Non-extremal black holes of N = 2, d = 4 supergravity

Abstract

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 $\left( {{{{{\text{SL}}\left( {2,\mathbb{R}} \right)}} \left/ {{{\text{U}}(1)}} \right.}} \right.$, ${\overline {\mathbb{C}\mathbb{P}}^n}$ and STU 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 super-potential, 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.