PDF《black holes》

arXiv:hep-th/0608044v2

28 Aug

2006 Preprint typeset in JHEP style - HYPER VERSION CERN-PH-TH/2006-158 Entropy function and attractors for AdS black holes Jose F.

Morales CERN, Theory Division, CH-1211, Geneva 23, Switzerland INFN, Universit` a di Roma "Tor Vergata",

00133 Rome, Italy E-mail: Francisco.Morales.Morera@cern.ch Henning Samtleben Zentrum f¨ ur Mathematische Physik, Universit¨ at Hamburg Bundesstrasse 55, D-20146 Hamburg, Germany E-mail: Henning.Samtleben@desy.de Abstract: We apply Sen's entropy formalism to the study of the near horizon geometry and the entropy of asymptotically AdS black holes in gauged supergravities. In particular, we consider non-supersymmetric electrically charged black holes with AdS2 * Sd?2 horizons in U(1)4 and U(1)3 gauged supergravities in d =

4 and d =

5 dimensions, respectively. We study several cases including static/rotating, BPS and non-BPS black holes in Einstein as well as in Gauss-Bonnet gravity. In all examples, the near horizon geometry and black hole entropy are derived by extremizing the entropy function and are given entirely in terms of the gauge coupling, the electric charges and the angular momentum of the black hole. Keywords: Black holes,attractors,AdS/CFT. Contents 1. Introduction

1 2. AdS4 static black holes

3 2.1 The solution

5 3. AdS5 static black holes

7 3.1 The solution

8 4. Rotating black holes in AdS5

10 4.1 BPS black holes

11 4.2 Non-extremal black holes

12 5. Higher derivative terms

13 6. Conclusions

15 A. Physical charge units

16 B. Black holes at T =

0 18 1. Introduction The study of black hole thermodynamics has played a central role in the development of our current notions of holography in gravity. In this line of thinking, black holes are viewed as thermodynamic objects at equilibrium with a temperature and an entropy. A simple analysis of this thermodynamic system leads to the remarkable Bekenstein- Hawking formula for the black hole entropy. This formula relates the entropy of the black hole to the area of its horizon and it suggests that the microscopic degrees of freedom of the black hole can be described by a "dual" quantum mechanics living on the horizon. This is further supported by the discovery of AdS/CFT dualities [1] that relate gravity on AdS spaces and gauge theories living on the AdS boundary. These observations drastically simplify the study of black hole physics, since the geometry of the horizon is typically much simpler than that of the full solution. C

1 C Even in theories with scalar ?elds and a large number of moduli C asymptotic values of massless scalars at in?nity C, scalars are attracted at the black hole horizon to special values and the full geometry is entirely determined in terms of the black hole charges. This is referred as the attractor mechanism [2C5]. Originally discussed in the context of N =

2 black holes the attractor mechanism has been recently extended in many directions, including non-supersymmetric and higher derivative gravity theories [6C19]. The results show that the attractor mechanism is a universal issue of any gravity theory. In [20], A. Sen introduced a unifying formalism, the entropy formalism, that de- scribes the attractor equations and black hole entropy in a general non-supersymmetric and higher derivative gravity theory. In this formalism, the near horizon geometry is determined by extremizing a single function F, the entropy function. The entropy of the black hole is given by the value of F at the extremum. The function F is de?ned by the Legendre transform with respect to the black hole charges of the gravity action evaluated at the horizon. More precisely, the gravity action is ?rst evaluated at a trial background geometry with volumes and scalar/gauge ?eld pro?les parametrized by a ?nite number of parameters. These parameters are then determined by extremizing the entropy function F. The formalism has been successfully applied to the study of general non-supersymmetric asymptotically ?at black holes in various supergravity settings [21C32]. The aim of this paper is to extend this analysis to the study of asymptotically AdS black holes in gauged supergravities. According to holography [1] the entropy of black holes in AdS spaces is related to the free energy of the dual gauge theory living on the AdS boundary, see [33C38]. To pursue the study of these holographic correspondences a detailed knowledge of the black hole near horizon data is required. To derive explicit formulas for the attractor geometry and for the entropy of AdS black holes is one of the main motivations of the present work. Black holes in gauged supergravities are di?erent from those in Poincar? e super- gravities in many respects. First, in the gauged theory the asymptotic values of the scalar ?elds at in?nity are typically ?xed at the minimum of a scalar potential. The moduli space is therefore reduced and often empty. Still once charges are placed on AdSd, even scalars ?xed at in?nity ?ow at the horizon to a di?erent ?xpoint speci?ed completely by the black hole charges. I.e. the attractor mechanism now describes a ?ow between two ?xpoint geometries. Second, it is well known that asymptotically AdS black hole solutions with regular horizons are always non-supersymmetric un- less a non-trivial angular momentum is turned on. This is very di?erent from the Minkowski case where BPS static solutions are quite common. Our analysis here explores both non-supersymmetric static and rotating black hole solutions. We apply the entropy formalism to non-supersymmetric black holes with near horizon geometry AdS2 * Sd?2 in d = 4,

