Fake Superpotential for Large and Small Extremal Black Holes

TL;DR

This paper explores the properties of fake superpotentials in supergravity black holes, demonstrating their stability, invariance, and analytical computability, especially for small non-BPS black holes.

Contribution

It clarifies the properties of fake superpotentials, including their stability and invariance, and provides analytical methods for small black holes, extending previous understanding.

Findings

Fake superpotential W acts as a Lyapunov function, ensuring stability of extremal black hole horizons.

W maintains flat directions along the radial flow, relevant for both large and small black holes.

Analytical expressions for W are obtainable for small non-BPS black holes, unlike large black holes.

Abstract

We consider the fist order, gradient-flow, description of the scalar fields coupled to spherically symmetric, asymptotically flat black holes in extended supergravities. Using the identification of the fake superpotential with Hamilton's characteristic function we clarify some of its general properties, showing in particular (besides reviewing the issue of its duality invariance) that W has the properties of a Liapunov's function, which implies that its extrema (associated with the horizon of extremal black holes) are asymptotically stable equilibrium points of the corresponding first order dynamical system (in the sense of Liapunov). Moreover, we show that the fake superpotential 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.