Spherically symmetric systems of fields and black holes. III. Positivity of energy and of a new type Euclidean action

Within the scope of the two-dimensional model of gravity which was defined and studied in the two preceding papers, we investigate the three famous positivity problems of general relativity: (1) energy, (2) Euclidean action, and (3) divergence identities. We show that the energy can be split in a unique way into a black-hole mass and a field energy. If there are no fields in the model which could discharge the hole, then the field energy itself is non-negative and has a zero minimum for the vacuum value of the fields. In the opposite case, the greatest lower bound for the total energy is the irreducible mass of the hole. We define a new type of Euclidean action for gravity theories, which is different from the action used currently in Euclidean quantum gravity. The new Euclidean action is obtained by a true analytical continuation of a reduced Lorentzian action so that the relation between the Euclidean and Lorentzian regimes is well defined. We prove that the new Euclidean action is positive definite without any additional "complexification." We show that the possibility of gravitational collapse leads to an unusual, saturation-curve-like form, of the Hamiltonian and of the new Euclidean action.