Influence of quantum matter fluctuations on geodesic deviation

We study the passive influence of quantum matter fluctuations on the expansion parameter of a congruence of timelike geodesics in a semiclassical regime. In particular, we show that the perturbations of this parameter can be considered to be elements of the algebra of observables of matter fields at all perturbative orders. Hence, once a quantum state for matter is chosen, it is possible to explicitly evaluate the behavior of geometric fluctuations. After introducing the formalism necessary to treat similar problems, in the last part of the paper, we estimate the approximated probability of having a geodesic collapse in a flat spacetime due to those fluctuations.