Metric fluctuations and the weak equivalence principle

We describe spacetime fluctuations by means of small fluctuations of the metric on a given background metric. From a minimally coupled Klein-Gordon equation we obtain a modified Schrodinger equation within a weak-field approximation up to second order and an averaging procedure over a finite spacetime scale given by the quantum particle in the nonrelativistic limit. The dominant modification consists of an anomalous inertial mass tensor which depends on the type of particle and the fluctuation scenario. The scenario considered in this paper is a most simple picture of spacetime fluctuations and gives an existence proof for an apparent violation of the weak equivalence principle and, in general, for a violation of Lorentz invariance.