Fake supersymmetry versus Hamilton-Jacobi

TL;DR

This paper investigates the conditions under which Hamilton-Jacobi equations for black holes in supergravity reduce to first-order equations from a fake superpotential, revealing a new constant of motion and extending the attractor mechanism.

Contribution

It introduces a new constant of motion linked to fake supersymmetry and extends the attractor mechanism to small black holes with specific scalar trajectories.

Findings

Fake supersymmetry is necessary for physically sensible extremal black holes.

Small black holes exhibit scaling solutions near the horizon.

Scalar fields follow geodesic curves determined by black hole charges.

Abstract

We explain when the first-order Hamilton-Jacobi equations for black holes (and domain walls) in (gauged) supergravity, reduce to the usual first-order equations derived from a fake superpotential. This turns out to be equivalent to the vanishing of a newly found constant of motion and we illustrate this with various examples. We show that fake supersymmetry is a necessary condition for having physically sensible extremal black hole solutions. We furthermore observe that small black holes become scaling solutions near the horizon. When combined with fake supersymmetry, this leads to a precise extension of the attractor mechanism to small black holes: The attractor solution is such that the scalars move on specific curves, determined by the black hole charges, that are purely geodesic, although there is a non-zero potential.