Thermal Fluctuations Effects on Reissner-Nordström-AdS Black Hole
TThermal Fluctuations Effects on Reissner-Nordstr¨om-AdS BlackHole ¨Ozg¨ur ¨Okc¨u ∗ and Ekrem Aydıner † Department of Physics, Faculty of Science,˙Istanbul University, ˙Istanbul, 34134, Turkey (Dated: April 8, 2019)
Abstract
In this paper, we study the effects of thermal fluctuations for Reissner-Nordstr¨om-AdS (RN-AdS) black hole. We obtain the corrected thermodynamic quantities such as entropy, temperature,equation of state and heat capacities in the presence of thermal fluctuations. We also investigatethe phase transition of RN-AdS black hole for thermal fluctuations. Finally we compute the criticalexponents for small thermal fluctuations. We show that critical exponents are the same as criticalexponents without thermal fluctuations.
PACS numbers: 04.70.-s, 05.70.Ce, 04.60.-m ∗ [email protected] † [email protected] a r X i v : . [ phy s i c s . g e n - ph ] M a r . INTRODUCTION Black holes behave as thermodynamic objects with a Bekenstein-Hawking entropy and aHawking temperature [1–7], and set deep connection between the laws of classical generalrelativity, thermodynamics, and quantum mechanics due to the presence of horizons [1–10].It is suggested that black holes have very massive entropy in the universe, i.e., their entropiesare greater than any other object of the same volume [4, 6]. This entropy is defined by S = A/
4. However, it is found that quantum fluctuations on the planck scale may change theentropy of the black holes with time. These quantum fluctuations become important whenthe black hole shrinks its size and therefore, entropy of the black holes need to get corrected.In order to get corrections on the black hole entropy different approaches such as non-perturbative quantum general relativity, the Cardy formula, the exact partition function,matter fields in backgrounds of a black hole, Rademacher expansion of the partition functionand string theoretical effects have been used [11–22]. On the other hand, recently thecorrections to the black hole entropy can be obtained through the the generalized uncertaintyprinciple [23–25].Quantum fluctuations may lead to thermal fluctuations in the black hole thermodynamics.Based on this assumption, in Ref. [26] shown that thermal fluctuations can also contributeto entropy of the black hole. They suggested that the logarithmic corrections to Bekenstein-Hawking entropy can be interpreted as corrections due to the thermal fluctuations of theblack hole around its equilibrium configuration. Recently, the effects of thermal fluctuationswere studied for different kind of black objects such as charged AdS black hole [27], chargedhairy BTZ black hole [28], black Saturn [29, 30], modified Hayward black hole [31], dyoniccharged AdS black hole [32] and Kerr-AdS black hole [33].It is shown that the RN-AdS black holes have a van der Waals like first order small-large black hole phase transition ending in a critical point [34, 35]. Moreover the phasetransition in RN-AdS black hole is remarkable coincidence with van der Waals liquid-gasphase transition, when the cosmological constant and its conjugate quantities are consideredas thermodynamical variables [36]. The cosmological constant corresponds to pressure P = − Λ8 π = 38 π l (1)and its conjugate quantity corresponds to thermodynamic volume, V = ∂M∂P . For the RN-2dS black hole, it is given by V = 43 πr h . (2)Variable cosmological constant notion has led to a rich structure of phase transitions andcritical phenomena. Indeed, in the recent years, thermodynamics and phase transitionproperties of the black holes in AdS space have been discussed in many studies [36–53].In this article, we investigate the thermodynamics and phase transition of RN-AdS blackhole with thermal fluctuations following the method in the Refs. [26, 36]. As mentionedbefore, authors studied effects of thermal fluctuations for RN-AdS black holes [27] and ourresults differ from Ref. [27] which follows different approaches and assumptions.The paper is organized as follows. In section 2, we briefly review the logarithmic correctionto black hole entropy due to thermal fluctuations [26]. In section 3, we obtain correctedthermodynamic quantities and investigate phase transition of the RN-AdS black hole in thepresence of thermal fluctuations. We also compute critical exponents of the phase transitionwith small thermal fluctuations. Finally, we discuss our results in section 4. (We use theunits G N = (cid:126) = k B = c = 1.) II. LOGARITMIC CORRECTIONS TO BLACK HOLE ENTROPY
