Critical behaviors of black hole in an asymptotically safe gravity with cosmological constant
We study the criticality and phase transition of quantum-corrected black hole in asymptotic safety (AS) gravity in the extended phase space. For the black hole, the cosmological constant is dependent on the momentum cutoff or energy scale, therefore one can naturally treat it as a variable and connect it with the thermodynamic pressure. We find that for the quantum-corrected black hole there is 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 to Ehrenfest’s classification. It is shown that they are second-order phase transition.
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 thermodynamicsHawking1 (). Therefore, we can treat black holes as thermodynamic systems. In fact, between black holes and the conventional thermodynamic systems, there are other similarities, such as phase transition and critical behaviors. The pioneering work of DaviesDavies () and the well-known Hawking-Page phase transitionHawking2 () are both proposed to elaborate these points. The phase transitions and critical phenomena in anti-de Sitter (AdS) black holes have been studied extensivelyHut (); Mazur (); Lousto (); Lemos (); Banerjee1 (); Banerjee2 (); Myung (). Some interesting works show that there exists phase transition similar to the van der Waals liquid/gas phase transition for some black holeschamblin (); chamblin1 (); Wu (); Kastor (); Cvetic (); Tian (); Banerjee3 (); LYX (); BM (). Even for the black holes in dS space critical behaviors can also be studied by considering the connections between the black hole horizon and the cosmological horizonZhao1 (); Zhao2 ().
Recently, some physicists reconsidered the critical phenomena of AdS black holes by treating the cosmological constant as a variable and connecting it with the thermodynamic pressureDolan1 (); RBM (); RBM2 (); RBM3 (); Cai (); Hendi (). In some models, the cosmological constant may be considered to be a time-variable quantityPeebles (); RH (), or as some thermodynamic quantities, such as thermodynamic pressureBrown1 (); Brown2 (), which should be a conjugate quantity of thermodynamic volume. Inclusion of the variation of can make the first law of black hole thermodynamics consistent with the Smarr formula for some black holes.
In this letter, we study the critical behaviors of a kind black hole derived in asymptotic safety gravity. The asymptotic safety scenario for quantum gravity was put forward by WeinbergWeinberg (). It is based on a nontrivial fixed point of the underlying renormalization group(RG) flow for gravity. This theory has been studied extensively and applied to several different subjects in quantum gravityReuter1 (); Niedermaier (); Reuter2 (). Bonanno and ReuterReuter3 () derived the renormalization group improved black hole metrics by replacing Newtonian coupling constant with a “ running ” one. Cai, et.al CYF () find a spherically symmetric vacuum solution to field equation derived from the AS gravity with higher derivative terms and with cosmological constant. In this theory, the cosmological constant is no longer constant but dependent on a momentum cutoff. Therefore it is reasonable to include the variation of in the first law of black hole thermodynamics as thermodynamic pressure . The quantum correction of AS gravity to the conventional Schwarzschild-AdS black hole 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 transition in the Van der Waals system. According to Ehrenfest’s classification we also consider the Gibbs free energy, the isothermal compressibility and the expansion coefficient. It is shown that the type of phase transition for the black hole at the critical point belongs to the second-order or continuous one.
The paper is arranged as follows: in the next section we simply introduce the AS gravity model and its quantum-corrected black hole solution. In section 3 we will study the criticality by considering the cosmological constant as thermodynamic pressure. We also calculate the critical exponents here. In section 4 we also analyze the type of the phase transition of the quantum-corrected black hole in the extended phase space according to Ehrenfest’s classification. 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 terms involving a momentum cutoff CYF () :
where is the determinant of the metric tensor , is the Ricci scalar, is the Ricci tensor and is the Riemann tensor. The coefficients are dimensionless coupling parameters and are functions of the dimensionful, UV cutoff. The couplings satisfy the RG equations:
Assuming a static spherically symmetric metric ansatz and choosing the Schwarzschild gauge
and then substituting it into the generalized Einstein field equations
one can derive a Schwarzschild-(anti)-de Sitter-like solution
where and are the gravitational coupling and the radius of the asymptotically (A)dS space, and both depend on the momentum cutoff .
