Summary
The kinematics implied by the VPG transformations is developed as the special caseg(β)=K(β)−1 of the kinematics implied by the asymmetric transformationsL g *(β)=g(β)L(β),L(β) being the ordinary 1-parametric Lorentz transformation. It is shown that the 1-parametric transformationsL g *, which do not form a group, give rise to two (formally identical) 2-parametric groups ĝ={Ĝ ik} and ğ={Ĝ ϱσ} connecting frames of two distinct classesĈ={Σ i} andČ={Σ ϱ}, respectively, eachΣ i being connected with anyΣ ϱ by an asymmetrical transformationL g*. It is concluded that this theory is the mathematical model of a hypothetical world possessingtwo uniform motion equivalences represented byĈ andČ, respectively. Some misconceptions that have arisen in connection with the work of Palacios and Gordon are resolved.
Riassunto
Si sviluppa la cinematica implicata dalle transformazioni VPG come caso specialeg(β)=K(β)−1 delle cinematiche implicate dalle trasformazioni asimmetricheL g*(β)==g(β)L(β), in cuiL(β) è la trasformazione di Lorentz monoparametrica ordinaria. Si mostra che le trasformazioni monoparametricheL g*, che non formano un gruppo, danno origine a due gruppi diparametrici (formalmente identici) ĝ={Ĝ ik} e ğ={Ĝ ϱσ} che connettono le strutture di due classiĈ={Σ i} eČ={Σ ϱ}, rispettivamente, ciascunΣ i essendo connesso con ogniΣ ϱ da una trasformazione asimmetricaL g*. Si conclude che questa teoria è il modello matematico di un ipotetico mondo che possiededue equivalenze di moto uniformi rappresentate rispettivamente daĈ eČ. Si spiegano alcun malintesi sorti in relazione al lavoro di Palacios e Gordon.
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Literatur
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The most general integrable point transformation between moving observers maintaining an invariant speedc found byCashmore (l. c.) contains 19 group parameters. Of these, four can be shown to be redundant in the sense that the manifold of transformations is not reduced if these four parameters are equated to constants. The resulting 15-parametric group is identical with the full conformal group in four dimensions.
For a more comprehensive critical review cf.M. Strauss:The Lorentz Group: Axiomatics, Generalizations, Alternatives. International Seminar on Relativistic Physics, Georgenthal (February 1965).
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Strauss, M. On the voigt-palacios-gordon transformation and the kinematics implied by it. Nuovo Cim 39, 658–666 (1965). https://doi.org/10.1007/BF02735831
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DOI: https://doi.org/10.1007/BF02735831