Hawking radiation, local temperatures, and nonequilibrium thermodynamics of the black holes with non-killing horizon
HHawking radiation, local temperatures, and nonequilibriumthermodynamics of the black holes with non-killing horizon
Ran Li , ∗ and Jin Wang , † School of Physics,Henan Normal University,Xinxiang 453007, China Department of Chemistry,SUNY at Stony Brook,Stony Brook, New York 11794, USA Department of Physics and Astronomy,SUNY at Stony Brook,Stony Brook, New York 11794, USA a r X i v : . [ g r- q c ] J a n bstract Recently, a class of stationary black hole solutions with non-killing horizon in the asymptotic AdSbulk space (i.e. non-equilibrium black funnel) was constructed to describe the far from equilibriumheat transport and particle transport from the boundary black holes via AdS/CFT correspondence.It is generally believed that the temperature of a black hole with non-killing horizon can not beproperly defined by the conventional methods used in the equilibrium black holes with killinghorizon. In this study, we calculate the spectrum of Hawking radiation of the non-equilibriumblack funnel using the Damour-Ruffini method. Our results indicate that the spectrum and thetemperatures as well as the chemical potentials of the non-equilibrium black funnel do dependon one of the spatial coordinates. This is different from the equilibrium black holes with killinghorizon, where the temperatures are uniform. Therefore, the black hole with non-killing horizon canbe overall in non-equilibrium steady state while the Hawking temperature of the black funnel can beviewed as the local temperature and the corresponding Hawking radiation can be regarded as beingin the local equilibrium with the horizon of the black funnel. By AdS/CFT, we discuss some possibleimplications of our results of local Hawking temperature for the non-equilibrium thermodynamicsof dual conformal field theory. We further discuss the nonequilibrium thermodynamics of the blackfunnel, where the first law can be formulated as the entropy production rate being equal to thesum of the changes of the entropies from the system (black funnel) and environments while thesecond law is given by the entropy production being larger than or equal to zero. We found thetime arrow emerged from the nonequilibrium black hole heat and particle transport dissipation.We also discuss how the nonequilibrium dissipation may influence the evaporation process of theblack funnel. ∗ [email protected] † Corresponding author: [email protected] . INTRODUCTION The uniqueness theorem [1–3] states that all regular stationary, asymptotically flat non-degenerated black hole solutions of the Einstein-Maxwell equations of gravitation and elec-tromagnetism in four dimensional general relativity can be uniquely characterized by theirmass, electric charge, and angular momentum and have compact horizon topology of S .The most general solution to the vacuum Einstein-Maxwell equations is characterized bythe 3-parameter Kerr-Newman family. It seems that all the information about the matterthat formed a black hole is inaccessible to the external observer. Wheeler used the phrase”black holes have no hair” to express the idea of the uniqueness theorem [4].In recent years, in the context of AdS/CFT correspondence [5–7], many hairy blackholes that evade the assumptions of uniqueness theorem have been constructed and used todescribe the strongly coupled properties of condensed matter systems [8, 9]. For example, inholographic superconductor models [10], scalar, vector, and tensor fields can be added to theEinstein-Hilbert action with negative cosmological constant to construct the bulk spacetimedual to the boundary superconducting system. These examples indicate that the uniquenesstheorem can be evaded in AdS asymptotics.Another way to evade the assumptions of uniqueness theorem is to construct the blackhole solutions with the bulk horizons that extend to the asymptotic regions where theboundary conditions can be imposed. This programm first initiated in [11] for the aim ofgaining some insight into the strong coupling properties of quantum field in curved spacetime,and continued in the following studies [12–15, 17–20, 36]. (See [21] for a review.) Accordingto the the connectedness of the bulk horizon(s), the solutions can roughly be divided intotwo classes: black funnels and black droplets.Black funnel solution has a single connected bulk horizon that extends to meet the con-formal boundary. Then the induced conformal boundary has smooth horizons as well. Inparticular, in [17], the black funnel solution in vacuum Einstein-Hilbert gravity with negativecosmological constant was numerically constructed, where the boundary spacetime containsa pair of black holes connected through the bulk by a tubular bulk horizon. The boundaryconditions that the boundary black holes have two different temperatures can be imposed.These are examples of the stationary bulk black hole with a non-Killing horizon, whichcan be used to describe the far from equilibrium transport of heat on the boundary from3he hotter boundary black hole to the cooler boundary black hole by using the holographicdictionary.The well known black hole thermodynamics told us that the surface gravity κ is uniformeverywhere on the killing horizon. This indicates that killing horizon can be viewed as beingin thermal equilibrium and the temperature is well-defined by Hawking’s result T = κ π . Thesurface gravity and the temperature of the black hole with the non-killing horizon can not beproperly defined by the conventional methods used in the black hole with killing horizon. Forexample, Visser has calculated the surface gravity of the black holes with non-killing horizon,which shows the different definitions of surface gravity can lead to different results [22]. Fromthe thermodynamics perspective, black holes with killing horizon can be described by theequilibrium thermodynamics. The temperature of the equilibrium black hole is related to thesurface gravity of the horizon while the entropy is found to be proportional to the area of thehorizon. The thermodynamics of equilibrium black hole can be formulated as the internalenergy change of the black hole being equal to the heat generated (product of temperatureand entropy change). For non-killing horizon, the temperature of the black hole can notbe treated as constant. The equilibrium description of the black hole is no longer valid inthis situation. An inhomogenous temperature is expected to lead