First-order flow equations for extremal and non-extremal black holes
arXiv:0810.1528 · doi:10.1088/1126-6708/2009/03/150
Abstract
We derive a general form of first-order flow equations for extremal and non-extremal, static, spherically symmetric black holes in theories with massless scalars and vectors coupled to gravity. By rewriting the action as a sum of squares a la Bogomol'nyi, we identify the function governing the first-order gradient flow, the `generalised superpotential', which reduces to the `fake superpotential' for non-supersymmetric extremal black holes and to the central charge for supersymmetric black holes. For theories whose scalar manifold is a symmetric space after a timelike dimensional reduction, we present the condition for the existence of a generalised superpotential. We provide examples to illustrate the formalism in four and five spacetime dimensions.
27 pages, v2: small changes, referencing and misprints corrected, v3: text updated and a reference added to match the JHEP version
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