Black Holes and First Order Flows in Supergravity

Abstract

We review the description of static, spherically symmetric, asymptotically- flat black holes in four dimensional supergravity in terms of an autonomous Hamiltonian system. A special role in this analysis is played by the so called fake superpotentialW, which is identified with a particular solution to the Hamilton-Jacobi equation. This function defines a first order, gradient-flow, description of the radial flow of the scalar fields, coupled to the solution, and of the red-shift factor. Identifying W with the Liapunov’s function, we can make the general statement that critical points of W are asymptotically stable equilibrium points of the corresponding first order dynamical system (in the sense of Liapunov). Such equilibrium points way only exist for extremal regular solutions and define their near horizon behavior. Thus the fake superpotential provides an alternative characterization of the attractor phenomenon. We focus on extremal black holes and deduce very general properties of the fake superpotential from its duality invariance. In particular we shall show that W has, along the entire radial flow, the same flat directions which exist at the attractor point. This allows to study properties of the ADM mass also for small black holes where in fact W has no critical points at finite distance in moduli space. In particular the W function for small non-BPS black holes can always be computed analytically, unlike for the large black hole case.