Abstract
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.
Highlights
There exists a one to one map between such solutions [3]
The most general form for the first-order equations for non-extremal solutions was found in [20], where it was emphasized that the flow equations do differ from those for extremal black holes in the sense that the black hole warp factor appears in a non-trivial way, different from extremal solutions
We have established that regular extremal flows must be fake supersymmetric. This is the usual assumption for the flow equations of regular extremal black holes and here we provided a proof that this is a necessary condition
Summary
Fake supersymmetry is a concept that is usually formulated on the level of effective onedimensional actions for black hole solutions, domain walls and, through the map between domain walls and FLRW cosmologies [7], for the latter, where it is referred to as pseudo-supersymmetry. Supersymmetric domain walls and black holes are a special subset of solutions that fulfill certain first-order differential equations that follow from the Killing spinor equations. These equations take the form of flow equations, derived from the superpotential function W : φi = ǫeaU Gij∂jW (φ) , 4U = aeaU W (φ) ,. Solutions that can be found from a flow governed by a fake superpotential, in a certain sense mimic supersymmetric solutions
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