  We investigate the implications of energy-dependence of the speed of photons,
one of the candidate effects of quantum-gravity theories that has been most
studied recently, from the perspective of observations in different reference
frames. We examine how a simultaneous burst of photons would be measured by two
observers with a relative velocity, establishing some associated conditions for
the consistency of theories. For scenarios where the Lorentz transformations
remain valid these consistency conditions allow us to characterize the
violations of Lorentz symmetry through an explicit description of the
modification of the quantum-gravity scale in boosted frames with respect to its
definition in a preferred frame. When applied to relativistic scenarios with a
deformation of Lorentz invariance that preserves the equivalence of inertial
observers, we find an insightful characterization of the necessity to adopt in
such frameworks non-classical features of spacetime geometry, e.g. events that
are at the same spacetime point for one observer cannot be considered at the
same spacetime point for other observers. Our findings also suggest that, at
least in principle (and perhaps one day even in practice), measurements of the
dispersion of photons in relatively boosted frames can be particularly valuable
for the purpose of testing these scenarios.
