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The isotropy mappings of Minkowski space-time generate the orthochronous poincaré group

Published online by Cambridge University Press:  17 February 2009

John W. Schutz
John W. Schutz
Affiliation:
Bendigo College of Advanced Eduction, Bendigo, Victoria 3550, Australia.
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Abstract

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Minkowski space-time is specified with respect to a single coordinate frame by the set of timelike lines. Isotropy mappings are defined as automorphisms which leave the events of one timelike line invariant. We demonstrate the existence of two special types of isotropy mappings. The first type of isotropy mapping induce orthogonal transformations in position space. Mappings of the second type can be composed to generate Lorentz boosts. It is shown that isotropy mappings generate the orthochronous Poincaré group of motions. The set of isotropy mappings then maps the single assumed coordinate frame onto a set of coordinate frames related by transformations of the orthochronous Poincaré group.

Type
Research Article
Information
The ANZIAM Journal , Volume 31 , Issue 4 , April 1990 , pp. 425 - 433
Copyright
Copyright © Australian Mathematical Society 1990

References

[1] Alexandrov, A. D., “On Lorentz transformations”, Usp. Mat. Nauk 5:3 (1950) 187.Google Scholar
[2] Alexandrov, A. D., “Mappings of Spaces with Families of Cones and Space-Time Transformations”, Annali di Mathematica Pura ed Applicata 103 (1975) 229–257.Google Scholar
[3] Borchers, H. J. and Hegerfeldt, G. C., “The structure of space-time transformations”, Commun. Math. Phys. 28 (1972) 259–266.Google Scholar
[4] Einstein, A., “Zur Elektrodynamic bewegter Körper”, Ann. der Phys. 17 (1905) 891–921. [“On the electrodynamics of moving bodies”, in A. Einstein, H. A. Lorentz, H. Minkowski and H. Weyl, ‘The Principle of Relativity’, translated by W. Perrett and G. B. Jeffrey (Methuen, 1923). (Reprinted. Dover, New York, 1958).].Google Scholar
[5] Guts, A. K., “Axiomatic relativity theory”, Usp. Mat. Nauk 37:2 (1982) 39–79. [Russian Math. Surveys 37:2 (1982) 41–89.].Google Scholar
[6] Lenard, A., “A characterization of Lorentz transformations”, J. Math. Phys. 19 (1978) 157.Google Scholar
[7] Lester, J. A. and McKiernan, M. A., “On null cone preserving mappings”, Math. Proc. Cambridge Philos. Soc. 81 (1977) 455–462.CrossRefGoogle Scholar
[8] Robb, A. A., The absolute relations of time and space (Cambridge University Press, Cambridge, 1921).Google Scholar
[9] Schutz, J. W., Foundations of special relativity, kinematic axioms for Minkowski space-time, Lecture Notes in Mathematics 361 (Springer, Berlin-Heidelberg-New York, 1973).Google Scholar
[10] Schutz, J. W., “Affine structure and isotropy imply Minkowski space-time and the orthochronous Poincaré group”, J. Math. Phys. 30 (1989) 635–638.Google Scholar
[11] Zeeman, E. C., “Causality implies the Lorentz group”, J. Math. Phys. 5 (1964) 490–493.Google Scholar