PLB33024S0370-2693(17)30575-010.1016/j.physletb.2017.07.019The Author(s)Theory

Fig. 1

The diagram of function f(x) with respect to x by choosing different constants y.

Fig. 1

Fig. 2

The T–E diagram in the Schwarzschild case.

Fig. 2

Fig. 3

The T–E diagram in the dRGT massive gravity with c1=0 and c2<0.

Fig. 3

Fig. 4

The T–E diagram in the dRGT massive gravity with c2m2+1=0 and c1>0.

Fig. 4

The effects of massive graviton on the equilibrium between the black hole and radiation gas in an isolated box

Ya-Peng

⁎

huyp@nuaa.edu.cn

Feng

Pan

fan_physics@126.com

Xin-Meng

wuxm@nuaa.edu.cn

aCollege of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, ChinaCollege of ScienceNanjing University of Aeronautics and AstronauticsNanjing

210016

ChinabInstituut-Lorentz for Theoretical Physics, Leiden University, Niels Bohrweg 2, Leiden 2333 CA, The NetherlandsInstituut-Lorentz for Theoretical PhysicsLeiden University

Niels Bohrweg 2

LeidenCA

2333

The NetherlandscKey Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, ChinaKey Laboratory of Theoretical PhysicsInstitute of Theoretical PhysicsChinese Academy of SciencesBeijing

100190

China⁎Corresponding author.

Editor: M. Trodden

Abstract

It is well known that the black hole can have temperature and radiate the particles with black body spectrum, i.e. Hawking radiation. Therefore, if the black hole is surrounded by an isolated box, there is a thermal equilibrium between the black hole and radiation gas. A simple case considering the thermal equilibrium between the Schwarzschild black hole and radiation gas in an isolated box has been well investigated previously in detail, i.e. taking the conservation of energy and principle of maximal entropy for the isolated system into account. In this paper, following the above spirit, the effects of massive graviton on the thermal equilibrium will be investigated. For the gravity with massive graviton, we will use the de Rham–Gabadadze–Tolley (dRGT) massive gravity which has been proven to be ghost free. Because the graviton mass depends on two parameters in the dRGT massive gravity, here we just investigate two simple cases related to the two parameters, respectively. Our results show that in the first case the massive graviton can suppress or increase the condensation of black hole in the radiation gas although the T–E diagram is similar as the Schwarzschild black hole case. For the second case, a new T–E diagram has been obtained. Moreover, an interesting and important prediction is that the condensation of black hole just increases from the zero radius of horizon in this case, which is very different from the Schwarzschild black hole case.

