Critical behaviors of black hole in an asymptotically safe gravity with cosmological constant
aa r X i v : . [ h e p - t h ] M a y Critical behaviors of black hole in an asymptotically safe gravitywith cosmological constant
Meng-Sen Ma a,b ∗ , Ya-Qin Ma c a Department of Physics, Shanxi Datong University, Datong 037009, China b Institute of Theoretical Physics, Shanxi Datong University, Datong 037009, China c Medical College, Shanxi Datong University, Datong 037009, China
We study the P − V /r + criticality and phase transition of quantum-corrected blackhole in asymptotic safety (AS) gravity in the extended phase space. For the blackhole, the cosmological constant is dependent on the momentum cutoff or energyscale, therefore one can naturally treat it as a variable and connect it with thethermodynamic pressure. We find that for the quantum-corrected black hole thereis a similar first-order phase transition to that of the van der Waals liquid/gas system.We also analyze the types of the phase transition at the critical points according toEhrenfest’s classification. It is shown that they are second-order phase transition. PACS numbers: 04.70.Dy, 05.70.Fh
I. INTRODUCTION
Like ordinary thermodynamic matter, black holes also have temperature, entropy and energy.The laws of black hole mechanics have the similar forms to the laws of thermodynamics[1].Therefore, we can treat black holes as thermodynamic systems. In fact, between black holes andthe conventional thermodynamic systems, there are other similarities, such as phase transitionand critical behaviors. The pioneering work of Davies[2] and the well-known Hawking-Pagephase transition[3] are both proposed to elaborate these points. The phase transitions andcritical phenomena in anti-de Sitter (AdS) black holes have been studied extensively[4–10].Some interesting works show that there exists phase transition similar to the van der Waalsliquid/gas phase transition for some black holes[11–19]. Even for the black holes in dS spacecritical behaviors can also be studied by considering the connections between the black holehorizon and the cosmological horizon[20, 21].Recently, some physicists reconsidered the critical phenomena of AdS black holes by treat-ing the cosmological constant Λ as a variable and connecting it with the thermodynamicpressure[22–27]. In some models, the cosmological constant may be considered to be atime-variable quantity[28, 29], or as some thermodynamic quantities, such as thermodynamicpressure[30, 31], which should be a conjugate quantity of thermodynamic volume. Inclusionof the variation of Λ can make the first law of black hole thermodynamics consistent with theSmarr formula for some black holes.In this letter, we study the critical behaviors of a kind black hole derived in asymptotic safetygravity. The asymptotic safety scenario for quantum gravity was put forward by Weinberg[32].It is based on a nontrivial fixed point of the underlying renormalization group(RG) flow forgravity. This theory has been studied extensively and applied to several different subjects inquantum gravity[33–35]. Bonanno and Reuter[36] derived the renormalization group improved ∗ Email: [email protected] black hole metrics by replacing Newtonian coupling constant with a “ running ” one. Cai, et.al[37] find a spherically symmetric vacuum solution to field equation derived from the AS gravitywith higher derivative terms and with cosmological constant. In this theory, the cosmologicalconstant is no longer constant but dependent on a momentum cutoff. Therefore it is reasonableto include the variation of Λ in the first law of black hole thermodynamics as thermodynamicpressure P . The quantum correction of AS gravity to the conventional Schwarzschild-AdS blackhole makes its thermodynamic quantities and critical behaviors very different. The quantum-corrected black hole can also show a phase transition analogous to the liquid-gas phase transitionin the Van der Waals system. According to Ehrenfest’s