  In this manuscript we provide a consistent way of describing a localized
non-relativistic quantum system undergoing a timelike trajectory in a
background curved spacetime. Namely, using Fermi normal coordinates, we
identify an inner product and canonically conjugate position and momentum
operators defined in the rest space of the trajectory for each value of its
proper time. This framework then naturally provides a recipe for mapping a
quantum theory defined in a non-relativistic background to a theory around a
timelike trajectory in curved spacetimes. This is done by reinterpreting the
position and momentum operators and by introducing a local redshift factor to
the Hamiltonian, which gives rise to new dynamics due to the curvature of
spacetime and the acceleration of the trajectory. We then apply our formalism
to particle detector models, that is, to the case where the non-relativistic
quantum system is coupled to a quantum field in a curved background. This
allows one to write a general definition for particle detector models which is
able to recover the previous models in the literature. Our framework also
allows one to estimate the regime of validity of these models, characterizing
the situations where particle detectors can be used to accurately probe quantum
fields.
