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Ben,
Will you be able to recover Minkowski spacetime and relativity in general from your new principles without additionally assuming the constancy of the speed of light (Einstein's 1905 light postulate)? Einsteinians sometimes claim that "the constant speed of light is unnecessary for the construction of the theories of relativity" but this is a fraud of course:
Jean-Marc Lévy-Leblond: "Supposez que demain un expérimentateur soit capable de vraiment mettre la main sur le photon, et de dire qu'il n'a pas une masse nulle. Qu'il a une masse de, mettons 10^(-60)kg. Sa masse n'est pas nulle, et du coup la lumière ne va plus à la "vitesse de la lumière". Vous pouvez imaginer les gros titres dans les journaux : "La théorie de la relativité s'effondre", "Einstein s'est trompé", etc. Or cette éventuelle observation ne serait en rien contradictoire avec la théorie de la relativité!"
Jean-Marc Lévy-Leblond "De la relativité à la chronogéométrie ou: Pour en finir avec le "second postulat" et autres fossiles": "Il se pourrait même que de futures mesures mettent en évidence une masse infime, mais non-nulle, du photon ; la lumière alors n'irait plus à la "vitesse de la lumière", ou, plus précisément, la vitesse de la lumière, désormais variable, ne s'identifierait plus à la vitesse limite invariante. Les procédures opérationnelles mises en jeu par le "second postulat" deviendraient caduques ipso facto. La théorie elle-même en serait-elle invalidée ? Heureusement, il n'en est rien..."
Tom Roberts: "If it is ultimately discovered that the photon has a nonzero mass (i.e. light in vacuum does not travel at the invariant speed of the Lorentz transform), SR would be unaffected but both Maxwell's equations and QED would be refuted (or rather, their domains of applicability would be reduced)."
Why Einstein was wrong about relativity, 29 October 2008, Mark Buchanan, NEW SCIENTIST: "A photon with any mass at all would imply that our understanding of electricity and magnetism is wrong, and that electric charge might not be conserved. That would be problem enough, but a massive photon would also spell deep trouble for the second postulate, as a photon with mass would not necessarily always travel at the same speed. Feigenbaum's work shows how, contrary to many physicists' beliefs, this need not be a problem for relativity."
Tom Roberts: "As I said before, Special Relativity would not be affected by a non-zero photon mass, as Einstein's second postulate is not required in a modern derivation (using group theory one obtains three related theories, two of which are solidly refuted experimentally and the third is SR). So today's foundations of modern physics would not be threatened.
Mitchell J. Feigenbaum: "In this paper, not only do I show that the constant speed of light is unnecessary for the construction of the theories of relativity, but overwhelmingly more, there is no room for it in the theory. (...) We can make a few guesses. There is a "villain" in the story, who, of course, is Newton."
Pentcho Valev