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The action of the four-dimensional massive gravity theory considered here is given by (in the units
c=G=ℏ=kB=4πε0=1 ) [76]S=116π∫d4x√−g[R+6l2−14F2+m2g∑4i=1ciUi(g,f)],
(1) where
mg is the graviton mass in the theory. In this equation, the last term represents the massive potentials associated with the graviton; these break the diffeomorphism invariance in the bulk theory and generate momentum relaxation in the dual boundary theory. The coupling parametersci are constants,f denotes the reference metric, andUi denote symmetric polynomials of the eigenvalues for the4×4 matrixKμν≡√gμαfαν , which are expressed asU1=[K],U2=[K]2−[K2],U3=[K]3−3[K][K2]+2[K3],U4=[K]4−6[K2][K]2+8[K3][K]+3[K2]2−6[K4],
(2) where
[K]=Kμμ and the square root inK is defined as(√K)μν(√K)νλ=Kμλ .The static spherical black hole solution of the above action yields [61]
ds2=−f(r)dt2+dr2f(r)+r2(dθ2+sin2θdφ2).
(3) Following [76], we take the reference metric as
fμν=diag(0,0,c20hij) . Then, combining this with the ansatz (Eq. (3)), we obtainU1=2c0/r ,U2=2c20/r2 , andU3=U4=0 ; thus, the field solutions areFμν=∂μAν−∂νAμ,A=−qrdt,
(4) f(r)=1+r2l2−2Mr+q2r2+c0c1m2g2r+c20c2m2g.
(5) According to the Hamiltonian approach, the parameters
M andq relate to the mass and charge of the black hole as [61]M=V24πM,Q=V24πq,
(6) where
V2 is the volume of the two-dimensional space. Usingf(r)|r=rh=0 , whererh denotes the horizon radius, we derive the mass asM=rh2+q22rh+12c20c2m2grh+14c0c1m2gr2h+r3h2l2.
(7) The Hawking temperature and entropy of the black hole are
T=f′(rh)4π= c0c1m2g4π+14πrh+c20c2m2g4πrh−q24πr3h+3rh4πl2,
(8) S=∫rh0dr1T(∂M∂r)q,l,ci=πr2h,
(9) respectively.
The thermodynamics of massive black holes have been studied in [61-75]. Here, we study the thermodynamics of massive gravity black holes using two methods: one in which the black hole absorbs a charged particle and one in which a shell of dust falls into the black hole. We primarily focus on the validity of the thermodynamical laws in both the normal and extended phase spaces.
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In this section, we study the thermodynamics of massive gravity black hole, by considering their absorption of a charged particle.
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If we assume that a charged, massive-gravity black hole is perturbed by absorbing a charged particle, then its mass, charge, and AdS radius (which relates to the thermodynamic pressure and volume) can be shifted. During the process, changes to the black hole's conserved quantities are identical to those of the charged particle. Thus, the relation between the conserved quantities of the black hole following particle absorption can be simultaneously obtained by analyzing the corresponding relations of the particle. We start with the Hamiltonian of the charged particle,
H=12gμν(Pμ−eAμ)(Pν−eAν),
(10) and the Hamiltonian-Jacobi action,
S=12m2λ−Et+Lϕ+Sr(r)+Sθ(θ),
(11) where the four-momentum
Pμ of the particle is defined asPμ=∂μS . Here,m2 ,e , andλ denote the mass and charge of the particle and an affine parameter, respectively. Because of the symmetry of the metric (Eq. (3)), the quantitiesE andL are conserved with respect tot andφ and denote the energy and angular momentum of the particle, respectively. To proceed, we combine Eq. (3) with Eqs. (10-11) to deducem2−f(r)−1(−E−eAt)2+f(r)(∂rSr(r))2+r−2((∂θSθ)2+sin−2θL2)=0.
(12) The angular term of the above expression can be defined separately as
K≡(∂θSθ(θ))2+sin−2θL2=−m2r2+r2f(r)(−E−eAt)2−r2f(r)(∂rSr(r))2.
(13) Thus, Eq. (11) is rewritten as
S=−Et+Lϕ+∫dr√R+∫dθ√Θ,
(14) where
Sr≡∫dr√R,Sθ≡∫dθ√Θ,Θ≡K−sin−2θL2,
(15) R=1r2f(r)(−K2−u2r2)+1r2f(r)(r2f(r)(−E−eAt)2).
(16) Subsequently, the radial momentum
Pr and angular momentumPθ can be computed asPr=grr∂rSr(r)=f(r)√−K−m2r2r2f(r)+(−E−eAt)2f2(r),
(17) Pθ=gθθ∂θSθ(θ)=1r2√K−1sin2θL2.
(18) Following [33], we assume that the charged particle is completely absorbed by the black hole upon passing through the outer horizon. As a result, it is impossible for an observer outside the horizon to distinguish the conserved quantities of the particle from those of the black hole. In particular, a relationship between the conserved quantities and momenta at any radial position can be derived. Then, the limit (Eq. (17)) at the outer horizon gives us the following relation between the conserved quantities and radial momenta:
E=qrhe+|Pr|.