Introduction

Among the studies on the black hole physics, a significant progress is the discovery that the black hole can have temperature and radiate the particles with black body spectrum, i.e. Hawking radiation [1]. Note that, before the discovery of Hawking radiation, there have already been several clues that the black hole may exist the thermodynamics like a thermal system. For example, the area A of the horizon of a general black hole could never decrease found by Hawking [2], which is even argued to relate to the entropy of black hole by Bekenstein [3]. Moreover, four mechanical laws can be found for the stationary black hole, which are very similar as the thermodynamical laws of thermal system [4]. After the discovery, one will find that the four mechanical laws are just the four laws of thermodynamics for a stationary black hole system, and hence the thermodynamics of a stationary black hole is constructed.It should be pointed out that the Hawking radiation is a quantum effect, since the black hole usually absorbs particles classically. Therefore, one simple consequent question is that whether there is an equilibrium between the black hole and radiated particles if the black hole is surrounded by an isolated box, i.e. an equilibrium between the absorbed and radiated particles [5]. For the radiated particles, usually they can be considered as the radiation gas whose energy and entropy are well known. Therefore, the above question can be also equivalent to whether there is a stable condensation of a black hole among the radiation gas in an isolated box. For the simple case, the condensation of a black hole is just the Schwarzschild black hole, which has been well investigated in detail in the pioneer work by G.W Gibbons and M.J Perry [5]. Their most impressive results are that indeed there is an equilibrium between the Schwarzschild black hole and radiation gas in an isolated box. Moreover, the equilibrium condition has been analytically obtained after taking the conservation of energy and principle of maximal entropy for the isolated system into account [5].In this paper, we would like to investigate the effect of massive graviton on this equilibrium. A simple motivation is that the massive gravity has attracted many attentions recently [6,7]. It has been found that the massive gravity can be not only a theoretical proposal, but also a possibility to interpret the dark matter and dark energy problem [6,7]. In addition, according to the AdS/CFT correspondence, the effect of massive graviton in the bulk gravity can be related to the effects from the lattice in the dual field theory, i.e. deducing the momentum dissipation of electrons [8–12]. It should be emphasized that the extension from Einstein's gravity to a massive gravity with massive graviton is difficult. The reason is that many massive gravity theories suffer from the instability problem of the Boulware–Deser ghost [6,7,13,14]. Recently, the so called de Rham–Gabadadze–Tolley (dRGT) massive gravity has been proposed [15,16], which is a nonlinear massive gravity theory and has been found to be ghost free [17–19]. Many studies about the dRGT massive gravity have been also done [8–12,17–34]. Therefore, we will use the dRGT massive gravity to investigate the effect of massive graviton on the equilibrium between the black hole and thermal radiation gas in an isolated box.Note that, there is a so-called reference metric fμν in the dRGT massive gravity. During the original proof of ghost-free in the dRGT massive gravity, the reference metric is assumed to be invertible [17]. However, the later studies show that the dRGT massive gravity with a degenerate reference metric is also ghost-free [8,18,19]. Furthermore, the dRGT massive gravity with a degenerate reference metric has been found to have many applications in the AdS/CFT correspondence. For example, the degenerate reference metric is usually chosen to break the diffeomorphism invariance of bulk spacetime in the spatial directions. According to the AdS/CFT correspondence, the dual field theory in the boundary has a conserved energy but no conserved momentum currents, which can further correspond to the dissipation of momentum to obtain a finite DC conductivity [8–10,30]. Therefore, the dRGT massive gravity with a degenerate reference metric has been also widely investigated recently. For instance, the Schwarzschild-like black hole solution and its thermodynamics were studied in [8,18,20,22,28,32]. The counterterm of this massive gravity with a degenerate reference metric has also been obtained in [29]. In our paper, for simplicity, we just take the Schwarzschild-like black hole in the dRGT massive gravity with a degenerate reference metric into account [20–27]. After considering the conservation of energy and principle of maximal entropy for the isolated system, we have investigated the effects of massive graviton on the equilibrium. Note that, since the graviton mass just depends on two parameters in the four dimensional spacetime case, and hence in our paper we only consider two simple cases which are related to the two parameters, respectively. Furthermore, these two simple cases can be analytically investigated. The main interesting results are that the T–E diagram is similar as the Schwarzschild black hole case in the first case, which is further predicted that the massive graviton can suppress or increase the condensation of black hole among the radiation gas, i.e. dependent on the value of the other parameter c2. For the second case, a new T–E phase diagram has been obtained. Moreover, an interesting and important prediction is that the condensation of black hole just increases from the zero radius of horizon in this case, which is very different from the first case or the Schwarzschild black hole case.The rest of our paper is organized as follows. In Sec. 2, we give a brief introduction to the pioneer work by G.W Gibbons and M.J Perry as a warmup to make the whole paper more readable and complete. In Sec. 3, we use the dRGT massive gravity to investigate the effects from massive graviton on the equilibrium, and consider two simple cases which can be analytically investigated in detail. Sec. 4 is devoted to the conclusion and discussion.2

Warmup: equilibrium between the Schwarzschild black hole and radiation gas in an isolated box

