Issues on Black Holes in Four Dimensional Supergravity

We discuss some issues related to spherically symmetric, asymptotically flat black hole solutions in four dimensional supergravity. Particular emphasis will be placed on their properties relative to the electric-magnetic duality symmetry of the theory. We first elaborate on the description of these solutions in terms of an autonomous Hamiltonian system. This will provide us with a useful framework where to study the description of specific classes of solutions in terms of a first order “gradient flow” dynamical system of equations defined by a superpotential W, solution to the Hamilton-Jacobi equation. Regular extremal solutions play a special role in this analysis sice the dynamical system exhibits an asymptotically stable equilibrium point for the scalar fields in correspondence to the horizon (attractor mechanism). We also comment on the relation between this formalism and the three-dimensional description of spherically symmetric black holes as geodesics on suitable pseudo-Riemannian manifolds.