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Thermodynamics of Conformal Anomaly Corrected Black Holes in Ads Space

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Thermodynamics of Conformal Anomaly Corrected Black Holes in Ads Space Physics Letters B 733 (2014) 183–189 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Thermodynamics of conformal anomaly corrected black holes in AdS space Rong-Gen Cai a,b a State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China b King Abdulaziz University, Jeddah, Saudi Arabia article info abstract Article history: We present exact analytical black hole solutions with conformal anomaly in AdS space and discuss the Received 14 February 2014 thermodynamical properties of these black hole solutions. These black holes can have a positive, zero and Received in revised form 12 April 2014 negative constant curvature horizon, respectively. For the black hole with a positive constant curvature Accepted 24 April 2014 horizon, there exists a minimal horizon determined by the coefficient of the trace anomaly, the black Available online 30 April 2014 hole with a smaller horizon is thermodynamically unstable, while it is stable for the case with a larger Editor: M. Cveticˇ horizon. The Hawking–Page transition happens in this case. For the black hole with a Ricci flat horizon, the black hole is always thermodynamically stable and there is no Hawking–Page transition. In the case of the black hole with a negative constant curvature horizon, there exists a critical value for the coefficient of the trace anomaly, under this critical value, the black hole is always thermodynamical stable and the Hawking–Page transition does not happen. When the coefficient is beyond the critical value, the black hole with a smaller horizon is thermodynamically unstable, but it becomes stable for the case with a larger horizon, the Hawking–Page transition always happens in this case. The latter is a new feature for the black holes with a negative constant curvature horizon. © 2014 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3. 1. Introduction It turns out that thermodynamical properties of AdS black holes crucially depend on horizon structure. For (positive constant cur- vature) spherical horizon black holes in AdS space, there exists a The AdS/CFT correspondence [1–3] conjectures that string/M minimal black hole temperature, under which there is no black theory in an anti-de Sitter (AdS) space (times some compact space) hole solution. The black hole is thermodynamically unstable for is dual to a strongly coupled conformal field theory (CFT) living on smaller black hole horizon, while it is thermodynamically stable the AdS boundary. In this sense, thermodynamics of black holes in for larger horizon. Therefore there exists a so-called Hawking– AdS space can be identified with the one of the dual strongly cou- Page phase transition [25] between the thermal gas and stable pled CFTs [4], while the latter is not able to be obtained through large black hole in AdS space. This phase transition can be iden- conventional perturbative quantum field theory. This is one of rea- tified with the confined/deconfined transition of gauge field in the sons for many recent studies of black holes in AdS space. On the dual CFT side [4]. For the black holes with (zero curvature) Ricci other hand, it is also of great interest to investigate black holes in flat horizon and (negative curvature) hyperbolic horizons, however, AdS space in its own right. Compared to its counterpart in asymp- they are always thermodynamically stable (see, for example, [14]). totically flat spacetime, the horizon of black holes in AdS space This means that the black hole phase is always dominated in the has much rich topological structure: AdS black holes can have pos- dual conformal field theory side. In this case, the Hawking–Page itive, zero and negative constant curvature horizon. These black phase transition will not happen. holes have been discussed extensively, for example, in Einstein– Conformal anomaly [26] plays an important role in quantum (Maxwell) theory [5–14], Einstein–Maxwell–Dilaton theory [8,15], field theories in curved spacetime and has various applications in Wely conformal gravity [16], Gauss–Bonnet gravity [17–20], and black hole physics, cosmology, string theory, and statistical me- more general Lovelook gravity [21–24],etc. chanics, etc. The effective action associated with the trace anomaly is non-local, but in four dimensional spacetime case, it can be written in a local form by introducing two auxiliary scalar fields E-mail address: [email protected]. (see [27,28] and reference therein). The quantum trace anomaly http://dx.doi.org/10.1016/j.physletb.2014.04.044 0370-2693/© 2014 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3. 