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 nontrivial relativistic invariants, have concerned the possibility of relativistically enforcing some nonlinear laws on momentum space. For the recently proposed "relative-locality framework" 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, here I identify a requirement necessary for "DSR compatibility"; i.e. when the requirement is not satisfied a preferred-frame formulation of theories on that momentum space inevitably emerges. I find that, within a natural parametrization of momentum-space geometry, the requirement takes the form of an elementary algorithm. By working out a few examples I provide evidence that my requirement might be not only necessary but also sufficient: when the requirement is 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 κ-Poincaré-inspired momentum spaces. © 2012 American Physical Society