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Thermodynamic extremality relations in massive gravity

  • A universal relation between the leading correction to the entropy and extremality was proposed in the work of Goon and Penco. In this paper, we extend this work to massive gravity and investigate thermodynamic extremality relations in a topologically higher-dimensional black hole. A rescaled cosmological constant is added to the action of the massive gravity as a perturbative correction. This correction modifies the extremality bound of the black hole and leads to shifts in the mass, entropy, etc. Regarding the cosmological constant as a variable related to pressure, we obtain the thermodynamic extremality relations between the mass and entropy, pressure, charge, and parameters ci by accurate calculations. Finally, these relations are verified by a triple product identity, which shows that the universal relation exists in black holes.
  • The string landscapes formed by effective quantum field theories are broad and complex. However, there are some theories that appear to be self-consistent but are not compatible with string theory. Thus, the swampland program was proposed [1-4]. Its aim is to find the subset of infinite space in effective field theories that arises at low energies from quantum gravity theories with specific constraints. These constraints were first proposed in [1]. As one of the constraints, the weak gravity conjecture (WGC) has attracted much attention. It asserts that, for the lightest charged particle along the direction of a basis vector in charge space, the charge-to-mass ratio is larger than those for extremal black holes [2]. This conjecture shows that extremal black holes are allowed to decay.

    A proof of the WGC is that it is mathematically equivalent to a certain property of a black hole entropy. In [5], the authors introduced the higher-derivative operators to the action to compute the shift in the entropy. Using these operators, the extremality condition of the black hole is modified, and the mass and entropy are shifted. These authors derived the relation between the ratio of charge-to-mass and the entropy shift, q/m1ΔS, where ΔS>0. The charge-to-mass ratio approaches unity asymptotically with increasing mass. Thus, a large extremal black hole is unstable and decays to a smaller extremal black hole with charge-to-mass ratios greater than unity. This phenomenon satisfies the requirement of the WGC. Subsequently, WGC behavior was found in a four-dimensional rotating dyonic black hole and other spacetimes [6, 7]. Other studies of the WGC have been reported in [8-23]; see also the references therein.

    In a recent study [24], Goon and Penco derived a universal extremality relation using perturbative corrections to the free energy of generic thermodynamic systems. This relation takes the form

    Mext(Q,ϵ)ϵ=limMMext(Q,ϵ)T(S(M,Q,ϵ)ϵ)M,Q,

    (1)

    where Mext(Q,ϵ) and S(M,Q,ϵ) are the extremality mass and entropy, respectively. Both of them are ϵdependent, and ϵ is a control parameter for the free energy. Q are additional quantities in thermodynamic systems, other than the mass. The above relation can be interpreted as a comparison between states in the classical and corrected theories. Meanwhile, an approximation relation ΔMext(Q)T0(M,Q)ΔS(M,Q)|MM0ext(Q) was proposed, where ΔMext(Q) and ΔS(M,Q) are the leading order corrections to the extremal bound and to the entropy of a state with fixed mass and Q, respectively. M0ext is the mass in the extremal case without corrections. The result shows that the mass of the perturbed extremal black hole is less than that of the unperturbed one with the same quantum numbers, if ΔS>0, which implies that the perturbation decreases the mass of the extremal black hole. Therefore, WGC-like behavior exists in the extremal black hole. In particular, the Goon-Penco relation (1) was verified in an AdS-Reissner-Nordstr ¨om black hole by rescaling the cosmological constant as a perturbative correction. The approximation relation was also verified using higher-derivative operators introduced in the action.

    To further explore the WGC behavior and the Goon-Penco relation, researchers have studied the thermodynamic corrections in specific spacetimes by introducing higher-derivative operators or perturbative parameters [25, 26]. The Goon-Penco relation was confirmed, and other extremality relations were obtained. In [25], Cremonini et al. computed the four-derivative corrections to thermodynamic quantities in the higher-dimensional AdS-Reissner-Nordstr¨om black hole and found the extremality relation between the mass and charge,

    limT0(Mϵ)Q,T=limT0Φ(Qϵ)M,T.

    (2)

    Extending this work to rotating anti-de Sitter spacetimes, Liu et al. derived the extremality relation between the mass and angular momentum in the BTZ and Kerr anti-de Sitter spacetimes [26],

    (Mextϵ)J,l=limMMextΩ(Jϵ)M,S,l.

    (3)

    Relations (2) and (3) are extensions of the Goon-Penco relation (1). These relations will shed light on theories of quantum gravity.

    In this paper, we extend the work of [24] to massive gravity and investigate the extremality relations between the mass and pressure, entropy, charge, and parameters ci of a charged topological black hole in higher-dimensional spacetime. Einstein's general relativity (GR) is a low energy effective theory. The UV completeness requires that GR be modified to meet physical descriptions in the high energy region. Massive gravity is a straightforward modification to GR. We introduce a perturbative correction by adding a rescaled cosmological constant to the action of massive gravity. This scenario is different from that in [24], where the cosmological constant was directly rescaled in the action and consistent with that in [26]. In our investigation, the cosmological constant is regarded as a variable related to pressure [27-31]. Its conjugate quantity is a thermodynamic volume. The black hole mass is naturally interpreted as an enthalpy. The first reason for this is that the cosmological constant, as a variable, can reconcile the inconsistency between the first law of thermodynamics of black holes and the Smarr relation, derived from the scaling method. The second reason is that physical constants, such as the gauge coupling constants, Newtonian constant, or cosmological constant, which are vacuum expectation values, are not fixed and vary in the more fundamental theories [32].

