Mario Trigiante, Thomas Van Riet, Bert Vercnocke
We discuss the existence of first-order flow equations for extremal black holes in ungauged supergravity and for domain walls in gauged supergravity. We explain when the first-order Hamilton--Jacobi equations 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.
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http://arxiv.org/abs/1203.3194
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