Reaching the Planck scale with muon lifetime measurements

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 loop quantum gravity and Ho\v{r}ava-Lifshitz 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 $\kappa$-Poincar\'e dispersion relation in the bicrossproduct basis with Planck scale sensitivity with help of the muon's lifetime.