Suppose that there is an isolated box of volume V containing the radiation gas, and the total energy for this isolated system is E (more details of this box can be seen in Ref. [5]). For the radiation gas, the energy and entropy are well known as(1)Eg=aVT4, Sg=43aVT3, where a is the Stefan's constant, and the subscript g is related to the radiation gas. For the Schwarzschild black hole,(2)ds2=−(1−2Mr)dt2+(1−2Mr)−1dr2+r2(dθ2+sin2θdφ2), its energy and Bekenstein–Hawking entropy are(3)Es=M, Ss=A4=4πM2, where A is the area of event horizon located at rh=2M, and c=G=ħ=kB=1 have been assumed for the convenience here and in the following. Therefore, if there is a condensation of the Schwarzschild black hole among the radiation gas, the total energy and entropy inside the isolated box will be(4)E=M+aVT4, S=4πM2+43aVT3, where the volume of black hole has been neglected.Note that, for this isolated system, the total energy E should be conserved, and hence the stable equilibrium state should have the maximal total entropy S. After setting x=M/E and y=(1/3π)(aV/E5)1/4, one will find that maximizing the total entropy S is equal to maximizing the function(5)f(x)=x2+y(1−x)3/4, where the range of independent variable x is [0,1].For convenience, we have drawn a simple diagram for this function f(x) with respect to x by choosing different constants y in the Fig. 1

The main conclusions are:(1) For y>1.4266, the function f(x) is monotonically decreasing and there are no turning points, thus the maximum value of f(x) is at x=0, which means that the equilibrium configuration is just the pure radiation and there is no black hole.(2) For 1.4266>y>yc=1.0144, there are two turning points, a local minimum for x<4/5 and a local maximum for x>4/5. However, the global maximum of f(x) is still given by x=0, which means that the equilibrium configuration is still the pure radiation.(3) For y<yc=1.0144, there are also two turning points, but the global maximum of f(x) is given by x>xc=0.97702, which means that the stable equilibrium configuration consists of the Schwarzschild black hole and black-body radiation, and the temperature of radiation will be same as the temperature of Schwarzschild black hole T=1/(8πM).In addition, if one fixes the volume V of the isolated box and the Stefan's constant a, the T–E diagram of this isolated system has been drawn by G.W Gibbons and M.J Perry in the Fig. 2

. In this figure, the solid line represents the stable equilibrium state, while the dotted lines represent unstable state and the dash line represents the pure black hole state.

From this figure, one can easily see that there is only the pure radiation if the total energy E is low, since the constant y=(1/3π)(aV/E5)1/4 is very high, which is also consistent with the physics that the black hole will not be formed if the energy density is not high enough. However, when the total energy E becomes more, the constant y will decrease down to the critical value yc=1.0144. If more energy E is added, the black hole will be condensed among the radiation gas, i.e., the stable equilibrium configuration consists of the Schwarzschild black hole and black-body radiation. Therefore, there is an upper temperature bound Tmax for this isolated system with fixed volume V(6)Tmax=1(3πycaV)1/5, while the upper temperature bound or critical temperature of black hole is(7)Tc=18πxc(3πyc)4/5(aV)−1/5. It should be emphasized that this temperature of black hole Tc is usually not equal to the temperature of the rest of radiation gas during its condensation, i.e. usually lower than the temperature of radiation gas, and hence this configuration state is in fact not stable. Therefore, the black hole will become bigger, and finally the isolated system will reach the stable configuration state, i.e., the temperature of condensed black hole is equal to the temperature of radiation gas. Note that, during this non-equilibrium process, this isolated system with fixed volume V will keep the same total energy Ec=aVTmax4, and its temperature will decrease to the final value Tcrit, i.e. the temperature of this isolated system in the stable configuration state and Tcrit<Tc<Tmax. After Ec<E, the temperature of this isolated system in the stable configuration state will be lower than Tcrit. Note that, an interesting prediction is that the radius of horizon is nonzero when the condensation of a Schwarzschild black hole occurs among the radiation gas.3

Effects from massive graviton on the equilibrium between the black hole and radiation gas in an isolated box