to heat flow and generatethermodynamic dissipation. Therefore, the thermodynamics of this kind of non-killing blackholes is important to describe the overall behavior and should have nonequilibrium nature,but has not been formulated yet. Furthermore, the nonequilibrium evaporation dynamicsof the non-killing black hole is also not discussed so far.In this study, we will address how to calculate the Hawking temperature of non-killinghorizon of stationary black hole solution. We take the previous mentioned black funnelsolution as an example. For the black funnel solution, the bulk horizon is non-compact,and can be extended to conformal boundary. From the thermodynamic point of view, theheat, which is originated from the boundary black holes with different temperatures, canbe transported on the bulk horizon. Then the horizon of the black funnel no longer hasa constant temperature. This means that the non-killing horizon should be taken as anon-equilibrium thermodynamic system.There are several methods to derive Hawking radiation in literatures, for example, quan-tum field in curved space used by Hawking [23], Damour-Ruffini method [24], anomalymethod [25], and quantum tunneling method [26], et al. The Damour-Ruffini method [24]4as been generalized by Zhao et al to study the Hawking radiation from the dynamicalblack holes and accelerating black holes [27]. We find that the Damour-Ruffini method isa convenient method to calculate the radiation spectrum of the non-killing horizon. Byproperly investigating the behavior of a scalar field near the horizon, we are able to derivethe local temperature and the corresponding radiation spectrum of the bulk horizon of theblack funnel. Our results indicate that the spectrum and the temperatures as well as thechemical potentials of non-equilibrium black funnel do depend on one of the spatial coordi-nates, which is different from the conventional case of the constant temperature equilibriumblack hole with killing horizon. Therefore, the black hole with non-killing horizon can beoverall in non-equilibrium steady state while the Hawking temperature of black funnel canbe viewed as the local temperature and the corresponding Hawking radiation can be re-garded as being in local equilibrium with the horizon of the black funnel. By AdS/CFT,we discuss some possible implications of our results of the local Hawking temperature forthe non-equilibrium thermodynamics of dual conformal field theory. We further discuss thenonequilibrium thermodynamics of the steady state black funnel. We suggest the first andsecond laws of the thermodynamics of the noequilibrium steady state black funnel. We dis-cuss the time arrow emerged from the nonequilibrium black hole heat and particle transportdissipation. We also consider the effect of the nonequilibrium thermal transport in additionto Hawking radiation on the evaporation process of black funnel. II. HAWKING RADIATION OF THE NONEQUILIBRIUM BLACK HOLE-BLACK FUNNEL
According to AdS/CFT correspondence [5–7], the conformal filed theory on the boundaryis equivalent to the AdS spacetime in the bulk. A conformal field theory with thermaltemperature is shown to be equivalent to an AdS black hole (black brane) in the bulk. Allthese are based on the equilibrium description. One can naturally ask the equation whatthe corresponding bulk spacetme is if the boundary conformal fluid is under nonequilibriumcondition (for example, driven by a temperature gradient). The earlier exploration focusedon initial temperature contrast setup and the subsequent evolution, where the bulk spacetimebecomes a boosted black brane [28]. More recent work leads to the more fixed nonequilibriumsetup by imposing two black holes on the boundary edges. As a result, the bulk spacetime5as two outcomes: steady state black hole with non-killing horizon named black funneland equilibrium black hole with killing horizon named black droplet. It is the purpose ofthis work to quantify the Hawking radiation and local temperature of the black hole withnon-killing horizon. We will also formulate the non-equilibrium thermodynamics and theevaporation dynamics of the non-killing black holes.
A. Nonequilibrium Black Hole: Black funnel
In this section, we want to calculate the spectrum of Hawking radiation from the bulkhorizon of the black funnel solution based on the Damour-Ruffini method [24]. For this aim,let us briefly introduce the construction of black funnel solution.In order to find a solution that describes the heat transport on the AdS boundary, onenaturally consider that there are two heat reservoirs on the AdS boundary [11]. The eventhorizon, which has a temperature from the thermodynamics of black hole, can play the roleof the heat reservoir. A sketch of the black funnel solution is presented in Fig.1. The bulkhorizon H of the black funnel solution meets the AdS conformal boundary ∂ at the blackpoints which are the induced horizons on the AdS boundary as shown on the left panel ofFig.1. In other words, there are two horizons (represented by two black points) on the AdSconformal boundary and the two horizons on the AdS boundary are connected through thehorizon H in the bulk.As shown on the left panel, the computational domain is not regular, which is not con-venient for the numerical calculation. However, as shown on the right panel of Fig.1, thepoints where the bulk horizon H meet the AdS conformal boundary ∂ can be blown up intotwo different hyperbolic black holes at the left and right boundaries [17]. In fact, this isthe case for the region where the bulk horizon meets the boundary horizon. In this way,the computational domain was transformed into a rectangle. Therefore, the black funnel isconstructed as a solution to the Einstein equations that has one bulk horizon H , approachestwo different hyperbolic black holes on either ends of this horizon, and has a boundary thatis conformal to the background metric of the field theory. In addition, the left or rightasymptotic hyperbolic black hole is fixed by the choice of the bulk horizon temperature andthe boundary horizon temperature. The two hyperbolic black holes with different tempera-tures can be regarded as the thermal reservoirs. Such a black funnel solution describes the6 ∂ ∂ℋ H H ω y FIG. 1: A sketch of black funnel solution. ∂ represents the AdS conformal boundary, H the bulkhorizon, and H the hyperbolic black hole. ω is the space coordinate along the bulk horizon and y is the AdS radial coordinate. heat flow between two reservoirs at different temperatures. In the following, we will showthat the Hawking temperature and the chemical potential of the bulk horizon vary with thecoordinate ω , or vary along the bulk horizon. This implies that there are heat flow as wellas particle flow driven by the difference in temperatures and chemical potentials along thebulk horizon.The metric of black funnel solution is given by [17] ds = l (1 − ω ) (1 − y ) (cid:40) − M ( y ) G ( ω ) (1 − ω ) y A (cid:20) l − dt + Q ( ω ) χ y dy (cid:21) + 4(1 − ω ) Bdy M ( y ) + y (cid:34) S (2 − ω ) (cid:18) dω + l − χ dt + F dyy (cid:19) + S dφ (cid:35)(cid:41) , (1)where A, B, F, S , S , χ and χ are all functions of y and ω . In addition, other metricfunctions are defined as G ( ω ) = 1 + β ω (5 − ω ) ,M ( y ) = 2 − y − (1 − y ) (1 − y ) y ,Q ( ω ) = 1 − M (0) G ( ω ) . (2)The parameter l is the length scale of AdS space. y is a parameter that controls thevalidity of the fluid approximation. Here y ranges over [0 ,