In this section, we will briefly review the thermal fluctuations correction to black holeentropy [26, 27]. First, we begin to consider partition function of a canonical ensemble Z ( β ) = ∞ (cid:90) ρ ( E ) e − β dE , (3)where β denotes the inverse of the temperature, β = T . One can obtain density of statesfrom Eq. (3), ρ ( E ) = 12 πi c + i ∞ (cid:90) c − i ∞ Z ( β ) e βE dβ = 12 πi c + i ∞ (cid:90) c − i ∞ e S ( β ) dβ , (4)where S ( β ) = lnZ ( β ) + βE . (5)Expanding S ( β ) around the equilibrium temperature β , one can obtain S = S + 12 ( β − β ) (cid:18) ∂ S ( β ) ∂β (cid:19) β = β + ... , (6)3here S = S ( β ). From Eqs. (4) and (6), density of state is given by ρ ( E ) = e S πi c + i ∞ (cid:90) c − i ∞ e ( β − β ) S (cid:48)(cid:48) dβ (7)where S (cid:48)(cid:48) = (cid:16) ∂ S ( β ) ∂β (cid:17) β . Substituting c = β and ( β − β ) = iz in Eq. (7), density of stateis given by ρ ( E ) = e S (cid:112) πS (cid:48)(cid:48) (8)and we can write S = S − lnS (cid:48)(cid:48) + ... . (9)This formula is valid for all thermodynamic system considered as canonical ensemble andit is also valid for black holes. S denotes the black hole entropy and thus we can replace T → T H . Now, we can determine S (cid:48)(cid:48) . We consider the entropy, which is suggested in Ref.[26] and based on Ref. [15], has the form, S = xβ m + yβ − n , (10)where x, y, m, n >
0. This entropy has an extremum at β = (cid:16) nymx (cid:17) m + n . (11)From Eq.(10) and Eq.(11), one can obtain S ( β ) = x (cid:16) nymx (cid:17) mm + n + y (cid:18) mxny (cid:19) nm + n , (12) S (cid:48)(cid:48) ( β ) = m ( m − x (cid:16) nymx (cid:17) m − m + n + n ( n + 1) y (cid:18) mxny (cid:19) n +2 m + n . (13)Using Eqs. (12) and (13) in Eq.(6), one can give the entropy as S ( β ) = A ( x n y m ) m + n + 12 B ( x n +2 y m − ) m + n ( β − β ) + ... (14)where A = (cid:16) nm (cid:17) mm + n + (cid:16) mn (cid:17) nm + n , B = ( m + n ) m n +2 m + n n m − m + n (15)are the constants. Comparing with Eq.(6), we can find S ( β ) = A ( x n y m ) m + n , S (cid:48)(cid:48) ( β ) = B ( x n +2 y m − ) m + n . (16)4e can obtain x and y in terms of S and S (cid:48)(cid:48) from Eq.(16) x = A m − B m ( S (cid:48)(cid:48) ) m S − m − , y = A − n +22 B − n S n +22 ( S (cid:48)(cid:48) ) − n . (17)Substituting x and y in Eq.(11), we can obtain β in terms of S , S (cid:48)(cid:48) β = (cid:16) nm (cid:17) m + n (cid:115) BA S S (cid:48)(cid:48) (18)and S (cid:48)(cid:48) is given by S (cid:48)(cid:48) = (cid:20)(cid:18) BA (cid:19) (cid:16) nm (cid:17) m + n (cid:21) S β − (19)The factor in the square brackets can be negligible, so we can write S (cid:48)(cid:48) = S β − . (20)Substituting Eq. (20) in Eq. (9), we can obtain S = S − lnS T H . (21)This is the generic correction formula for black hole entropy. We will discuss, in the nextsection, the corrections to thermodynamic quantities due to thermal fluctuations. III. REISSNER-NORDST ¨ORM-ADS BLACK HOLE
RN-AdS black hole in four dimensional space is given by the metric ds = − f ( r ) dt + f − ( r ) dr + r dθ + r sin ( θ ) dφ , (22)with f ( r ) = 1 − Mr + Q r + r l (23)where l , M and Q are the AdS radius, mass and charge of the black hole, respectively. Blackhole event horizon r h is given by as a largest root of f ( r h ) = 0. The mass of black hole canbe obtained in terms of r h , Q and l from Eq. (23), M = r h (cid:18) Q r h + r h l (cid:19) . (24)Hawking temperature is given by T H = (cid:18) f (cid:48) ( r h )4 π (cid:19) r = r h = 14 πr h (cid:18) r h l − Q r h (cid:19) (25)5nd expression for the entropy can be written S = A πr h , A = 4 πr h . (26)In Ref. [27] to track corrections coming from the thermal fluctuations, adding a parameterinto Eq. (21), entropy is generalized as S = S − a lnS T H . (27)In the case of a = 1, it is assumed that thermal fluctuations contribution is maximum,however, by setting a = 0, we