On backreaction of matter fields on the spacetime metric

We will consider several problems within the framework of classical Einstein gravity and quasi-classical gravity, which consider the influence of classical particles and quantum fields on the background metric of spacetime. First, the problem of a classical particle falling into extremal black holes will be considered, and it will be shown that metric perturbations for extremal black holes undergo instabilities. Next, within the framework of quantum field theory in curved spacetime, the behavior of Gaussian quantum fields in a thermal state with an arbitrary temperature will be studied in the background of spacetimes with horizons. It will be shown that the Wightman function changes its behavior if both points lie on the horizon, and that the regularized energy-momentum tensor diverges at the horizon when the field temperature does not coincide with the canonical temperature of the horizon. The nonstationary dynamics of interacting fields in the expanding Friedmann-Lemaitre-Robertson-Walker geometry will also be considered and the question of the influence of quantum effects on the geometry will be investigated.