Comparison of relativity theories with observer - independent scales of both velocity and length / mass

There are now in the literature two proposals of relativity theories with observer-independent scales of both velocity and length/mass: one, which we here call “DSR1”, proposed by Amelino-Camelia in gr-qc/0012051,hep-th/0012238 and one, which we here call “DSR2”, proposed by Magueijo and Smolin in hep-th/0112090. We argue that a key aspect of such theories are the kinematical conditions for particle production in collision processes. We derive these kinematical conditions for DSR2, which was originally formulated only in the one-particle sector. This result is then used to characterize the differences between the two proposals; in particular, we find that, while experiments such as AMS and GLAST can test some of the predictions of DSR1, the effects predicted by DSR2 are much weaker and none of them should be testable in the foreseeable future. We also show that if one interprets, as done in hep-th/0112090, the DSR2-boost parameter v as the relative velocity between the observers connected by the boost one is then forced to give up the “Hamiltonian” relation vparticle = dE/dp between the speed of a particle and its momentum. We argue that, adopting an accordingly modified interpretation of the DSR2-boost parameter v, one can maintain the relation vparticle = dE/dp in DSR2.