(19) The above equation implies that both the momentum and electric charge of the particle contribute to the energy. For simplicity, we hereafter denote
|Pr| as the radial momentum at the horizon. We note thatq/rh is the electromagnetic potential at the event horizon. To ensure that the particle flows into the black hole in a positive flow of time,|Pr| should be positive. When electrical attraction acts upon the particles, their total energy can be negative. Here, we simply choose the signs in front ofE and|Pr| to be positive, as was performed in [33, 36, 37, 77].The relation concerns the interactions between the particle and the black hole, and we assume that no energy is lost during the process. Namely, the charge of the particle
e is equal to the charge variationdQ of the black hole. In Eq. (19), the energy of the particle is expressed in terms ofe and|Pr| near the horizon. We must find a corresponding thermodynamical term that also contains the variation ofe and|Pr| . Following Gwak [33], we assume that the energy of the charged particle changes the internal energy of the black hole as it is absorbed. Here, we check the validity of the thermodynamical laws in both the normal and extended phase spaces, under the assumption thatE=dU . -
We first study the validity of the thermodynamical laws for massive black holes in the normal phase space; in such spaces, the mass of the black hole represents its internal energy; thus, with the assumption of
E=dU , Eq. (19) can be rewritten asE=dU=dM=qrhe+|Pr|.
(20) To analyze the first law under particle absorption, we rewrite the variation of entropy from the Bekenstein's area law as
dSh=2πrhdrh . The event horizon variationdrh is determined by the charge, energy, and radial momentum of the absorbed particle and directly contributes to the changes of the functionf(r) . Then, the change off(r) results in a relocated event horizonrh+drh . More explicitly, we can express the difference between the initial statef(M,Q,rh) and final statef(M+dM,Q+dQ,rh+drh) asf(M+dM,Q+dQ,rh+drh)−f(M,Q,rh)=∂fh∂MdM+∂fh∂QdQ+∂fh∂rhdrh,
(21) where
dM,dQ,drh represent the variation of the charge, energy, and horizon of the black hole, respectively. At the event horizon,f(M,Q,rh)=0 andf(M+dM,Q+dQ,rh+drh)=0 should be satisfied; thus, we deduce thatdf(rh)=dfh=0 ; that is,dfh=∂fh∂MdM+∂fh∂QdQ+∂fh∂rhdrh=0.
(22) Using Eqs. (20) and (22), we can obtain
drh and therebydSh , asdSh=24π|Pr|r3h3c0c1m2gr3h+12Mrh+12r4h/l2−12q2.
(23) Combining Eqs. (8) and (23) gives us the relation
TdSh=|Pr| . With this, Eq. (19) becomesE=eq/rh+TdSh . Subsequently, becauseE=dM ande=dQ , we obtain the form of the variation asdM=TdSh+ΦdQ,
(24) which is the first law of thermodynamics for massive black holes in normal phase space.
Next, we use Eq. (23) to check the validity of the second law of the thermodynamics, which states that the entropy must increase after particle absorption.
First, for the case of the extremal massive black hole (in which the temperature vanishes), applying
T=0 in Eq. (8) gives us the marginal massMe=re+c20c2m2gre+34c0c1m2gr2e+2r3e/l2,
(25) where
re denotes the horizon of the extremal black hole and can be readily reduced as a function ofq,l,c0,c1 , andc2 ; we omit these steps here for simplicity. Substituting Eq. (25) into (23), we deduce thatdSh=∞ holds for the extremal black hole.In the case of a non-extremal black hole,
dSh remains positive for anyci(i=1,2) . We plot the relation betweendSh and the values ofci in Fig. 1. It can be seen that the value ofdSh always exceeds zero. More explicit relations betweendSh andc1,c2 are shown in Fig. 2.Figure 1. (color online)
dSh near the horizon with respect toc1,c2 for a non-extremal black hole in normal phase space.Figure 2. (color online) (left) the relationship between
dSh andc2 with fixedc1=5 ; (right) the relationship betweendSh andc1 with fixedc2=5 .Thus, the variation of entropy is always positive in the normal phase space. Expressed otherwise, the second law of thermodynamics is valid in the normal phase space under particle absorption. In the next subsection, we consider the case for the
PV term. -
The PV criticality, characterized by treating the pressure as
P=3/8πl2 , has been studied in [62-64]. Here, we briefly review the main properties of the extended phase space; then, we check the validity of the thermodynamical laws under the absorption of a charged particle.In the extended phase space, the corresponding conjugate quantity of the pressure
P=3/8πl2 is treated as volumeV [9,10]. Then, the mass (Eq. (7)) and temperature (Eq. (8)) are respectively rewritten asM=rh2+q22rh+12c20c2m2grh+14c0c1m2gr2h+43Pπr3h,
(26) T= c0c1m2g4π+14πrh+c20c2m2g4πrh−q24πr3h+2Prh,S=πr2h.
(27) The other conjugate quantities of the intensive parameters
P,Q,c1 , andc2 areV=∂M∂P=43πr3h,Φ=∂M∂Q=qrh,A=∂M∂c1=c0m2gr2h4,B=∂M∂c2=c20m2grh2,
(28) respectively, where we consider the coupling constants
c1 andc2 as thermodynamical variables. Here, the conjugate variablesV ,A andB are defined by the differential of the enthalpyM . It is noticed that Kastor and Ray defined the thermodynamic conjugate variable in terms of a Killing potential in [9]; And the Hamiltonian method was applied to study the normal thermodynamics in massive gravity in [61]. It would be interesting to develop an alternative method to define those thermodynamical variables and understand their deep physics in our model. We hope to address this topic in the future. All thermodynamical quantities satisfy the extended first law of thermodynamicsdM=TdS+ΦdQ+VdP+Adc1+Bdc2,
(29) and the generalized Smarr relation is
M=2TS−2VP+ΦQ−Ac1.