1] and ω ranges over [ − , y = 0 being the bulk horizon and y = 1 the conformal boundary. The black funnel solution7s a solution that has a horizon, approaches two different hyperbolic black holes on eitherends of this horizon, and has a boundary that is conformal to the background metric ofthe field theory. With the proper boundary conditions, the induced boundary metric hastwo horizons (two black holes). The above set of considerations is graphed in Fig. 1. Theparameter β controls the temperature difference between the two boundary black holes. ByAdS/CFT, this black funnel solution is used to describe the non-equilibrium field theory onthe boundary.The above metric of the black funnel is considered to solve the vacuum Einstein equationswith negative cosmological constant in the form of R µν + 3 l g µν = 0 . (3)However, boundary conditions should be imposed when solving the Einstein equations. Forour purpose, we only give the boundary conditions that imposed on the bulk horizon sincethe Hawking radiation is relevant to the near horizon properties of black hole spacetime.By demanding the regularity of the black funnel metric in ingoing Eddington-Finkelsteincoordinates (which cover the future horizon), the boundary conditions at the bulk horizon y = 0 are taken as F (0 , ω ) = χ (0 , ω ) , B (0 , ω ) = 14 M (0) G ( ω ) A (0 , ω ) [1 − Q ( ω ) χ (0 , ω )] ,∂ y A (0 , ω ) = ∂ y S (0 , ω ) = ∂ y S (0 , ω ) = ∂ y χ (0 , ω ) = ∂ y χ (0 , ω ) = 0 . (4)With the metric ansatz and boundary conditions of the black funnel, the numerical solutioncan be found by using the Einstein-De Turk method [9]. It should be noted that theseboundary conditions will be supplemented to simplify the equation of motion of a scalarfield near the horizon.Let g denote the determinant of the metric of the black funnel. It can be calculated that √− g = 4 l y y (1 − ω ) (1 − y ) (cid:115) ABS S G ( ω )(2 − ω ) . (5)It is obvious that the determinant of the metric is degenerate at the bulk horizon y = 0. Inthe following, we set l = 1 without the loss of generality.8 . Local Hawking temperature and local chemical potential of the nonequilibriumblack hole-black funnel To derive the spectrum of Hawking radiation, for simplicity, we consider the dynamics ofa massive scalar field which is govern by the Klein-Gorden equation ∇ µ ∇ µ ψ = 1 √− g ∂ µ (cid:0) √− gg µν ∂ ν ψ (cid:1) = m ψ , (6)where m denotes the mass of scalar field. The black funnel metric g µν are independent of thecoordinates t and φ , which means that ∂ t and ∂ φ are killing vector respectively. Accordingto these symmetries, the Klein-Gorden equation can be rewritten in the following form g tt ∂ t ψ + 2 g ty ∂ t ∂ y ψ + 2 g tω ∂ t ∂ ω ψ + 1 √− g ∂ y (cid:2) √− gg ty (cid:3) ∂ t ψ + 1 √− g ∂ y (cid:2) √− gg yy ∂ y ψ (cid:3) + 1 √− g ∂ y (cid:2) √− gg yω (cid:3) ∂ ω ψ + 2 g yω ∂ y ∂ ω ψ + 1 √− g ∂ ω (cid:2) √− gg tω (cid:3) ∂ t ψ + 1 √− g ∂ ω (cid:2) √− gg yω (cid:3) ∂ y ψ + 1 √− g ∂ ω (cid:2) √− gg ωω (cid:3) ∂ ω ψ + g ωω ∂ ω ψ + g φφ ∂ φ ψ − m ψ = 0 . (7)We can introduce the generalized tortoise coordinate in the form of [27] y ∗ = ln | y | = ln y, y > − y ) , y < y > y < y , and taking the limit y → ω → ω , where ω is an arbitrary spatial location in the ω direction, the resulting equationin generalized tortoise coordinate can be written as C tt ∂ t ψ + C t ∂ t ψ + C ty ∂ r ∂ y ∗ ψ + C tω ∂ r ∂ ω ψ + C yy ∂ y ∗ ∂ y ∗ ψ + C y ∂ y ∗ ψ + C yω ∂ y ∗ ∂ ω ψ = 0 , (9)9here the coefficients are given by C tt = 14 M (0) (cid:20) M (0) Q ( ω ) χ (0 , ω ) B (0 , ω ) − A (0 , ω ) G ( ω ) (cid:21) ,C t = χ (0 , ω ) (cid:16) ∂ y ln S (0 , ω ) + ∂ y ln S (0 , ω ) + ω (9 − ω )2 − ω + ω (cid:17) + 2 ∂ y χ (0 , ω )2 A (0 , ω ) G ( ω ) M (0) (1 − Q ( ω ) χ (0 , ω )) C ty = − M (0) Q ( ω ) χ (0 , ω )2 B (0 , ω ) ,C tω = M (0) χ (0 , ω ) (1 − Q ( ω ) χ (0 , ω ))2 B (0 , ω ) ,C yy = M (0)4 B (0 , ω ) ,C y = − C t ,C yω = − C tω . (10)The limit y → ω → ω means that we only concentrate on the local behavior ofthe scalar field near the horizon and at the location of ω = ω . In this limit, the resultingradiation spectrum and Hawking temperature will depend on the spatial location ω , whichis very different from the results of the equilibrium black hole with killing horizon. This isan example of local temperature. We will show that Damour-Ruffini method is robust inderiving the Hawking radiation from the non-killing horizon of the black funnel.By performing the separation of variables, we take the ansatz of the scalar field as ψ = e − iEt R ( y ∗ ) e ik ω ω + ik φ φ . (11)Substituting this expression into the reduced scalar equation, we can obtain the followingradial equation after some tedious algebra L ∂ y ∗ R + L ∂ y ∗ R + L R = 0 , (12)with L = M (0)4 B (0 , ω ) ,L = iEM (0) Q ( ω ) χ (0 , ω )2 B (0 , ω ) − ik ω M (0) χ (0 , ω ) (1 − Q ( ω ) χ (0 , ω ))2 B (0 , ω ) − C t ,L = E M (0)(1 − Q ( ω ) χ (0 , ω ))4 B (0 , ω ) + Ek ω M (0) χ (0 , ω ) (1 − Q ( ω ) χ (0 , ω ))2 B (0 , ω ) − iEC t . (13)10t is obvious that this equation has two independent solutions, which are given explicitlyas R in = e − iEy ∗ , (14) R out = e ( iE (1 − Q ( ω ) χ (0 ,ω )) − ik ω χ (0 ,ω )(1 − Q ( ω ) χ (0 ,ω )) − B (0 ,ω ) C t /M (0)) y ∗ . (15)Then, the waves corresponding, respectively, to in-going wave and out-going wave on thehorizon are ψ in = e − iEv e ik ω ω + ik φ φ , (16) ψ out = e − iEv e i ( E − k ω χ (0 ,ω ))(1 − Q ( ω ) χ (0 ,ω )) y ∗ e − B (0 ,ω ) C t /M (0) y ∗ e ik ω ω + ik φ φ , (17)with v = t + y ∗ being defined as the ingoing Eddington-Finkelstein coordinate.The ingoing wave is analytic while the outgoing wave is not at the bulk horizon. There isa logarithmic singularity for the out-going wave solution at the bulk horizon. Damour andRuffini suggested that the outgoing wave can be analytically extended through the lowerhalf complex y plane into the inside of the horizon [24]. In our present case, the analyticalcontinuation to extend the outgoing wave outside the bulk horizon to the outgoing waveinside the horizon can be realized by making the replacement of y (cid:55)→ | y | e − iπ = ( − y ) e − iπ . (18)Then, the outgoing wave inside the bulk horizon by analytical continuation can be given by ψ out ( y > (cid:55)→ ψ out ( y <
