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Annali di Matematica Pura ed Applicata

, Volume 103, Issue 1, pp 229–257 | Cite as

Mappings of spaces with families of cones and space-time transformations

  • A. D. Alexandrov
Article

Summary

Let A, A′ be affine spaces withdim A⩾3,dim A=∞ being not excluded. Let C, C′ be cones in A and A′; Cx, C′v denote the cones with the vertices x, y got from C, C′ by translation. Such one-to-one mappings f:A → A′ are considered that for every x ∈ A there exists such an y ∈ A′ that f(Cx)=C′v. It is shown that f is affine provided C, C′ belong to one of certain classes of cones: those of the second order and « strictly convex » ones, which are suitably defined, especially in infinitely dimensional space. Cones of the second order in n-space are determined by conditions of the following types: (1) ϕ(x) ≡ ≡x 1 2 +...+x m 2 −x m+1 2 −...−x n 2 =0, (2) ϕ(x)⩾0, (3) ϕ(x)>0 plus the vertex (0, ...,0). Strictly convex cones are of6 types, the simplest representatives of which in4-spaces are3 « double cones »: (I)\(x^2 \equiv x_0^2 - \sum\limits_1^3 {x_i^2 = 0} \), (II) x2⩾0, (III) x2>0 plus the vertex (0, ...,0), and three « ordinary » ones got from(I)–(III) by adding the condition x00. In space-time these cones correspond to different kinds of connection of events, e.g.(II) consists of points x causally connected with0=(0, ..., 0), an event at x influences that at0 or vice versa. Thus, the above result implies that a1−1 mapping of space-time onto itself preserving causal connection is affine. Therefore mere preservation of causal connection implies Lorentz group.

Keywords

Dimensional Space Convex Cone Lorentz Group Causal Connection Double Cone 
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.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1974

Authors and Affiliations

  • A. D. Alexandrov
    • 1
  1. 1.Novosibirsk

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