(30) We note that in the extended phase space,
M plays the role of enthalpy rather than the internal energy of the thermodynamical system; that is, we haveM=U+VP+Ac1+Bc2.
(31) In this case, the energy flux entering the event horizon will alter the internal energy of the black hole, which can be given as a function
U(Q,S,V,c1,c2) . From Eq. (31), the conserved energy and charge of the particle areE=dU=d(M−PV−Ac1−Bc2),e=dQ.
(32) Consequently, the energy relation near the horizon (Eq. (19)) can be expressed as
dU=qrhdQ+|Pr|.
(33) Similar to the analysis of the previous subsection, the slight variations of the redshift function are
dfh=∂fh∂MdM+∂fh∂QdQ+∂fh∂PdP+∂fh∂rhdrh+∂fh∂c1dc1+∂fh∂c2dc2=0.
(34) By combining Eqs. (31) and (34), the contributions of the
dP,dM,dc1 , anddc2 terms can be directly eliminated; thus, the above expression gives usdrh=12(c1dA+c2dB+|Pr|)rh−12M+rh(12+12c20c2m2g+9c0c1m2grh+16Pπr2h).
(35) Subsequently, we obtain the change of entropy and volume as
dS=24π(c1dA+c2dB+|Pr|)r2h−12M+rh(12+12c20c2m2g+9c0c1m2grh+16Pπr2h),
(36) dV=16π(c1dA+c2dB+|Pr|)r3h−12M+rh(12+12c20c2m2g+9c0c1m2grh+16Pπr2h).
(37) From the aforementioned formulas and related thermodynamical quantities, we obtain the relation
TdS−PdV−c1dA−c2dB=|Pr|.
(38) Thus, the expression of the internal energy in Eq. (32) becomes
dU=ΦdQ+TdS−PdV−c1dA−c2dB,
(39) then, combining Eqs. (31) and (39) gives
dM=TdS+ΦdQ+VdP+Adc1+Bdc2,
(40) which is simply the first law of thermodynamics (Eq. (29)) for a massive black hole in the extended phase space. Expressed otherwise, the first law of thermodynamics holds in the extended phase space under particle absorption.
The next step is to analyze the sign of Eq. (36), to verify that the second law is valid in the extended phase space.
Taking the derivatives of
A andB with respect torh in Eq. (28) gives usdA=c0m2grh/2drh,dB=c20m2g/2drh ; then, substituting these into Eq. (36) gives usdS=24π|Pr|r2h−12M+rh(12+6c20c2m2g+3c0c1m2grh+16Pπr2h).
(41) Because the numerator of the above expression for
dS is always positive, we must consider only the sign of the denominator, which we label asΔS1 . The value ofΔS1 depends on the model parametersc1 andc2 . As a trial, we setc0=100,mg=0.01,p=1,Pr=1 and list the values ofΔS1 for differentc2 under a fixed value ofc1=5 in Table 1. It can be seen that certain values ofc2 convert the sign ofΔS1 from positive to negative as it increases. Moreover, in Fig. 3, we plot the boundary lines(c1c,c2c) for whichΔS1→0 . Hence, for parameters below the lines,ΔS1 is always positive, which indicates that the second law of thermodynamics is maintained; however, for the parameters above each line, one always obtainsΔS1<0 , which implies a violation of the second law. It is also evident from the figure that the borderline depends uponM andQ , and that the parameter range in which the the second law is maintained is narrower for smallerM but biggerQ .M Q c1 c2 rh ΔS1 2 0.5 5 3.6931 0.5099 0.1171 2 0.5 5 3.7931 0.5049 0.0587 2 0.5 5 3.8931 0.5 2.88058×10−6 2 0.5 5 3.9931 0.4951 -0.0589 2 0.5 5 4.0931 0.4903 -0.1181 Table 1. Numerical results of
ΔS1 for differentc2 under fixedc1=5 .Figure 3. (color online) The diagrams of critical
c1 andc2 values for differentM (left) andQ values (right).Notably,
ΔS1=0 indicates that the change of entropydS diverges at the points(c1c,c2c) on the critical line. Expressed otherwise, for a givenc1 , there exists a certainc2c that brings about the divergence ofdS , as shown in Fig. 4. However, this is a trivial result because the important information is contained within the change of sign indS . Furthermore, the number of degrees of freedom is maximized asc2 approachesc2c , which is denoted by the vertical line in the figures. In fact, the critical point ofc2 is non-physical; as such, the black hole will not reach this point because it violates the second law of thermodynamics.Here, we consider the case of the extremal massive black hole. We substitute the marginal mass (Eq. (25)) into Eq. (41); thus, we obtain
dSextremal=−4π|Pr|rec0c1m2gre+c20c2m2g+8πPr2e.
