P - V Criticality of Logarithmic Corrected Dyonic Charged AdS Black Hole
aa r X i v : . [ g r- q c ] A ug P - V Criticality of Logarithmic CorrectedDyonic Charged AdS Black Hole
J. Sadeghi a ∗ B. Pourhassan b † , M. Rostami c ‡ a Department of Physics, University of Mazandaran, Babolsar, Iran b School of Physics, Damghan University, Damghan, Iran c Department of Physics, Tehran Northem Branch, Islamic Azad University, Tehran, Iran
October 9, 2018
Abstract
In this paper, we consider dyonic charged AdS black hole which is holographic dualof a van der Waals fluid. We use logarithmic corrected entropy and study thermo-dynamics of the black hole and show that holographic picture is still valid. Criticalbehaviors and stability also discussed. Logarithmic corrections arises due to thermalfluctuations which are important when size of black hole be small. So, thermal fluc-tuations interpreted as quantum effect. It means that we can see quantum effect of ablack hole which is a gravitational system.
Keywords:
Dyonic charged AdS black hole; Quantum Gravity; Thermodynamics;Holography.
Pacs Number:
We know that the black hole entropy is proportional to the horizon aria A , known as entropy-area law of the black hole [1]. On the other hand, thermal fluctuations will happen for anythermodynamical system like black holes. Specially, thermal fluctuations are more importantfor the small objects, because one can see that thermal fluctuations arise due to quantumfluctuations in the geometry of space-time. So, one can neglect such fluctuations for the ∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected] S = A − α AT , (1)where α is a constant usually fixed as unity. We can track effect of thermal fluctuationsusing α and obtain ordinary thermodynamics by setting α = 0. Recently effect of thermalfluctuations considered for several kinds of black objects like charged AdS black hole [4],black Saturn [5, 6] and modified Hayward black hole [7]. Moreover, in the Ref. [8] black holeentropy corrections due to thermal fluctuations in the black hole extensive parameters forvarious AdS black holes have been studied for both canonical and grand canonical ensembles.Some asymptotic AdS black holes like BTZ, D=4 KN-AdS and R-charged black holes invarious dimensions considered in that paper and found an universality in the logarithmiccorrections to charged AdS black holes entropy in various dimensions, which are expressedin terms of the black hole response coefficients via fluctuation moments.In this paper, we would like to consider an interesting kind of black hole which is calleddyonic charged AdS black hole in four dimensions which has both electric and magneticcharges [9, 10]. In the Ref. [11] using critical point investigation it is found that chargedAdS black holes are corresponding to van der Waals fluid. Also, it has been argued thatdyonic charged AdS black hole may be holographic dual of van der Waals fluid with chemicalpotential [12]. It is also found that van der Waals fluid is holographic dual of RN AdS blackhole [13]. So, by using holographic principles one can study dyonic charged AdS black holevia a van der Waals. We use this motivation to study logarithmic corrected thermodynamicsof dyonic charged AdS black hole to see such quantum gravity effect.The paper is organized as follows. In the next section we give brief review of dyonic chargedAdS black hole and recall necessary relations which need to study its thermodynamics underthermal fluctuation effect. In section 3 we obtain several thermodynamics relations and seethe effect of logarithmic correction. In section 4 we use holographic picture which is a vander Waals fluid and study about critical behavior. In section 5 we discuss about stability ofblack hole, and finally in section 6 we give conclusion. Dyonic charged AdS black hole is solution of Einstein-Maxwell theory with negative cosmo-logical constant in four dimensions which described by the following action [10, 12], I = 116 π Z d x √− g (cid:18) − R − l + 14 F µν F µν (cid:19) , (2)where R is the Ricci scalar, F µν is the strength of electromagnetic field and l is the curvatureradius of AdS space, also G = 