    The rest of this paper is organized as follows. In the next section, the solution of the higher-dimensional black hole in massive gravity is given, and its thermodynamic properties are discussed. In section III, we introduce a perturbative correction to the action and derive the extremality relations between the mass and pressure, entropy, charge, and parameters ci. Section IV is devoted to our discussion and conclusion.

    The action for an (n+2)-dimensional massive gravity is [33]

    S=116πdxn+2g[R+n(n+1)l2F24+m24i=1ciui(g,f)],

    (4)

    where the terms including m2 represent the massive potential associated with the graviton mass, f is a fixed symmetric tensor called the reference metric, ci are constants, and ui are symmetric polynomials of the eigenvalues of the (n+2)×(n+2) matrix Kμν=fμαgαν:

    u1=[K],u2=[K]2[K2],u3=[K]33[K][K2]+2[K3],u4=[K]46[K2][K]2+8[K3][K]+3[K2]26[K4].

    (5)

    The square root in K denotes (A)μν(A)νλ=Aμλ and [K]=Kμμ.

    The solution of the charged black hole with the spacetime metric and reference metric is given by [34]

    ds2=f(r)dt2+1f(r)dr2+r2hijdxidxj,

    (6)

    fμν=diag(0,0,c20hij),

    (7)

    where

    f(r)=k+r2l216πMnΩnrn1+(16πQ)22n(n1)Ω2nr2(n1)+c0c1m2rn+c20c2m2+(n1)c30c3m2r+(n1)(n2)c40c4m2r2,

    (8)

    l2 is related to the cosmological constant Λ as l2=n(n+1)2Λ. M and Q are the mass and charge of the M black hole, respectively. Ωn is the volume spanned by coordinates xi, and c0 is a positive integral constant. hijdxidxj is the line element for an Einstein space with the constant curvature n(n1)k. k=1, 0, or 1 denotes spherical, Ricci flat, or hyperbolic topology black hole horizons, respectively. The thermodynamics in the extended phase space of massive gravity have been studied in [35-41]. The event horizon r+ is determined by f(r)=0. A general formula for the Hawking temperature can be given as T=κ2π, where κ=12limrr+g11g00ln(g00)r is the surface gravity. For this black hole, the Hawking temperature is

    T=f(r+)4π=14πr+[(n+1)r2+l2+(16πQ)22nΩ2nr2(n1)++c0c1m2r++(n1)c20c2m2+(n1)k+(n1)(n2)c30c3m2r++(n1)(n2)(n3)c40c4m2r2+].

    (9)

    The mass expressed by the horizon radius and charge is

    M=nΩnrn1+16π[k+r2+l2+(16πQ)22n(n1)Ω2nr2(n1)++c0c1m2r+n+c20c2m2+(n1)c30c3m2r++(n1)(n2)c40c4m2r2+].

    (10)

    The cosmological constant was seen as a fixed constant in the past. In this paper, it is regarded as a variable related to pressure, P=Λ8π=n(n+1)16πl2, and its conjugate quantity is a thermodynamic volume V. The entropy,volume, and electric potential at the event horizon are given by

    S=Ωnrn+4,V=Ωnrn+1+n+1,Φe=16πQ(n1)Ωnrn1+,

    (11)

    respectively. Because of the appearance of pressure, the mass is no longer interpreted as the internal energy but as an enthalpy. c1, c2, c3, and c4 are seen as extensive parameters for the mass. Their conjugate quantities are

    Φ1=Ωnc0m2rn+16π,Φ2=nΩnc20m2rn1+16π,Φ3=n(n1)Ωnc30m2rn2+16π,Φ4=n(n1)(n2)Ωnc40m2rn3+16π,

    (12)

    respectively. It is easy to verify that these thermodynamic quantities obey the first law of thermodynamics,

    dM=TdS+VdP+ΦedQ+4i=1Φidci.

    (13)

    When the cosmological constant is fixed, the term VdP disappears, and the mass is interpreted as the internal energy. When a perturbative correction is introduced, the related thermodynamic quantities are shifted, which is discussed in the next section.

    In this section, we derive the extremality relations between the mass and entropy, charge, pressure, and parameters ci by adding a rescaled cosmological constant to the action as a perturbative correction. The rescaled parameter is ϵ. Here, the black hole is designated as an extremal one.

    We first introduce the correction

    ΔS=116πdxn+2gn(n+1)ϵl2,

    (14)

    to the action (4). The corrected action is S+ΔS. The action (4) is recovered when ϵ=0. A black hole solution is obtained from the corrected action and takes the form of Eqs. (6) and (8), but there is a shift. With the correction, the Hawking temperature is also shifted, and it is given by

    T=14πr+[(n+1)r2+ϵl2+(n+1)r2+l2+(16πQ)22nΩ2nr2(n1)++c0c1m2r++(n1)c20c2m2+(n1)k+(n1)(n2)c30c3m2r++(n1)(n2)(n3)c40c4m2r2+].