184 R.-G. Cai / Physics Letters B 733 (2014) 183–189 can give rise to macroscopic effects. To study the back reaction of spacetime with two additional assumptions: (i) The stress–energy μν the trace anomaly, one has to solve the equations of motion for tensor is covariant conserved, namely, ∇μT =0; (ii) The time– these scalar fields in some curved or topological nontrivial back- time component of the stress–energy tensor and the radial–radial t r grounds. It turns out that at the horizon, the quantum fluctuations component are equal in the spacetime, that is, T t (r)=T r (r). of the scalar degrees of freedom would be large [28,29].Thesetwo The assumption (i) manifestly holds because the left hand side of scalar fields are determined by boundary conditions, which brings the equations are of the same property, while the assumption (ii) back the non-local effect of the trace anomaly. Furthermore, the is imposed in order to find an exact analytical solution of the Ein- equations of motion from the effective action of the trace anomaly stein equations. Here we would like to stress that introducing the are quite complicated, therefore in general it is impossible to ob- assumption (ii) is just for easily solving the equations, it is not a tain exact analytical solutions of Einstein equations with the trace physical requirement. We now generalize the solution presented in anomaly. In [30], with an additional assumption the authors are [30] into the case with a cosmological constant. able to find an exact analytical black hole solution to the Einstein Consider the static spherically symmetric spacetime of the form equations with the trace anomaly. There some interesting proper- 1 ties of the trance anomaly corrected black hole are observed. Some 2 =− 2 + 2 + 2 2 ds f (r)dt dr r dΩ2k, (4) further discussions can be found, for example, in [31–40]. g(r) In this note we are going to generalize the black hole solution where f (r) and g(r) are two functions of the radial coordinate r, presented in [30] to the case with a negative cosmological con- 2 and dΩ2k is the line element of a two-dimensional Einstein con- stant and to study the thermodynamical properties of the trace stant curvature space with scalar curvature 2k.Withoutlossof anomaly corrected black holes in AdS space. This note is organized generality, one may take k to be 1, 0 and −1, respectively. In that as follows. To be self-contained, in Section 2 we will briefly re- case, they correspond to a sphere, Ricci flat space, and negative view the procedure to solve the Einstein equations with the trace curvature space, respectively. According to the symmetry of met- anomaly and present the exact analytical black hole solutions. In ric (4),wedefine Section 3 we study the thermodynamical properties of these black t r θ φ holes with positive, zero and negative constant curvature horizon, T t =−ρ(r), T r = p(r), T θ = T φ = p⊥(r), (5) respectively. There we will see that the trace anomaly changes and other components must vanish. Under the assumption (ii), one qualitatively thermodynamical properties of the black holes, com- can easily show that f (r) = g(r) (see, for example, [42]). Further- pared to the case without the trace anomaly. Section 4 is devoted more, combining the covariant conservation of the stress–energy to the conclusions and further discussions. tensor with the trace anomaly (1),wehave 2. Black hole solutions 2α −ρ + p + 2p⊥ =− (1 − f )2 , r2 In four dimensional spacetime, one loop quantum correction 4 f (p − p⊥) + (ρ + p)rf + 2rfp = 0, (6) leads to a trace anomaly of the stress–energy tensor of conformal field theory. The trance anomaly has the form [26,41] where we have set β = 0 and the prime stands for the derivative with respect to the radial coordinate r. With the assumption (ii), μν g Tμν=β I4 − αE4, (1) ρ =−p, we can have from (6) that where I = C Cμνλσ and E = R2 − 4R Rμν + R Rμνλσ , 2α 4 μνλσ 4 μν μνλσ rp + 4p + (1 − f )2 = 0. (7) β and α are two positive constants related to the content of the r2 conformal field theory: This equation can be easily solved with the solution 1 2 β = (n0 + 6n1/2 + 12n1), 2α q 120(4π)2 p = (1 − f ) 1 − f + 2rf − , (8) r4 r4 1 2 α = (n0 + 11n1/2 + 62n1), (2) where q is an integration constant. Thus we are able to obtain the 360(4π)2 other nonvanishing components of the stress–energy tensor asso- where n0, n1/2 and n1 are the number of scalar, Dirac fermion and ciated with the trance anomaly as vector fields, respectively, in the conformal field theory. For exam- α 2 ple, for the N = 4 SU(N) Supersymmetric Yang–Mills theory, one ρ =−p, p⊥ =−2p − (1 − f ) .
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