51 . Black holes with these type of horizons

1 More precisely, in the case of rotating black holes the horizons are described by a "squashed C

2 C have always zero temparature (with coinciding inner and outer horizons) but they are in general non-supersymmetric. For concreteness we focus on the U(1)4 and U(1)3 gauged supergravities in d =

4 and d = 5, respectively. These theories can be embed- ded into the maximal gauged supergravities with gauge groups SO(8) and SO(6), respectively, following from compacti?cations of M-theory and type IIB theory on AdS4 * S7 and AdS5 * S5 , respectively. Black holes in these gauged supergravities have been extensively studied and classi?ed in full generality in the literature [39C50] (see [51] for a review and a list of references). In the case of Einstein gravity, the solutions derived here via the entropy formalism follow from these general solutions by taking the zero temperature limit. Our focus here is on the near horizon geometry and black hole entropy. We test the entropy formalism in a number of examples, including static/rotating black holes with or without supersymmetry in Einstein as well as Gauss-Bonnet gravity. In each case we show that the attractor geometry follows from extremization of the entropy function. In the case of Einstein gravity the entropy function output will be shown in agreement with the Bekenstein-Hawking formula as expected. The entropy formalism is particularly e?cient in the study of black holes in higher derivative gravity. Higher derivative corrections to black hole entropies in rigid supergravities were ?rst studied in [52C55]. Higher derivative corrections to asymptotically AdS black holes in Gauss-Bonnet gravity were studied in [56]. More recently in [57] the authors consider several examples of higher derivative terms and derive the ?rst corrections to the Schwarzschild AdS black holes. Here we consider the Einstein-Maxwell system in the presence of a Gauss-Bonnet term and derive exact expressions for the near horizon geometry and the black hole entropy. The paper is organized as follows: In sections

2 and

3 we consider non-rotating asymptotically AdS black holes in U(1)4 and U(1)3 gauged supergravities in d =

4 and d = 5, respectively. In section

4 we apply the entropy formalism to rotating black holes in d =

5 gauged supergravity. The study of higher derivative corrections is sketched in section

5 for the Gauss-Bonnet type of interactions in the Maxwell- Einstein system in d = 4,

5 dimensions. In Section

6 we summarize our results and draw some conclusions. Appendix A contains a discussion on the normalization of the physical charges used in the main text. Appendix B presents the link between our AdS2 * Sd?2 solutions and zero temperature limits of the general black hole solutions. 2. AdS4 static black holes We start by considering U(1)4 gauged supergravity in four dimensions. This theory follows from a truncation of the maximal N = 8, SO(8) gauged supergravity [58] AdS2 * Sd?2 " rather than a tensor product geometry. C

3 C down to the Cartan subgroup of SO(8). The bosonic action can be written as [41]: S =

1 16πG4 d4 x √ ?g R ?

1 4 X2 I FI ?νF?νI ?

1 2 X?2 I ??XI ?? XI ? V , (2.1) with I = 1,4, and FI ?ν = 2?[?AI ν] , V = ?4 g2 I