    (15)

    The corrected mass is

    M=nΩnrn1+16π[r2+ϵl2+k+r2+l2+(16πQ)22n(n1)Ω2nr2(n1)++c0c1m2r+n+c20c2m2+(n1)c30c3m2r++(n1)(n2)c40c4m2r2+],

    (16)

    which is a function of parameters r+,ϵ, Q,l,c1,c2,c3, and c4. Our interest is focused on the thermodynamic extremality relation. The Hawking temperature (15) in the extremal case is zero, which leads to a solution r+=r+(ϵ). Inserting this solution into the above equation yields an expression regarding the mass, Mext=Mext(ϵ). Carrying out the differential on Mext(ϵ), we have

    (Mextϵ)Q,l,c1,c2,c3,c4=nΩnrn+1+16πl2.

    (17)

    Because the expression of the differential expressed by ϵ is very complex, we adopted the expression of r+in the above derivation. In fact, this relation can also be derived by the following calculation. For convenience, we use c to denote all parameters Q,l,c1,c2,c3,c4, except for r+ and ϵ. From Eq. (16), the differential of M to ϵ is obtained as follows:

    (Mϵ)c=(Mr+)c,ϵ(r+ϵ)c+(Mϵ)c,r+=(MS)c,ϵ(Sr+)c,ϵ(r+ϵ)c+(Mϵ)c,r+=14TΩnrn1+(r+ϵ)c+(Mϵ)c,r+.

    (18)

    In the extremal case, the first term in the last line of the above equation disappears, and the mass can be rewritten as M=Mext. Therefore, Eq. (17) is readily recovered.

    The entropy S, pressure P, charge Q, c1, c2, c3, and c4 are usually regarded as a complete set of extensive parameters for the mass. Their conjugate quantities can be derived from the mass and take the same form as those given in section II, except for the temperature and volume. We first verify the extremality relation between the mass and entropy.

    The expression for ϵ is obtained from Eq. (16) and takes the form

    ϵ=[16πMnΩnrn+1+kr2+(16πQ)22n(n1)Ω2nr2n+c0c1m2nr+c20c2m2r2+(n1)c30c3m2r3+(n1)(n2)c40c4m2r4+]l21.

    (19)

    Using the relation between the entropy and horizon radius given in Eq. (11), the above equation is a function ϵ(S), and r+S=4nΩnrn1+. Carrying out the differential calculation on this function yields

    (ϵS)M,Q,l,c1,c2,c3,c4=4l2nΩnrn1+[(n+1)16πMnΩnrn+2++2kr3++(16πQ)2(n1)Ω2nr2n+1++c0c1m2nr2++2c20c2m2r3++3(n1)c30c3m2r4++4(n1)(n2)c40c4m2r5+].

    (20)

    To evaluate the value, we insert the expression of the mass into the above equation and obtain

    (ϵS)M,Q,l,c1,c2,c3,c4=4l2nΩnrn1+[(n1)kr3+(n+1)(1+ϵ)r+l2+(16πQ)22nΩ2nr2(n+1)+c0c1m2r2+(n1)c20c2m2r3+(n1)(n2)c30c3m2r4+(n1)(n2)(n3)c40c4m2r5+].

    (21)

    Combining the inverse of the above differential with the expression of the temperature given in Eq. (15), we have

    T(Sϵ)M,Q,l,c1,c2,c3,c4=nΩnrn+1+16πl2.

    (22)

    Compared with relation (17), it is easy to see that

    (Mextϵ)Q,l,c1,c2,c3,c4=limMMextT(Sϵ)M,Q,l,c1,c2,c3,c4,

    (23)

    where S is a function of M, Q, l, c1, c2, c3, c4, and ϵ. Therefore, the Goon-Penco relation is verified in the higher-dimensional black hole.

    In this paper, the cosmological constant is regarded as a variable related to pressure. The entropy, pressure, charge, c1, c2, c3, and c4 are usually regarded as extensive parameters for the mass. Because the entropy satisfies the thermodynamic extremality relation, it is natural to ask whether other extensive quantities also satisfy corresponding relations. The goal of the following investigation is to determine these relations. Let us first derive the extremality relation between the mass and pressure. The pressure can be expressed by the constant l2 as P=n(n+1)16πl2. Then, Pl2=16πl4n(n+1). Using Eqs. (16) and (19), we get the differential of ϵ with respect to the pressure,

    (ϵP)M,r+,Q,c1,c2,c3,c4=16πl2(1+ϵ)n(n+1).

    (24)

    The perturbation parameter ϵ exists in the above differential relation as an explicit function. The reason for this is that the perturbation correction is introduced by adding the rescaled cosmological constant to the action, and this constant is related to the pressure. Because of the shift in the mass, the thermodynamic volume is also shifted, and its expression is different from that given given in Eq. (11). The volume is

    V=ϵ+1n+1Ωnrn+1+.

    (25)

    Using Eq. (25) and the inverse of the differential of ϵ to P yields

    V(Pϵ)M,r+,Q,c1,c2,c3,c4=nΩnrn+1+16πl2.

    (26)

    Comparing the above equation with Eq. (17), we obtain the extremality relation between the mass and pressure,

    (Mextϵ)Q,l,c1,c2,c3,c4=limMMextV(Pϵ)M,r+,Q,c1,c2,c3,c4,

    (27)

    where P is a function of M, r+, Q, c1, c2, c3, c4, and ϵ. This relation is an extension of the Goon-Penco relation.