(42) Because the denominator of
dSextremal is always positive, we follow the strategy used in the non-extremal case and define the numerator asΔS2 . For the fixed valuesc0=100,mg=0.01,p=1,q=1/2 , we list the numerical result ofΔS2 in Table 2. The result is similar to the case described above, which states that the violation of the second law of thermodynamics depends on the model parameters.c1 c2 re ΔS2 −2 −6.07319 0.493109 −0.0833 −2 −6.17319 0.49656 −0.0414 −2 −6.27319 0.5 1.93433×10−6 −2 −6.37319 0.50343 0.0410 −2 −6.47319 0.506849 0.0816 Table 2. Numerical results of
ΔS2 for differentc2 under fixedc1=−2 in the extremal case.Here, we briefly summarize the results obtained for the absorption of a charged particle by a massive black hole. (i) Both the first and second laws of thermodynamics hold in the normal phase space. (ii) When we consider the extended thermodynamics case by introducing the
PV term, the first law is always saved but the second law can be violated depending on the model parameters. This violation may be brought about by the assumption that the internal energy of the black hole is equal to the energy of the particle.In the next section, as an alternative strategy to particle absorption, we consider a thin shell falling into the massive black hole. We calculate the mass of the shell when it arrives at the horizon of the black hole and then revisit the validity of the laws of thermodynamics in both the extended and normal phase spaces.
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By considering a thin shell approaching the horizon and using the techniques presented in Appendix A, we obtain the external surface of the shell in terms of
(M,Q,P,r,c1,c2) n+r=(1+8πP3r2−2r+Q2r2+c0c1m2g2r+c20c2m2g+˙R)121+8πP3r2−2Mr+q2r2+c0c1m2g2r+c20c2m2g,
(43) and the internal surface of the shell is denoted in terms of
(M+dM,Q+dQ, P+dP,r+dr,c1+dc1,c2+dc2) nr−=−(1−2(M+dM)r+(Q+dQ)2r2+8π(P+dP)3r2+c0(c1+dc1)m2g2r+c20(c2+dc2)m2g+˙R2)12.
(44) Subsequently, as the shell approaches the horizon, its mass can be calculated using
μ=−R(nr+−nr−)=−R(1+8πP3r2−2Mr+Q2r2+c0c1m2g2r+c20c2m2g+˙R2)12−R(1−2(M+dM)r+(Q+dQ)2r2+8π(P+dP)3r2+c0(c1+dc1)m2g2r+c20(c2+dc2)m2g+˙R2)12,
(45) where
dP denotes the variation of the shell's pressure. To vary the values ofdc1 anddc2 , we can interpret them as special charges, labeledQc1 andQc2 , respectively. The terms containingc1 andc2 in the above equation above can be treated as a special form of energy, similar to the chargeQ .To ensure that the particle does not recoil to outside the horizon and falls into the black hole successfully, the absorption condition, stipulating that the energy of the thin shell of dust must exceed a minimum threshold value, should be satisfied. Using the condition
nr+−nr−⩾0 , the minimum energy can be derived asdM⩾qdQr+d2Q2r2,
(46) where the second term is the negligible self-interaction of the particle. In Eq. (45), we have
˙R=dR/dτ=ur ; thus, we can solve forM and calculate the variation asdM=12c20m2grdc2+14c0m2gr2dc1+43πr3dP+(dQ)22r+QrdQ−μ22r+μnr.
(47) Here,
nr=√1+c20c2m2g+12c0c1m2gr−2Mr+83πPr2+Q2r2+˙R2,
(48) where
μ andnr denote the rest mass and radial velocity of the shell, respectively. We take the latter as a positive value, to ensure that the shell arrives at the black hole horizon. The higher orders ofμ anddQ represent the self-interactions of particles and can be neglected in the current context. Thus, Eq. (45) can be re-written asdM=43πr3dP+QdQr+14c0m2gr2dc1+12c20m2grdc2+μnr.
(49) Here, we discuss the thermodynamical properties under a variation of mass, as calculated using the shell in the extended phase space. The increase in energy relates to the black hole's mass. Hence, we can discuss the violation of thermodynamics using the expression of
drh , which can be found by inserting Eq. (49) into (34). Then, we can deletedM directly. Interestingly,dQ,dc1,dc2 can be simultaneously cancelled out; thus, we obtain the expression fordrh near the horizon, asdrh=12μnrr−12M+r(12+12c20c2m2g+9c0c1m2gr+64πPr2).
(50) Using the relationships connecting the entropy and volume with the radius (i.e.,
dS=2πrhdrh anddV=4/3πr2hdrh , respectively), we obtaindSh=24πμnrr2−12M+r(12+12c20c2m2g+9c0c1m2gr+64πPr2),
(51) dVh=16πμnrr3−12M+r(12+12c20c2m2g+9c0c1m2gr+64πPr2).
(52) From these formulas, we obtain
dM=TdSh+ΦdQ+VdP+Adc1+Bdc2 , which implies that the first law of thermodynamics is recovered for the case of infalling dust.Here, we discuss the validity of the second law via the entropy variation expressed in Eq. (52). For an extremal black hole in which
T=0 , by substituting Eq. (25) into (52), we can deduce thatdSh=∞ . For a non-extremal black hole, the event horizon should exceedrh . We plot the denominator ofdSh for the fixed parametersc1=5,c0=100,mg=0.01,p=1,M=2,Q=0.5 for different values ofc2 ; from this, we conclude that the denominator ofdSh always exceeds zero, as shown in Fig. 5.From Fig. 6, we can see that
dSh remains positive, regardless of the values ofc1 andc2 ; thus, the entropy of the black hole will not decrease. This demonstrates that the second law of thermodynamics is also valid.From the above discussion, we concluded that for infalling dust (considered in the extended phase space), the second law of thermodynamics is always valid. This conclusion significantly differs from that of Section 3, where we treated the black hole mass as an enthalpy that is related to the internal energy via
U=M−PV−Ac1−Bc2 ; there, the entropy variations were determined by the value ofc1 andc2 . This indicates that the results differ considerably between the two methods. However, in the normal phase space, the conclusions are very similar because the variation of energy in both cases relates to the black hole's mass without thePV term. For simplicity, we omit the detailed calculations here. -
In this section, we briefly discuss the WCCC under charged particle absorption. The WCCC states that no naked singularity can be seen by an observer at the future null infinity. This means that the singularity should be hidden by the event horizon of the black hole. We consider the variation that occurs as a charged particle is absorbed by a massive black hole, to ensure that the event horizon exists and that the black hole is not destroyed. We primarily follow the strategy used in [33].