In this section, we will investigate the effects from massive graviton on the equilibrium between the black hole and radiation gas in an isolated box. Note that, since the graviton is massive, the condensed black hole among the radiation gas is usually not the Schwarzschild black hole, which should be considered under the massive gravity theory. For the massive gravity theory, here we just use the dRGT massive gravity which has been proven to be ghost free. Therefore, we will first give a short introduction to the dRGT massive gravity and its Schwarzschild-like black hole solution, i.e. the spherically static black hole solution in the vacuum. Then, also using the conservation of energy and principle of maximal entropy for an isolated system, we will investigate the equilibrium between the Schwarzschild-like black hole and radiation gas in an isolated box. Since there are two parameters related to the graviton mass in the Schwarzschild-like black hole solution, here we just consider two simple cases related to the two parameters, which can be found to be analytically investigated. Moreover, we will find that these two simple cases can also express the interesting results. For example, the T–E diagram is similar as the Schwarzschild case in the first case, while it is different in the second case.3.1

The dRGT massive gravity and its Schwarzschild-like black hole solution

The action of the dRGT massive gravity in an (n+2)-dimensional spacetime with the cosmological constant Λ=−(n+1)n2ℓ2 is usually read as [8,15,16,20](8)S=116πG∫dn+2x−g[R+n(n+1)L2+m2∑i4ciUi(g,f)], where f is a fixed symmetric tensor usually called the reference metric, L is the radius of AdSn+2 spacetime; ci are constants, m stands for the graviton mass parameter, and Ui are symmetric polynomials of the eigenvalues of the (n+2)×(n+2) matrix Kνμ≡gμαfαν:(9)U1=[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]. The square root in K means (A)νμ(A)λν=Aλμ and [K]=Kμμ=gμαfαμ. After making variation of the action with respect to the metric, the equations of motion (EoM) turns out to be(10)Rμν−12Rgμν−n(n+1)2L2gμν+m2χμν=8πGTμν, where(11)χμν=−c12(U1gμν−Kμν)−c22(U2gμν−2U1Kμν+2Kμν2)−c32(U3gμν−3U2Kμν+6U1Kμν2−6Kμν3)−c42(U4gμν−4U3Kμν+12U2Kμν2−24U1Kμν3+24Kμν4). Since the background we are going to consider is (3+1) dimension, and thus a general black hole solution in the vacuum has been found in [20](12)ds2=−f(r)dt2+dr2f(r)+r2hijdxidxj,(13)f(r)=k+r2L2−m0r+c1m2r2+c2m2, where hijdxidxj is the line element for the 2-dimensional spherical, flat or hyperbolic space with k=1, 0 or −1 respectively, and m0 is related to the mass of the black hole. The reference metric now can have a special choice(14)fμν=diag {0,0,hij}. The Hawking temperature of this black hole solution can be easily found(15)TBH=(f(r))′4π|r=rh=14πrh(k+3rh2L2+c1m2rh+c2m2), where rh is the event horizon of the black hole, i.e. the largest root of equation f(rh)=0.In our case, we are just interested in the Schwarzschild-like black hole solution with zero cosmological constant, therefore, the corresponding solution in (13) is(16)f(r)=1−2Mr+c1m2r2+c2m2, while its temperature, energy and entropy are(17)TBH=14πrh(1+c1m2rh+c2m2),(18)E=M, S=πrh2. Note that, there are two parameters c1 and c2 related to the graviton mass in (16), which can deduce the expression of entropy to be complicated. This complication will be further amplified during maximizing the total entropy of the isolated system, which can deduce the difficulty to analytically investigate the condition of phase transition, i.e. condensation of the Schwarzschild-like black hole among the radiation gas in an isolated box. Therefore, we will just consider two simple cases in the following, which have been found to be analytically investigated and also contained the interesting results.3.2

