  We consider the coupling of a scalar field to linearised gravity and derive a
relativistic gravitationally induced decoherence model using Ashtekar
variables. The model is formulated at the gauge invariant level using suitable
geometrical clocks in the relational formalism, broadening existing gauge
invariant formulations of decoherence models. For the construction of the Dirac
observables we extend the known observable map by a kind of dual map where the
role of clocks and constraints is interchanged. We also discuss a second choice
of geometrical clocks existing in the ADM literature. Then we apply a reduced
phase space quantisation on Fock space and derive the final master equation
choosing a Gibbs state for the gravitational environment and using the
projection operator technique. The resulting master equation is not
automatically of Lindblad type, a starting point sometimes assumed for
phenomenological models, but still involves a residual time dependence at the
level of the effective operators in the master equation due to the form of the
correlation functions that we express in terms of thermal Wightman functions.
Furthermore, we discuss why in the model analysed here the application of a
second Markov approximation in order to obtain a set of time independent
effective system operators is less straightforward than in some of the quantum
mechanical models.