will recover entropy expression in Eq. (26) without anycorrections. Indeed, if we can use Eqs. (25) and (26) into Eq. (27), we can obtain thefollowing expression, S = πr h − a ln (cid:18) l ( r h − Q ) + 3 r h √ πl r h (cid:19) (28)which refers to new entropy definition of black hole due to thermal fluctuations. For a = 0,entropy in Eq. (26) is recovered. Corrected temperature can be obtained from T = ∂M∂S T = (3 r h + l ( r h − Q )) l r h (3 πr h + ( πl − a ) r h − πl Q r h − l Q a ) . (29)Specific heat can be obtained from C = T ∂S∂T and thus specific heats at constant volume andpressure can be given by C V = a πr h + ( πl − a ) r h − πl Q r h − al Q − πr h + (3 a − πl ) r h + l ( πQ − a ) r h + 3 al Q (30)and C p = 2 9 π r h + 6 π l r h + π l ( l − Q ) r h − π l Q r h + π l Q r h + χ ( a )9 πr h + πl (6 Q − l ) r h + 4 πl Q r h − πl Q r h + χ ( a ) (31)where χ ( a ) = − aπr h + 3 a (3 a − πl ) r h + 2 al Q (3 a − πl Q ) r h + 2 aπl Q r h + a l Q and χ ( a ) = 3 a ( − r h + l r h − Q l r h − l Q r h ).Local thermodynamic stability of canonical ensemble is given by C P > C P < A. Equation of State and Phase Transition
In this section, we analyze the P − V criticality of RN-AdS black holes in the presenceof thermal fluctuations. We use the critical point investigation method which is defined in6ef. [36]. From Eqs. (1) and (29), one can obtain the equation of state P = Q πr h − πr h + T r h − aT πr h ± (cid:112) π T r h − W ( a )4 πr h (32)where W ( a ) = aT r h (2 πT r h − r h − aT r h + 2 Q ). The positive sign has been taken before thesquare root so we can obtain the equation of state in the a → ∂P∂r h = 0 , ∂ P∂r h = 0 . (33)It is possible to obtain critical points for the small values of a . For small thermal fluctuations,equation of state can be expanded as P = Q πr h − πr h + T r h + a (cid:18) π r h − T πr h − Q π r h (cid:19) + O ( a ) . (34)Specific volume v should be identified with v = 2 r h (35)for the small thermal fluctuations. Now one can obtain the critical points for small thermalfluctuations v c = 2 √ Q − √ πQ a + O ( a ) , (36) T c = √ πQ + √ π Q a + O ( a ) , (37) FIG. 1. The P − r h diagram. The temperatures of the isotherms decreases from top to bottomand correspond to 1 . T c , T c , 0 . T c . We fix a = 10 − and Q = 1. c = 196 πQ + a π Q + O ( a ) . (38)Universal number can be calculated from Eqs. (36), (37) and (38) P c T c v c = 38 − a πQ + O ( a ) . (39)As it can be seen from Eqs. (36), (37), (38) and (39) for a = 0, the results in Ref. [36] areobtained i.e, when a = 0, the critical points are v c = 2 √ Q , T c = √ πQ , P c = πQ and P c T c v c = . Critical points change due to thermal fluctuations. For example critical specificvolume decreases while the critical temperature and pressure increase. Universal number ofphase transition depends on Q and a .In order to investigate the phase transition behavior of RN-AdS black hole, P − r h diagramhas been plotted in Fig. 1 for small values of thermal fluctuations. As it can be seen from thefigure that under the critical temperature P − r h diagram has characteristic behavior of vander Waals phase transition. It is also important to investigate the phase transition behaviorfor large values of thermal fluctuations since effects of thermal fluctuations are importantfor small size. When thermal fluctuations are increased, the phase transition deviates fromvan der Waals behavior. In Figs. 2 and 3, we have plotted P − r h diagrams for thermalfluctuations. It can easily be seen from figures that thermal fluctuations affect the phasetransition. Moreover, in contrast to Fig. 1, small black hole branch is unstable and pressureis imaginary under a certain event horizon radius. It may be interpreted as black holesdo not exist under a certain value of event horizon. This interpretation is also useful to FIG. 2. The P − r h diagram. The temperatures of the isotherms decreases from top to bottomand correspond to 1 . T c , T c , 0 . T c . We fix a = 1 and Q = 1. . T c isotherm in Fig. 3 is imaginary for r h < . r h < . B. Critical Exponents