The first simple case: c1=0

For the case c1=0, we can obtain the location of horizon(19)rh=2Mc2m2+1. After the assumption of the condition c2m2+1>0, a positive radius of the black hole horizon can be obtained, and the total energy and entropy of the isolated system consisting of a Schwarzschild-like black hole and radiation gas can be easily calculated(20)S=πrh2+43aVT3=4πM2(c2m2+1)2+43aVT3,(21)E=M+aVT4. After introducing two parameters: x=M/E, y=(aV/E5)1/4(c2m2+1)2/3π, we will find that maximizing entropy S is also equal to maximizing the function(22)f(x)=x2+y(1−x)3/4. Note that, this function is just same as the Schwarzschild case in (5), and the only difference is that the parameter y has been affected by the graviton mass. Therefore, the main conclusions are also same as the Schwarzschild case. But yet, just due to the same function and conclusions, an interesting result can be further found, i.e. effects from graviton mass on the equilibrium by comparison with the Schwarzschild case for the critical energy Ec.For this case c1=0 in the massive gravity, its corresponding critical energy is(23)Ec′={[(c2m2+1)23πyc]4aV}15, while the critical energy in the Schwarzschild case is(24)Ec={[13πyc]4aV}15. Obviously, if c2<0 and 0<c2m2+1<1, we will obtain Ec′<Ec, which means that the graviton mass can increase the condensation of black hole with the same isolated box with fixed volume V. On the other hand, if c2>0, then c2m2+1>1 and Ec′>Ec, which means that the graviton mass can suppress the condensation of black hole. Note that, these interesting predictions can be also seen in the T–E diagram under the fixed volume V of the isolated box, where we have drawn the diagram in the case c1=0,c2<0 in the Fig. 3

. In this figure, Tmax′ and Tcrit′ represent the two different turning points respectively compared with the Schwarzschild case, while the T–E diagram in the other case c1=0,c2>0 has not been drawn since it is very similar with this diagram.

3.3

The second simple case: c2m2+1=0

For this case with c2m2+1=0, the radius of horizon rh is also simple(25)rh=4Mc1m2, where c1>0 has been assumed to obtain a positive radius of horizon. Then, the total energy and entropy of the isolated system can be calculated(26)S=πrh2+43aVT3=4πMc1m2+43aVT3,(27)E=M+aVT4. From Eq. (27), we can obtain T=(E−MaV)1/4 which is substituted into Eq. (26). Hence, we will have(28)S=4πMc1m2+43aVT3=4πEc1m2[ME+13π(aVE)1/4c1m2(1−ME)3/4]. Similarly, after introducing two parameters: x=M/E, y=(aV/E)1/4c1m2/(3π), we will find that maximizing entropy S is equal to maximizing the function(29)f(x)=x+y(1−x)3/4. For this function f(x) in (29), the main conclusions are:(1) For y>43, the function f(x) is monotonically decreasing, thus the maximum value of f(x) is at x=0, which means that the stable equilibrium configuration is just the pure radiation and there is no black hole.(2) For 0<y<43, there is just one turning point, a maximum at specific xc=1−(34y)4, which means that the stable equilibrium configuration consists of the Schwarzschild-like black hole and black-body radiation.If the volume V of the isolated box is fixed, the above conclusions can also have the corresponding physical meaning in the following. Suppose that we add more energy E into the isolated box with a fixed volume V from E=0, the stable equilibrium configuration is just the pure radiation until the parameter y=(aV/E)1/4c1m2/3π decreases to the critical value yc=43. After adding more energy, the y will become smaller than yc=43, and then the black hole will be condensed among the radiation gas, i.e., the stable equilibrium configuration consists of the Schwarzschild-like black hole and black-body radiation. Therefore, there is also a critical energy(30)Ec=aV(c1m24π)4, and the corresponding temperature for this isolated system is(31)Tmax=c1m24π.Note that, in this case with c2m2+1=0, a very interesting and impressive result is that the above temperature is just same as the temperature of Schwarzschild-like black hole condensed in (17):(32)Th=14πrh(1+c1m2rh+c2m2)=c1m24π, which means that the temperature of this isolated system will keep same after more energy E is added into, and there is no non-equilibrium process after the condensation of the black hole in this case. Therefore, after the total energy Ec<E, the stable equilibrium configuration consists of the Schwarzschild-like black hole and black-body radiation, and the main change is just that the maximum point xc=1−(34y)4 will become bigger and closer to 1 from 0, i.e., the percentage of black hole's energy in the total energy will become bigger. A most interesting and impressive prediction is that the radius of horizon will become bigger from zero when the Schwarzschild-like black hole is condensed among the radiation gas, which is very different from the Schwarzschild case or the above case. This difference can be also clearly seen in the corresponding new T–E diagram with fixed volume V in this case, which has been drawn in the Fig. 4

