Linearity of space-time transformations without the one-to-one, line-onto-line, or constancy-of-speed-of-light assumptions

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Using a version of the fundamental theorem of geometry without the 1-to-1 assumption, recently obtained by the authors, the following is proved: Let n ≥ 2 and T be a mapping of Rn onto itself which maps every timelike line ℓ into an arbitrary line so that the image of every future ray of ℓ contains at least two distinct points or the same holds for every past ray of ℓ. Then T is affine. A version of the Pappus theorem under minimal assumptions is also given, which is then used as a tool in this paper. Related results have been obtained by Borchers and Hegerfeldt.

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Communications in Mathematical Physics