  Planck scale modified dispersion relations are one way how to capture the
influence of quantum gravity on the propagation of fundamental point particles
effectively. We derive the time dilation between an observer's or particle's
proper time, given by a Finslerian length measure induced from a modified
dispersion relation, and a reference laboratory time. To do so, the Finsler
length measure for general first order perturbations of the general
relativistic dispersion relation is constructed explicitly. From this we then
derive the time dilation formula for the $\kappa$-Poincar\'e dispersion
relation in several momentum space bases, as well as for modified dispersion
relations considered in the context of string theory and loop quantum gravity.
Most interestingly we find that the momentum Lorentz factor in the present and
future colliders can, in principle, become large enough to constrain the
Finsler realization of the $\kappa$-Poincar\'e dispersion relation in the
bicrossproduct basis as well as a string theory inspired modified dispersion
relation, at Planck scale sensitivity with the help of the muon's lifetime.