0) = e − iEv e i ( E − k ω χ (0 ,ω ))(1 − Q ( ω ) χ (0 ,ω )) y ∗ e − B ((0 ,ω )) C t /M (0) y ∗ e ik ω ω + ik φ φ e π ( E − k ω χ (0 ,ω ))(1 − Q ( ω ) χ (0 ,ω )) e iπB (0 ,ω ) C t /M (0) , (19)The relative probability of the scattered outgoing wave at the bulk horizon is given by P = (cid:12)(cid:12)(cid:12)(cid:12) ψ out ( y > ψ out ( y < (cid:12)(cid:12)(cid:12)(cid:12) = e − π ( E − k ω χ (0 ,ω ))(1 − Q ( ω ) χ (0 ,ω )) . (20)According to the derivation of Sannan [29], we obtain the distribution function of the out-going energy flux (i.e. the spectrum of Hawking radiation from the bulk horizon) N E = 1 e ( E − µ ) /T − , (21)where the Hawking temperature of the black funnel is given by T = 14 π (1 − Q ( ω ) χ (0 , ω )) , (22)11nd the chemical potential is given by µ = k ω χ (0 , ω ) . (23)As seen, both the local temperature T and chemical potential µ are not constants alongthe bulk horizon. This indicates black funnel with non-killing horizon can not be in ther-modynamic equilibrium state from the nonuniform temperature and chemical potential. Infact, the nonuniform temperature and chemical potential will generate heat flow and par-ticle flow, respectively. This will leads to nonequilibrium thermodynamic dissipation. Thecorresponding nonequilibrium thermodynamics and evaporation dynamics of black funnelwill be discussed in the next section. C. Asymptotic behavior of the local Hawking temperature along the bulk horizon
Now, we consider the asymptotic behavior of the local Hawking temperature in the ω direction and compare with the temperatures of the asymptotic black holes. By taking thelimit of ω → ±
1, the Hawking temperature takes the forms of T | ω →± = M (0) G ( ± π . (24)In the following, we can check this result by deriving the temperatures from the metric ofthe black funnel near the left and right boundaries. The left and right boundaries lie at ω = ±
1. There, the boundary conditions are imposed as A ( ± , y ) = B ( ± , y ) = S ( ± , y ) = S ( ± , y ) = χ ( ± , y ) = 1 ,F ( ± , y ) = χ ( ± , y ) = 0 . (25)The metric in Eq.(1) is reduced to ds | ω →± = 1(1 − y ) (cid:40) − M ( y ) G ( ± y (cid:20) dt + Q ( ± dyy (cid:21) + 4 dy M ( y )+ y (1 ∓ ω ) (cid:18) dω + dφ (cid:19)(cid:27) . (26)By coordinate transformation, it can be shown that this metric is asymptotic to the hyper-bolic black hole. The temperature of this metric can be found by using the conventional12ethod for the black hole with killing horizon (for example, using the definition of surfacegravity and the relation between temperature and surface gravity T = κ π ) T ± = M (0) G ( ± π , (27)which is exactly the limit of the Hawking temperature of the black funnel. Therefore weshow that the limits of the local Hawking temperature of bulk horizon are consistent withthe Hawking temperatures of black holes on the left and right boundaries. D. Implications of the nonequilibrium thermodynamics for the boundary CFT
At last, let us discuss some implications of the local temperature of black funnel for thenonequilibrium thermodynamics of dual conformal field theory. The nonequilibrium blackfunnel solution was proposed to describe the far from equilibrium transport of heat on theboundary field theory [17]. If the local Hawking temperature of the bulk horizon is identifiedwith the local temperature of boundary field theory, the entropy production of the boundaryfield theory can be computed from the point view of AdS/CFT correspondence. In [17], dueto the absence of the definition of the local temperature of the boundary field theory, theentropy production of the boundary conformal field was not calculated. However, the entropycurrent and its divergence can be computed by using the hydrodynamic approximation.With the local Hawking temperature given by Eq.(22), it is possible to calculate the entropycurrent (cid:126)J and the local entropy production Θ for the conformal fluid by identifying the localtemperature of the conformal fluid as the local temperature of the black funnel horizon.In non-equilibrium thermodynamics, the entropy current and entropy production rate aregiven by (cid:126)J S = (cid:126)J q T ,
Θ = (cid:126)J q · ∇ (cid:18) T (cid:19) . (28)where the energy current (cid:126)J q can be determined by the metric of AdS funnel solution byemploying the holographic renormalization technique. The obtained numerical results can becompared with the analytical results under the hydrodynamic approximation. This questiondepending on the numerics deserves further study in the future.13 II. NONEQUILIBRIUM THERMODYNAMICS OF BLACK HOLE: BLACKFUNNEL
The conventional black holes with the killing horizons can be described by the equilib-rium thermodynamics for its macroscopic emergent state. As discussed, the black funnelsolution, which has a temperature distribution on its bulk horizon, should be treated asa non-equilibrium thermodynamic system. Therefore, the non-equilibrium thermodynam-ics is expected to describe the black funnel solution. It should be noted that the blackfunnel solution is time independent. This implies that black funnel is in a steady state.Due to the temperature variation, black funnel is in nonequilibrium steady state. Then thenonequilibrium thermodynamics of the black funnel should be dramatically different fromthe equilibrium thermodynamics of black hole solution with the killing horizon. In ther-modynamics of the non-equilibrium steady state, the energy current, the entropy current,and the local entropy production are the characteristic quantities that describe the physicalnatures of the system. These quantities turn out to be closely related with the first and thesecond laws of non-equilibrium thermodynamics. The energy current, entropy current, aswell as the local entropy production of the bulk horizon of the black funnel solution can alsobe quantified.