1 assumed. Solving Einstein equations yields to the following2etric, ds = − f ( r ) dt + dr f ( r ) + r dθ + r sin θdφ , (3)where, f ( r ) = 1 + r l − Mr + q e + q m r . (4)where q e , q m and M are electric charge, magnetic charge and mass of the black hole respec-tively. Electric and magnetic potential Φ defined by the following equation,Φ e = q e r + , Φ m = q m r + , (5)where the black hole horizon r + is given by, f ( r ) = 1 + r l − Mr + q e + q m r = 0 , (6)which has at least two real and positive roots r ± . So, one can write the mass of the blackhole as follow, M = r + r l + q e + q m r + . (7) f ( r ) in terms of r , and black hole mass as a function of r + plotted by the Fig. 1. We willdiscuss about them in the next section when study the logarithmic corrected thermodynamicsof the solution discussed above.Before end of this section it is useful to note that 4 dimensional dyonic charged AdS blackhole in asymptotically AdS space-time is conjectured to be dual to a 3 dimensional CFTliving on the boundary of the AdS space [11, 12]. The bulk gauge field is dual to a global U (1) current operator J µ , while conserved global charge of CFT side given by, J t = q e πG , (8)where G is four dimensional Newtonian constant related to degree of the gauge group N ofthe CFT using the holographic dictionary,14 G = √ N b , (9)where b related to the strength of the magnetic field Bb = q m .3 Logarithmic corrected thermodynamics
We know that a black hole may be considered as thermodynamic systems characterized bya Hawking temperature T and the corresponding logarithmic corrected entropy S given bythe equation (1), with A = 4 πr , hence we have, S = πr − α πr T ) , (10)where T = (cid:18) f ′ ( r )4 π (cid:19) r = r + = 14 π (cid:18) r + l + 1 r + − q e + q m r (cid:19) , (11)where we used the equation (7) in the last equality.Using the entropy and temperature we can find the Helmholtz function, F = − Z SdT = − r l + r + F ( q ) + F ( α ) , (12)where F ( q ) = 3( q e + q m )4 r + ,F ( α ) = α ( q e + q m )8 πr (cid:20)
43 + ln 8 πl r ( l ( q e + q m ) − l r − r ) (cid:21) − αr + πl (cid:20) πl r ( l ( q e + q m ) − l r − r ) (cid:21) − α πr + (cid:20) ln 8 πl r ( l ( q e + q m ) − l r − r ) (cid:21) . (13)Neglecting thermal fluctuation ( α = 0) equation (13) vanishes. Also, the first term of r.h.s ofthe equation (12) is corresponding to ordinary AdS black hole with thermodynamics pressurerelated to the cosmological constant [14], P ( α = q e = q m = 0) = − Λ2 l = 316 π l , (14)and thermodynamic volume, V = 43 πr . (15)Hence, in presence of electric and magnetic charge and thermal fluctuations, at least on ofthe above equations should be modified including charge and α dependent terms. We willassume thermodynamic volume given by the equation (15) and obtain modified pressure.Let us now discuss about Helmholtz free energy given by the equation (12). We will fixparameters as l = 1 and q e + q m = 1. In that case from the Fig. 1 (a) we can see that if0 . ≤ r + ≤ . . < M <
2. So, from the Fig. 1 (b) one can see extremal limit given4igure 1: (a) Black hole mass in terms of event horizon. (b) Horizon radius with variationof black hole mass M = 1 .
23 (thin), M = 1 . M = 1 . M = 1 . l = q e + q m = 1.Figure 2: Left: Helmholtz free energy in terms of horizon radius with l = 1. Right: Internalenergy in terms of horizon radius with l = 1. Q ≡ q e + q m = 1, α = 1 (solid), α = 0 (dotted). Q = 0, α = 1 (dashed), α = 0 (dash dotted).by M = 1 .
23, while r + = 1 is corresponding to M = 1 . M = 1 .
68 is corresponding to r + = 1 . r + ≫ F hasnegative large value and E has positive large value. Effect of thermal fluctuations on theblack hole with small size is obvious and important. Solid (red) and dashed (yellow) linesrepresent logarithmic corrected cases (charged and uncharged respectively) where Helmholtzfunction goes to negative infinity at r + → α = 0 we have F → ∞ (charged) and F → r + → M = 1 .
68 where F = 0 near the outer horizon ( r + ≈ .
1) and F = F max near the inner horizon (see Fig. 1 (b)). Outside of the black hole, free energy5s negative for the mentioned black hole mass. We can see critical radius r + c ≈ .