There always exists a minimum value of the function
f(r) for radiusrm . If the event horizon exists, the conditionf(rm)⩽0 should be satisfied. Atrm , we havef|r=rm≡fm=δ⩽0,∂rf|r=rm≡f′m=0,(∂r)2f|r=rm≡fm″
(53) For extremal conditions,
\delta = 0 should satisfied; otherwise,\delta is a small quantity. After the charged particle has been absorbed by the black hole, the minimum point becomesr_m+{\rm d}r_m . Correspondingly, the variation off(r) atr = r_m is\partial_r f|_{r = r_m+{\rm d}r_m} = f_m'+{\rm d}f_m' = 0.
(54) In the extended thermodynamical phase space, we have
\begin{split} {\rm d}f_m' =& \frac{\partial{f_m'}}{\partial M}{\rm d}M+\frac{\partial{f_m'}}{\partial Q}{\rm d}Q+\frac{\partial{f_m'}}{\partial P}{\rm d}P\\&+\frac{\partial{f_m'}}{\partial r_m}{\rm d}r_m+\frac{\partial{f_m'}}{\partial c_1}{\rm d}c_1+\frac{\partial{f_m'}}{\partial c_2}{\rm d}c_2 = 0. \end{split}
(55) The physical parameters of the black hole vary from the initial state
(M,Q,P,c_1,c_2,r_m) to the final state(M+{\rm d}M,Q+{\rm d}Q,P+{\rm d}P, c_1+{\rm d}c_1,c_2+{\rm d}c_2, r_m+{\rm d}r_m) ; this final state can be expressed as\begin{split} f(r_m+{\rm d}r_m) =& f_m+{\rm d}f_m = \delta+\frac{\partial{f_m}}{\partial M}{\rm d}M+\frac{\partial{f_m}}{\partial Q}{\rm d}Q\\&+\frac{\partial{f_m}}{\partial P}{\rm d}P+\frac{\partial{f_m}}{\partial c_1}{\rm d}c_1+\frac{\partial{f_m}}{\partial c_2}{\rm d}c_2, \end{split}
(56) where
\begin{split} \frac{\partial{f_m}}{\partial M}\Big|_{r = r_m} = -\frac{2}{r_m},\; \; \; \frac{\partial{f_m}}{\partial Q}\Big|_{r = r_m} = \frac{2Q}{r_m^2},\; \; \; \frac{\partial{f_m}}{\partial P}\Big|_{r = r_m} = \frac{8\pi r_m^2}{3}, \end{split}
\begin{split} \frac{\partial{f_m}}{\partial c_1}\Big|_{r = r_m} = \frac{c_0 m_g^2r_m}{2},\; \; \; \frac{\partial{f_m}}{\partial c_2}\Big|_{r = r_m} = c_0^2m_g^2. \end{split}
(57) The crucial information is contained in the sign of Eq. (56). First, we discuss the case for the extremal black hole, where we have
\delta = 0 . We deform{\rm d}M using the conditionf_m' = 0 ; thus,\begin{split} {\rm d}M =& \frac{\partial{M}}{\partial P}{\rm d}P+\frac{\partial{M}}{\partial Q}{\rm d}Q+\frac{\partial{M}}{\partial c_1}{\rm d}c_1\\&+\frac{\partial{M}}{\partial c_2}{\rm d}c_2+\frac{\partial{M}}{\partial r_m}{\rm d}r_m. \end{split}
(58) By combining Eq. (58) with (32),
{\rm d}r_m can be written as{\rm d}r_m = -\frac{r(-2Q {\rm d}Q +2c_1 r {\rm d}A+2 r c_2 {\rm d}B+2 r |P^r|+c_0^2 m_g^2 r^2 {\rm d}c_2+c_0 m_g^2 r^3 {\rm d}c _1+8\pi r^4 {\rm d}P)}{2 Q^2+c_0 c_1 m_g^2 r^3+24 P\pi r^4}.
(59) Then, inserting Eq. (58) into (55), we deduce that
{\rm d}r_m = 0 . Applying Eq. (59), we have that{\rm d}P = \frac{2Q {\rm d}Q -r(2c_1 {\rm d}A+2c_2 {\rm d}B+2 |P^r|+c_0^2 m_g^2 r {\rm d}c_2+c_0 m_g^2 r^2 {\rm d}c _1)}{8 P\pi r^4}.
(60) Substituting Eqs. (60) and (59) into (56), the transformation
f(r_m+{\rm d}r_m) can be expressed asf_m+{\rm d}f_m = -\frac{2 |P^r|}{r_m}.
(61) The negative value following the absorption of a charged particle implies that the final state of the extremal black hole is a non-extremal black hole. This means that the event horizon always exists, which implies that the WCCC is valid in the extended phase space for the extremal black hole.
