First-order corrected thermodynamic geometry of a static black hole in f(R) gravity
aa r X i v : . [ g r- q c ] J a n First-order corrected thermodynamic geometry of a staticblack hole in f ( R ) gravity Sudhaker Upadhyay, a Saheb Soroushfar b,c and Reza Saffari d a Department of Physics, K.L.S. College, Nawada-805110, India b Faculty of Technology and Mining, Yasouj University, Choram 75761-59836, Iran c Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha,Iran d Department of Physics, University of Guilan, 41335-1914, Rasht, Iran
E-mail: [email protected] , [email protected] , [email protected] Abstract:
In this paper, we consider a static black hole in f ( R ) gravity. We recapitulate theexpression for corrected thermodynamic entropy of this black hole due to small fluctuationsaround equilibrium. Also, we study the geometrothermodynamics (GTD) of this black holeand investigate the adaptability of the curvature scalar of geothermodynamic methods withphase transition points of this black hole. Moreover, we point out the effect of α parameteron thermodynamic behavior of this black hole. As a result, we see that, the singular point ofthe curvature scalar of Ruppeiner metric is completely coincided with zero point of the heatcapacity and by increasing α , this match will be less. ontents f ( R ) gravity 4 In recent years, much attention has been paid to the physics and thermodynamics of blackholes. Thermodynamic quantities of black holes such as temperature, entropy and heat ca-pacity; thermodynamic properties such as phase transitions and thermal stability; geometricalquantities such as horizon area and surface gravity; thermodynamical variable such as cosmo-logical constant have been studied in several papers [1–25]. Geometrical thermodynamics isother applied and important method to study phase transition of black holes. For present dif-ferent implications of geometry in to usual thermodynamics, many endeavor are presented invarious articles [26–32]. It is well-known that the larger black holes in comparison to the Planckscale have entropy proportional to its horizon area, so it is important to investigate the formof entropy as one reduces the size of the black hole [33–38]. It is found that the area-law ofentropy of any thermodynamic system due to small statistical fluctuations around equilibriumgets correction [38]. These corrections are evaluated through both microscopically and using astringy embedding of the Kerr/CFT correspondence [39]. The leading-order correction to theentropy area-law is also estimated through the logarithmic corrections to the Cardy formula[40]. Various check-ups confirm that the entropy originated due to the statistical fluctuationsaround thermal equilibrium of the black hole has logarithmic corrections.The study of quantum effects to the black holes thermodynamics is a subject of currentinterests. For instance, the thermal fluctuations corrects the thermodynamics of higher dimen-sional AdS black hole, where it has been found that the Van der Waals black hole is completelystable in presence of the logarithmic correction [41]. The leading-order corrections to the Gibbsfree energy, charge and total mass densities of charged quasitopological and charged rotating qu-asitopological black holes due to thermal fluctuation are derived, where the stability and boundpoints of such black holes under effect of leading-order corrections are also discussed [42]. The– 1 –ffect of thermal fluctuations on the thermodynamics of a black geometry with hyperscalingviolation is also studied recently [43]. The quantum corrections on the thermodynamics andstability of Schwarzschild-Beltrami-de Sitter black hole [44], G¨odel black hole [45], and mas-sive black hole [46] have been investigated recently. Also, the effects of thermal fluatuationon the Hoˇrava-Lifshitz black hole thermodynamics and their stability are discussed [47]. Theleading-order correction to modified Hayward black hole is derived and found that correctionterm reduces the pressure and internal energy of the Hayward black hole [48].On the other hand, in order to introduce concepts of geometry into ordinary thermodynamicsmany efforts have been made. In this regard, the implication of thermodynamic phase space as adifferential manifold is defined, which embeds a special subspace of thermodynamic equilibriumstates [49]. Many different perspective with different goals and reviews in f ( R ) gravity havebeen studied in the past decade until now [50–59]. Keeping importance of f ( R ) gravity inmind, our motivation is to study the effect of thermal fluctuation on the thermodynamics andthermodynamic geometry of a static black hole in f ( R ) gravity.In this paper, we obtain corrected thermodynamic entropy and investigate thermodynamicquantities and thermodynamic geometric methods for a black holes in f ( R ) gravity. We findthat the Hawking temperature is a decreasing function of horizon radius. In order to evaluatethe correction to the entropy of a static black hole in f ( R ) gravity due to thermal fluctuation,we exploit the expressions of Hawking temperature and uncorrected specific heat. Moreover,utilizing standard thermodynamical relation, we compute first-order corrected mass and heatcapacity. Here, we observe that the uncorrected mass of the black hole has a maximum value at r + = r m = 2 .