Conclusion and discussion

In this paper, we mainly investigate the effects from massive graviton on the equilibrium between the black hole and radiation gas in an isolated box. Since the graviton is massive, the underlying gravity theory should be massive gravity, and hence the condensed black hole among the radiation gas will usually not be the Schwarzschild black hole. Therefore, we have taken the dRGT massive gravity into account, i.e. a ghost free massive gravity, while the Schwarzschild-like black hole solution in this dRGT massive gravity has already been found and investigated much. After using the conservation of total energy and maximizing the total entropy for this isolated system, we find that two parameters c1 and c2 are crucial to investigate the effects from massive graviton on the equilibrium analytically. For simplicity, we just consider two simple cases which can be found to have analytical investigations on the conditions of phase transition or the T–E phase diagram. Moreover, some interesting and new results have been obtained, too. For example, in the first case with c1=0, although the T–E phase diagram is similar as the Schwarzschild case, however, we can further conclude that the graviton mass can suppress or increase the condensation of black hole which is dependent on the value of c2. For the other case with c2m2+1=0, a new T–E phase diagram turns out. Moreover, an interesting and important prediction is that the condensation of black hole just increases from the zero radius of horizon in this case, which is very different from the Schwarzschild black hole case.A direct interesting question is whether there are other analytical investigations for different choices of c1 and c2. In addition, since the Birkhoff's theorem may be broken in the dRGT massive gravity, more cases can be considered, i.e. other static black hole condensed among the radiation gas [25–27]. While the information of the reference metric fμν needs be also carefully investigated, since the different reference metric may also deduce the different static black hole in the massive gravity [25–27]. Furthermore, the c1 and c2 are just proposed as two constants in the dRGT massive gravity, thus there is no further information for these two parameters in this gravity. Therefore, maybe some experiments or astronomical phenomena can be related to test the dRGT massive gravity and further extract the information of these two parameters. For example, the Laser Interferometer Gravitational-Wave Observatory (LIGO) experiment detects the gravitational waves, and the gravitational wave will not travel as the light if the graviton is massive. Another possible effect from the massive graviton comes from the accretion disk of black hole-a common astronomical phenomenon. On the other hand, the cosmological constant can be also taken into account during the investigations on the equilibrium. For the positive cosmological constant, since our universe may be the de Sitter spacetime during its inflation time or late time with acceleration, and hence the cosmological horizon can be naturally considered as the wall of the isolated box. For the negative cosmological constant, the static black hole solutions are usually asymptotical to the Anti-de Sitter spacetime, which have been found to have many interesting results, such as the well-known Hawking–Page transition [35]. Moreover, according to the AdS/CFT correspondence, the static black hole solutions asymptotical to the Anti-de Sitter spacetime can have many interesting results on the dual field theory, such as the holographic superconductor in the holographic duality in the condensed matter physics [36]. In addition, the isolated box may be considered as the cutoff surface related to the energy scale of renormalization group (RG) flow for the strongly coupling holographic fluid [37–41]. Therefore, during the investigations on the equilibrium between the black hole and radiation gas in an isolated box, it will be also an interesting issue to be further studied by taking the cosmological constant into account.

