  Observables which would indicate a modified vacuum dispersion relations,
possibly caused by quantum gravity effects, are a four momentum dependence of
the cosmological redshift and the existence of a so called lateshift effect for
massless or very light particles. Existence or non-existence of the later is
currently analyzed on the basis of the available observational data from gamma
ray bursts and compared to predictions of specific modified dispersion relation
models. We consider the most general perturbation of the general relativistic
dispersion relation of freely falling particles on homogeneous and isotropic
spacetimes and derive the red- and lateshift to first order in the
perturbation. Our result generalizes the existing formulae in the literature
and we find that there exist modified dispersion relations causing both, one or
none of the two effects to first order.