It is shown in CYF () that there are a Gaussian fixed point in the IR limit and a non-Gaussian fixed point in the UV limit. A central result is
where and are the values of the gravitational coupling and the cosmological constant in the IR limit which should be determined by astronomical observations. The parameters and are both related to the running couplings at the non-Gaussian fixed point. The coefficient has the familiar logarithmic form which approaches asymptotic freedom
where is a fixed value of the coefficient at the Planck scale, and is the Planck mass.
It is shown that the running gravitational coupling is related the Newtonian gravitational coupling constant by
At high energy scale,
which is singular-free at . However, the curvature singularity still exists due to divergent .
At low energy scale, the momentum cutoff drop to the infrared(IR) limit, and . At this time,
where has been set for simplicity. The parameter represents the quantum correction to the conventional Schwarzschild-AdS black hole. Obviously, when , the corrected metric will return back to the Schwarzschild-AdS one. Below we will study the thermodynamics of the quantum-corrected black holes based on Eq.(12). It is shown that owing to the correction, the thermodynamic quantities will also be corrected.
Iii criticality of the quantum-corrected black hole
In this paper we only concern with the asymptotic AdS black hole. Firstly, we identify the pressure with
From Eq.(12), one can easily obtain the mass
where is the radius of the black hole event horizon.
The first law of black hole thermodynamics should written as
where the conjugate thermodynamic volume . Here, the mass of black hole is no more internal energy, but should be interpreted as the thermodynamic enthalpy, namely Kastor (); Dolan1 (); Dolan2 (); Dolan3 (). The first law of black hole thermodynamics represented by the internal energy reads
The Hawking temperature of the black hole can be easily derived
When , it will give the temperature of Schwarzschild-AdS black hole. According to the first law, Eq.(15), one can derive the entropy
where is an integration constant which can be decided by the boundary conditions. The additional logarithmic term in the expression of the entropy indicates the quantum gravitational correction. As the parameter , the standard Bekenstein-Hawking area law will return. It is interesting that no exists in the expression of although it is included in and .
The heat capacity at constant pressure can be given by
The qualitative behaviors of the temperature and the heat capacity are depicted in Fig.1. Obviously, owing to the existence of , the temperature will not blow up as the radius of the event horizon approaches zero, but tends to zero at a finite radius where the black hole will become extremal one. For a fixed pressure there is a critical value of , below which there will be both local maximum and minimum for the temperature, and above which no local extremum exists. At the critical value, the maximum and minimum will coincide. From Fig.1(b), one can see that, when , suffers discontinuities at two points , which can be identified as the critical points for phase transition in the quantum-corrected black hole. The divergences of the heat capacity appear precisely at the extrema of the temperature. The small and large black holes with positive heat capacity can be stable. While the intermediate black hole with negative heat capacity is instable.
From Eq.(17), one can derive the equation of state of the black hole
One can take the specific volume as , with counting the number of degrees of freedom associated with the black hole horizonRBM3 (). here is the Planck length. If we take , the specific volume can be expressed as
Obviously, when , it will give the result similar to that in RBM (); RBM2 (). Replacing in Eq.(20) with , one can obtain the equation of motion, . As is done in RBM2 (), one can also expand in powers of in the small limit and take the first several terms approximately.
One can see that the above equation of state is similar to that of Kerr-AdS black hole. It indeed exhibits criticality. However, in this paper we want to treat Eq.(20) exactly. We will use the horizon radius in the equation of the state instead of the specific volume hereafter.
The critical point can be obtained according to
which lead to
where the constant , which is a real number. Numerically . These critical values can lead to the following universal ratio
Obviously, it is independent of the quantum-corrected constant . Note that for the van der Waals gas, the universal ratio is , while for some actual gas, such as water, it is .