classification we also consider the Gibbsfree energy, the isothermal compressibility and the expansion coefficient. It is shown that thetype of phase transition for the black hole at the critical point belongs to the second-order orcontinuous one.The paper is arranged as follows: in the next section we simply introduce the AS gravitymodel and its quantum-corrected black hole solution. In section 3 we will study the P − V /r + criticality by considering the cosmological constant as thermodynamic pressure. We alsocalculate the critical exponents here. In section 4 we also analyze the type of the phase transitionof the quantum-corrected black hole in the extended phase space according to Ehrenfest’sclassification. We make some concluding remarks in section 5. II. QUANTUM-CORRECTED BLACK HOLE IN AS GRAVITY
We start with a generally covariant effective gravitational action with higher derivative termsinvolving a momentum cutoff p [37] :Γ p [ g µν ] = Z dx √− g [ p g ( p ) + p g ( p ) R + g a ( p ) R + g b ( p ) R µν R µν + g c ( p ) R µνσρ R µνσρ + O ( p − R ) + ... ] , (2.1)where g is the determinant of the metric tensor g µν , R is the Ricci scalar, R µν is the Riccitensor and R µνσρ is the Riemann tensor. The coefficients g i ( i = 0 , , a, ... ) are dimensionlesscoupling parameters and are functions of the dimensionful, UV cutoff. The couplings satisfythe RG equations: dd ln p g i ( p ) = β i [ g ( p )] (2.2)Assuming a static spherically symmetric metric ansatz and choosing the Schwarzschild gauge ds = − f ( r ) dt + f ( r ) − dr + r d Ω , (2.3)and then substituting it into the generalized Einstein field equations˜ G µν ≡ δ Γ p [ g µν ] δg µν = 0 , (2.4)one can derive a Schwarzschild-(anti)-de Sitter-like solution f ( r ) = 1 − G p Mr ± r l p , (2.5)where G p and l p are the gravitational coupling and the radius of the asymptotically (A)dSspace, and both depend on the momentum cutoff p .It is shown in [37] that there are a Gaussian fixed point in the IR limit and a non-Gaussianfixed point in the UV limit. A central result is g ≃ − (Λ IR + ηp G N )(1 + ξp G N )8 πp G N (2.6) g ≃ ξp G N πp G N , (2.7)where G N and Λ IR are the values of the gravitational coupling and the cosmological constantin the IR limit which should be determined by astronomical observations. The parameters ξ and η are both related to the running couplings λ ( p ) g a , λ ( p ) g b , λ ( p ) g c at the non-Gaussianfixed point. The coefficient λ has the familiar logarithmic form which approaches asymptoticfreedom λ ( p ) = λ π λ ln p/M p , (2.8)where λ is a fixed value of the coefficient λ at the Planck scale, and M p is the Planck mass.It is shown that the running gravitational coupling G p is related the Newtonian gravitationalcoupling constant G N by G p = G N ξp G N . (2.9)At high energy scale, p ( r ) ≃ . M | λ | ) / r − / . (2.10)thus, f ( r ) ≃ − π | λ | / ( M r ) / (2.11)which is singular-free at r = 0. However, the curvature singularity still exists due to divergent R µν R µν , R µνρσ R µνρσ .At low energy scale, the momentum cutoff drop to the infrared(IR) limit, and p ≃ r . Atthis time, f ( r ) ≃ − M rr + ξ ± r l p (2.12)where G N = 1 has been set for simplicity. The parameter ξ represents the quantum correctionto the conventional Schwarzschild-AdS black hole. Obviously, when ξ = 0, the corrected metricwill return back to the Schwarzschild-AdS one. Below we will study the thermodynamics of thequantum-corrected black holes based on Eq.(2.12). It is shown that owing to the correction,the thermodynamic quantities will also be corrected. III. P − V CRITICALITY OF THE QUANTUM-CORRECTED BLACK HOLE
