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

Reaching the Planck scale with muon lifetime measurements