BLACK HOLES IN SUPERGRAVITY AND HAMILTON-JACOBI FORMALISM

In my thesis we have addressed the issue of the first order description of generic stationary axisymmetric black holes in supergravity. To this the end we extended the extend the Hamilton-Jacobi formalism from mechanical models, whose degrees of freedom depend on just one variable, to field theories where the degrees of freedom depend on two or more variables. This problem was addressed and developed in generality in field theory, but not much was known in the context of gravitational field theories. An important issue in this thesis was to apply such extended formalism to the study of black holes. We have worked with the so-called De Donder-Weyl-Hamilton-Jacobi (DWHJ) theory, which is the simplest extension of the classical Hamilton-Jacobi approach in mechanics. One important difference with respect to the case of classical mechanics consists in the replacement of the Hamilton principal function S, directly related to the fake-superpotential of static black holes, with a Hamilton principal 1-form, which is a covariant vector Si. The application of this formalism to the description of axisymmetric solutions black holes required working out the general form of the principal functions Sm associated with the corresponding effective 2D sigma-model in the DWHJ setting. We have also given a characterization of the general properties of such solutions with respect to the global symmetry group of the effective 2D sigma-model which describes them. This was done by introducing, aside from the Nöther charge matrix, a further characteristic constant matrix Qψ, in the Lie algebra of G(3), the global symmetry properties of affine solutions 2D model, associated with the rotational motion of the black hole.