31, and takes zero value at two points, r + = 0 and r + = 4. For the corrected masswith positive correction coefficient α , the number of zero points of mass are not changing, butthey appear with larger horizon radius. In case of heat capacity, we find that the uncorrectedheat capacity is in the negative region (unstable phase) for 0 < r + < r m and it takes type onephase transition at r + = r m for which C ( r + = r m ) = 0. For r + > r m , it becomes positivevalued (stable). So, without considering thermal fluctuation, this black hole has a type onephase transition. However, as long as we turn on effects of thermal fluctuation, the numberof zero points of the heat capacity changes to the three zero points. Furthermore, we analysethe thermodynamic geometry of such black holes. In order to do so, we plot thermodynamicquantities and curvature scalar of Weinhold, Ruppiner and GTD methods in terms of horizonradius. We find that the singular points of curvature scalar of Weinhold and GTD methods, forboth with and without thermal fluctuations, are not coinciding with zero point of heat capacity(the phase transition points) which suggests that we are unable to get any physical informationabout the system with these two methods. However, without considering quantum effects, wefind that the heat capacity has only one zero, and the singular point of the curvature scalarof Ruppeiner metric is completely coinciding with it. The heat capacity under the effects ofthermal fluctuation has three zero points and not all the singular points of the curvature scalarof Ruppeiner metric are not completely coinciding with zero points of such black hole but onlyone of the singular point of the curvature scalar of Ruppeiner metric is completely coincidewith one of the zero point of the heat capacity. This suggests that, by increasing α , the above– 2 –daptation is reducing.This paper is organized as follows, in Sec. 2, we recapitulate the expression for correctedthermodynamic entropy of black hole due to small fluctuations around equilibrium, Then, inSec. 3, we briefly review the static f ( R ) black hole solution and its thermodynamics. Also,we study modified thermodynamics due to thermal fluctuations and thermodynamic geometrymethods for this black hole. We conclude our results in Sec. 4. In this section, we recapitulate the expression for corrected thermodynamic entropy of blackholes due to small fluctuations around equilibrium. To do so, let us begin by defining thedensity of states with fixed energy as [60, 61] ρ ( E ) = 12 πi Z c + i ∞ c − i ∞ e S ( β ) dβ, (2.1)where the exact entropy, S ( β ) = log Z ( β ) + βE , depends on temperature T (= β − ) explicitly.So, this (exact entropy) is not just its value at equilibrium. The exact entropy corresponds to thesum of entropies of subsystems of the thermodynamical system. This thermodynamical systemsare small enough to be considered in equilibrium. To investigate the form of exact entropy, wesolve the complex integral (2.1) by considering the method of steepest descent around the saddlepoint β (= T − H ) such that ∂ S ( β ) ∂β (cid:12)(cid:12)(cid:12) β = β = 0. Now, performing Taylor expansion of exact entropyaround the saddle point β = β leads to S ( β ) = S + 12 ( β − β ) (cid:18) ∂ S ( β ) ∂β (cid:19) β = β + (higher order terms) . (2.2)Here S (= S ( β )) is the leading-order entropy. With this value of S ( β ), the density of states(2.1 becomes ρ ( E ) = e S πi Z c + i ∞ c − i ∞ exp "
12 ( β − β ) (cid:18) ∂ S ( β ) ∂β (cid:19) β = β dβ. (2.3)This further leads to [38] ρ ( E ) = e S r π (cid:16) ∂ S ( β ) ∂β (cid:17) β = β , (2.4)where c = β and ∂ S ( β ) ∂β (cid:12)(cid:12)(cid:12) β = β > S = S −
12 log (cid:18) ∂ S ( β ) ∂β (cid:19) β = β + (sub-leading terms) . (2.5)– 3 –ne can determine the form of ∂ S ( β ) ∂β (cid:12)(cid:12)(cid:12) β = β by considering the most general form of the exactentropy density S ( β ). Das et al. in [38] have found the form of ∂ S ( β ) ∂β (cid:12)(cid:12)(cid:12) β = β = C T H , whichleads to the leading-order corrected entropy as S = S −
12 ln( C T H ) , (2.6)where C = (cid:0) ∂E∂T (cid:1) T H . By considering an arbitrary (positive) correction parameter α , we write ageneric expression for leading-order corrected entropy as following: S = S − α ln( C T H ) . (2.7)Now, we would like to study the effects of such correction term on the thermodynamic geometryof black holes in f ( R ) gravity. f ( R ) gravity In this section we briefly review the static f ( R ) black hole solution and its thermodynamics.The generic form of the action is: I = 12 k Z d x √− gf ( R ) + S mat , (3.1)where S mat refers to the matter part of the action. The spherically symmetric solution of thefield equations of the action (3.1) is ds = − B ( r ) dt + B ( r ) − dr + r ( dθ + sin θdϕ ) , (3.2)where B ( r ) = 1 − mr + β r −
