  Recently a model of metric fluctuations has been proposed which yields an
effective Schr\"odinger equation for a quantum particle with a modified
inertial mass, leading to a violation of the weak equivalence principle. The
renormalization of the inertial mass tensor results from a local space average
over the fluctuations of the metric over a fixed background metric. Here, we
demonstrate that the metric fluctuations of this model lead to a further
physical effect, namely to an effective decoherence of the quantum particle. We
derive a quantum master equation for the particle's density matrix, discuss in
detail its dissipation and decoherence properties, and estimate the
corresponding decoherence time scales. By contrast to other models discussed in
the literature, in the present approach the metric fluctuations give rise to a
decay of the coherences in the energy representation, i. e., to a localization
in energy space.