    We continue to investigate the extremality relation between the mass and charge. The calculation process is similar. From Eq. (19), the differential of ϵ with respect to Q takes the form

    (ϵQ)M,r+,l,c1,c2,c3,c4=(16π)2Ql2n(n1)Ω2nr2n+.

    (28)

    Multiplying the electric potential Φe=16πQ(n1)Ωnrn1+ by the inverse of the above differential yields

    Φe(Qϵ)M,r+,l,c1,c2,c3,c4=nΩnrn+1+16πl2.

    (29)

    Obviously, there is a minus sign difference between Eqs. (17) and (29). Therefore,

    (Mextϵ)Q,l,c1,c2,c3,c4=limMMextΦe(Qϵ)M,r+,l,c1,c2,c3,c4,

    (30)

    which is the extremality relation between the mass and charge. Now, Q is a function of M, r+, l, c1, c2, c3, c4, and ϵ. This relation is also an extension of the Goon-Penco relation.

    For the extremality relations between the mass and parameters c1, c2, c3, and c4, the calculations are parallel. Their differential relations are

    (ϵc1)M,r+,Q,l,c2,c3,c4=c0m2l2nr+,

    (31)

    (ϵc2)M,r+,Q,l,c1,c3,c4=c20m2l2r2+,

    (32)

    (ϵc3)M,r+,Q,l,c1,c2,c4=(n1)c30m2l2r3+,

    (33)

    (ϵc4)M,r+,Q,l,c1,c2,c3=(n1)(n2)c0m2l2r4+.

    (34)

    The conjugate quantities of c1, c2, c3, and c4 are

    Φ1=Ωnc0m2rn+16π,Φ2=nΩnc20m2rn1+16π,Φ3=n(n1)Ωnc30m2rn2+16π,Φ4=n(n1)(n2)Ωnc40m2rn3+16π,

    respectively. Using these quantities, it is not difficult to obtain

    (Φ1c1ϵ)M,r+,Q,l,c2,c3,c4=(Φ2c2ϵ)M,r+,Q,l,c1,c3,c4=(Φ3c3ϵ)M,r+,Q,l,c1,c2,c4=(Φ4c4ϵ)M,r+,Q,l,c1,c2,c3=nΩnrn+1+16πl2.

    (35)

    Thus, the extremality relations between the mass and extensive parameters ci are

    (Mextϵ)Q,l,c1,c2,c3,c4=limMMextΦi(ciϵ)M,r+,Q,l,cj,ck,cu,

    (36)

    where i,j,k,u=1,2,3,4, and ijku. Therefore, the Goon-Penco relation is extended to the case of the extensive parameters ci of the higher-dimensional black hole.

    In the above investigation, the thermodynamic extremality relations between the mass and entropy, pressure, charge, and parameters ci were obtained by accurate calculations. They are expressed as Eqs. (22), (27), (30), and (36), respectively. The values of these relations are equal. In fact, these relations can be derived uniformly using the triple product identity

    (MXi)ϵ,T(Xiϵ)M,T(ϵM)T,Xi=1,

    (37)

    which yields

    (Mϵ)T,Xi=(MXi)ϵ,T(Xiϵ)M,T=Φi(Xiϵ)M,T.

    (38)

    In the above derivation, (MXi)ϵ,T were identified as Φi , which are the conjugate quantities of Xi. Here, Xi are chosen as S, Q, P, c1, c2, c3, and c4. M and T are the corrected mass and temperature given in (16) and (15), respectively. In the extremal case, T0 and MMext. The above relation becomes

    (Mextϵ)M,Xi=limMMextΦi(Xiϵ)M,Xj,

    (39)

    where XiXj, and Xj are parameters S, Q, P, c1, c2, c3, or c4, except for Xi. This relation implies that the universal extremality relation exists in black holes. The relation (39) is easily reduced to (22), (27), (30), and (36) when Xi are the entropy, charge, parameters ci, and pressure, respectively. In the calculation, because of the shift in the mass, the expression of the volume V=ϵ+1n+1Ωnrn+1+ is different from that given in Eq. (11). In [26], the authors derived the extremality relation between the mass and angular momentum in BTZ and Kerr anti-de Sitter spacetimes and suggested that a general formula of the extremality relation existed in black holes. Our result provides verification of this conjecture.

    In this paper, we extended the work of Goon and Penco to massive gravity and investigated the thermodynamic extremality relations in a higher-dimensional black hole. The extremality relations between the mass and pressure, entropy, charge, and parameters ci were derived by accurate calculations. The values of these extremality relations are equal, which may be due to the first law of thermodynamics. In the calculation, the cosmological constant was treated as a variable related to pressure. A perturbative correction was introduced by adding the rescaled cosmological constant to the action, but this addition does not affect the form of the extremality relation between the mass and pressure.