13 Λ r , (3.3)here parameter m is related to the mass of the black hole, Λ is the cosmological constant and β is a real constant [62, 63]. If r + denotes the radius of the event horizon, by setting B ( r ) = 0,the mass of the black hole, using the relation between entropy S and event horizon radius r + ,( S = πr ), is given by M ( S , l, β ) = l π / β S + l π S / − S / l π / , (3.4)where parameter l is related to the cosmological constant Λ as follows [64],Λ = 3 l . (3.5)The Hawking temperature ( T H = ∂M ∂ S ) and heat capacity ( C = T H ∂ S ′ ∂T H ) are calculated as [65] T H = 2 β l π / S / − S + l π π / l S / , (3.6) C = − β l π / S / − S + 2 l π S l π + 3 S . (3.7)– 4 –e shall utilize these expressions of Hawking temperature and heat capacity to compute thecorrected entropy due to quantum fluctuations. The behavior of Hawking temperature in termsof horizon radius r + can be seen in Fig. 1. From this plot, it is obvious that the temperature Figure 1 : Variation of temprature of the black hole in terms of horizon radius r + . l = 4 . β = 10 − . is a decreasing function of r + and takes positive value only in a particular range of r + , then at r + = r m , it reaches into zero, and after that, it falls into negative region. Exploiting relations (2.7), (3.6) and (3.7), the leading-order corrected entropy due to thermalfluctuations is given by S = S − α log " − β l π / S / − S + 2 l π S l π + 3 S − α log " β l π / S / − S + l π π / l S / . (3.8)Using standard relation (cid:0) M = R T H dS (cid:1) , the corrected mass is calculated as M = l π / β S + l π S / − S / l π / + α l π + 3 S l π / S / + √ αlπ cot − lπ / √ S ! − αβ π [3 log S − l π + 3 S )] − α β l π / S / − S + l π π / l S / . (3.9)The corrected heat capacity, using relation (cid:16) C = T H ∂S∂T H (cid:17) , is computed as C = − β l π / S / − S + 2 l π S l π + 3 S − α − α (cid:0) π l + 3 β √ πl √S (cid:0) πl + S (cid:1) − πl S − S (cid:1) (cid:16) l β √ π S / + πl − S (cid:17) S ( πl + 3 S ) (cid:0) l β √ π √S + πl − S (cid:1) . (3.10)– 5 – a) (b) Figure 2 : Variation of mass in terms of horizon radius of a static black hole r + for l = 4 . β = 10 − and α = 0, α = 0 . Figure 3 : Variation of heat capacity in terms of horizon radius of a static black hole r + for l = 4 . β = 10 − and α = 0, α = 0 . These thermodynamic parameters are plotted in terms of horizon radius r + (see Figs. 2 and 3).From Fig. 2(a)(for α = 0), it can be seen that, mass of the black hole has a maximum valueat r + = r m = 2 .
31, and it become zero at two points, r + = r = 0 and r + = r = 4. Also, ascan be seen from Fig. 2(b) (for α > α , the number of zero points of mass arenot changing, but they shift towards larger values. In addition, plot of the heat capacity canbe seen in Fig. 3. In this case, we find that, for the case α = 0 (Fig. 3(a)), for 0 < r + < r m ,the heat capacity is in the negative region (unstable phase), then, at r + = r m , it takes type onephase transition, in which C ( r + = r m ) = 0), after that, for r + > r m , it becomes positive valued– 6 –stable). So, for the case α = 0, this black hole has a type one phase transition. However, (for α > α , the number of zero points of the heatcapacity are changing to the three zero points. In other words, for the case α = 0, in additionto one zero two other zeroes also appear. In this section, using the geometric technique of Weinhold, Ruppiner and GTD metrics of thethermal system, we construct the geometric structure for a static black hole in f ( R ) gravity.In this case, the extensive variables are, N r = ( l, β ). The Weinhold metric is defined as theHessian in the mass representation as follows [27] g Wij = ∂ i ∂ j M ( S, N r ) . (3.11)We can write the Weinhold metric for this system as follows g Wij = ∂ i ∂ j M ( S, l, β ) . (3.12)The line element corresponding to Weinhold metric is given by ds W = M SS dS + M ll dl + M β β dβ M Sl dSdl + 2 M Sβ dSdβ + 2 M lβ dldβ , (3.13)therefore g W = M SS M Sl M Sβ M lS M ll M lβ M β S M β l . (3.14)The components of above matrix can be found using the expression of M , given in Eq. (3.9).Since the equation of the curvature scalar of the Weinhold metric is too large, so we demonstrateit in Fig. (4).It can be observed from Fig. 4 that the singular points of curvature scalar of Weinholdmetric do not coincide with zero points of heat capacity, so, in this case we can’t find anyphysical information about the system from the Weinhold method. In following, we use Ruppinermethod, which is conformaly transformed to the Weinhold metric. The