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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.
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2007
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P. K. Townsend, Hamilton-Jacobi Mechanics from Pseudo-Supersymmetry
2008
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K. Skenderis and P. K. Townsend, Pseudo-supersymmetry and the domain-wall / cosmology correspondence
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J. Sonner and P. K. Townsend, Axion-Dilaton Domain Walls and Fake Supergravity
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