Moduli and (un)attractor black hole thermodynamics

TL;DR

This paper explores the thermodynamics and attractor behavior of four-dimensional black holes with scalar fields, showing how moduli influence thermodynamic laws and how the entropy remains moduli-independent in extremal cases, linking microscopic and macroscopic descriptions.

Contribution

It provides a detailed analysis of moduli effects on black hole thermodynamics and demonstrates the attractor mechanism using multiple formalisms, connecting microscopic and macroscopic entropy calculations.

Findings

Scalar charges appear in the first law when moduli vary.

In extremal black holes, entropy is independent of asymptotic moduli.

Attractor mechanism explains entropy matching for extremal non-BPS black holes.

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.