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Deyou Chen, Jun Tao and Peng Wang. Thermodynamic extremality relations in the massive gravity[J]. Chinese Physics C. doi: 10.1088/1674-1137/abcf21
Deyou Chen, Jun Tao and Peng Wang. Thermodynamic extremality relations in the massive gravity[J]. Chinese Physics C.  doi: 10.1088/1674-1137/abcf21 shu
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Thermodynamic extremality relations in massive gravity

  • 1. School of Science, Xihua University, Chengdu 610039, China
  • 2. Center for Theoretical Physics, College of Physics, Sichuan University, Chengdu 610064, China

Abstract: A universal relation between the leading correction to the entropy and extremality was proposed in the work of Goon and Penco. In this paper, we extend this work to massive gravity and investigate thermodynamic extremality relations in a topologically higher-dimensional black hole. A rescaled cosmological constant is added to the action of the massive gravity as a perturbative correction. This correction modifies the extremality bound of the black hole and leads to shifts in the mass, entropy, etc. Regarding the cosmological constant as a variable related to pressure, we obtain the thermodynamic extremality relations between the mass and entropy, pressure, charge, and parameters ci by accurate calculations. Finally, these relations are verified by a triple product identity, which shows that the universal relation exists in black holes.

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    I.   INTRODUCTION
    • The string landscapes formed by effective quantum field theories are broad and complex. However, there are some theories that appear to be self-consistent but are not compatible with string theory. Thus, the swampland program was proposed [1-4]. Its aim is to find the subset of infinite space in effective field theories that arises at low energies from quantum gravity theories with specific constraints. These constraints were first proposed in [1]. As one of the constraints, the weak gravity conjecture (WGC) has attracted much attention. It asserts that, for the lightest charged particle along the direction of a basis vector in charge space, the charge-to-mass ratio is larger than those for extremal black holes [2]. This conjecture shows that extremal black holes are allowed to decay.

      A proof of the WGC is that it is mathematically equivalent to a certain property of a black hole entropy. In [5], the authors introduced the higher-derivative operators to the action to compute the shift in the entropy. Using these operators, the extremality condition of the black hole is modified, and the mass and entropy are shifted. These authors derived the relation between the ratio of charge-to-mass and the entropy shift, q/m1ΔS, where ΔS>0. The charge-to-mass ratio approaches unity asymptotically with increasing mass. Thus, a large extremal black hole is unstable and decays to a smaller extremal black hole with charge-to-mass ratios greater than unity. This phenomenon satisfies the requirement of the WGC. Subsequently, WGC behavior was found in a four-dimensional rotating dyonic black hole and other spacetimes [6, 7]. Other studies of the WGC have been reported in [8-23]; see also the references therein.

      In a recent study [24], Goon and Penco derived a universal extremality relation using perturbative corrections to the free energy of generic thermodynamic systems. This relation takes the form

      Mext(Q,ϵ)ϵ=limMMext(Q,ϵ)T(S(M,Q,ϵ)ϵ)M,Q,

      (1)

      where Mext(Q,ϵ) and S(M,Q,ϵ) are the extremality mass and entropy, respectively. Both of them are ϵdependent, and ϵ is a control parameter for the free energy. Q are additional quantities in thermodynamic systems, other than the mass. The above relation can be interpreted as a comparison between states in the classical and corrected theories. Meanwhile, an approximation relation ΔMext(Q)T0(M,Q)ΔS(M,Q)|MM0ext(Q) was proposed, where ΔMext(Q) and ΔS(M,Q) are the leading order corrections to the extremal bound and to the entropy of a state with fixed mass and Q, respectively. M0ext is the mass in the extremal case without corrections. The result shows that the mass of the perturbed extremal black hole is less than that of the unperturbed one with the same quantum numbers, if ΔS>0, which implies that the perturbation decreases the mass of the extremal black hole. Therefore, WGC-like behavior exists in the extremal black hole. In particular, the Goon-Penco relation (1) was verified in an AdS-Reissner-Nordstr ¨om black hole by rescaling the cosmological constant as a perturbative correction. The approximation relation was also verified using higher-derivative operators introduced in the action.

      To further explore the WGC behavior and the Goon-Penco relation, researchers have studied the thermodynamic corrections in specific spacetimes by introducing higher-derivative operators or perturbative parameters [25, 26]. The Goon-Penco relation was confirmed, and other extremality relations were obtained. In [25], Cremonini et al. computed the four-derivative corrections to thermodynamic quantities in the higher-dimensional AdS-Reissner-Nordstr¨om black hole and found the extremality relation between the mass and charge,

      limT0(Mϵ)Q,T=limT0Φ(Qϵ)M,T.

      (2)

      Extending this work to rotating anti-de Sitter spacetimes, Liu et al. derived the extremality relation between the mass and angular momentum in the BTZ and Kerr anti-de Sitter spacetimes [26],

      (Mextϵ)J,l=limMMextΩ(Jϵ)M,S,l.

      (3)

      Relations (2) and (3) are extensions of the Goon-Penco relation (1). These relations will shed light on theories of quantum gravity.

      In this paper, we extend the work of [24] to massive gravity and investigate the extremality relations between the mass and pressure, entropy, charge, and parameters ci of a charged topological black hole in higher-dimensional spacetime. Einstein's general relativity (GR) is a low energy effective theory. The UV completeness requires that GR be modified to meet physical descriptions in the high energy region. Massive gravity is a straightforward modification to GR. We introduce a perturbative correction by adding a rescaled cosmological constant to the action of massive gravity. This scenario is different from that in [24], where the cosmological constant was directly rescaled in the action and consistent with that in [26]. In our investigation, the cosmological constant is regarded as a variable related to pressure [27-31]. Its conjugate quantity is a thermodynamic volume. The black hole mass is naturally interpreted as an enthalpy. The first reason for this is that the cosmological constant, as a variable, can reconcile the inconsistency between the first law of thermodynamics of black holes and the Smarr relation, derived from the scaling method. The second reason is that physical constants, such as the gauge coupling constants, Newtonian constant, or cosmological constant, which are vacuum expectation values, are not fixed and vary in the more fundamental theories [32].

