We construct interpolating solutions describing single-center static extremal non-supersymmetric black holes in four-dimensional N = 2 supergravity theories with cubic prepotentials. To this end, we derive and solve first-order flow equations for rotating electrically charged extremal black holes in a Taub-NUT geometry in five dimensions. We then use the connection between five-and four-dimensional extremal black holes to obtain four-dimensional flow equations and we give the corresponding solutions.
We derive a general form of first-order flow equations for extremal and nonextremal, static, spherically symmetric black holes in theories with massless scalars and vectors coupled to gravity. By rewriting the action as a sum of squaresà 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.
We propose a generic recipe for deforming extremal black holes into non-extremal black holes and we use it to find and study the static non-extremal black-hole solutions of several N = 2, d = 4 supergravity models (SL(2, R)/U (1), CP n and ST U with four charges). In all the cases considered, the non-extremal family of solutions smoothly interpolates between all the different extremal limits, supersymmetric and not supersymmetric. This fact can be used to explicitly find extremal non-supersymmetric solutions also in the cases in which the attractor mechanism does not completely fix the values of the scalars on the event horizon and they still depend on the boundary conditions at spatial infinity. We compare (supersymmetry) Bogomol'nyi bounds with extremality bounds, we find the first-order flow equations for the non-extremal solutions and the corresponding superpotential, which gives in the different extremal limits different superpotentials for extremal black holes. We also compute the entropies (areas) of the inner (Cauchy) and outer (event) horizons, finding in all cases that their product gives the square of the moduli-independent entropy of the extremal solution with the same electric and magnetic charges. a
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