A. Nonequilibrium thermodynamics
To proceed with the nonequilibrium thermodynamics description of the black holes withnon-killing horizon (black funnels), let us divide the black hole into smaller parts witheach part still in equilibrium characterized by the local temperature and chemical potentialalthough the overall system is in non-equilibrium. If the relaxation time of the parts aremuch faster than that of the whole system, the parts can be regarded as in local equilibrium.The local equilibrium can also be understand as follows. Between the microscopic andmacroscopic emergent black hole, there is a mesoscopic scale with many microscopic degreesof freedom but not macroscopic degrees of freedom. So mesoscopic degrees of freedom ismuch larger than the microscopic degrees of freedom but much smaller than the macroscopicdegrees of freedom. The local equilibrium is achieved in the mescopic scale. The wholemacroscopic emergent scale of black hole is in non-equilibrium state with many local patches14t their own local temperatures and chemical potentials on the horizon. The local entropytherefore refers to the mesoscopic degrees of freedom of the local patch.In this situation, the temperature, chemical potential, internal energy, entropy, and parti-cle number can have the definitive meanings. We can assume that these local thermodynamicquantities still satisfy the thermodynamic first law. Dividing by volume, we can reach [30]
T ds = du − Σ µ i dn i , (29)where s, u, n i are the entropy density, local energy density, and particle density, and T and u are local temperature and chemical potential respectively. Here, we should explain themeaning of the volume. For Schwarzchild black hole, we have dV = rdA where dV is thesmall volume element while r is the horizon radius and A is the area of the horizon. For theblack funnel, dV = rdA , everything has the same meaning except that r now is varying andnot a constant any more as in Schwarzchild black hole.In a fixed volume element, the change of the particle number density satisfies a localconservation equation of matter ∂n∂t = −∇ · J n . (30)The physical meaning is clear: the change of the particle number density is equal to the netparticle flow or flux density in or out of the volume. Similarly, the change of the internalenergy density also satisfies the local conservation equation of energy ∂u∂t = −∇ · J u . (31)The physical meaning is also clear: the change of the internal energy density is equal to thenet energy flux density in or out of the volume.From the first law of the thermodynamics of the local volume, T ds = du − Σ µ i dn i , weknow that when particle number density is increased as dn , the internal energy change is µdn . Therefore, when the particle flow is present, the internal energy density flux can bequantified as J u = J q + µJ n . (32)Thus, the internal energy flux density is the sum of the heat flux density and particle fluxdensity. By substituting the above formula into the local conservation law of the internal15nergy density, we reach ∂u∂t = −∇ · J q − ∇ · ( µJ n ) . (33)On the other hand, from the first law of equilibrium thermodynamics of the local volume T ds = du − µdn , we obtain ∂s∂t = 1 T ∂u∂t − µ ∂n∂t . (34)By substituting the local conservation law of the particle number density into the aboveexpression, we get the change of the system entropy [30] ∂s∂t = − T ∇ · J q − T ∇ · ( µJ n ) + µT ∇ · J n = −∇ · (cid:18) J q T (cid:19) + J q · ∇ T − J n T · ∇ µ . (35)The physical meaning is also clear: ∂s∂t represents the change of the entropy of the system; −∇ · (cid:16) J q T (cid:17) represents the change of the local entropy density due to the heat flow; J q · ∇ T represents the local entropy density production rate due to the temperature gradient whichleads to the heat transport; − J n T · ∇ µ represents the local entropy production rate due tothe chemical potential gradient which leads to the particle transport. It is well knownfromthe classical general relativity that only the photons can propagate along the event horizon.Here, the particles that transport along the horizon are the bosons with both temperatureand chemical potentials. They are not like photons completely but rather act like Bose-Einstein gas (temperature and chemical potential). These particles will be moving on thehorizon surface under the temperature and chemical potential gradient creating energy orheat flows driven by temperature gradient as well as the particle flows driven by the chemicalpotential gradient.We can define the entropy flux density and the total entropy production rate as J s = J q T , epr = J q · ∇ T − J n T · ∇ µ . (36)We can also define the driving force generated by the nonuniform temperature and chemicalpotential as X q = ∇ T , X n = − T ∇ µ , (37)16here X q is called the driving force of the heat flow while X n ia called the driving force ofthe particle flow. Thus the local entropy density production rate can be expressed as thesum of the product of the driving force and the flux of the two kinds, one for the heat andthe other for the particle.The equation for the entropy change in time as: ∂s∂t = −∇ · (cid:16) J q T (cid:17) + J q · ∇ T − J n T · ∇ µ = −∇ · (cid:16) J q T (cid:17) + epr has a clear physical meaning: ∂s∂t represents the change of the entropy ofthe system; −∇ · (cid:16) J q T (cid:17) represents the change of the local entropy density due to the heatflow or from the environment; epr = J q · ∇ T − J n T · ∇ µ represents the total local entropyproduction generated from the energy and the particle flows as well as the temperature andthe chemical potential difference (system plus environment). In fact, this formulates thefirst law of the nonequilibrium thermodynamics. It can be shown that epr ≥
0. That is thetotal local entropy density change is always greater or equal to zero. This formulates thesecond law of the nonequilibrium thermodynamics.For a system, although the total local entropy density of the system and the environmenttogether always prefer to increase leading to disorder of the whole system and environment,the local entropy density of the system can be decreased in the process. This is becausethat the heat or entropy flow to the system from the environment can be negative or thenet positive heat or entropy flow from the system to the environment. The possibility of thedecrease of the system entropy can create order. In fact, for living system, the entropy ofthe system is relatively low giving rise to order while the total entropy change still increasesleading to disorder.
B. Nonequilibrium black hole thermodynamics
As discussed, if the black hole can be described by the normal thermodynamics forits macroscopic emergent state, then from the studies of the time dependent evaporationprocess and dynamical Vaidya black hole accretion process, it is natural to suggest that thenonequilibrium thermodynamics can also be used to describe the nonequilibrium process ofthe black hole.Notice that the nonequilibrium open system is very different from the equilibrium closedsystem where the entropy of the system always increases. For black hole with the non-killing horizon, such as black funnel in our study, we can write down the first law of its17lobal nonequilibrium thermodynamics as ∂S∂t = − (cid:90) J q T · dA + (cid:90) J q · ∇ T dV − (cid:90) J n T · ∇ µdV , (38)where S is the total entropy of the black funnel, dV is the volume element, and the integralis over the whole volume. J q is the heat energy flow density for the masslesss particles under chemical potential,and J n is the particle flux under the chemical potential gradient. Here, we will presentthe formulas of particle flux of quantum gases driven by chemical potential gradient. Thenull nature of the event horizon of black holes restricts that only the massless particles canpropagate along the horizon. In the present paper, we are particularly interested in theenergy and particle fluxes along the horizon of black hole. Therefore, we only consider thecase of massless particles. If including the fermionic field for black funnel, we expect Fermi-Dirac statistics under chemical potential for the Hawking radiation. In the following, wewill consider the massless Bosons as well as the massless Fermions.For the massless particles, the energy can be expressed as (cid:15) = (cid:126) ω , where ω is the frequency.The number of states available to a single particle with frequency between ω and ω + dω is g ( ω ) dω = V ω dωπ c . (39)According to Bose-Einstein/Fermi-Dirac distribution a l = g l e ( (cid:15) l − µ ) /kT ± , (40)where ± is for fermion/Boson and g l is the degeneracy of the energy level (cid:15) l , we can easilycompute the number density of the massless particles in the gas n ( µ ) = NV = 1 π c (cid:90) + ∞ ω dωe ( (cid:126) ω − µ ) /kT ±
1= 1 π (cid:18) kT (cid:126) c (cid:19) (cid:90) + ∞ x dxe − µ/kT e x ± , (41)and the energy density for the massless particles u ( µ ) = EV = 1 π c (cid:90) + ∞ ω dωe ( (cid:126) ω − µ ) /kT ±
1= 1 π c (cid:18) kT (cid:126) (cid:19) (cid:90) + ∞ x dxe − µ/kT e x ± , (42)18 dA FIG. 2: Energy and particle flux of Bose particles.
