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Relativity without light: a new proof of Ignatowski’s theorem. (English) Zbl 1473.22018

A theorem formulated by Ignatowski for two dimensional space-times, and generalized by Corini to higher dimensions, states that based on simple assumptions (not including the assumption that the speed of light is invariant), the symmetry group of the space-time is one of only a few possibilities. A simplified proof is given for this theorem.

MSC:

22E70 Applications of Lie groups to the sciences; explicit representations
83A05 Special relativity

References:

[1] Berger, Marcel, Geometry I (1987), Springer-Verlag: Springer-Verlag Berlin · Zbl 0606.51001
[2] Besse, Arthur L., Einstein Manifolds (1987), Springer-Verlag: Springer-Verlag Berlin · Zbl 0613.53001
[3] Bourbaki, Nicolas, Groupes et algèbres de Lie. Chapitre II: Algèbres de Lie libres, Chapitre III: Groupes de Lie (1972), Hermann: Hermann Paris · Zbl 0244.22007
[4] Bourbaki, Nicolas, Algèbre. Chapitre 8: Modules et anneaux semi-simples (2012), Springer-Verlag: Springer-Verlag Berlin · Zbl 1245.16001
[5] Einstein, Albert, Zur Elektrodynamik bewegter Körper, Ann. Phys. (4), 17, 891-921 (1905) · JFM 36.0920.02
[6] Fell, J. M.G.; Doran, Robert S., Representations of \({}^\ast \)-Algebras, Locally Compact Groups, and Banach \({}^\ast \)-Algebraic Bundles (1988), Academic Press: Academic Press Boston · Zbl 0652.46050
[7] Frank, Philipp G., Das Relativitätsprinzip der Mechanik und die Gleichungen für die elektromagnetischen Vorgänge in bewegten Körpern, Ann. Phys. (4), 27, 897-902 (1908) · JFM 39.0910.03
[8] Godement, Roger, Introduction to the Theory of Lie Groups (2017), Springer: Springer Cham · Zbl 1367.22001
[9] Goldhaber, Alfred S.; Nieto, Michael M., Photon and graviton mass limits, Rev. Modern Phys., 82, 939-979 (2010)
[10] Gorini, Vittorio, Linear kinematical groups, Comm. Math. Phys., 21, 150-163 (1971) · Zbl 0224.22019
[11] Gorini, Vittorio, Derivation of the Lorentz and Galilei groups from rotational invariance, (Barut, Asim O., Studies in Mathematical Physics (Istanbul, August, 1970). Studies in Mathematical Physics (Istanbul, August, 1970), NATO Advanced Study Institutes Series C: Mathematical and Physical Sciences, vol. 1 (1973), Reidel: Reidel Dordrecht, Boston), 179-196 · Zbl 0275.22017
[12] Hahn, Emil, Grundlagen zu einer Theorie der Lorentztransformationen, Arch. Math. Phys. (3), 21, 1-42 (1913) · JFM 44.0777.03
[13] Hegerfeldt, Gerhard C., The Lorentz transformations: Derivation of linearity and scale factor, Nuovo Cimento A (11), 10A, 257-267 (1972)
[14] Hilgert, Joachim; Neeb, Karl-Hermann, Structure and Geometry of Lie Groups (2012), Springer: Springer New York · Zbl 1229.22008
[15] Ignatowski, Vladimir S., Einige allgemeine Bemerkungen zum Relativitätsprinzip, Verh. Dtsch. Phys. Ges., 12, 788-796 (1910), (Reprint: Physik. Z. 11 (1910) 972-976) · JFM 41.0766.01
[16] Ignatowski, Vladimir S., Das relativitätsprinzip, Arch. Math. Phys. (3), 17, 1-24 (1911) · JFM 41.0766.02
[17] Jordan, Camille, Traité des substitutions et des équations algébriques (1870), Gauthier-Villars: Gauthier-Villars Paris · JFM 03.0042.02
[18] Lalan, Victor, Sur les postulats qui sont à la base des cinématiques, Bull. Soc. Math. France, 65, 83-99 (1937) · JFM 63.0347.02
[19] Lévy-Leblond, Jean-Marc, Une nouvelle limite non-relativiste du groupe de Poincaré, Ann. Inst. Henri Poincaré A, 3, 1-12 (1965) · Zbl 0143.22601
[20] Lorentz, Hendrik A., Deux mémoires de Henri Poincaré sur la physique mathématique, Acta Math., 38, 293-308 (1921)
[21] Miller, Arthur I., Albert Einstein’s Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905-1911) (1981), Addison-Wesley Publishing Co.: Addison-Wesley Publishing Co. Reading, Mass
[22] Minkowski, Hermann, Raum und zeit, Jber. Dtsch. Math.-Ver., 18, 75-88 (1909), (Translations: Ann. Sci. École Norm. Sup. (3) 26 (1909) 499-517. Space and time, in The Principle of Relativity, pp. 73-91. Methuen, London, 1923)
[23] Pauli, Wolfgang, Relativitätstheorie, Encykl. Math. Wiss. V D, 19, 539-775 (1921), (Translation: Theory of Relativity. Pergamon Press, London, 1958) · JFM 48.0977.01
[24] Poincaré, Henri, Sur la dynamique de l’électron, Rend. Circ. Mat. Palermo, 21, 129-176 (1906) · JFM 37.0886.01
[25] Rossmann, Wulf, Lie Groups: An Introduction Through Linear Groups (2002), Oxford University Press: Oxford University Press Oxford · Zbl 0989.22001
[26] Souriau, Jean-Marie, Structure des Systèmes Dynamiques (1970), Dunod: Dunod Paris, (Reprint: Éditions Jacques Gabay, Sceaux, 2008. Translation: Structure of Dynamical Systems. Birkhäuser, Boston, 1997) · Zbl 0186.58001
[27] Whittaker, Edmund T., A History of the Theories of Aether and Electricity, Vol. II. The Modern Theories, 1900-1926 (1953), Thomas Nelson and Sons: Thomas Nelson and Sons London, Edinburgh · Zbl 0052.40802
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