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An axiomatic system for Minkowski space–time
1.A. G. Walker, Proc. R. Soc. Edinburgh Sect. A 62, 319 (1948).
2.A. G. Walker, “Axioms for Cosmology,” in The Axiomatic Method, edited by L. Henkin, P. Suppes, and A. Tarski (North‐Holland, Amsterdam, 1959).
3.G. Szekeres, J. Aust. Math. Soc. 8, 134 (1968).
4.J. W. Schutz, Foundations of Special Relativity: Kinematic Axioms for Minkowski Space‐Time, Lecture Notes in Mathematics (Springer, Berlin‐Heidelberg‐New York, 1973), Vol. 361. This will be referred to subsequently as KA.
5.D. Hilbert, Grundlagen der Geometrie (Teubner, Leipzig, 1913).
5.Hilbert describes plane geometry in terms of two sets of undefined elements, points and lines, and an undefined relation of incidence. An alternative formal approach which describes a geometry as a set of points, with certain subsets called lines, is given by K. Borsuk and W. Szmielew, Foundations of Geometry (North‐Holland, Amsterdam, 1960). In the present axiomatic system, we regard Minkowski space‐time as a set of paths, each path being a set of undefined elements called “events”.
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11.Details of all proofs are given in the following research report: J. W. Schutz, “An Axiomatic System for Minkowski Space‐Time,” MPI‐PAE/Astro 181, April 1979 published by (and available on request from) Max‐Planck‐Institut für Physik und Astrophysik, Föhringer Ring 6, 8 München 40, Federal Republic of Germany. In this report the temporal order relation has been defined as a relation on the set of events, rather than as a relation on each path as in the present paper, and the axiom of connectedness (axiom IV, 1.31) is stated differently. The axiom of uniqueness (axiom II, 1.12) ensures the consistency of the two alternative definitions of temporal order.
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15.The significance of the statements and is that there should be paths joining the pairs of events d, a and d,f.
16.The term SPRAY in this context should not be confused with the different meaning assigned to the same word in the theory of second order differential equations as given by, for example, S. Lang, Introduction to Differentiable Manifolds (Wiley, New York, 1962).
17.CSP〈R, S〉 is a linearly ordered set, so any of its members separates it into two sides. A more detailed definition of the concept of left and right sides is given in Theorem 1.512.
18.Also see Theorem 1.73 for a more complete definition.
19.L. Redei, “Foundation of Euclidean and Non‐Euclidean Geometries According to F. Klein,” International Series of Monographs in Pure and Applied Mathematics (Pergamon, Oxford‐New York‐Toronto, 1968), Vol. 97.
20.It would be sufficient to consider a mapping o/ of into itself. Then the statement of (ii) would have to be modified because o/ would, in general, send subsets of paths onto subsets of paths.
21.The axiomatic system could be restated by rephrasing each theorem with the conditional statement “If there are at least two distinct paths…” prior to the statement of the theorem as given. Then with a simple restatement of the axiom of dimension (axiom IX, 2.1) a separate axiom of existence would not be required and the axiomatic system would appear to have one less axiom.
22.Veblen also states an axiom of uniqueness of parallels which is required for Euclidean geometry, but not for ordered geometry.
23.With the following substitutions for quoted propositions, where the firstmentioned 2 is to KA and the second is to the present treatment: [Axiom I, 2.2 |1.72], [Theorem 1, 2.5 |1.72], [Axiom, VIII, 2.10 |Axiom IX (2.1)], [Axiom X, 2.12|Axiom IV (1.31)], [Corollary 1 to Theorem 33, 6.4 |Corollary 1.512], [Theorem 56, 8.4 11.9], [Theorem 57, 9.1 |2.33].
24.M. A. Castagnino, J. Math. Phys. 12, 2203 (1971).
25.W. Kundt and B. Hoffmann, “Determination of Gravitational Standard Time,” in Recent Developments in General Relativity, (Warszawa, Warszaw, 1962), p. 63.
26.R. F. Marzke and J. A. Wheeler, “Gravitation as Geometry I,” in Gravitation and Relativity, edited by H. Y. Chiu and W. F. Hoffman (Benjamin, New York, 1964).
27.F. A. E. Pirani, Building Space‐Time from Light Rays and Free Particles, Symposia Mathematica (Academic, London, 1973), Vol. XII, p. 67.
28.J. L. Synge, Relativity: The General Theory (North‐Holland, Amsterdam, 1960).
29.H. Reichenbach, Axiomatik der relativistischen Raum‐Zeit (Vieweg, Braunschweig, 1924).
30.H. Weyl, Raum, Zeit, Materie (Springer, Berlin, 1918);
30.and H. Weyl, Mathematische Analyse des Raumproblems (Springer, Berlin, 1913).
31.J. Ehlers, F. A. Pirani, and A. Schild, “The Geometry of Free Fall and Light Propagation,” in General Relativity, Papers in Honour of J. L. Synge (Oxford University, Oxford, 1972).
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