On the other hand, in radiation theory, there are formulas for energy flux and particleflux. As shown in Fig.2, the energy flow per unit time in certain direction through area dA is equal to cu d Ω4 π cos θdA , where d Ω represents the solid angle and θ is the angle between thepropagation and area dA . Integrating over all propagating directions, one can obtain thetotal energy flow as J q dA = cu π (cid:90) cos θd Ω = 14 cudA , (43)Therefore, the well known formula for the energy flux of the radiation passing through aunit area in a unit time [37] is given by J q = 14 cu , (44)where c is the speed of light and u is the energy density.Correspondingly, the particles flow per unit time in certain direction through dA is equalto cn d Ω4 π cos θdA , where n is the particle number density. Integrating over all the directions,we have the particle flux defined as the particle number passing through a unit area in aunit time as J n = 14 cn , (45)where n is the number density. If we treat the neighboring two volumes with differentchemical potential µ and µ + δµ are in local equilibrium states respectively, one can derivethe particle flux between the two neighboring volumes as J n = 14 c ( n ( µ + δµ ) − n ( µ )) . (46)19n principle, with the number density of the particles in hand, one can calculate the particleflux driven by the chemical potential gradient by using this formula.In Eq.(38), ∇ represents the gradient with respect to the location coordinates ( ω here) onthe black hole (black funnel) horizon. dA represents the area element on the horizon. (cid:82) dA represents the integral over the area of the horizon while dV represents the integral overthe black hole volume. ∇ T represents the heat flow driving force in analogy to the voltagein the electric circuit where the heat flow density is in analogy to the electric current flow.Then (cid:82) J q · ∇ T dV is the entropy production rate which represents the thermodynamic costor dissipation due to the heat flow as the product of the heat flow and the heat flow drivingforce in analogy to the electric power generated by the product of the electric current andvoltage. Since the temperature of the black funnel is given in Eq.(22) as T ( ω ) = 14 π (1 − Q ( ω ) χ (0 , ω )) , (47)depending only on one local coordinate, the heat driving force takes the gradient to ω . Inthis case of black funnel, the local temperature only varies on the horizon surface. Thus theentropy production of (cid:82) J q · ∇ T dV becomes an effective surface integral of (cid:82) J q · ∇ T RdA where R is the radius, in general not a constant on the black funnel horizon.Similarly, J n represents the particle flux while ∇ µ is the gradient of the chemical potentialgiving rise to the driving force for the particle flow. The product of the particle flow andthe associated driving force − (cid:82) J n T · ∇ µdV represents the entropy production rate givingrise to the thermodynamic cost or the dissipation due to the particle flow. Similarly, theparticle driving force ∇ µ also depends only on the local coordinate ω on the horizon asin Eq.(23). Again, in this case of black funnel, the local chemical potential only varies onthe horizon surface. Therefore, the entropy production − (cid:82) J n T · ∇ µdV becomes an effectivesurface integral.The first law of the nonequilibrium thermodynamics of the black hole can thus be for-mulated explicitly as the total entropy production being the sum of the entropy productionof the system and the environment EP R = dS tot dt = dSdt + dS env dt , (48)where EP R = dS tot dt = (cid:82) J q · ∇ T dV − (cid:82) J n T · ∇ µdV , dSdt is the entropy change of the system,and dS env dt = (cid:82) J q T · dA is the entropy change of the environment.20ince the total entropy production rate EP R = dS tot dt ≥ , (49)this formulates the second law of the nonequilibrium thermodynamics of the black funnel.Note that at the steady state, the entropy change of the system is equal to zero, i.e. dSdt = 0.Then at the steady state, one has EP R = dS tot dt = dS env dt . (50)This indicates that at steady state the change of the total entropy is from the environment.In other words, to maintain a steady state of the nonequilibrium black hole, there is athermodynamic cost to pay which is caused by the heat dissipation against the environments.We can see that EP R can be used to measure the degree of nonequilibriumness away fromthe equilibrium.
