Spacetime curvature from ultrarapid measurements of quantum fields

We write the curvature of spacetime in terms of the excitation probability of particle detectors ultrarapidly coupled to a quantum field. More precisely, we provide an expansion for the excitation probability of a smeared Unruh-DeWitt detector delta-coupled to a real scalar quantum field in a curved background. Using a short-distance expansion for the Wightman function, we express the excitation probability of a detector as the transition probability in Minkowski spacetime plus correction terms written as a function of the curvature tensors and the detector size. Comparing the excitation probability in curved spacetimes with its flat analog, we are able to express the components of the Ricci and Riemann curvature tensors as a function of physically measurable excitation probabilities of different-shaped detectors.