On the fate of Lorentz symmetry in relative-locality momentum spaces

The most studied doubly-special-relativity scenarios, theories with both the speed-of-light scale and a length/inverse-momentum scale as non-trivial relativistic invariants, have concerned the possibility of enforcing relativistically some nonlinear laws on momentum space. For the "relative-locality framework" recently proposed in arXiv:1101.0931 a central role is played by nonlinear laws on momentum space, with the guiding principle that they should provide a characterization of the geometry of momentum space. Building on previous doubly-special-relativity results I here identify a criterion for establishing whether or not a given geometry of the relative-locality momentum space is "DSR compatible", i.e. compatible with an observer-independent formulation of theories on that momentum space. I find that given some chosen parametrization of momentum-space geometry the criterion takes the form of an elementary algorithm. I show that relative-locality momentum spaces that fail my criterion definitely "break" Lorentz invariance, i.e. theories on such momentum spaces necessarily are observer-dependent "ether" theories. By working out a few examples I provide evidence that when the criterion is instead satisfied one does manage to produce a relativistic formulation. The examples I use to illustrate the applicability of my criterion also have some intrinsic interest, including two particularly noteworthy cases of $\kappa$-Poincar\'e-inspired momentum spaces.