C. Linear nonequilibrium thermodynamics of black hole-black funnel
As known, many nonequilibirum processes are due to certain inhomogeneity of the system.Temperature gradient can induce to the heat flow or energy flow while the concentration ordensity gradient can lead to particle transport or diffusion. Empirically, the current gener-ated by such inhomogeneities is often relatively small, that is not very far from equilibrium: J = LX (51)where J represents the transport current flow (mass, charge, momentum, heat or energy) perunit time per unit cross section area while X represents the driving force (density gradient,voltage gradient, velocity gradient, temperature gradient).When several current flows and driving forces simultaneously are present, coupling be-haviors emerge. The more general empirical linear nonequilibrium law can be formulatedas J k = (cid:88) l L kl X l (52)where L kl is the Onsager coefficient represents intensity of k th current caused by the l thdriving force. The above relationship is often called Onsager relationship [30, 31]. In our21ase of the black funnel, both temperature gradient and chemical potential gradient on thehorizon are present. As a result, both particle and thermal current emerge on the horizon.Therefore, assuming the linear relationshio between the current and the force not very farfrom equilibrium, the currents are approximately determined by the driving force as follows: J q = − L T ∇ µ + L ∇ TJ n = − L T ∇ µ + L ∇ T (53)According to Onsager reciprocal relationship [31], L = L . As we see, for black funnel,the temperature gradient on the horizon not only can lead to thermal or energy current onthe horizon, but also the particle transport on the horizon.In the same way, the chemical potential gradient on the horizon not only can induce theparticle flow but also the thermal or energy flow on the horizon of the black funnel. This isdue to the coupling revealed in the generalized Onsager relations for linear nonequilibriumthermodynamics not too far from equilibrium. Therefore, L represents the intensity ofthe thermal induced diffusion on the horizon. The temperature gradient on the horizon canlead to particle transport on the horizon through L . This phenomena is called Soret effect[30, 32]. L represents the intensity of current induced by the thermal flow on the horizon.The chemical gradient can give rise to the density or concentration gradient and thus leadto the thermal current on the horizon through L . This phenomena is called Dufour effect[30, 33]. When the temperature gradient or the chemical gradient is small, the thermalinduced diffusion or the diffusion induced thermal flow on the horizon will usually be small.However, when the temperature gradient or the chemical potential gradient becomes large,the thermal induced diffusion or the diffusion induced thermal current on the horizon canbe significant. Since the total entropy production is given by epr = J q · ∇ T − J n T ∇ µ . (54)We then have four contributions for the entropy production epr = (cid:18) − L T ∇ µ + L ∇ T (cid:19) · ∇ T − (cid:18) − L T ∇ µ + L ∇ T (cid:19) ∇ µT = L (cid:18) ∇ µT (cid:19) − L ∇ (cid:18) T (cid:19) · ∇ µ + L (cid:18) ∇ T (cid:19) (55)22 (cid:0) ∇ µT (cid:1) represents the contribution to the entropy production rate from the chemical po-tential gradient induced particle flow on the horizon. L (cid:0) ∇ T (cid:1) represents the contributionto the entropy production rate from the temperature gradient induced thermal flow on thehorizon. − L ∇ T · ∇ µT representd the contribution to the entropy production rate from thethermal gradient induced particle current or diffusion on the horizon. − L ∇ µ · T repre-sents the contribution to the entropy production rate from the chemical potential gradientinduced thermal current flow on the horizon. Note that the second law of the thermody-namics guarantees that the total entropy production rate epr of the nonequilibrium blackhole (the black funnel here) is larger than zero.When the temperature gradient and the chemical potential gradient are relatively small,the contributions to epr from L and L parts are often less significant than the onesfrom L and L parts on the black funnel horizon. When the temperature and chemicalpotential gradients becomes high, contributions to epr from L and L parts may not beignored compared to the ones from L and L parts on the black funnel horizon. D. Time arrow from the nonequilibrium black hole
The entropy production rate not only provides a measure of thermodynamic cost ordissipation for nonequilibrium black hole, but also gives a qualitative predictor of the degreeof nonequilibriumness.One important implication of the non-zero entropy production rate is the time irreversibil-ity or time arrow. From the fluctuation theorem, the ratio of the distribution of the statevariable C along the forward in time path C ( t ) and backward in time path ˜ C ( t ) is given by[34, 35] P [ C ( t )] P [ ˜ C ( t )] = exp[∆ S tot ] , (56)where ∆ S tot is the entropy production. If there is a time reversal symmetry between back-ward and forward in time path, then the entropy production is zero. This is the conclusionfor the equilibrium state with time reversal symmetry such as the conventional black holewith killing horizon. On the other hand, the nonzero entropy production implies the for-ward in time path has a different probability than than that in backward in time path. Thismeans that the time reversal symmetry is broken by the nonzero entropy production. The23onzero entropy production thus gives rise to the time arrow. Therefore, we see that fromthe nonequilibrium boundary CFT, the corresponding bulk black hole is in the nonequi-librium steady state due to AdS/CFT correspondence. This nonequilibrium black hole inthe bulk has a temperature and chemical potential variation or gradient on the horizon. Itbreaks the detailed balance and leads to the entropy production or dissipation cost. Thisdissipation provides the thermodynamic origin of the time asymmetry or time arrow. IV. EVAPORATION DYNAMICS OF THE NONEQUILIBRIUM BLACK HOLE-BLACK FUNNEL
The radiation spectrum of the black funnel as presented in Eq.(21) should be treated asthe local property of the radiation field. The global spectrum from black funnel horizonthat is in non-equilibrium state is expected to be different from the equilibrium Hakwingradiation spectrum. Consequently, the lifetime of the black funnel due to the non-equilibriumradiation will be very different from that of the equilibrium black hole.Let us consider the black hole evaporation dynamics. In the conventional equilibriumblack hole evaporation approach, the energy loss which leads to the evaporation is purelyfrom the Hawking radiation. In contrast, for the nonequilibrium black holes, such as blackfunnel, the evaporation process is not only determined by the Hawking radiation but also bythe dissipation from the nonequilibriumness. By using the first law of the nonequilibriumthermodynamics of the black holes we just formulated, we can see dSdt == − (cid:90) J q T · dA + (cid:90) J q · ∇ T dV − (cid:90) J n T · ∇ µdV , (57)where S is the entropy of the black hole which is in turn given by the area of the black holehorizon.For the conventional equilibrium black hole, there is an uniform temperature and chemicalpotential on the horizon. Therefore the entropy productions involving the gradients oftemperature and chemical potential vanishes. For the Schwarzschild black hole, S ∼ A ∼ M , T ∼ M , and J q ∼ T ∼ M . The J q ∼ T assumes the flux in the cavity volume of theblack body, it works for the photon energy flux. In principle the radiation can be directedto any directions not limited to the horizon