Imitación

For the non-supersymmetric solution (4.3.22) the Hessian of the black hole potential has two distinct eigenvalues, which exhibit complicated behavior as T varies (Fig. 4.1, right). For very small T the eigenvalues have opposite signs, indicating a saddle point of the potential (the solution is unstable), but in the range approximately T ∈[0.005,0.06] the eigenvalues are both positive, so the solution becomes an attractor (provided that the prepotential in this region can be still reliably described by (4.2.1)).

4.5

Extrema of the entropy in the moduli space

In anticipation of chapter 7, where we will discuss the entropic principle [133, 92], let us display the black hole entropy as a function on the moduli space, rather than, as otherwise common, a function of the charges. (We ignore at this point the question whether a solution to the attractor equations with integral charges can be found for an arbitrary point in the moduli space.)

Inserting the supersymmetric solution (4.3.21) into (4.3.9) yields the entropy

S = 2(Y0)2 2πRea−βT2(logT + 1), (4.5.1) in agreement with (3.4.37),

SBPS =π|Y0|2e−G(z,z¯) = 2|Y0|2 2πRea−β|T|2log|T| −β(ReT)2

. (4.5.2)

The entropy S, regarded as a function ofz (or T) for constant Y0_{, has a local maximum}

at the conifold point T = 0 [31], as shown in Fig. 4.2. The left graph corresponds to our explicit solution (4.5.1) for two charges (constrained to the positive T semi-axis), while the right represents the general formula (4.5.2), without restrictions on the charges.

Inserting the non-supersymmetric solution (4.3.22) into (4.3.9) yields the entropy

S π(Y0₎2 = 1 4π(βT2_{(logT + 1)−2πRea)}4 " 1 βT2_{+ 4πRea} βT 2_{(2 logT + 3)}3 _βT2_{+ 4πRea}5

−β2T4 −9βT2+ 4 aπ−βT2(logT)2+ 4 2aπ−3βT2logT + 4aπ

−8πβRea T2 5βT2(logT)2+ 2 7βT2+aπlogT + 2 5βT2+aπ

−32π3(Rea)3+ 16π2(Rea)2 −4βT2(logT)2−7βT2 −10βT2logT +aπ

+ 4π¯a βT2(logT + 1)−2πRea2 2 ! −8 βT2(logT + 1)−2πRea5 # = 4 Rea+ β πT 2

32(logT)3+ 144(logT)2+ 214 logT + 106+O(T3),

again in agreement with (3.4.37), Snon-BPS =π|Y0|2e−G(z,z¯) 1 + 4 g 3 z¯z |C111|2 = 2|Y0|2 _{2πRea−β|T|}2_{log|T| −β(ReT)}2

1 +β|T|2(β|T|2(1−2 log|T|) + 2β(ReT)2(1 + 2 log|T|) + 4πRea(3 + 2 log|T|)) 3

4 (2πRea−β|T|2_{log|T| −β(ReT)}2₎4

!

.

(4.5.4)

In contrast to the supersymmetric case, S attains for constant Y0 _{a local minimum at the}

conifold point. There exists, however, also a local maximum around T ≈0.05 (Fig. 4.3). As mentioned at the end of the previous section, this point is an attractor, provided that one can still trust the prepotential (4.2.1) there. If we keep Y0 _{constant as in [92], the maximal}

4.5 Extrema of the entropy in the moduli space 45

Figure 4.2: S/π as a function of T for Y0 = 1, β =−1/2, a = 1 in the supersymmetric case. The left graph is a cross section along the positive T semi-axis through the surface in the right graph.

Figure 4.3: S/π as a function of T forY0 = 1, β= −1/2, a= 1 in the non-supersymmetric case, similarly to Fig. 4.2.

Chapter 5

Entropy function for five-dimensional

rotating black holes

5.1

Extremal black holes in five and four dimensions

Extremal black holes in five dimensions can be related to extremal black holes in four dimensions. This connection is implemented by placing the five-dimensional black hole in a Taub-NUT geometry, and by using the modulus of the Taub-NUT space to interpolate between the five and the four-dimensional description. In the vicinity of the NUT charge, spacetime looks five-dimensional, whereas far away from the NUT the spacetime looks four-dimensional. This connection was first established in [77, 76] for supersymmetric black holes in the context ofN = 2 supergravity theories that in four dimensions are based on cubic prepotentials, and was further discussed in [14].

In the following, we focus on rotating extremal black holes in five dimensions which are connected to static extremal black holes in four dimensions in the way described above. We use this link to define the entropy function for these rotating black hole solutions in the context of N = 2 supergravity theories with cubic prepotentials. In four dimensions, the static extremal black holes we consider carry charges (PI_{, Q}

I), whereP0 6= 0 corresponds to the NUT charge in five dimensions (in this chapter we distinguish the charges as measured in four and five dimensions by using capital and small letters, respectively). These four- dimensional black holes are connected to rotating five-dimensional black holes with one independent angular momentum parameter. The five-dimensional N = 2 supergravity theories contain Chern–Simons terms for the abelian gauge fields, so that the definition of the entropy function given in [146, 6] cannot be directly applied whenever these terms play a role for the given background. Therefore, we define the entropy function for these rotating five-dimensional black holes to equal the entropy function of the associated static black holes in four dimensions. The latter was computed for N = 2 supergravity theories in [144, 29]. Then, we specialize to the case of black holes with non-vanishing charges (P0_{, Q}

I), which in five dimensions correspond to rotating electrically charged extremal black holes in a Taub-NUT geometry. Extremization of the entropy function yields a set

of attractor equations for the various parameters characterizing the near-horizon solution. We check that these attractor equations are equivalent to the equations of motion in five dimensions evaluated in the black hole background. We construct two types of solutions to the attractor equations and we compute their entropy.

Our approach for defining the entropy function in the presence of Chern–Simons terms is based on dimensional reduction, and is therefore similar to the approach used in [143] for defining the entropy function of the three-dimensional BTZ black hole. Related results for rotating AdS5 black holes have appeared in [124].