      The rest of this paper is organized as follows. In the next section, the solution of the higher-dimensional black hole in massive gravity is given, and its thermodynamic properties are discussed. In section III, we introduce a perturbative correction to the action and derive the extremality relations between the mass and pressure, entropy, charge, and parameters ci. Section IV is devoted to our discussion and conclusion.

    II.   BLACK HOLE SOLUTION IN MASSIVE GRAVITY
    • The action for an (n+2)-dimensional massive gravity is [33]

      S=116πdxn+2g[R+n(n+1)l2F24+m24i=1ciui(g,f)],

      (4)

      where the terms including m2 represent the massive potential associated with the graviton mass, f is a fixed symmetric tensor called the reference metric, ci are constants, and ui are symmetric polynomials of the eigenvalues of the (n+2)×(n+2) matrix Kμν=fμαgαν:

      u1=[K],u2=[K]2[K2],u3=[K]33[K][K2]+2[K3],u4=[K]46[K2][K]2+8[K3][K]+3[K2]26[K4].

      (5)

      The square root in K denotes (A)μν(A)νλ=Aμλ and [K]=Kμμ.

      The solution of the charged black hole with the spacetime metric and reference metric is given by [34]

      ds2=f(r)dt2+1f(r)dr2+r2hijdxidxj,

      (6)

      fμν=diag(0,0,c20hij),

      (7)

      where

      f(r)=k+r2l216πMnΩnrn1+(16πQ)22n(n1)Ω2nr2(n1)+c0c1m2rn+c20c2m2+(n1)c30c3m2r+(n1)(n2)c40c4m2r2,

      (8)

      l2 is related to the cosmological constant Λ as l2=n(n+1)2Λ. M and Q are the mass and charge of the M black hole, respectively. Ωn is the volume spanned by coordinates xi, and c0 is a positive integral constant. hijdxidxj is the line element for an Einstein space with the constant curvature n(n1)k. k=1, 0, or 1 denotes spherical, Ricci flat, or hyperbolic topology black hole horizons, respectively. The thermodynamics in the extended phase space of massive gravity have been studied in [35-41]. The event horizon r+ is determined by f(r)=0. A general formula for the Hawking temperature can be given as T=κ2π, where κ=12limrr+g11g00ln(g00)r is the surface gravity. For this black hole, the Hawking temperature is

      T=f(r+)4π=14πr+[(n+1)r2+l2+(16πQ)22nΩ2nr2(n1)++c0c1m2r++(n1)c20c2m2+(n1)k+(n1)(n2)c30c3m2r++(n1)(n2)(n3)c40c4m2r2+].

      (9)

      The mass expressed by the horizon radius and charge is

      M=nΩnrn1+16π[k+r2+l2+(16πQ)22n(n1)Ω2nr2(n1)++c0c1m2r+n+c20c2m2+(n1)c30c3m2r++(n1)(n2)c40c4m2r2+].

      (10)

      The cosmological constant was seen as a fixed constant in the past. In this paper, it is regarded as a variable related to pressure, P=Λ8π=n(n+1)16πl2, and its conjugate quantity is a thermodynamic volume V. The entropy,volume, and electric potential at the event horizon are given by

      S=Ωnrn+4,V=Ωnrn+1+n+1,Φe=16πQ(n1)Ωnrn1+,

      (11)

      respectively. Because of the appearance of pressure, the mass is no longer interpreted as the internal energy but as an enthalpy. c1, c2, c3, and c4 are seen as extensive parameters for the mass. Their conjugate quantities are

      Φ1=Ωnc0m2rn+16π,Φ2=nΩnc20m2rn1+16π,Φ3=n(n1)Ωnc30m2rn2+16π,Φ4=n(n1)(n2)Ωnc40m2rn3+16π,

      (12)

      respectively. It is easy to verify that these thermodynamic quantities obey the first law of thermodynamics,

      dM=TdS+VdP+ΦedQ+4i=1Φidci.

      (13)

      When the cosmological constant is fixed, the term VdP disappears, and the mass is interpreted as the internal energy. When a perturbative correction is introduced, the related thermodynamic quantities are shifted, which is discussed in the next section.

    III.   EXTREMALITY RELATIONS IN MASSIVE GRAVITY
    • In this section, we derive the extremality relations between the mass and entropy, charge, pressure, and parameters ci by adding a rescaled cosmological constant to the action as a perturbative correction. The rescaled parameter is ϵ. Here, the black hole is designated as an extremal one.

      We first introduce the correction

      ΔS=116πdxn+2gn(n+1)ϵl2,

      (14)

      to the action (4). The corrected action is S+ΔS. The action (4) is recovered when ϵ=0. A black hole solution is obtained from the corrected action and takes the form of Eqs. (6) and (8), but there is a shift. With the correction, the Hawking temperature is also shifted, and it is given by

      T=14πr+[(n+1)r2+ϵl2+(n+1)r2+l2+(16πQ)22nΩ2nr2(n1)++c0c1m2r++(n1)c20c2m2+(n1)k+(n1)(n2)c30c3m2r++(n1)(n2)(n3)c40c4m2r2+].

