Hyperboloid preservation implies the Lorentz and Poincaré groups without dilations

An analogue of the Alexandrov-Zeeman theorem, based on hyperboloid preservation, as opposed to light cone preservation, is provided. This characterizes exactly the Poincaré group, as opposed to the Poincaré group extended by dilations. The hyperbolic analogue, as opposed to the cone-based Alexandrov-Zeeman theorem, is also valid in the case of a single space dimension. An orthochronous version also holds, based on the preservation of forward hyperboloid shells.

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