For the case of a near-extremal black hole, the minimum value of
\delta is a small quantity, which we label asf(M,P,Q,c_1,c_2,r_m) = \delta_\epsilon . Besides this, the value off'(r_m) is very close to zero. Thus, we user_m = r_0(1-\epsilon) to ensure a near-extremal case, because Eq. (32) cannot be used directly; here,0<\epsilon\ll 1 . Then, we expand Eq. (32) near the minimum point as\begin{split} {\rm d}M =& \frac{Q}{r_m}{\rm d}Q+\frac{c_0^2 m_g^2 r_m}{2} {\rm d}c_2+\frac{c_0 m_g^2 r_m^2}{4} {\rm d}c_1 +\frac {4\pi r_m^3}{3} {\rm d}P\\ &+ \left (\frac12+\frac{c_0^2 c_2 m_g^2 }{2} -\frac{Q^2}{2r_m^2}+\frac{c_0 c_1 m_g^2 r_m}{2} +4p\pi r_m^2 \right){\rm d}r_m \\& + \left(\frac{c_0^2 m_g^2 }{2} {\rm d}c_2+\frac{c_0 c_1 m_g^2}{2} {\rm d}r_m+\frac{Q^2}{r_m^3}{\rm d}r_m-\frac{Q}{r_m^2}{\rm d}Q\right. \end{split}
\begin{split}&\left.+\frac{c_0 m_g^2 r_m}{2} {\rm d}c_1+8 p\pi r_m {\rm d}r_m+4\pi r_m^2 {\rm d}P\right)\epsilon\\& + \left(\frac{c_0 m_g^2}{4} {\rm d}c_1+4 P \pi {\rm d}r_m -\frac{3Q^2}{2r_m^4}{\rm d}r_m+\frac{Q}{r_m^3}{\rm d}Q \right. \\&+4\pi r_m {\rm d}P \Bigg)\epsilon^2+{\cal{O}}(\epsilon)^3. \end{split}
(62) Inserting Eq. (62) into (56), we obtain
\begin{split} {\rm d}f_m =& \left(-\frac{c_0^2 c_2 m_g^2}{r_m}-c_0 c_1 m_g^2-8 \pi P r_m+\frac{Q^2}{r_m^3}-\frac{1}{r_m}\right){\rm d}r_m\\& + \left(-c_0 m_g^2 {\rm d}c_1-\frac{c_0^2 m_g^2}{r_m} {\rm d}c_2+\frac{2 Q}{r_m^3} {\rm d}Q-8 \pi r_m {\rm d}P\right.\\&\left.-\left(\frac{ c_0 c_1 m_g^2}{r_m}+\frac{2 Q^2}{ r_m^4}+16 \pi P \right) {\rm d}r_m\right)\epsilon \\& + \left(-\frac{c_0 m_g^2}{2 r_m}{\rm d}c_1-\frac{2Q}{r_m^4}{\rm d}Q-8 \pi {\rm d}P -\left(\frac{8 \pi P}{r_m}-\frac {3Q^2}{r_m^5}\right){\rm d}r_m\right)\epsilon^2\\&+{\cal{O}}(\epsilon)^3. \end{split}
(63) We solve the expression for
P usingf'(r_m) = 0 , obtainingP = \frac{Q^2-r_m^2-c_0^2 c_2 m_g^2 r_m^2 -c_0 c_1 m_g^2 r_m^3}{8\pi r_m^4}.
(64) Thus, the expression of
{\rm d}P reads\begin{split} {\rm d}P =& \frac{Q}{4\pi r_m^4}{\rm d}Q-\frac{c_0^2 m_g^2 }{8\pi r_m^2}{\rm d}c_2-\frac{c_0 m_g^2}{8\pi r_m}{\rm d}c_1\\& + \left(\frac{-2r_m-2c_0^2 c_2 m_g^2 r_m-3 c_0 c_1 m_g^2 r_m^2}{8\pi r_m^4} \right.\\&\left.-\frac{Q^2-r_m^2-c_0^2 c_2 m_g^2 r_m^2-c_0 c_1 m_g^2 r_m^3}{2\pi r_m^5}\right){\rm d}r_m. \end{split}
(65) Substituting Eqs. (64) and (65) into (63), we obtain
\begin{split} {\rm d}f_m =& \left(\frac{c_0 m_g^2}{2 r_m}{\rm d}c_1+\frac{c_0^2 m_g^2}{ r_m^2}{\rm d}c_2-\frac{4Q}{r_m^4}{\rm d}Q+\left(\frac{6Q}{r_m^5}-\frac{1}{r_m^3}\right.\right.\\&\left.\left.-\frac{c_0^2 c_2 m_g^2}{r_m^3}\right){\rm d}r_m \right)\epsilon^2+ {\cal{O}}(\epsilon)^3, \end{split}
(66) such that
\begin{split} f_m+{\rm d}f_m =& \delta_\epsilon+\left(\frac{c_0 m_g^2}{2 r_m}{\rm d}c_1+\frac{c_0^2 m_g^2}{ r_m^2}{\rm d}c_2-\frac{4Q}{r_m^4}{\rm d}Q\right.\\&\left.+\left(\frac{6Q}{r_m^5}-\frac{1}{r_m^3}-\frac{c_0^2 c_2 m_g^2}{r_m^3}\right){\rm d}r_m \right)\epsilon^2+{\cal{O}}(\epsilon)^3 \\ =& -\left(8P\pi+\frac{Q^2}{r_m^4}+\frac{c_0c_1m_g^2}{2r_m}\right)\epsilon^2+\left(\frac{c_0 m_g^2}{2 r_m}{\rm d}c_1+\frac{c_0^2 m_g^2}{ r_m^2}{\rm d}c_2\right.\\&\left.-\frac{4Q}{r_m^4}{\rm d}Q+\left(\frac{6Q}{r_m^5}-\frac{1}{r_m^3}-\frac{c_0^2 c_2 m_g^2}{r_m^3}\right){\rm d}r_m \right)\epsilon^2+{\cal{O}}(\epsilon)^3, \end{split}
(67) where in the second line we have introduced
\delta_\epsilon = -(8P\pi+ Q^2/r_m^4+c_0c_1m_g^2/2r_m)\epsilon^2+{\cal{O}}(\epsilon)^3 for the near-extremal black hole. Evidently,\delta_\epsilon is always negative, its leading term is\delta_\epsilon\sim\epsilon^2 , and the leading order of{\rm d}f_m is in the form of{\rm differential\; quantities}\, \times \epsilon^2 . In Eq. (67), the coefficients expressed as differential quantities in the second term cannot be compared with the negative finite coefficient in the first term; hence, its total value is always negative, which implies that weak cosmic censorship remains valid for near-extremal black holes.Thus, under particle absorption, although the second law of thermodynamics may be violated for certain model parameters, the WCCC remains valid in the extended phase space.