7, beforeit internal energy vanishes (Fig. 1 (b)) and Helmholtz energy have different behavior. Aswe will see later it relate to stability of black hole, ie. black hole is stable for r + > r + c . Itmeans that the logarithmic effect is more important for small black hole.In order to obtain internal energy we used well-known thermodynamics relation, E = F + ST. (16)Effect of thermal fluctuation on the internal energy is decreasing its value for the large blackholes. On the other hand for the infinitesimal black hole ( r + →
0) internal energy will beinfinite (see dashed yellow line of right plot of the Fig. 2). In the case of α = q e + q m = l = 1the internal energy will be imaginary for r + < .
66, which may be interpreted as criticalvalue of horizon radius where black hole can exist. There is also a minimum for the internalenergy corresponding to minimum mass shown in the Fig. 1 (a).As we told already, modified pressure due to thermal fluctuation can be obtained using thederivative of Helmholtz function with respect to the volume, P = − (cid:18) ∂F∂V (cid:19) T . (17)By using the equations (13) and (15) one can obtain pressure of dyonic charged AdS blackholes as follow, P = 316 πl − πr + 3( q e + q m )16 πr + 3 α π l r ( r − l r + ( q e + q m ) l ) ln ( 4 √ πl r Ql − l r − r ) . (18)The first term of r.h.s is corresponding to ordinary AdS black hole pressure given by theequation (14).Then, we can calculate enthalpy using the equations (15), (16), (18) and the followingrelation, H = E + P V = 3 r l + 512 r + + 3( q e + q m ) l )4 r + + H ( α ) , (19)where H ( α ) is contribution of thermal fluctuations. Enthalpy of the black hole may be con-sidered as black hole mass [15, 16], from the left plot of the Fig. 3 we can see as the α = 0limit (dotted line) enthalpy behaves as black hole mass which illustrated by the Fig. 1 (a).As expected we can see that enthalpy of large black hole has similar behavior for α = 0, α = 1, q e + q m = 1 and q e + q m = 0. For the uncharged black hole we can see that effect ofthermal fluctuation is some negative enthalpy (negative mass) which is forbidden. Hence, asbefore, there is a minimum value for the horizon radius of the black hole. There is a criticalvalue for the horizon radius r c + , effect of thermal fluctuations is decreasing H for r + > r c + and is increasing H for r + < r c + . 6igure 3: Left: Enthalpy in terms of horizon radius with l = 1. Right: Gibbs free energy interms of horizon radius with l = 1. Q ≡ q e + q m = 1, α = 1 (solid), α = 0 (dotted). Q = 0, α = 1 (dashed), α = 0 (dash dotted).Then we can obtain Gibbs free energy using the following relation, G = H − T S = r + Qr + + G ( α ) , (20)where G ( α ) is contribution of thermal fluctuations. In the right plot of the Fig. 3 we can seebehavior of Gibbs free energy with variation of α and black hole charge. We can see similarbehavior with Helmhotz free energy. Apparent kink in the right plot and initial value of leftplot is about r + c = 0 . dM = T ds + Φ e dq e + Φ m dq m . (21)It is clear that the case of α = 0 satisfied first law of black hole thermodynamics. It wouldbe interesting to investigate validity of above equation in presence of logarithmic correctionand obtain appropriate condition to satisfy the equation (21). After a bit calculation we findthat the following condition q e dq e dr + + q m dq m dr + = (1 + 3 r l ) r + , (22)is necessary to have valid first law of black hole thermodynamics. Dyonic charged AdS black hole in absence of thermal fluctuations is holographic dual of afluid with van der Waals equation state given by [11], (cid:16) P + aV (cid:17) ( V − b ) = kT. (23)7here k is the Boltzmann constant. The constant b > a > (cid:18) ∂P∂V (cid:19) T = T c = 0 , (cid:18) ∂ P∂V (cid:19) T = T c = 0 . (24)At T = T c direction of P-V line is zero but it is not minimum or maximum. Below it wehave some region with negative compressibility corresponding to dual van der Waals fluid,which means instability of black hole.Figure 4: (a) P-V diagram of van der Waals fluid with b = 0 . k = 0 .