In this paper we only concern with the asymptotic AdS black hole. Firstly, we identify thepressure with P = 38 πl p . (3.1)From Eq.(2.12), one can easily obtain the mass M = (cid:0) πP r + 3 (cid:1) (cid:0) ξ + r (cid:1) r + , (3.2)where r + is the radius of the black hole event horizon.The first law of black hole thermodynamics should written as dM = T dS + V dP (3.3)where the conjugate thermodynamic volume V = ∂M∂P (cid:12)(cid:12)(cid:12) S = πr + (cid:0) ξ + r (cid:1) . Here, the mass ofblack hole is no more internal energy, but should be interpreted as the thermodynamic enthalpy,namely H = M ( S, P )[14, 22, 38, 39]. The first law of black hole thermodynamics representedby the internal energy U ( S, V ) reads dU = T dS − P dV (3.4)where U = H − P V .The Hawking temperature of the black hole can be easily derived T = f ′ ( r + )4 π = − ξ + 24 πP r + 8 πξP r + 3 r πr + 12 πξr + . (3.5)When ξ = 0, it will give the temperature of Schwarzschild-AdS black hole. According to thefirst law, Eq.(3.3), one can derive the entropy S = Z dMT = Z T ∂M∂r + (cid:12)(cid:12)(cid:12)(cid:12) P dr + = πr + 2 πξ ln r + p ξ + S (3.6)where S is an integration constant which can be decided by the boundary conditions. Theadditional logarithmic term in the expression of the entropy indicates the quantum gravitationalcorrection. As the parameter ξ →
0, the standard Bekenstein-Hawking area law will return. Itis interesting that no P exists in the expression of S although it is included in M and T . Ξ= Ξ= Ξ= Ξ= r + T (a) r + - C P (b) FIG. 1:
The temperature and heat capacity at constant pressure as functions of r + . We set l p = 5 here. The three black dots represent the local extrema of the temperature. The dot-dashed line in (b)corresponds to the temperature of the Schwarzschild-AdS black hole. The heat capacity at constant pressure can be given by C P = ∂M∂T (cid:12)(cid:12)(cid:12)(cid:12) P = 2 π (cid:0) ξ + r (cid:1) (cid:0) − ξ + 24 πP r + 8 πξP r + 3 r (cid:1) ξ + 24 πP r + 64 πξP r + 8 πξ P r − r + 12 ξr (3.7)The qualitative behaviors of the temperature T and the heat capacity C P are depicted in Fig.1.Obviously, owing to the existence of ξ , the temperature will not blow up as the radius of theevent horizon approaches zero, but tends to zero at a finite radius where the black hole willbecome extremal one. For a fixed pressure there is a critical value of ξ , below which there willbe both local maximum and minimum for the temperature, and above which no local extremumexists. At the critical value, the maximum and minimum will coincide. From Fig.1(b), onecan see that, when ξ < ξ c , C P suffers discontinuities at two points , which can be identified asthe critical points for phase transition in the quantum-corrected black hole. The divergencesof the heat capacity appear precisely at the extrema of the temperature. The small and largeblack holes with positive heat capacity can be stable. While the intermediate black hole withnegative heat capacity is instable.From Eq.(3.5), one can derive the equation of state of the black hole P = 3 (cid:0) ξ + 4 πr T − r + 4 πξr + T (cid:1) πr ( ξ + 3 r ) (3.8)One can take the specific volume as v ∝ V /N , with N = A/l P counting the number of degreesof freedom associated with the black hole horizon[25]. l P here is the Planck length. If we take v = 6 V /N , the specific volume can be expressed as v = 2( r + + ξr + ) . (3.9)Obviously, when ξ = 0, it will give the result similar to that in [23, 24]. Replacing r + inEq.(3.8) with v , one can obtain the equation of motion, P = P ( v, T, ξ ). As is done in [24], onecan also expand P ( v, T, ξ ) in powers of a in the small a limit and take the first several termsapproximately. P = Tv − πv + 4(5 πT v − ξ πv + 8(68 πT v − ξ πv + O [ ξ ] (3.10)One can see that the above equation of state is similar to that of Kerr-AdS black hole. It indeedexhibits P − v criticality. However, in this paper we want to treat Eq.(3.8) exactly. We willuse the horizon radius in the equation of the state instead of the specific volume hereafter.The critical point can be obtained according to ∂P∂r + = 0 , ∂ P∂r = 0 (3.11)which lead to r c = p cξ, T c = 3 c − c − π (3 c + 8 c + 1) √ cξ , P c = 3 ( c − c − πc (3 c + 8 c + 1) ξ (3.12)where the constant c = 3 + 23 / (cid:2) (39 + i √ / + (39 − i √ / (cid:3) , which is a real number.Numerically c ≈ .