Acknowledgements

Y.P Hu thanks a lot for Profs. Hong-Sheng Zhang and Hai-Qing Zhang's valuable comments, and the Sino-Dutch scholarship program under the China Scholarship Council (CSC). This work is supported by the Fundamental Research Funds for the Central Universities under grant No. NS2015073.

References

[1]

S.W.

HawkingNature

248

1974

S.W.

HawkingCommun. Math. Phys.

1975

199

Erratum:Commun. Math. Phys.

1976

206

[2]

S.W.

HawkingPhys. Rev. Lett.

1971

1344

[3]

J.D.

BekensteinPhys. Rev. D

1973

2333

J.D.

BekensteinPhys. Rev. D

1974

3292

[4]

J.M.

Bardeen

Carter

S.W.

HawkingCommun. Math. Phys.

1973

161

[5]

G.W.

Gibbons

M.J.

PerryProc. R. Soc. Lond. A

358

1978

467

[6]

HinterbichlerRev. Mod. Phys.

2012

671

arXiv:1105.3735 [hep-th]

[7]

de RhamLiving Rev. Relativ.

2014

arXiv:1401.4173 [hep-th]

[8]

Vegh

arXiv:1301.0537 [hep-th]

[9]

Blake

Tong

VeghPhys. Rev. Lett.

112

2014

071602

arXiv:1310.3832 [hep-th]

Blake

TongPhys. Rev. D

2013

106004

arXiv:1308.4970 [hep-th]

[10]

R.A.

DavisonPhys. Rev. D

2013

086003

arXiv:1306.5792 [hep-th]

R.A.

Davison

Schalm

ZaanenPhys. Rev. B

2014

245116

arXiv:1311.2451 [hep-th]

[11]

Amoretti

Braggio

Maggiore

Magnoli

MussoJ. High Energy Phys.

1409

2014

160

arXiv:1406.4134 [hep-th]

[12]

Y.P.

H.F.

H.B.

Zeng

H.Q.

ZhangPhys. Rev. D

2016

104009

arXiv:1512.07035 [hep-th]

Y.P.

X.X.

Zeng

H.Q.

ZhangPhys. Lett. B

765

2017

120

arXiv:1611.00677 [hep-th]

[13]

Fierz

PauliProc. R. Soc. Lond. A

173

1939

211

[14]

D.G.

Boulware

DeserPhys. Rev. D

1972

3368

[15]

de Rham

GabadadzePhys. Rev. D

2010

044020

arXiv:1007.0443 [hep-th]

[16]

de Rham

Gabadadze

A.J.

TolleyPhys. Rev. Lett.

106

2011

231101

arXiv:1011.1232 [hep-th]

[17]

S.F.

Hassan

R.A.

RosenPhys. Rev. Lett.

108

2012

041101

arXiv:1106.3344 [hep-th]

S.F.

Hassan

R.A.

Rosen

Schmidt-MayJ. High Energy Phys.

1202

2012

026

arXiv:1109.3230 [hep-th]

[18]

Y.P.

ZhangPhys. Rev. D

2015

024006

arXiv:1502.00069 [hep-th]

[19]

Zhang

X.Z.

LiPhys. Rev. D

2016

124039

arXiv:1510.03204 [gr-qc]

[20]

R.G.

Cai

Y.P.

Q.Y.

Pan

Y.L.

ZhangPhys. Rev. D

2015

024032

arXiv:1409.2369 [hep-th]

[21]

S.H.

Hendi

Panahiyan

Eslam PanahJ. High Energy Phys.

1601

2016

129

arXiv:1507.06563 [hep-th]

S.H.

Hendi

Eslam Panah

PanahiyanJ. High Energy Phys.

1605

2016

029

arXiv:1604.00370 [hep-th]

S.H.

Hendi

Riazi

Panahiyan

arXiv:1610.01505 [hep-th]

[22]

S.H.

Hendi

B.E.

Panah

PanahiyanJ. High Energy Phys.

1511

2015

157

arXiv:1508.01311 [hep-th]

S.H.

Hendi

B.E.

Panah

Panahiyan

arXiv:1510.00108 [hep-th]

[23]

Y.P.

