First Order Description of Black Holes in Moduli Space

Abstract

We show that the second order field equations characterizing extremal solutions for spherically symmetric, stationary black holes are in fact implied by a system of first order equations given in terms of a prepotential W. This confirms and generalizes the results in [14]. Moreover we prove that the squared prepotential function shares the same properties of a c-function and that it interpolates between M^2_{ADM} and M^2_{BR}, the parameter of the near-horizon Bertotti-Robinson geometry. When the black holes are solutions of extended supergravities we are able to find an explicit expression for the prepotentials, valid at any radial distance from the horizon, which reproduces all the attractors of the four dimensional N>2 theories. Far from the horizon, however, for N-even our ansatz poses a constraint on one of the U-duality invariants for the non-BPS solutions with Z \neq 0. We discuss a possible extension of our considerations to the non extremal case.Comment: Some points clarified, a comment on the interpretation of the prepotential W in terms of c-function added, typos corrected. Version to appear on JHE