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Nonextremal black holes in gauged supergravity and the real formulation of special geometry. II. (English) Zbl 1267.83108

Summary: In [the authors, “Nonextremal black holes in gauged supergravity and the real formulation of special geometry” (2012), arXiv:1207.2679], a new prescription for finding nonextremal black hole solutions to \(\mathcal N = 2, D = 4\) Fayet-Iliopoulos gauged supergravity was presented, and explicit solutions of various models containing one vector multiplet were constructed. Here, we use the same method to find new nonextremal black holes to more complicated models. We also provide a general recipe to construct non-BPS extremal solutions for an arbitrary prepotential, as long as an axion-free condition holds. These follow from a set of first-order conditions, and are related to the corresponding supersymmetric black holes by a multiplication of the charge vector with a constant field rotation matrix \(S\). The fake superpotential driving this first-order flow is nothing else than Hamilton’s characteristic function in a Hamilton-Jacobi formalism, and coincides in the supersymmetric case (when \(S\) is plus or minus the identity) with the superpotential proposed by G. Dall’Agata and A. Gnecchi [“Flow equations and attractors for black holes in \(\mathcal N=2\,U(1)\) gauged supergravity” (2011), arXiv:1012.3756]. For the nonextremal black holes that asymptote to (magnetic) AdS, we compute both the mass coming from holographic renormalization and the one appearing in the superalgebra. The latter correctly vanishes in the BPS case, but also for certain values of the parameters that do not correspond to any known supersymmetric solution of \(\mathcal N = 2\) gauged supergravity. We finally show that the product of all horizon areas depends only on the charges and the asymptotic value of the cosmological constant.

MSC:

83E50 Supergravity
83C57 Black holes
83C15 Exact solutions to problems in general relativity and gravitational theory
70H20 Hamilton-Jacobi equations in mechanics