      (15)

      The corrected mass is

      M=nΩnrn1+16π[r2+ϵl2+k+r2+l2+(16πQ)22n(n1)Ω2nr2(n1)++c0c1m2r+n+c20c2m2+(n1)c30c3m2r++(n1)(n2)c40c4m2r2+],

      (16)

      which is a function of parameters r+,ϵ, Q,l,c1,c2,c3, and c4. Our interest is focused on the thermodynamic extremality relation. The Hawking temperature (15) in the extremal case is zero, which leads to a solution r+=r+(ϵ). Inserting this solution into the above equation yields an expression regarding the mass, Mext=Mext(ϵ). Carrying out the differential on Mext(ϵ), we have

      (Mextϵ)Q,l,c1,c2,c3,c4=nΩnrn+1+16πl2.

      (17)

      Because the expression of the differential expressed by ϵ is very complex, we adopted the expression of r+in the above derivation. In fact, this relation can also be derived by the following calculation. For convenience, we use c to denote all parameters Q,l,c1,c2,c3,c4, except for r+ and ϵ. From Eq. (16), the differential of M to ϵ is obtained as follows:

      (Mϵ)c=(Mr+)c,ϵ(r+ϵ)c+(Mϵ)c,r+=(MS)c,ϵ(Sr+)c,ϵ(r+ϵ)c+(Mϵ)c,r+=14TΩnrn1+(r+ϵ)c+(Mϵ)c,r+.

      (18)

      In the extremal case, the first term in the last line of the above equation disappears, and the mass can be rewritten as M=Mext. Therefore, Eq. (17) is readily recovered.

      The entropy S, pressure P, charge Q, c1, c2, c3, and c4 are usually regarded as a complete set of extensive parameters for the mass. Their conjugate quantities can be derived from the mass and take the same form as those given in section II, except for the temperature and volume. We first verify the extremality relation between the mass and entropy.

      The expression for ϵ is obtained from Eq. (16) and takes the form

      ϵ=[16πMnΩnrn+1+kr2+(16πQ)22n(n1)Ω2nr2n+c0c1m2nr+c20c2m2r2+(n1)c30c3m2r3+(n1)(n2)c40c4m2r4+]l21.

      (19)

      Using the relation between the entropy and horizon radius given in Eq. (11), the above equation is a function ϵ(S), and r+S=4nΩnrn1+. Carrying out the differential calculation on this function yields

      (ϵS)M,Q,l,c1,c2,c3,c4=4l2nΩnrn1+[(n+1)16πMnΩnrn+2++2kr3++(16πQ)2(n1)Ω2nr2n+1++c0c1m2nr2++2c20c2m2r3++3(n1)c30c3m2r4++4(n1)(n2)c40c4m2r5+].

      (20)

      To evaluate the value, we insert the expression of the mass into the above equation and obtain

      (ϵS)M,Q,l,c1,c2,c3,c4=4l2nΩnrn1+[(n1)kr3+(n+1)(1+ϵ)r+l2+(16πQ)22nΩ2nr2(n+1)+c0c1m2r2+(n1)c20c2m2r3+(n1)(n2)c30c3m2r4+(n1)(n2)(n3)c40c4m2r5+].

      (21)

      Combining the inverse of the above differential with the expression of the temperature given in Eq. (15), we have

      T(Sϵ)M,Q,l,c1,c2,c3,c4=nΩnrn+1+16πl2.

      (22)

      Compared with relation (17), it is easy to see that

      (Mextϵ)Q,l,c1,c2,c3,c4=limMMextT(Sϵ)M,Q,l,c1,c2,c3,c4,

      (23)

      where S is a function of M, Q, l, c1, c2, c3, c4, and ϵ. Therefore, the Goon-Penco relation is verified in the higher-dimensional black hole.

      In this paper, the cosmological constant is regarded as a variable related to pressure. The entropy, pressure, charge, c1, c2, c3, and c4 are usually regarded as extensive parameters for the mass. Because the entropy satisfies the thermodynamic extremality relation, it is natural to ask whether other extensive quantities also satisfy corresponding relations. The goal of the following investigation is to determine these relations. Let us first derive the extremality relation between the mass and pressure. The pressure can be expressed by the constant l2 as P=n(n+1)16πl2. Then, Pl2=16πl4n(n+1). Using Eqs. (16) and (19), we get the differential of ϵ with respect to the pressure,

      (ϵP)M,r+,Q,c1,c2,c3,c4=16πl2(1+ϵ)n(n+1).

      (24)

      The perturbation parameter ϵ exists in the above differential relation as an explicit function. The reason for this is that the perturbation correction is introduced by adding the rescaled cosmological constant to the action, and this constant is related to the pressure. Because of the shift in the mass, the thermodynamic volume is also shifted, and its expression is different from that given given in Eq. (11). The volume is

      V=ϵ+1n+1Ωnrn+1+.

      (25)

      Using Eq. (25) and the inverse of the differential of ϵ to P yields

      V(Pϵ)M,r+,Q,c1,c2,c3,c4=nΩnrn+1+16πl2.