We proceed to verify the WCCC in the normal phase space. To this end, we write the changes in the black hole's conserved quantities as
(M+{\rm d}M,Q+{\rm d}Q) ; thus, the locations of the minimum value and event horizon can be written asr_m+{\rm d}r_m andr_h+{\rm d}r_h , respectively. Subsequently, the variation off(r_m+{\rm d}r_m) is obtained asf(r_m+{\rm d}r_m) = f_m+{\rm d}f_m = \delta+\frac{\partial{f_m}}{\partial M}{\rm d}M+\frac{\partial{f_m}}{\partial Q}{\rm d}Q.
(68) At
r_m+{\rm d}r_m , the variation off(r) is defined as{\rm d}f_m ; then, consideringf'_m = 0 , we have{\rm d}f'_m = \frac{\partial{f'_m}}{\partial M}{\rm d}M+\frac{\partial{f'_m}}{\partial Q}{\rm d}Q+\frac{\partial{f'_m}}{\partial r_m}{\rm d}r_m .
(69) Furthermore, using the condition
f'_m = 0 , we obtainM and subsequently{\rm d}M as{\rm d}M = {\rm d}R \left(-\frac{1}{2} c_0 c_1 m_g^2 r-8 \pi P r^2-\frac{Q^2}{r^2}\right)+\frac{2 Q }{r}{\rm d}Q.
(70) Inserting Eq. (70) into (69), we get
{\rm d}R = \frac{2 r (|P^r| r-Q{\rm d}Q )}{c_0 c_1 m_g^2 r^3+16 \pi P r^4+2 Q^2}.
(71) Combining Eqs. (70) and (68), we obtain an expression for the minimum point, as
f_m+{\rm d}f_m = -\frac{2 |P^r|}{r},
(72) which is identical to the result (Eq. (61)) for the extended phase space. We do not repeat the discussions here.
From the above analysis, we can conclude that under particle absorption, weak cosmic censorship holds in both the extended and normal phase spaces, even though the second law of thermodynamics may be violated for certain model parameters. Thus, the violation of the second law of thermodynamics (under this assumption) does not affect the WCCC. Hence, we argue that the assumption in the first method may be the cause of second law's violation and requires careful further treatment.
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In this paper, we studied the thermodynamical laws of massive gravity black holes, using two methods. First, we investigated their thermodynamics properties by considering a charged particle being absorbed by a black hole, and we checked the validity of the thermodynamical laws in both the normal and extended phase spaces. Under this method, the first law remained valid, whereas the expression of the entropy could be negative depending on the model parameters; furthermore, the second law of thermodynamics could be violated in an extended thermodynamical phase space in which the cosmological constant is treated as the pressure of the black hole. For the case with normal thermodynamics (i.e., without the
PV term), this violation does not occur.Then, we applied a second method that considers a shell of dust falling into a massive gravity black hole; here, the mass of the shell could be directly calculated as it approached the black hole horizon. We found that the expressions for the horizon and entropy always exceeded zero, which implies that the first and second laws of thermodynamics are always valid in both cases, with or without the
PV term. We summarize our results in Table 3.Thermodynamics of black hole Thermodynamic laws Method with charged particle absorbed Method with shell of dust falling Normal First Law Satisfied Satisfied Second Law Satisfied Satisfied Extended First Law Satisfied Satisfied Second Law Entropy could be negative depending on model parameters Satisfied Table 3. Summary of the validity of the first and second laws of thermodynamics in the normal space and extended spaces.
We argued that the violation of the second law in the extended thermodynamical phase space may be a consequence of the assumption in the first method, which states that the particle affects the internal energy of the black hole. Thus, this assumption should be treated carefully.
Finally, we verified the WCCC for the first method. Our analysis showed that weak cosmic censorship is always valid in both the normal and extended phase spaces. This further supports our hypothesis that the aforementioned assumption may contribute to the violation of the second law of thermodynamics.
In [78], it was argued that under this assumption, when the particle energy increases
M , it simultaneously increasesV ; as a result, the increase in internal energy is less than the increase in enthalpy, and the assumption is not appropriate. Moreover, the second law of thermodynamics is known to play a fundamental role in physics. Thus, its violation in extended black hole thermodynamics deserves careful investigation of several questions; for instance, what are the underlying physics of the violation? Is it natural to treat the cosmological constant as a variable in extended thermodynamics? Further investigations pertaining to these issues are called for.We are thankful to De-Chang Dai,Yen Chin Ong, and Jian-Pin Wu for helpful discussions.