surface. The reason why the entropy productionis not effective outside the horizon is that the gradient of temperature or chemical potential24s zero outside the horizon. So effectively the integration for the entropy production is overthe horizon surface. Then the first law becomes M dMdt ∼ − M . (58)This gives M ( t ) ∼ (1 − λt ) which is exactly the same evaporation dynamics of the initialequilibrium black hole.We can see clearly that for an initially nonequilibrium black hole, in addition to theHawing radiation leading to the energy or entropy loss, there is also a significant contributionfrom the nonequilibrium thermodynamic dissipation in term of the entropy production dueto the non-uniform temperature and chemical potential. Since the entropy production isalways positive, it is expected that the evaporation process of the nonequilibrium black holetends to be slower than the initial equilibrium black hole with the Hawking radiation alone.One can estimate this conclusion by using the first law of the nonequilibrium thermo-dynamics for the black funnel. We naively make a crude approximate assumption that theentropy and the mass of the black funnel still satisfy the relation of the entropy and themass of the the Schwarzschild black hole, i.e. S ∼ M . The second term (cid:82) J q · ∇ T dV on theright side of Eq.(57) is the entropy production due to temperature gradient ∇ T , which is apositive number denoted as C . On the other hand, the entropy production has a contribu-tion C from the chemical gradient − (cid:82) J n · ∇ µdV which is also a positive number. Let usdefine C = C + C . This term represents the total entropy production, which implies that C >
0. If we consider the contribution of the heat flow from temperature gradient and theparticle flow from chemical potential gradient to the entropy production, the first law givesthe evolution equation for the black funnel as follows
M dMdt ∼ − M + C . (59)Based on the Eq. (58) and (59), we have plotted the mass of the Schwarzschild black holeas a function of time t as well as the mass of the black funnel in Fig. 3. It is seen thatthe evaporation time of the nonequilibrium black hole appears to be longer than that ofthe equilibrium black hole. In other words, the nonequilibrium effect is to slow down theevaporation process of the black funnel through the thermodynamic dissipation by the heatcurrent flow from the temperature gradient and the particle current flow from the chemicalpotential gradient. 25 .0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.00.20.40.60.81.0 t M ( t ) FIG. 3: Illustration of Hawking evaporation process of the Schwarzschild black hole (blue line)and the black funnel (red line). The two curves are plotted based on the Eq. (50) and (52). Theparameter C is chosen as 0 . V. CONCLUSION
In summary, we have derived the Hawking radiation from the bulk horizon of the non-equilibrium black funnel solution based on the Damour-Ruffini method. This provides thefirst derivation of Hawking temperature of the non-killing horizon as far as we know. Ourresults indicate that the spectrum, the temperature, and the chemical potential of the non-equilibrium black funnel do depend on one of the spatial coordinates. This is differentfrom the conventional case of the constant temperature equilibrium black hole with killinghorizon. Therefore, the black hole with non-killing horizon can be overall in nonequilibriumsteady state while the Hawking temperature of this black funnel can be viewed as thelocal temperature and the corresponding Hawking radiation can be regarded as being inlocal equilibrium with the horizon of black funnel. Our results indicate that the blackfunnel horizon itself, which is a non-killing horizon, should be viewed as a non-equilibriumthermodynamics system.We further discuss the nonequilibrium thermodynamics of the black funnel, where the firstlaw can be formulated as the entropy production rate being equal to sum of the changes ofthe entropies from the system and environments while the second law is given by the entropyproduction being larger than or equal to zero. We show that the nonequilibrium black holeleads to a time arrow due to the inhomogeneous distribution of the temperatures and thechemical potentials on the horizon giving rise to heat and particle transport dissipation.We also discuss how the nonequilibrium dissipation influences on the evaporation process of26lack funnel.
Appendix A: Hawking radiation of black droplet
In this appendix, we will compute the Hawking temperature of black droplet solutionby using the Damour-Ruffini method. Black droplet solution is another type of solutionconjectured by Marolf et al [11] to describe the gravity dual of heat transport between theboundary black hole. However, unlike the black funnel solution, the bulk horizon of blackdroplet is disconnected with the horizon of the boundary black hole, which suggests that thecoupling between the boundary black hole and the heat bath at infinity is suppressed. Theblack droplet solution describes the boundary CFT in equilibrium state and the temperatureof droplet horizon should be constant. Now, we performed the computation of spectrum ofHawking radiation to verify this point.The metric of black droplet is given by [36] ds = 1 yh (cid:26) − f y f ρ f y + f ρ − f y f ρ T c dt + A c y dy + B c (1 − x ) (cid:20) dx + x (1 − x ) F c x + (1 − x ) y dy (cid:21) + x S c (1 − x ) d Ω (cid:41) , (60)where T c , A c , B c , F c , and S c are functions of the Cartesian coordinates x and y , and d Ω is the metric of two-dimensional unit sphere. f y and f ρ which are functions of y and ρ respectively, are given by f y ( y ) = f ( y )(1 − λy ) ,f ρ ( ρ ) = (1 − ρ ) (1 + ρ ) , (61)with f ( y ) being a smooth and positive definite function and the coordinate transformation y = R ρ (cid:18) − x x + (1 − x ) y (cid:19) . (62)The function h is also a smooth and positive definite function of y . h ( y ) and f ( y ) can bedetermined numerically by solving the Einstein equations. The conformal boundary locatesat y = 0, while the bulk horizon locates at y = 1 /λ . The boundary conditions at the bulkhorizon y = 1 /λ are given by T c | y =1 /λ = A c | y =1 /λ , F c | y =1 /λ = 0 ,∂ y T c | y = 1 /λ = ∂ y A c | y = 1 /λ = ∂ y B c | y = 1 /λ = ∂ y S c | y = 1 /λ = 0 . (63)27he determinant of the matric is given by √− g = x S c sin θ y (1 − x ) h (cid:115) A c B c T c f y f ρ h ( f y + f ρ − f y f ρ ) . (64)We consider the Klein-Gorden equation for the scalar field in this background. It can beexplicitly written as g tt ∂ t ψ + 1 √− g ∂ y (cid:2) √− gg yy ∂ y ψ (cid:3) + 1 √− g ∂ y (cid:2) √− gg yx ∂ x ψ (cid:3) + 1 √− g ∂ x (cid:2) √− gg xy ∂ y ψ (cid:3) + 1 √− g ∂ x (cid:2) √− gg xx ∂ x ψ (cid:3) + 1 √− g ∂ θ (cid:2) √− gg θθ ∂ θ ψ (cid:3) + g φφ ∂ φ ψ − m ψ = 0 . (65)We introduce the generalized tortoise coordinate in the form of y ∗ = 12 κ ln | y − /λ | = κ ln(1 /λ − y ) , < y < /λ κ ln( y − /λ ) , y > /λ (66)where κ is an adjustable parameter and will be determined in the following, 0 < y < /λ denotes the spacetime outside the bulk horizon and y > /λ denotes the spacetime insidethe bulk horizon.By multiplying the Klein-Gorden equation by (1 /λ − y ) , and taking the limit y → /λ and x → x , where x is an arbitrary spatial location in the x direction, the resultingequation in generalized tortoise coordinate can be written as C tt ∂ t ψ + C yy κ ∂ y ∗ ψ = 0 , (67)with C tt = h (1 /λ ) λ f (1 /λ ) T c (1 /λ, x ) , C yy = 4 h (1 /λ ) λ A c (1 /λ, x ) . (68)If choosing the parameter κ = (cid:113) C yy C tt = (cid:112) λf (1 /λ ), the above equation can be reducedto the standard wave equation as ∂ y ∗ ψ − ω ψ = 0 . 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