Ruppiner metric isdefined by [28, 29, 66] ds R = 1 T ds W . (3.15)The matrix corresponding to the metric components of Ruppiner method is as following: g R = l π S l π S β − S + l π M SS M Sl M Sβ M lS M ll M lβ M β S M β l . (3.16)Plot of the scalar curvature of the Ruppiner metric is shown in Fig. 5(a). Also, plots of thecurvature scalar of the Ruppiner metric and the heat capacity, in terms of r + , are demonstrated– 7 – igure 4 : Curvature scalar variation of Weinhold metric (orange continuous line) and the heat capacity of a static blackhole (green dash line) in terms of the radius of the horizon r+, for α = 0 . l = 4 . β = 10 − . in Fig. 5(b), (c) and (d). As one can see from Fig. 5(b), for the case α = 0, the heat capacity hasonly one zero (the phase transition point), however, the singular point of the curvature scalar ofRuppeiner metric completely coincides with zero point of the heat capacity. On the other hand,for the case α > α , this adaptation is reduced.In the next step, we will investigate the GTD method. The general form of the metric inGTD method is as following [31, 67]: g = (cid:18) E c ∂ Φ ∂E c (cid:19) (cid:18) η ab δ bc ∂ Φ ∂E c ∂E d dE a dE d (cid:19) , (3.17)where ∂ Φ ∂E c = δ cb I b , (3.18)in which Φ is the thermodynamic potential, I b and E a are the intensive and extensive thermo-dynamic variables. So, according to Eq. (3.17), the metric for this thermodynamic system is asfollowing: g GT D = ( SM S + lM l + βM β ) − M SS M ll M lβ M β l . (3.19)The plot of the scalar curvature of GTD metric is shown in Fig. (6). It can be observed fromFig. 6, the singular point of curvature scalar of GTD metric does not coincide with zero pointof heat capacity, so, in this case we can’t find any physical information about the system fromthe GTD method. – 8 – a) α = 0 . α = 0(c) α = 0 . Figure 5 : Curvature scalar variation of Ruppeiner metric (orange continuous line) and the heat capacity of a static blackhole (green dash line) in terms of r + for l = 4 . β = 10 − . In this paper, we studied small statistical fluctuations around equilibrium for a static black holein f ( R ) gravity and analyzed thermodynamic quantities of this black hole according to correctedthermodynamic entropy. We have found that the Hawking temperature is a decreasing functionof horizon radius. We have utilized the expressions of Hawking temperature and uncorrectedspecific heat to evaluate the correction to the entropy of a static black hole in f ( R ) gravity due tothermal fluctuation. Also, we have computed the first-order corrected mass and heat capacity byutilizing the standard thermodynamical definitions. From graphical study, we observed that theuncorrected mass of the black hole gets a maximum value at r + = r m = 2 .
31, and takes zero valueat two points, r + = 0 and r + = 4. However, for the corrected mass with positive correction– 9 – igure 6 : Curvature scalar variation of GTD metric (orange continuous line) and the heat capacity of a static black hole(green dash line) in terms of the radius of the horizon r+, for α = 0 . l = 4 . β = 10 − . coefficient α , the number of zero points of mass do not changing, but they appear at largerhorizon radius. For the case of heat capacity, we have found that the uncorrected heat capacityhas negative values (unstable phase) for 0 < r + < r m and type one phase transition takes placefor black holes undergo at r + = r m as C ( r + = r m ) = 0. For r + > r m , the heat capacity takespositive value (stable). So, one may conclude that without considering thermal fluctuation, thisblack hole has a type one phase transition. However, due to e thermal fluctuation, the numberof zero points of the heat capacity changes to the three zero points.Also, we investigated the thermodynamic geometry of this black hole and plotted thermo-dynamic quantities and curvature scalar of Weinhold, Ruppiner and GTD methods in terms ofhorizon radius r + . We observed that, the singular points of curvature scalar of Weinhold andGTD methods, for both α = 0 and α > α = 0, we realized that theheat capacity has only one zero, and the singular point of the curvature scalar of Ruppeinermetric is completely coinciding with it, moreover for the case α >
0, the heat capacity hasthree zero points, and the singular points of the curvature scalar of Ruppeiner metric are notcompletely coinciding with zero points of the heat capacity of this black hole and only one ofthe singular point of the curvature scalar of Ruppeiner metric is completely coincide with oneof the zero point of the heat capacity, that’s mean by increasing α , this adaptation is reduced. Acknowledgement
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