The thermodynamic behavior of a system near the phase transition point has been classi-fied by using critical exponents. It is supposed that the critical exponents to be universal andare the independent of the details of the interaction. On the other hand, different physicalsystems may share the same critical exponents, which indicates that they are in the sameuniversal class. In this section, we discuss critical exponents of the phase transition at nearthe critical point for the RN-AdS black hole. The critical exponents are given as follows:The exponent α determines the behavior of specific heat at constant volume C V = T (cid:18) ∂S∂T (cid:19) ∝ | t | − α . (40)The exponent β determines the behavior of the order parameter on the given isotherm η = V l − V s ∝ | t | β . (41) FIG. 3. The P − r h diagram. The temperatures of the isotherms decreases from top to bottomand correspond to 1 . T c , T c , 0 . T c . We fix a = 0 . Q = 1. γ determines the behavior of the isothermal compressibility κ T κ T = − V (cid:18) ∂V∂P (cid:19) ∝ | t | − γ . (42)The exponent δ determines the following behavior on critical isotherm T = T c | P − P c | ∝ | V − V c | δ . (43)Recently, a more general method is suggested to obtain critical exponents in Ref. [53].However, in our case, it is convenient to use method in Ref. [36]. Now following method inRef. [36], we can obtain critical exponents of phase transition for RN-AdS black hole. Wecannot obtain the exact critical point due to the complexity of equation of state in Eq. (32)so we cannot obtain the law of corresponding states from Eq. (32) the critical exponent maydiffer from Ref. [36]. On the other hand, we can compute the critical exponents for smallthermal fluctuations. In the presence of small thermal fluctuations, we can obtain specificheat at constant volume, C V = − a + O ( a ) and this is independent of T so exponent α = 0.Defining p = PP c , τ = TT c , ν = VV c , (44)we can obtain the so called law of corresponding states with small thermal fluctuations.Expanding around the critical point t = τ − , ω = ν − p = 1 + (cid:18) − a πQ (cid:19) t + (cid:18) −
89 + 32 a πQ (cid:19) tω + (cid:18) −
481 + 16 a πQ (cid:19) ω + O ( tω , ω ) . (46)Differentiating Eq.(46) for a fixed t < dP = − P c (cid:20)(cid:18) − a πQ (cid:19) t + (cid:18) − a πQ (cid:19) ω (cid:21) dω . (47)Employing Maxwells equal area law, one can get the following two equations: p = 1 + (cid:16) − a πQ (cid:17) t + ( − + a πQ ) tω l + ( − + a πQ ) ω l = 1 + (cid:16) − a πQ (cid:17) t + ( − + a πQ ) tω s + ( − + a πQ ) ω s , (cid:72) ωdP = (cid:82) ω s ω l ω (cid:104)(cid:16) − a πQ (cid:17) t + (cid:16) − a πQ (cid:17) ω (cid:105) dω . (48)10here ω s and ω l denotes volume of small and large black holes. From these equation, onecan obtain ω l = − ω s = 3 √− t . (49)So we find the exponent βη = V c ( ω l − ω s ) = 2 V c √− t ⇒ β = 12 . (50)We compute the isothermal compressibility exponent γκ T = − V (cid:18) ∂V∂P (cid:19) T ∝ πQ P c (9 πQ − a ) t ⇒ γ = 1 . (51)Finally we obtain the exponent δ for t = 0 p − (cid:18) −
481 + 16 a πQ (cid:19) ω ⇒ δ = 3 . (52)We computed the critical exponents for small thermal fluctuations. All critical exponentsthat were computed in the presence of small thermal fluctuations coincide critical exponentsof van der Waals fluid and RN-AdS black hole without thermal fluctuations [36]. IV. CONCLUSION
In this paper, we studied effects of thermal fluctuations for RN-AdS black hole. We ob-tained some corrected thermodynamic quantities and investigated P − V criticality in thepresence of thermal fluctuations. We obtained critical points for small thermal fluctuationsand numerically studied phase transition for large values of thermal fluctuations. Thermalfluctuations effects are remarkable for small size black hole and also affect the phase tran-sition. Black hole may not exist under a certain value of event horizon. This minimumevent horizon condition provides us to avoid ill-defined regions of temperature and entropy.Finally we obtained the critical exponent for small thermal fluctuations. We showed thatcritical exponents are the same as critical exponent without thermal fluctuations. V. CONFLICT OF INTEREST
The authors declare that there is no conflict of interest regarding the publication of thispaper. 11
CKNOWLEDGMENTS
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