1. Dotted green: T = 0, a = 1. Dashed blue: T = 0 . a = 1. Solid red: T = 1, a = 1. Dash dottedblack: T = 1, a = 0 .
2. (b) P-V diagram of dyonic charged AdS black hole with l = 1. Q ≡ q e + q m = 0 . α = 1 (dash dotted black), α = 0 (dotted blue). Q = 0, α = 1 (dashedyellow). Q ≡ q e + q m = 0 . α = 1 (solid red).The typical behavior of P-V diagram corresponding to van der Waals fluid plotted in Fig.4 (a).We can use equation (15) and (18) to investigate P-V diagram of dyonic charged AdS blackhole in presence of thermal fluctuations. Our numerical analysis illustrated by Fig. 4 (b).We can see solid red lines in both (a) and (b) plots of Fig. 4 to find that dyonic charged AdSblack hole in presence of thermal fluctuations with suitable value of electric and magneticcharges is also dual of van der Waals fluid. We find that value of charge is many importantto have dual van der Waals fluid. Therefore, thermodynamics quantities obtained in theprevious section use to see thermal fluctuation and hence quantum gravity effects.We will discuss about critical points (corresponding to black lines (dash dot) of both Fig. 4(a) and (b)) and stability of the model in the next section.8 Critical points and stability
Critical points of the model are corresponding to the dash dotted black lines of Fig. 4 (a)and (b). At α = 0 limit and using both conditions given by (24) one can obtain the followingequation, r + 3 r − q e + q m ) r − q e + q m ) = 0 . (25)It gives us critical horizon radius as follow, r + c ( α = 0) = f ( Q ) − f ( Q ) + 2( q e + q m ) + 1 f ( Q ) , (26)where we defined, f ( Q ) = h q e + q m ) − p q e + q m ) − q e + q m ) − q e + q m ) i , (27)which means that 0 . ≤ ( q e + q m ) ≤ .
27. However we expect that effect of thermalfluctuations change charge interval.Specific heat is an important measurable physical quantity which can determine stability ofsystem. It has the following relation with internal energy E , entropy S and temperature T , C = T (cid:18) ∂S∂T (cid:19) = ∂E∂T . (28)If C >
C < C = 0is corresponding to phase transition of van der Waals fluid, similar to the critical pointdiscussed above. By using the equation (16) we can obtain specific heat as following, C = 2(3 r + l r − ( q e + q m ) l ) πr r − l r + ( q e + q m ) l ) + C ( α ) , (29)where C ( α ) is logarithmic corrected terms. In the Fig. 5 we can see effect of thermalfluctuations on the stability of black hole. We can see that there is a minimum size for theblack hole as expected. Below minimum size r + < r + c , black hole in unstable and it meansthat van der Waals fluid changes phase. Fortunately, it is possible to see thermal fluctuationeffects for r + ≥ r + c where black hole is stable and we have van der Waals fluid. Reducingelectric and magnetic charges help to obtain smaller minimum size, therefore we have morechance to see effects of thermal fluctuations. For the case of Q = 1 one may have r + c ≈ . C in terms of horizon radius with l = 1. Q ≡ q e + q m = 0 . α = 1(dashed blue), α = 0 (solid blue). Q = 0 . α = 1 (dashed red), α = 0 (solid red). Q = 0, α = 1 (dashed green), α = 0 (solid green). In present work we considered special solutions of Einstein-Maxwell theory with negativecosmological constant in four dimensions which is dyonic charged AdS black hole. Our mainwork is finding the effect of thermal fluctuations on the thermodynamics quantities.Thermal fluctuations exist for any black object, but they are important for small black holeand negligible for the large black hole. Advantage of dyonic charged AdS black hole back-ground is its holographic picture which is a van der Waals fluid. We have shown that, inpresence of thermal fluctuations there is still a van der Waals fluid as dual picture. Onlywe should fix black hole charge which is corresponding to the electric charges of a van derWaals fluid. We obtained some thermodynamics quantities like Gibbs and Helmholts freeenergies and shown that thermal fluctuations have not important effects for the large blackhole. On the other hand for the small black holes they are important and have crucial role.We found that thermal fluctuations reduced stable regions of the black hole, however thereare enough stable regions to see quantum gravity effects before phase transition of a vander Waals fluid. It means that there is a minimum radius, we called critical radius, whereblack hole is stable in presence of thermal fluctuations, and in this region dyonic chargedAdS black hole in presence of thermal fluctuations is dual of van der Waals fluid.
Acknowledgments
It is pleasure to thanks Prof. Robert B. Mann for introducing originof holographic dual of charged AdS black hole. Also we would like to thanks S. Mahapatraand Yu Tian for useful comments. 10 eferenceseferences