The TolmanEhrenfest criterion of thermal equilibrium for a static fluid in a static spacetime is generalized to stationary heat conduction in the approximation in which backreaction is negligible. Applying this generalized criterion to the Hawking radiation in the Schwarzschildde Sitter geometry shows that the two horizons which act as thermostats remain in thermal equilibrium. The temperature of the radiation fluid interpolates between the temperatures at the horizons with a static analytic profile that is given explicitly.
['General relativity', 'alternative theories of gravity', 'General Relativity and Quantum Cosmology']
How the Schwarzschildde Sitter horizons remain in thermal equilibrium at vastly different temperatures
{'How the Schwarzschild-de Sitter horizons remain in thermal equilibrium at vastly different temperatures': "Marcello Miranda ID , 1, 2, ∗ Massimiliano Rinaldi ID , 3, 4, † and Valerio Faraoni ID 5, ‡ 1 Scuola Superiore Meridionale, Largo San Marcellino 10, I-80138, Napoli, Italy 2 INFN Sez. di Napoli, Compl. Univ. Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy 3  \nDepartment of Physics, University of Trento, Via Sommarive 14, 38122 Trento, Italy  \n4 Trento Institute for Fundamental Physics and Applications TIFPA-INFN, Via Sommarive 14, 38122 Trento, Italy 5 Department of Physics & Astronomy, Bishop's University,  \n2600 College Street, Sherbrooke, Québec, Canada J1M 1Z7  \nThe Tolman-Ehrenfest criterion of thermal equilibrium for a static fluid in a static spacetime is generalized to stationary heat conduction, in the approximation in which backreaction is negligible. Applying this generalized criterion to the Hawking radiation in the Schwarzschild-de Sitter geometry shows that the two horizons (which act as thermostats) remain in thermal equilibrium. The temperature of the radiation fluid interpolates between the temperatures at the horizons, with a static analytic profile that is given explicitly.", 'I. INTRODUCTION': "It is well established that black holes are not black, but radiate Hawking radiation when quantum fields are present [1]. A Schwarzschild black hole with mass m and line element 1  \nds 2 = -( 1 -2 m r ) dt 2 + dr 2 1 -2 m/r + r 2 d Ω 2 (2) (1.1)  \n(where d Ω 2 (2) ≡ dϑ 2 +sin 2 ϑdϕ 2 is the metric on the unit 2-sphere) has a horizon at r = 2 m radiating at the Hawking temperature T H = 1 / (8 πm ) [1]. Similarly, the de Sitter geometry  \nds 2 = -( 1 -Λ r 2 3 ) dt 2 + dr 2 1 -Λ r 2 / 3 + r 2 d Ω 2 (2) (1.2)  \n(where Λ > 0 and H = √ Λ / 3 are, respectively, the cosmological and the Hubble constants) has a null horizon at r = 1 /H , radiating at the Gibbons-Hawking temperature T GH = H/ (2 π ) [3]. A geometry interpolating between the two above is the so-called Schwarzschild-de Sitter/Kottler (SdS) spacetime, with metric  \nds 2 = -f ( r ) dt 2 + dr 2 f ( r ) + r 2 d Ω 2 (2) , (1.3)  \nf ( r ) = 1 -2 m r -Λ r 2 3 . (1.4)  \nThe metric is a solution of the vacuum Einstein equation with cosmological constant Λ and can have two horizons, according to the values of m and Λ . Here, we assume that the black hole radius is much smaller than the cosmological horizon size, i.e., 2 m /lessmuch H -1 = √ 3 / Λ . In this approximation, T H ≈ 1 / (8 πm ) /greatermuch T GH ≈ H/ (2 π ) . Then the temperatures of the two horizons are vastly different, with the black hole horizon being much hotter than the de Sitter one. It is often stated in the literature that, because of the temperature difference between the black hole and cosmological horizons, there cannot be thermal equilibrium in Schwarzschild-de Sitter (SdS) space-except for  \nthe degenerate Nariai case [4], where the two