Furthermore, one can analyze the Gibbs free energy: . As is shown in Fig.3, the Gibbs free energy develops a “ swallow tail” for , which is a typical feature in a first-order phase transition. Above the critical pressure , the “ swallow tail” disappears.
Next we will calculate the critical exponents at the critical point for the quantum corrected black hole. For a van der Waals liquid/gas system, the critical behaviors can be characterized by the critical exponents as followsStanley ():
Here and refer to the specific volume for gas phase and liquid phase respectively. For the quantum-corrected black hole, we use and instead. Defing
and replacing in Eq.(20) with the new dimensionless parameters and then expanding the equation near the critical point approximately, one can obtain
where are all complicated expressions composed of the . Numerically, . Eq.(28) has the same form as that for the van der Waals system and the RN-AdS black holeRBM (). Therefore, we can derive the critical exponents in the same way. In addition, according to Eq.(18), the entropy is independent of . Thus, . Therefore, we also have the critical exponent . Obviously, they obey the scaling symmetry like the ordinary thermodynamic systems:
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 black hole at the critical points. It should be noted that the critical points depend on the values of the pressure or the temperature for a positive . When or , there is only one critical point; when or there will be two critical points; no critical point exists when or .
Ehrenfest had ever attempted to classify the phase transitions. Phase transitions connected with an entropy discontinuity are called discontinuous or first order phase transitions, and phase transitions where the entropy is continuous are called continuous or second/higher order phase transitions. More precisely, for the first-order phase transition the Gibbs free energy should be continuous and its first derivative with respect to the external fields:
are discontinuous at the phase transition points.
For the second-order phase transition the Gibbs free energy and its first derivative are both continuous, but the second derivative of will diverge at the phase transition points, such as the specific heat , the compressibility , the expansion coefficient :
One can easily calculate the and :
They will diverge when the denominator vanishes. It is clear that the denominators of the are the same as that of . As shown in Fig.5, there will be two divergent points for both and for . Only one divergent point left when . Owing to the divergence of , phase transitions at these critical points are all second-order.
When , the quantum-corrected black hole return to the Schwarzschild-AdS black hole, for which there is still critical point where diverge. However, in this case, only one critical point exists. One can also analyze the types of the phase transition at the critical point by means of Ehrenfest scheme employed in Banerjee4 (). That can give the same result. Generally, thermodynamic geometry can also be employed to study the phase transitionRuppeiner (); Quevedo1 (); Quevedo2 (). However, for the quantum-corrected black hole it does not work. Because the mass/enthalpy is linear in the pressure , 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 derivatives and cosmological constant. The asymptotic safety scenario includes the scale dependent Newtonian “ constant” and cosmological “ constant”. leads to the correction to the conventional Schwarzschild-AdS black hole. The running cosmological “ constant” can be treated as a variable naturally. We can identify it with the thermodynamic pressure and include its variation in the first law of black hole thermodynamics.
Based on the quantum-corrected black hole, we studied the criticality at the critical point and plotted the isotherm curves. It is shown that the phase diagram is the same as that of the van der Waals liquid/gas system. Furthermore, we calculated the critical exponents at the critical point, which all coincide with that of the van der Waals system and RN-AdS black hole. From the critical parameters we can also construct the universal ratio . We analyzed the types of phase transition at the critical points using Ehrenfest’s classification. The Gibbs free energy and entropy are both continuous functions of temperature. The heat capacity at constant pressure , the compressibility and the expansion coefficient all suffer discontinuities at some points when the pressure or the temperature is not larger than their critical values. Therefore, we conclude that the phase transitions at these points belong to the second-order one.
MSM thanks Prof. Ren Zhao for useful discussion. This work is supported in part by NSFC under Grant Nos.(11247261;11175109;11075098;11205097).
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