Large-scale non-locality in «doubly special relativity» with an energy-dependent speed of light

There are two major alternatives for violating the (usual) Lorentz invariance at large (Planckian) energies or momenta – either not all inertial frames (in the Planck regime) are equivalent (e.g., there is an effectively preferred frame) or the transformations from one frame to another are (non-linearly) deformed («doubly special relativity»). We demonstrate that the natural (and reasonable) assumption of an energy-dependent speed of light in the latter method goes along with violations of locality/separability (and even translational invariance) on macroscopic scales.