X.X.

Zeng

H.Q.

ZhangPhys. Lett. B

765

2017

120

arXiv:1611.00677 [hep-th]

[24]

Y.P.

X.M.

ZhangPhys. Rev. D

2017

084002

arXiv:1611.09042 [gr-qc]

[25]

M.S.

VolkovClass. Quantum Gravity

2013

184009

arXiv:1304.0238 [hep-th]

[26]

Tasinato

Koyama

NizClass. Quantum Gravity

2013

184002

arXiv:1304.0601 [hep-th]

[27]

Babichev

BritoClass. Quantum Gravity

2015

154001

arXiv:1503.07529 [gr-qc]

[28]

L.M.

Cao

Y.P.

HuPhys. Rev. D

2015

124033

arXiv:1506.03578 [gr-qc]

[29]

L.M.

Cao

Peng

arXiv:1509.08738 [hep-th]

L.M.

Cao

Peng

Y.L.

Zhang

arXiv:1511.04967 [hep-th]

[30]

Blake

TongPhys. Rev. D

2013

106004

arXiv:1308.4970 [hep-th]

[31]

R.A.

Davison

Schalm

ZaanenPhys. Rev. B

2014

245116

arXiv:1311.2451 [hep-th]

[32]

Adams

D.A.

Roberts

SaremiPhys. Rev. D

2015

046003

arXiv:1408.6560 [hep-th]

[33]

T.Q.

DoPhys. Rev. D

2016

104003

arXiv:1602.05672 [gr-qc]

T.Q.

DoPhys. Rev. D

2016

044022

arXiv:1604.07568 [gr-qc]

[34]

ArrautEurophys. Lett.

109

2015

10002

arXiv:1405.1181 [physics.gen-ph]

ArrautInt. J. Mod. Phys. D

2015

1550022

arXiv:1311.0732 [gr-qc]

Arraut

ChelabiEurophys. Lett.

115

2016

31001

arXiv:1603.06341 [hep-th]

[35]

S.W.

Hawking

D.N.

PageCommun. Math. Phys.

1983

577

[36]

Zaanen

Y.W.

Sun

Liu

SchalmHolographic Duality in Condensed Matter Physics2015

Cubrovic

Zaanen

SchalmScience

325

2009

439

arXiv:0904.1993 [hep-th]

[37]

Bredberg

Keeler

Lysov

StromingerJ. High Energy Phys.

1103

2011

141

10.1007/JHEP03(2011)141

arXiv:1006.1902 [hep-th]

Bredberg

Keeler

Lysov

StromingerJ. High Energy Phys.

1207

2012

146

10.1007/JHEP07(2012)146

arXiv:1101.2451 [hep-th]

[38]

R.-G.

Cai

Y.-L.

ZhangJ. High Energy Phys.

1107

2011

027

arXiv:1104.3281 [hep-th]

R.-G.

Cai

Z.-Y.

Nie

Y.-L.

ZhangNucl. Phys. B

864

2012

260

arXiv:1202.4091 [hep-th]

[39]

Niu

Tian

X.-N.

LingPhys. Lett. B

711

2012

411

arXiv:1107.1430 [hep-th]

T.-Z.

Huang

Ling

W.-J.

Pan

Tian

X.-N.

WuJ. High Energy Phys.

1110

2011

079

arXiv:1107.1464 [gr-qc]

C.-Y.

Zhang

Ling

Niu

Tian

X.-N.

WuPhys. Rev. D

2012

084043

[40]

Bai

Y.P.

B.H.

Lee

Y.L.

ZhangJ. High Energy Phys.

1211

2012

054

arXiv:1207.5309 [hep-th]

[41]

Y.-P.

Tian

X.-N.

WuPhys. Lett. B

732

2014

298

arXiv:1311.3891 [hep-th]

Y.-P.

Tian

X.-N.

H.-F.

H.-S.

Zhang

arXiv:1408.2173 [hep-th]