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In this appendix, we present the technique used for calculating the mass
\mu of the shell, which is equal to the change in mass{\rm d}M of the black hole when the shell of dust infalls near the horizon, as proposed in [79]. Considering\tau to be the proper time measured by an observer at rest in the dust, and specifying the shell motion asr(\tau) , the metric of the shell can be expressed as{\rm d}s^2 = -{\rm d}\tau^2+R^2(\tau)({\rm d}\theta^2+\sin^2{\theta}{\rm d}\phi^2), \tag{A1}
(73) where
R(\tau) = r(\tau) and4\pi R^2 denotes the surface area of the shell at\tau . We consider the continuity equation,0 = \frac{{\rm d}\sigma}{{\rm d}\tau}+\sigma u^i|_i = (\sigma u^i)|_i = (\sigma(^{(3)}g)^{\frac{1}{2}}u^i),_i/(^{(3)}g)^{\frac{1}{2}}, \tag{A2}
(74) where
\sigma is the surface mass density and^{(3)}g is the determinant of the 3-dimensional metric. Combining this withu^\tau = 1 and^{(3)}g = R^4(\tau) , we obtain that(\sigma R^2)_{,\tau} = 0 , which implies that the rest mass of the shell is a constant with4\pi R^2 \sigma = \mu .The discontinuity in extrinsic curvature
K is defined as[K_{ij}] = 8\pi\sigma(u_i u_j+\frac{1}{2}^{(3)}g_{ij}) (see [79] for more details), whereu denotes the 4-velocity. To proceed, we calculate the[K_{\theta\theta}] component as[K_{\theta\theta}] = 8\pi\sigma\left(u_\theta u_\theta+\frac{1}{2}^{(3)}g_{\theta\theta}\right) = 4\pi \sigma^{(3)}g_{\theta\theta} = \mu.\tag{A3}
(75) On the other hand, the component
K_{\theta\theta} can be computed from the definition asK_{\theta\theta} = -n_{\theta;\theta} = n_\alpha \Gamma_{\theta\theta}^\alpha = -\frac{1}{2}n^rg_{\theta\theta,r} = -r n^r. \tag{A4}
(76) Then, the discontinuity term can be derived as
[K_{\theta\theta}] = \mu = -r(n^{r+}-n^{r-}), \tag{A5}
(77) where
n^{r+} andn^{r-} denote the radial components of the normal evaluated in the exterior and interior geometries, respectively.When the shell falls into the black hole, we can calculate the external components
n^{r+} in the exterior geometry as well as the internal componentsn^{r-} in the interior geometry. More specifically, by employingu.n = 0 andn.n = -u.u = 1 , we can evaluate the components ofu andn . Thus, on the shell's exterior we have1 = f(r)(u^t)^2-(u^r)^2/f(r), \tag{A6}
(78) 0 = n_r u^r + n_t u^t,\tag{A7}
(79) 1 = -f(r)^{-1}(n_t)^2+f(r)(n_r)^2.\tag{A8}
(80) After eliminating
u^t,n_t , we obtainn_r^+ = \left[\frac{1+(u^r)^2/f(r)}{f(r)}\right]_{s+}^\frac{1}{2}. \tag{A9}
(81) Moreover, we have
r = R(\tau) andu^r = {\rm d}R/{\rm d}\tau\equiv\dot{R} on the shell; thus, outside the shell, the contravariant component ofn isn^{r+}\equiv g^{rr}n_r^+ = \left(f(r)+\dot{R}^2\right)^\frac{1}{2}\mid_{s+}. \tag{A10}
(82) With the same method, one can obtain
n^{r-} = \left(f(r)+\dot{R}^2\right)^\frac{1}{2}\mid_{s-} . Therefore, Eq. (77) can be re-evaluated as\mu = -R(n^{r+}-n^{r-}) = -R\left[(f(r)+\dot{R}^2)^\frac{1}{2}\mid_{s+}-(f(r)+\dot{R}^2)^\frac{1}{2}|_{s-}\right], \tag{A11}
(83) which is treated as the
{\rm d}M , the change of the mass when the shell approaches the horizon.
Revisiting black hole thermodynamics in massive gravity: charged particle absorption and infalling shell of dust
- Received Date: 2020-02-07
- Accepted Date: 2020-06-11
- Available Online: 2020-10-01
Abstract: In this study, we apply two methods to consider the variation of massive black holes in both normal and extended thermodynamic phase spaces. The first method considers a charged particle being absorbed by the black hole, whereas the second considers a shell of dust falling into it. With the former method, the first and second laws of thermodynamics are always satisfied in the normal phase space; however, in the extended phase space, the first law is satisfied but the validity of the second law of thermodynamics depends upon the model parameters. With the latter method, both laws are valid. We argue that the former method's violation of the second law of thermodynamics may be attributable to the assumption that the change of internal energy of the black hole is equal to the energy of the particle. Finally, we demonstrate that the event horizon always ensures the validity of weak cosmic censorship in both phase spaces; this means that the violation of the second law of thermodynamics, arising under the aforementioned assumption, does not affect the weak cosmic censorship conjecture. This further supports our argument that the assumption in the first method is responsible for the violation and requires deeper treatment.