The stress-energy tensor of an Unruh-DeWitt detector

We propose a model for a finite-size particle detector, which allows us to derive its stress-energy tensor. This tensor is obtained from a covariant Lagrangian that describes not only the quantum field that models the detector, ${\phi }_{\text{d}}$, but also the systems responsible for its localization: a complex scalar field, ${\psi }_{\text{c}}$, and a perfect fluid. The local interaction between the detector and the complex field ensures the square integrability of the detector modes, while the fluid serves to define the spatial profile of ${\psi }_{\text{c}}$, localizing it in space. We then demonstrate that, under very general conditions, the resulting energy tensor – incorporating all components of the system – is physically reasonable and satisfies the energy conditions.