53. These critical values can lead to the following universal ratio ρ c = P c r c T c = 3 ( c − c − c − c − ≈ . . (3.13) T = c T = T c T = c r + P FIG. 2: P − r + diagram for the AS improved black hole. We choose ξ = 0 . here. The critical radius,temperature and pressure are respectively r c = 0 . , T c = 0 . , P c = 0 . . Obviously, it is independent of the quantum-corrected constant ξ . Note that for the van derWaals gas, the universal ratio is ρ c = 3 /
8, while for some actual gas, such as water, it is ρ c = 0 . G = G ( T, P ) = H − T S = M − T S .As is shown in Fig.3, the Gibbs free energy develops a “ swallow tail” for
P < P c , which is atypical feature in a first-order phase transition. Above the critical pressure P c , the “ swallowtail” disappears. P = c P = P c P = c T G FIG. 3:
The Gibbs free energy as function of temperature for different pressure for the quantum-corrected black hole. We also choose ξ = 0 . here. Next we will calculate the critical exponents at the critical point for the quantum correctedblack hole. For a van der Waals liquid/gas system, the critical behaviors can be characterizedby the critical exponents as follows[40]: C v ∼ (cid:18) − T − T c T c (cid:19) − α , v g − v l v c ∼ (cid:18) − T − T c T c (cid:19) β , κ T ∼ (cid:18) − T − T c T c (cid:19) − γ , P − P c ∼ ( v − v c ) δ . (3.14)Here v g and v l refer to the specific volume for gas phase and liquid phase respectively. For thequantum-corrected black hole, we use r g and r l instead. Defing t = TT c − , x = r + r c − , p = PP c (3.15)and replacing r + , T, P in Eq.(3.8) with the new dimensionless parameters x, t, p and thenexpanding the equation near the critical point approximately, one can obtain p = 1 + At + Btx + Cx + O ( tx , x ) , (3.16)where A, B, C are all complicated expressions composed of the c = 3 +23 / (cid:2) (39 + i √ / + (39 − i √ / (cid:3) . Numerically, A ≈ . , B ≈ − . , C ≈ − . β = 1 / , γ = 1 , δ = 3 in the same way.In addition, according to Eq.(3.6), the entropy is independent of T . Thus, C V = T ∂S∂T (cid:12)(cid:12)(cid:12) V = 0.Therefore, we also have the critical exponent α = 0. Obviously, they obey the scaling symmetrylike the ordinary thermodynamic systems: α + 2 β + γ = 2 , α + β ( δ + 1) = 2 γ (1 + δ ) = (2 − α )( δ − , γ = β ( δ − . (3.17) IV. THE SECOND-ORDER PHASE TRANSITION AT THE CRITICAL POINT
In this section, we study the types of the phase transition for the quantum-corrected blackhole at the critical points. It should be noted that the critical points depend on the values ofthe pressure P or the temperature T for a positive ξ . When P = P c or T = T c , there is onlyone critical point; when P < P c or T < T c there will be two critical points; no critical pointexists when P > P c or T > T c .Ehrenfest had ever attempted to classify the phase transitions. Phase transitions connectedwith an entropy discontinuity are called discontinuous or first order phase transitions, and phasetransitions where the entropy is continuous are called continuous or second/higher order phasetransitions. More precisely, for the first-order phase transition the Gibbs free energy G ( T, P, ... )should be continuous and its first derivative with respect to the external fields: S = − ∂G∂T (cid:12)(cid:12)(cid:12)(cid:12) ( P,... ) , V = ∂G∂P (cid:12)(cid:12)(cid:12)(cid:12) ( T,... ) (4.1)are discontinuous at the phase transition points.For the second-order phase transition the Gibbs free energy G ( T, P, ... ) and its first derivativeare both continuous, but the second derivative of G will diverge at the phase transition points,such as the specific heat C P , the compressibility κ , the expansion coefficient α P : C P = T ∂S∂T (cid:12)(cid:12)(cid:12)(cid:12) P = − T ∂ G∂T (cid:12)(cid:12)(cid:12)(cid:12) P , κ T = − V ∂V∂P (cid:12)(cid:12)(cid:12)(cid:12) T = − V ∂ G∂P (cid:12)(cid:12)(cid:12)(cid:12) T , α P = − V ∂V∂T (cid:12)(cid:12)(cid:12)(cid:12) P = − V ∂ G∂P ∂T (4.2)According to Eq.(3.5) and Eq.(3.6), one can easily obtain the S − T plot, as is shown inFig.4. Obviously, the entropy is a continuous function of temperature. T S FIG. 4:
