Moduli and (un)attractor black hole thermodynamics

TL;DR

<3-5 sentence high-level summary>This paper examines four-dimensional, spherically symmetric black holes in theories with massless scalars non-minimally coupled to gauge fields. It shows that for non-extremal solutions, moduli induce scalar charges that enter the first law, while in the extremal limit the entropy becomes moduli-independent due to the AdS2×S2 near-horizon geometry and the attractor mechanism. The authors compare the effective-potential and Sen's entropy-function formalisms, derive exact non-extremal solutions for specific couplings, and elucidate how the attractor behavior underpins microscopic/macroscopic entropy matching for extremal non-BPS Kaluza-Klein black holes. They also discuss the interpretation of scalar hair as secondary hair and the role of attractors within the broader framework of special geometry and string theory.

Abstract

We investigate four-dimensional spherically symmetric black hole solutions in gravity theories with massless, neutral scalars non-minimally coupled to gauge fields. In the non-extremal case, we explicitly show that, under the variation of the moduli, the scalar charges appear in the first law of black hole thermodynamics. In the extremal limit, the near horizon geometry is $AdS_2\times S^2$ and the entropy does not depend on the values of moduli at infinity. We discuss the attractor behaviour by using Sen's entropy function formalism as well as the effective potential approach and their relation with the results previously obtained through special geometry method. We also argue that the attractor mechanism is at the basis of the matching between the microscopic and macroscopic entropies for the extremal non-BPS Kaluza-Klein black hole.