First order description of D=4 static black holes and the Hamilton-Jacobi equation

In this Note we discuss the application of the Hamilton-Jacobi formalism to the first order description of four-dimensional spherically symmetric and static black holes. In particular we show that the prepotential characterizing the flow coincides with the Hamilton principal function associated with the one-dimensional effective Lagrangian. This implies that the prepotential can always be defined, at least locally in the radial variable and in the moduli space, both in the extremal and non-extremal cases and allows us to conclude that it is duality invariant. We also give, in this framework, a general definition of the “Weinhold metric” in terms of which a necessary condition for the existence of multiple attractors is given. The Hamilton-Jacobi formalism can be applied both to the restricted phase space where the electromagnetic potentials have been integrated out as well as in the case where the electromagnetic potentials are dualized to scalar fields using the so-called three-dimensional Euclidean approach. We give some examples of application of the formalism, both for the BPS and the non-BPS black holes.