The entropy as functions of temperature for the choices of ξ = 0 . , l p = 5 . One can easily calculate the κ T and α P : κ T = − V ∂V∂r + ∂r + ∂P (cid:12)(cid:12)(cid:12)(cid:12) T = 8 πr (cid:0) ξ + 3 r (cid:1) ξ + 24 πP r + 64 πξP r + 8 πξ P r − r + 12 ξr (4.3) α P = 1 V ∂V∂r + ∂r + ∂T (cid:12)(cid:12)(cid:12)(cid:12) P = 12 πr + (cid:0) ξ + r (cid:1) (cid:0) ξ + 3 r (cid:1) ξ + 24 πP r + 64 πξP r + 8 πξ P r − r + 12 ξr (4.4)They will diverge when the denominator vanishes. It is clear that the denominators of the κ T , α P are the same as that of C P . As shown in Fig.5, there will be two divergent pointsfor both κ T and α P for P < P c . Only one divergent point left when P = P c . Owing to thedivergence of κ T , α P , phase transitions at these critical points are all second-order. r + - - Κ T r + - - Α P FIG. 5:
The compressibility κ and the expansion coefficient α P as functions of r + for the quantum-corrected black hole. The solid (blue) curve corresponds to P = 0 . P c , and the dashed (red) curvecorresponds to P = P c . We also set ξ = 0 . . When ξ = 0, the quantum-corrected black hole return to the Schwarzschild-AdS black hole,for which there is still critical point where C P , κ, α P diverge. However, in this case, only onecritical point exists. One can also analyze the types of the phase transition at the critical pointby means of Ehrenfest scheme employed in [41]. That can give the same result. Generally,thermodynamic geometry can also be employed to study the phase transition[42–44]. However,for the quantum-corrected black hole it does not work. Because the mass/enthalpy is linear inthe pressure P , which will lead to degenerate thermodynamic metric. V. CONCLUDING REMARKS
In this paper we studied the thermodynamics and critical behaviors of a kind of quantum-corrected black hole obtained in the asymptotically safe gravity theory with higher derivativesand cosmological constant. The asymptotic safety scenario includes the scale dependent Newto-nian “ constant” G p and cosmological “ constant”. G p leads to the correction to the conventionalSchwarzschild-AdS black hole. The running cosmological “ constant” can be treated as a vari-able naturally. We can identify it with the thermodynamic pressure and include its variationin the first law of black hole thermodynamics.Based on the quantum-corrected black hole, we studied the P − V /r + criticality at the criticalpoint and plotted the isotherm curves. It is shown that the P − V /r + phase diagram is thesame as that of the van der Waals liquid/gas system. Furthermore, we calculated the criticalexponents at the critical point, which all coincide with that of the van der Waals system andRN-AdS black hole. From the critical parameters we can also construct the universal ratio ρ c = P c r c T c ≈ . C P , the compressibility κ T and theexpansion coefficient α P all suffer discontinuities at some points when the pressure or thetemperature is not larger than their critical values. Therefore, we conclude that the phasetransitions at these points belong to the second-order one. Acknowledgements
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