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


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.


Dimensional Space Convex Cone Lorentz Group Causal Connection Double Cone 
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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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