique analytique et Mécanique céleste), 5e année(8):1–16, 1962.Google Scholar
  39. 39.
    Alfred A. Robb. Optical Geometry of Motion: A New View of the Theory of Relativity. W. Heffer & Sons Ltd., Cambridge, 1911.Google Scholar
  40. 40.
    Roman U. Sexl and Helmuth K. Urbantke. Relativity, Groups, Particles. Springer Verlag, Wien, first edition, 2001. First english edition, succeeding the 1992 third revised german edition.zbMATHGoogle Scholar
  41. 41.
    Arnold Sommerfeld. Über die Zusammensetzung der Geschwindigkeiten in der Relativtheorie. Physikalische Zeitschrift, 10:826–829, 1909.Google Scholar
  42. 42.
    Llewellyn Hilleth Thomas. The kinematics of an electron with an axis. Philosophical Magazine, 3:1–22, 1927.Google Scholar
  43. 43.
    Andrzej Trautman. Foundations and current problems of general relativity. In A. Trautman, F.A.E. Pirani, and H. Bondi, editors, Lectures on General Relativity, volume 1 of Brandeis Summer Institute in Theoretical Physics, pages 1–248. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964.Google Scholar
  44. 44.
    Abraham Ungar. Thomas rotation and the parametrization of the Lorentz transformation group. Foundations of Physics Letters, 1(1):57–89, 1988.MathSciNetCrossRefADSGoogle Scholar
  45. 45.
    Abraham Ungar. Beyond Einstein’s Velocity Addition Law, volume 117 of Fundamental Theories of Physics. Kluwer Academic, Dordrecht, 2001.Google Scholar
  46. 46.
    Abraham Ungar. Analytic Hyperbolic Geometry: Mathematical Foundations and Applications. World Scientific, Singapore, 2005.zbMATHCrossRefGoogle Scholar
  47. 47.
    Helmuth Urbantke. Physical holonomy: Thomas precession, and Clifford algebra. American Journal of Physics, 58(8):747–750, 1990. Erratum ibid. 59(12), 1991, 1150–1151.MathSciNetCrossRefADSGoogle Scholar
  48. 48.
    Helmuth Urbantke. Lorentz transformations from reflections: Some applications. Foundations of Physics Letters, 16:111–117, 2003. ArXiv math-ph/0212038.MathSciNetCrossRefGoogle Scholar
  49. 49.
    Vladimir Varičcak. Anwendung der Lobatschefskijschen Geometrie in der Relativtheorie. Physikalische Zeitschrift, 11:93–96, 1910.Google Scholar
  50. 50.
    Vladimir Varičak. Über die nichteuklidische Interpretation der Relativtheorie. Jahresberichte der Deutschen Mathematikervereinigung (Leipzig), 21:103–127, 1912.Google Scholar
  51. 51.
    Erik Christopher Zeeman. Causality implies the Lorentz group. Journal of Mathematical Physics, 5(4):490–493, 1964.MathSciNetCrossRefzbMATHADSGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • D. Giulini
    • 1
  1. 1.Physikalisches Institut der Universität Freiburg Hermann–Herder–Straße 3FreiburgGermany

Personalised recommendations