  Within all approaches to quantum gravity small violations of the Einstein
Equivalence Principle are expected. This includes violations of Lorentz
invariance. While usually violations of Lorentz invariance are introduced
through the coupling to additional tensor fields, here a Finslerian approach is
employed where violations of Lorentz invariance are incorporated as an integral
part of the space-time metrics. Within such a Finslerian framework a modified
dispersion relation is derived which is confronted with current high precision
experiments. As a result, Finsler type deviations from the Minkowskian metric
are excluded with an accuracy of 10^{-16}.
