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In this manuscript we provide a consistent way of describing a localizednon-relativistic quantum system undergoing a timelike trajectory in abackground curved spacetime. Namely, using Fermi normal coordinates, weidentify an inner product and canonically conjugate position and momentumoperators defined in the rest space of the trajectory for each value of itsproper time. This framework then naturally provides a recipe for mapping aquantum theory defined in a non-relativistic background to a theory around atimelike trajectory in curved spacetimes. This is done by reinterpreting theposition and momentum operators and by introducing a local redshift factor tothe Hamiltonian, which gives rise to new dynamics due to the curvature ofspacetime and the acceleration of the trajectory. We then apply our formalismto particle detector models, that is, to the case where the non-relativisticquantum system is coupled to a quantum field in a curved background. Thisallows one to write a general definition for particle detector models which isable to recover the previous models in the literature. Our framework alsoallows one to estimate the regime of validity of these models, characterizingthe situations where particle detectors can be used to accurately probe quantumfields.

Localized non-relativistic quantum systems in curved spacetimes: a general characterization of particle detector models