      (26)

      Comparing the above equation with Eq. (17), we obtain the extremality relation between the mass and pressure,

      (Mextϵ)Q,l,c1,c2,c3,c4=limMMextV(Pϵ)M,r+,Q,c1,c2,c3,c4,

      (27)

      where P is a function of M, r+, Q, c1, c2, c3, c4, and ϵ. This relation is an extension of the Goon-Penco relation.

      We continue to investigate the extremality relation between the mass and charge. The calculation process is similar. From Eq. (19), the differential of ϵ with respect to Q takes the form

      (ϵQ)M,r+,l,c1,c2,c3,c4=(16π)2Ql2n(n1)Ω2nr2n+.

      (28)

      Multiplying the electric potential Φe=16πQ(n1)Ωnrn1+ by the inverse of the above differential yields

      Φe(Qϵ)M,r+,l,c1,c2,c3,c4=nΩnrn+1+16πl2.

      (29)

      Obviously, there is a minus sign difference between Eqs. (17) and (29). Therefore,

      (Mextϵ)Q,l,c1,c2,c3,c4=limMMextΦe(Qϵ)M,r+,l,c1,c2,c3,c4,

      (30)

      which is the extremality relation between the mass and charge. Now, Q is a function of M, r+, l, c1, c2, c3, c4, and ϵ. This relation is also an extension of the Goon-Penco relation.

      For the extremality relations between the mass and parameters c1, c2, c3, and c4, the calculations are parallel. Their differential relations are

      (ϵc1)M,r+,Q,l,c2,c3,c4=c0m2l2nr+,

      (31)

      (ϵc2)M,r+,Q,l,c1,c3,c4=c20m2l2r2+,

      (32)

      (ϵc3)M,r+,Q,l,c1,c2,c4=(n1)c30m2l2r3+,

      (33)

      (ϵc4)M,r+,Q,l,c1,c2,c3=(n1)(n2)c0m2l2r4+.

      (34)

      The conjugate quantities of c1, c2, c3, and c4 are

      Φ1=Ωnc0m2rn+16π,Φ2=nΩnc20m2rn1+16π,Φ3=n(n1)Ωnc30m2rn2+16π,Φ4=n(n1)(n2)Ωnc40m2rn3+16π,

      respectively. Using these quantities, it is not difficult to obtain

      (Φ1c1ϵ)M,r+,Q,l,c2,c3,c4=(Φ2c2ϵ)M,r+,Q,l,c1,c3,c4=(Φ3c3ϵ)M,r+,Q,l,c1,c2,c4=(Φ4c4ϵ)M,r+,Q,l,c1,c2,c3=nΩnrn+1+16πl2.

      (35)

      Thus, the extremality relations between the mass and extensive parameters ci are

      (Mextϵ)Q,l,c1,c2,c3,c4=limMMextΦi(ciϵ)M,r+,Q,l,cj,ck,cu,

      (36)

      where i,j,k,u=1,2,3,4, and ijku. Therefore, the Goon-Penco relation is extended to the case of the extensive parameters ci of the higher-dimensional black hole.

      In the above investigation, the thermodynamic extremality relations between the mass and entropy, pressure, charge, and parameters ci were obtained by accurate calculations. They are expressed as Eqs. (22), (27), (30), and (36), respectively. The values of these relations are equal. In fact, these relations can be derived uniformly using the triple product identity

      (MXi)ϵ,T(Xiϵ)M,T(ϵM)T,Xi=1,

      (37)

      which yields

      (Mϵ)T,Xi=(MXi)ϵ,T(Xiϵ)M,T=Φi(Xiϵ)M,T.

      (38)

      In the above derivation, (MXi)ϵ,T were identified as Φi , which are the conjugate quantities of Xi. Here, Xi are chosen as S, Q, P, c1, c2, c3, and c4. M and T are the corrected mass and temperature given in (16) and (15), respectively. In the extremal case, T0 and MMext. The above relation becomes

      (Mextϵ)M,Xi=limMMextΦi(Xiϵ)M,Xj,

      (39)

      where XiXj, and Xj are parameters S, Q, P, c1, c2, c3, or c4, except for Xi. This relation implies that the universal extremality relation exists in black holes. The relation (39) is easily reduced to (22), (27), (30), and (36) when Xi are the entropy, charge, parameters ci, and pressure, respectively. In the calculation, because of the shift in the mass, the expression of the volume V=ϵ+1n+1Ωnrn+1+ is different from that given in Eq. (11). In [26], the authors derived the extremality relation between the mass and angular momentum in BTZ and Kerr anti-de Sitter spacetimes and suggested that a general formula of the extremality relation existed in black holes. Our result provides verification of this conjecture.

    IV.   CONCLUSION
    • In this paper, we extended the work of Goon and Penco to massive gravity and investigated the thermodynamic extremality relations in a higher-dimensional black hole. The extremality relations between the mass and pressure, entropy, charge, and parameters ci were derived by accurate calculations. The values of these extremality relations are equal, which may be due to the first law of thermodynamics. In the calculation, the cosmological constant was treated as a variable related to pressure. A perturbative correction was introduced by adding the rescaled cosmological constant to the action, but this addition does not affect the form of the extremality relation between the mass and pressure.

Reference (41)

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