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Affine structure and isotropy imply Minkowski space‐time and the orthochronous Poincaré group
1.H. Busemann, The Geometry of Geodesies (Academic, New York, 1955). Busemann states and proves the proposition: “(16.11) Let be a closed convex surface in and z an interior point of If for any two points p and q of an affinity exists that leaves z fixed, maps on itself and p on q, then is an ellipsoid with center z.” The convex surfaced of Busemann’s proposition corresponds to the boundary of the convex body of our Theorem 1.
2.H. Freudenthal, “Lie groups in the foundations of geometry,” Adv. Math. 1, 145 (1965).
3.H. Busemann, “Timelike spaces,” Dissertationes Math. (Rozprawy Mat.) 53, 1 (1967).
4.Busemann considers “timelike Minkowski spaces,” which have the additional structure of a pseudometric “gauge function.” Accordingly, the usual Minkowski space‐time (Lorentz space in Busemann’s terminology) is characterized by conditions on timelike and lightlike lines, with an additional condition of isometry, or “triplewise transitivity.” In the present paper, Minkowski space‐time is considered to be described by the structure of its timelike (and lightlike) lines; the squared interval and the pseudometric are derived concepts.
5.H. Freudenthal, “Das Helmholtz‐Liesche raumproblem bei indefiniter metrik,” Math. Ann. 156, 263 (1964).
6.D. Mayr, “A constructive‐axiomatic approach to physical space and space‐time geometries of constant curvature by the principle of reproducibility,” in Space‐Time and Mechanics, edited by D. Mayr and G. Sussmann (Reidel, Dordrecht, 1983).
7.A. D. Aleksandrov, “Cones with a transitive group,” Dokl. Akad. Nauk SSSR 189, 695 (1969)
7.[A. D. Aleksandrov, Sov. Math. Dokl. 10, 1460 (1969)].
8.Aleksandrov (Ref. 7) considers a property of “transitivity” on the rays of the future light cone; the corresponding symmetry group is the orthochronous Lorentz group with dilatations as shown by Aleksandrov (Ref. 9) and Zeeman (Ref. 10). The same result is implied by a result of Busemann (Ref. 3) (§7, p. 43, Theorem 9). An alternative characterization in terms of transitivity on the lines of the light cone can be based on Busemann’s characterization (Ref. 3) (§6, p. 34, Proposition 2) of ellipsoids in projective space; then the corresponding symmetry group is the Lorentz group with dilatations, as obtained by Alexandrov (Ref. 9) and Borchers and Hegerfeldt (Ref. 11).
9.A. D. Alexandrov [sic, same author as Aleksandrov (Ref. 7)], “On Lorentz transformations,” Usp. Mat. Nauk 5, 187 (1950);
9.A. D. Alexandrov, “Mappings of spaces with families of cones and space‐time transformations,” Ann. Mat. Pura Appl. 103, 229 (1975).
10.E. C. Zeeman, “Causality implies the Lorentz group,” J. Math. Phys. 5, 490 (1964).
11.H. J. Borchers and G. C. Hegerfeldt, “The structure of space‐time transformations,” Commun. Math. Phys. 28, 259 (1972).
12.R. I. Pimenov, “Kinematic spaces,” Zap. Nauc̆. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 6, 3 (1968).
12.[Seminars in Mathematics (V. A. Steklova Mathematical Institute, Leningrad) 6, 1 (1970)].
12.Pimenov defines an “affine space‐time” to have a partial order relation < in addition to the affine properties used in the present paper.
13.A. D. Alexandrov [sic, same author as Aleksandrov (Ref. 7)], “A contribution to chronogeometry,” Can. J. Math. 19, 1119 (1967).
14.J. W. Schutz, “Foundations of special relativity: Kinematic axioms for Minkowski space‐time,” Lecture Notes in Mathematics, Vol. 361 (Springer, Berlin, 1973).
15.J. W. Schutz, “An axiomatic system for Minkowski space‐time,” J. Math. Phys. 22, 293 (1981).
16.G. Szekeres, “Kinematic geometry: An axiomatic system for Minkowski space‐time,” J. Aust. Math. Soc. 8, 134 (1968).
17.A. G. Walker, “Axioms for cosmology,” in The Axiomatic Method, edited by L. Henkin, P. Suppes, and A. Tarski (North‐Holland, Amsterdam, 1959).
18.A. K. Guts, “Axiomatic relativity theory,” Usp. Mat. Nauk 37, 39 (1982)
18.[A. K. Guts, Russian Math. Surveys 37, 41 (1982)].
19.G. C. Hegerfeldt, “The Lorentz transformations: Derivation of linearity and scale factor,” Nuovo Cimento A 10, 257 (1972).
19.The result of linearity is stated for mappings of lines parallel to the lines within a light cone. However, the proof applies even more generally to the mappings of straight lines which satisfy our conditions (i) and (iii). In the present situation an even simpler proof may be used, since events on one path are invariant and conditions (ii) and (iii) imply that each plane which contains the invariant path contains a set of timelike lines bounded by two lightlike lines.
20.H. Kneser, “Eine Erweiterung des Begriffes ‘konvexer Körper,’” Math. Ann. 82, 287 (1921). Satz 5 (Theorem 5) on p. 296 includes the stated result as a special case. Alternatively the result may be obtained directly, using elementary methods of projective geometry.
21.Since we are considering affinities, we may also use the well‐known and less general result which is given, for example, by W. Rindler, Special Relativity (Oliver and Boyd, Edinburgh, 1960), p. 21.
22.The results of Sec. III are obtained directly using the same property of isotropy by J. W. Schutz, “The isotropy mappings of Minkowski space‐time generate the orthochronous Poincaré group,” to be published.
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