horizons coincide (a scenario which is not physically interesting). Consequently, multiple temperatures must be introduced, and some authors argue that the area law or other fundamental relations of black hole thermodynamics must be modified (e.g., [5-12]). These conflicting proposals point to an incomplete understanding of multi-horizon spacetimes in general, and of SdS in particular. Here we propose an alternative view in which the two horizons remain in thermal equilibrium although their temperatures are different. Thermal equilibrium (defined as the timeindependence of temperature and heat flux in the static frame) is achieved by letting the temperature vary with the radial position, interpolating between the Hawking and the Gibbons-Hawking temperatures (we provide explicitly the analytical temperature profile T ( r ) between the two horizons in the approximation of negligible backreaction). Here by 'temperature' we mean the temperature of the blackbody Hawking radiation between the two horizons, which will be treated as a radiation fluid at rest in the locally static geometry between these horizons. Since backreaction is neglected in the calculation of both the Hawking and the Gibbons-Hawking temperatures [1, 3], the test fluid approximation for the blackbody radiation is legitimate in the 'usual' regime of moderate emission. The temperature profile is derived from a generalization of the Tolman-Ehrenfest criterion [13-15] for the thermal equilibrium of a test fluid at rest in a static spacetime. This generalized criterion applies in the presence of heat sources, which are absent in the 'standard' situation conceived by Tolman.  \nTo begin with, we recall that, long ago, Tolman [13, 14], and then Tolman and Ehrenfest [15], showed that a test fluid at rest in a static spacetime in general relativity cannot be in thermal equilibrium at constant temperature. There must necessarily be a temperature gradient (unless g 00 is constant), and the Tolman-Ehrenfest criterion for thermal equilibrium is [13-15]  \nT √ -g 00 = const. (1.5)  \nin coordinates adapted to the time symmetry. The physical interpretation of this result is that heat is a form of energy; therefore, it sinks in a gravitational field and spacetime regions where gravity is stronger are hotter than those where gravity is weaker. The modern derivation of the Tolman-Ehrenfest criterion [16-19] uses Eckart's law of heat conduction (one of the three constitutive relations assumed in Eckart's first-order thermodynamics [20-22])  \nfor dissipative fluids. For the reader's convenience, we report this derivation in Appendix A.  \nArelativistic dissipative fluid has a stress-energy tensor of the form [21]  \nT ab = ρu a u b + Ph ab + q a u b + q b u a + π ab , (1.6)  \nwhere u a is the timelike fluid 4-velocity (normalized to u a u a = -1 ),  \nh ab ≡ g ab + u a u b (1.7)  \nis the metric induced onto the 3-space orthogonal to u c ,  \nρ = T ab u a u b (1.8)  \nis the energy density,  \nP = 1 3 T ab h ab (1.9)  \nis the isotropic pressure,  \nq a = -T cd u c h d a (1.10)  \nis the heat flux density,  \nπ ab = h a c h b d T cd -Ph ab (1.11)  \nis the (traceless) anisotropic stress tensor. h ab , q a , and π ab are purely spatial according to the observer u c ,  \nh ab u a = h ab u b = q a u a = π ab u a = π ab u b = 0 . (1.12)  \nAccording to Eckart's law (generalizing the Fourier law familiar from pre-relativistic physics), the heat flux density is [21]  \nq a = -K h ab ( ∇ b T + T ˙ u b ) , (1.13)  \nwhere K is the thermal conductivity. In the standard treatment, thermal equilibrium is defined by the absence of heat flow q a = 0 .  \nThe conditions for the validity of the Tolman-Ehrenfest criterion are quite restrictive, and it is not surprising that the latter has seen very limited applications. There are, however, applications to material physics [23] and to neutron star cores [24-27], see [17-19] for a revisitation from first principles and [28, 29] for further discussion. More recently, the Tolman-Ehrenfest criterion has been extended to conformally static spacetimes [30] and to scalar-tensor gravity [31].  \nEquation (1.5) makes it clear that a temperature gradient is necessary for thermal equilibrium and therefore thermal equilibrium between the two horizons of the SdS spacetime is not ruled out a priori . In fact, we will show that it is possible. To this end, we require thermal equilibrium for the radiation fluid generated by the two static horizons, and we apply a generalized Tolman-Ehrenfest criterion to it. The criterion (1.5) must be extended because it was derived in the absence of sources, while in the SdS case the two horizons act as thermostats keeping these two boundaries at fixed temperatures. To understand the physics at play, it is useful to make an analogy with heat conduction in a metal rod of finite length (indeed, Eckart's law is just a generalization of the non-  \n/vectorq  \n=  \n/vector  \n).  \nrelativistic Fourier law of heat conduction -K ∇T In the absence of heat sources, if the ends of the rod are initially at different temperatures heat flows from the  \nhotter to the cooler regions until the temperature gradient /vector ∇T disappears, thermal equilibrium is established, the heat flux density /vector q = -K /vector ∇T vanishes, and the two ends reach the same temperature, constant across the rod and time-independent. (In the relativistic case, the gravitational field instead maintains a temperature gradient, as discussed above.) But one can also consider stationary heat conduction , in which the two ends of the rod are kept at different temperatures T 1 and T 2 (constant in time) by thermostats. Then, heat flows from the hotter to the cooler end establishing a linear temperature profile, with T ( x ) and /vector q independent of time. The one-dimensional heat equation  \n∂ T ∂t = a ∂ 2 T ∂x 2 (1.14)  \n(where a is the Fourier coefficient), degenerates into d 2 T /dx 2 = 0 . This situation is more similar to the case of the radiation fluid between the two horizons of SdS, which act as thermostats. To make the analogy more appropriate, instead of a one-dimensional rod one should consider stationary heat conduction in a three-dimensional medium with spherical symmetry, purely radial heat flow, and spherical thermostats.  \nThe next section discusses the 'standard' Tolman-Ehrenfest criterion, which is generalized to stationary heat conduction and applied to the SdS geometry in Sec. III, where the temperature profile between the two horizons is also computed. Section IV discusses the regularization of this temperature at the horizons, while Sec. V contains the conclusions.", 'II. STANDARD TOLMAN-EHRENFEST CRITERION AND DISSIPATIVE FLUIDS': "In the derivation of the 'standard' Tolman-Ehrenfest condition for the thermal equilibrium of a static test fluid in a static spacetime, in the absence of heat sources or sinks [16-19], one uses a dissipative fluid at rest. The velocity gradient is decomposed as  \n∇ b u a = σ ab + Θ 3 h ab + ω ab -˙ u a u b , (2.1)  \nwhere an overdot denotes differentiation along the fluid 4-velocity, ˙ u a ≡ u c ∇ c u a . The kinematic quantities of the fluid are the expansion scalar  \nΘ ≡ ∇ a u a , (2.2)  \nthe trace-free shear tensor  \nσ ab ≡ ( h ac h bd -1 3 h cd h ab ) ∇ ( c u d ) , (2.3)  \nand the vorticity tensor  \nω ab ≡ h ac h bd ∇ [ d u c ] . (2.4)  \nThe velocity gradient is decomposed into symmetric and antisymmetric parts, and the former into trace-free and pure trace parts. The covariant derivative operator can be split into a contribution parallel to the 4-velocity and one perpendicular to it,  \n∇ a = h a b ∇ b -u a u b ∇ b . (2.5)  \nThe dissipative nature of the fluid is described by the heat flux, the viscous pressure, and the anisotropic stresses.  \nIn Eckart's first-order thermodynamics [21], an imperfect fluid at rest reduces to a perfect fluid because Eckart's theory for dissipative fluids is built assuming the three constitutive relations:  \nq a = -K h ab ( ∇ b T + T ˙ u b ) , (2.6)  \nP = P perfect + P visc = P perfect -ζ Θ , (2.7)  \nπ ab = -2 η σ ab , (2.8)  \nwhere ζ and η are the bulk and shear viscosity coefficients, respectively. Apart from the inertial term -KT ˙ u a in the heat flux density (2.6) [21], these constitutive relations are familiar from the 3-dimensional hydrodynamics of non-relativistic dissipative fluids with Newtonian fluid behavior. When the spatial velocity u i ( i = 1 , 2 , 3 ) vanishes, the dissipative fluid trivially reduces to a perfect one. In the absence of heat sources/sinks, thermal equilibrium is understood as the absence of heat flow q a = 0 , from which one deduces the Tolman-Ehrenfest condition of thermal equilibrium T √ -g 00 = const. beginning from Eckart's law (2.6) (Appendix A).", 'III. TOLMAN-EHRENFEST CRITERION GENERALIZED TO STATIONARY HEAT CONDUCTION': "/negationslash  \nHere we generalize the Tolman-Ehrenfest criterion for thermal equilibrium to stationary heat conduction, in which q a = 0 but q a and the fluid temperature T does not depend on time in the static frame.  \nNeglecting the backreaction of the quantum radiation onto the black hole and the de Sitter horizon, the equation of motion ∇ b T ab = 0 of the dissipative fluid (1.6) is projected 2 onto the time direction u a , giving [32]  \n˙ ρ +( P + ρ ) Θ + π ab σ ab + ∇ a q a + q a ˙ u a = 0 , (3.1)  \nwhile the projection onto the 3-space orthogonal to u a yields  \n( P + ρ ) ˙ u a + h a c ( ˙ q c + ∇ c P + ∇ b π bc ) + ( ω a b + σ a b ) q b + 4Θ 3 q a = 0 . (3.2)  \nFor a static fluid at rest in a static spacetime, in coordinates adapted to the time symmetry, we have ˙ ρ ≡ u c ∇ c ρ = ∂ρ/∂t = 0 , ˙ P = 0 , π ab = σ ab = ω ab = 0 , and  \nΘ = 1 √ -g ∂ a ( √ -g u a ) = 1 √ -g ∂ t ( √ -g u 0 ) = ∂ t u 0 = 0 , (3.3)  \nwhere we used the fact that u 0 = √ -g 00 (from the normalization u c u c = -1 ) and the time-independence of the metric components in adapted coordinates. As our new definition of thermal equilibrium for stationary heat conduction, we assume a steady heat flow, then h a b ˙ q b = 0 (see Appendix B) and Eqs. (3.1) and (3.2) reduce to  \n∇ c q c + q a ˙ u a = 0 , (3.4)  \n( P + ρ ) ˙ u a + h a c ∇ c P = 0 . (3.5)  \nUsing the Buchdahl relation ˙ u a = ∇ a (ln √ -g 00 ) valid in static spacetimes [33] and ∇ a q a = ∂ a ( √ -g q a ) / √ -g , we have  \n1 √ -g ∂ a ( √ -g q a ) = -q a ∇ a ( ln √ -g 00 ) , (3.6)  \n∇ a ( ln √ -g 00 ) = -∇ a P P + ρ . (3.7)  \n(3.8)  \nSubstituting Eq. (3.7) into (3.4) gives  \n1 √ -g ∂ i ( √ -g q i ) -q j ∇ j P P + ρ = 0 ( i, j = 1 , 2 , 3) .  \nFor a static and spherically symmetric geometry with q a = (0 , q r , 0 , 0) , Eq. (3.8) reads  \n1 r 2 q r d dr ( r 2 q r ) = 1 P + ρ dP dr = w (1 + w ) ρ dρ dr , (3.9)  \nwhere we used the linear equation of state P = wρ . Equation (3.9) has the general solution  \nq r ( r ) = C √ -g/ sin 2 ϑ ρ w/ (1+ w ) , (3.10)  \nwith C an integration constant. Notice that sin 2 ϑ removes the angular dependence of the metric determinant.  \nEquation (3.7) with the linear equation of state gives  \n∇ a ( ln √ -g 00 ) + w ( w +1) ρ dρ dr = 0 , (3.11)  \nwhich is immediately integrated to  \nρ ( r ) = ρ 0 ( 1 √ -g 00 ) ( w +1) /w . (3.12)  \nwith ρ 0 an integration constant, and, comparing with Eq. (3.10), one obtains  \nq r ( r ) = q 0 √ g 00 g/ sin 2 ϑ . (3.13)  \nwhere q 0 is an integration constant. Notice that g/ sin 2 ϑ only depends on the radial coordinate. Now, let us specify the discussion to the SdS geometry (1.3) with g = -r 2 sin 2 ϑ , which produces  \nq r ( r ) = q 0 r 2 √ -g 00 = q 0 r 2 √ 1 -2 m/r -H 2 r 2 . (3.14)  \nq r diverges on the horizons due to the adopted coordinate system. However, the Tolman criterion is necessarily formulated in this coordinate system, which is adapted to  \nthe time symmetry. The radial heat flux must be outgoing, which implies that q 0 > 0 .  \nImposing now Eckart's constitutive relation (2.6) gives  \nq r = K g 00 [ d T dr + T d dr ( ln √ -g 00 ) ] (3.15)  \nwhich, combined with Eq. (3.14), yields  \nd dr ( T √ -g 00 ) -q 0 K r 2 g 00 = 0 . (3.16)  \nAssuming K = const., this equation integrates to  \nT √ -g 00 = T 0 -q 0 K ∫ dr r 2 (1 -2 m/r -Λ r 2 / 3) . (3.17)  \nbeing T 0 a positive integration constant. As a check, this solution reduces to the 'standard' Tolman-Ehrenfest criterion (1.5) if the heat flux vanishes, q 0 = 0 . The integral on the right-hand side is computed explicitly, giving the static local temperature profile  \nT ( r ) = 1 √ 1 -2 m r -Λ 3 r 2 [ T 0 + q 0 2 m K ( ln x -∑ i ( Λ R 2 i -3 ) ln ( x -x i ) 3 (Λ R 2 i -1) )] , (3.18)  \nwhere x ≡ r/r 0 , r 0 is a constant, R i are the roots of the equation f ( r ) = 0 , and x i = R i /r 0 .", 'A. Schwarzschild black hole': 'For the pure Schwarzschild geometry with Λ = 0 and f ( r ) = 1 -2 m/r , the temperature profile (3.18) reads  \nT ( r ) = 1 √ 1 -2 m r [ T 0 -q 0 2 m K ln ( 1 -2 m r ) ] . (3.19)  \nSince the argument of the logarithm is smaller than unity, the requirement of a radially outgoing heat flow q 0 > 0 guarantees that the temperature T ( r ) is positive everywhere. As r → + ∞ , T must reduce to the Hawking temperature seen by the static distant observer, which identifies the integration constant T 0 as  \nT 0 = 1 8 πm . (3.20)  \nThe temperature T ( r ) diverges at the Schwarzschild horizon r → 2 m . This well-known fact is discussed in the next section. Similarly, the temperature (3.18) diverges at both the black hole and the de Sitter horizons.', 'B. de Sitter space': 'In the pure de Sitter case m = 0 , the formal solution is  \nT dS ( r ) = 1 √ 1 -H 2 r 2 { T 0 -q 0 K [ 1 r + H arctanh( Hr ) ]} . (3.21)  \nIn this formal solution one must choose the integration constant q 0 to vanish to avoid a divergent temperature (3.21) at r = 0 . This choice is physically wellmotivated. In fact, in this case, the heat flux from the de Sitter horizon travels radially inward toward r = 0 , but it cannot accumulate there and must continue travelling outward to larger radii and cross the horizon, to never come back to the interior region. Ingoing and outgoing fluxes balance out, resulting in a zero net flux, and one must set q 0 = 0 . As a check, in this case the  \nsolution (3.21) gives back the Tolman-Ehrenfest criterion (1.5)  \nT ( r ) = T 0 √ 1 -H 2 r 2 . (3.22)  \nThis temperature diverges on the de Sitter horizon, as is well known.', 'IV. TEMPERATURE REGULARIZATION AT THE HORIZONS': 'The divergence of the Tolman-Ehrenfest temperature at the black hole horizon and its relation with the Hawking temperature are well known in standard horizon thermodynamics [35, 42, 43]. Let us consider the Schwarzschild black hole: divergences are not new in thermodynamics and are related to the vacuum state of the quantum field used to compute the temperature (for example, the Boulware vacuum is singular on the black hole horizon). The Hawking temperature is T H = κ/ (2 π ) , where κ is the surface gravity, 3 and it represents the temperature seen by a static distant observer ( r → ∞ ). To remain at fixed radius, a detector is accelerated with uniform acceleration a and, in the inertial vacuum state, detects blackbody radiation at the Unruh temperature T U = a/ (2 π ) . In the Unruh vacuum the modes that, according to this observer, have propagated from the region near the black hole are populated (unlike the modes propagating from spatial infinity, which are not), at the temperature T = κ/ ( 2 π √ -k a k a ) , where k a is the local timelike Killing vector. As the Schwarzschild horizon is approached, in the limit r → r S = 2 m , the surface gravity is obtained as [2, 42]  \nTherefore,  \nκ = lim r → 2 m √ -k c k c a. (4.1)  \nT = a 2 π → κ 2 π √ -k c k c , (4.2)  \nand, since k c k c = g 00 in coordinates adapted to the local time symmetry,  \nT √ -g 00 /similarequal a 2 π √ -g 00 /similarequal κ 2 π (4.3)  \nfor the Unruh observer. Therefore, the local temperature of the Hawking radiation obeys the Tolman-Ehrenfest criterion (as remarked explicitly, e.g., in [35, 42, 43]) and applying this criterion to the radiation fluid generated from the horizons, as we did in the previous section, is not far-fetched nor new.  \nThere is an even more important aspect regarding the apparent divergence of the temperature T . There is increasing consensus that the Hawking radiation does not arise at the horizon but in a quantum atmosphere surrounding it, which is macroscopic [35-40]. That is, this atmosphere is not confined to a Planck-sized near the horizon, but it extends to radii r /similarequal 3 √ 3 4 r S /similarequal 1 . 3 r S , as indicated by the fact that the Hawking flux indicates an emission area larger than the horizon area [34-40]. Likewise, the wavelength of the typical Hawking quantum is not r S , but /similarequal 79 r S [35, 42]. Presumably, similar considerations hold for the Gibbons-Hawking radiation associated with the de Sitter horizon.  \nAnother possibility for regularizing the divergence of the Tolman temperature consists of including the trace anomaly for the radiation [41]. As a conclusion, the divergence of the temperature profile (3.18) can be eliminated in more nuanced descriptions of the relevant vacuum state and emission process, or is simply not occurring because of the macroscopic quantum atmosphere associated with horizons.', 'V. CONCLUSIONS': "Thermal equilibrium (or the lack thereof) between the two horizons of the SdS spacetime exhibits some similarities with heat conduction in a rod of finite length whose ends are kept at constant, different, temperatures. The temperature gradient maintains a steady heat flow and there is thermal equilibrium in the sense that both the heat flux density /vector q and the linear temperature profile T ( x ) are time-independent. In the SdS case, it is the blackbody radiation between the two horizons (a perfect fluid) that is in thermal equilibrium with a radial temperature profile T ( r ) and the heat flux density (which is purely spatial in Eckart's description). This feature, which implies superluminal heat flow and is clearly unrealistic, is corrected in more refined causal thermodynamics formalisms [44-50], but the Eckart model is still the simplest and widely adopted model of dissipative fluid in relativity [51, 52].  \nThe SdS spacetime is locally static between the horizons, which remain at the Hawking and the Gibbons-Hawking temperatures, respectively. A stationary heat flux with purely radial density q r ( r ) > 0 is established and the resultant temperature profile T ( r ) is timeindependent. Heat flows from the hotter black hole horizon to the cooler de Sitter horizon, from which it escapes forever the locally static region between these two causal boundaries. There is no breakdown of thermodynamic laws, as suggested in the literature, and the explanation proposed here of the two-temperature puzzle is both simple and economical in its assumptions. Our main point is that the horizons being at vastly different temperatures  \ndoes not preclude thermal equilibrium in the SdS spacetime, as is often stated. Quite the opposite, the Tolman criterion of thermal equilibrium (1.5) in a static spacetime requires a temperature gradient: the temperature cannot be constant.  \nThe main limitation of this work consists of the noncausal character of the dissipative fluid model adopted and of the heat flux. It remains an open question whether more refined, causal thermodynamic formalisms provide the same answer to the puzzle of the two temperatures of SdS and thermal equilibrium.", 'ACKNOWLEDGMENTS': 'M. M. is grateful for the support of Istituto Nazionale di Fisica Nucleare (INFN) iniziativa specifica MOONLIGHT2. V. F. is supported by the Natural Sciences & Engineering Research Council of Canada (grant 202303234).', "Appendix A STANDARD TOLMAN-EHRENFEST CRITERION FROM ECKART'S LAW": "First, we need to prove the Buchdahl relation [33] expressing the fact that in a static spacetime, in coordinates adapted to the time symmetry, the 4-acceleration of a particle is a a ≡ ˙ u a = ∇ a ln ( √ -g 00 ) .  \nLet us consider the coordinate frame adapted to k a , in which k c k c = g 00 . An observer (e.g., a fluid element) following the Killing trajectory is at rest in these coordinates and has a 4-velocity  \nLet k a be the timelike Killing vector satisfying the Killing equation ∇ ( a k b ) = 0 . Its norm is conserved along the integral curve of k a , k a ∇ a ( g bc k b k c ) = 0 , and ∇ a k a = 0 .  \nu a = k a √ -k c k c (B.1)  \nand 4-acceleration  \n˙ u a ≡ u b ∇ b u a = -k b ∇ b k a k c k c . (B.2)  \nThe Killing equation ∇ ( a k b ) = 0 yields  \n˙ u a = -k b ∇ a k b k c k c = 1 2 ∇ a ( -k b k b ) -k c k c = ∇ a ( ln √ -k b k b ) = ∇ a ( ln √ -g 00 ) . (B.3)  \nTo complete the derivation of the Tolman-Ehrenfest criterion from Eckart's law  \nq a = -K h ab ( ∇ b T + T ˙ u b ) , (B.4)  \nconsider a viscous fluid with a stress-energy tensor of the form (1.6). The spatial components of the fluid's 4velocity and, consequently, the quantities describing dissipation vanish for a fluid at rest. In static coordinates the temperature T does not depend on time, u a ∇ a T = 0 . Then,  \nh ab ∇ b T = ( g ab + u a u b ) ∇ b T = ∇ a T . (B.5)  \nThe 'standard' definition of thermal equilibrium corresponds to the absence of heat flow, q a = 0 , in which case Eckart's law reads  \n∇ a T + T ˙ u a = 0 . (B.6)  \nThe Buchdahl relation now gives  \n∇ a T + T ∇ a ( ln √ -g 00 ) = 0 , (B.7)  \nor  \n∇ a ln ( T √ -g 00 ) = 0 (B.8)  \nyielding the 'standard' Tolman-Ehrenfest criterion  \nT √ -g 00 = T 0 . (B.9)", 'Appendix B STATIC SPACETIME AND STATIC FLUID': "Static spacetimes are characterized by a timelike Killing vector and, in coordinates adapted to the time symmetry, the metric coefficients are time-independent, ∂ t g µν = 0 and g 0 i = 0 . In spherical symmetry, the metric is also diagonal.  \nA static fluid in this coordinate system has 4-velocity u a orthogonal to the hypersurfaces of constant time, with  \n- [1] S. W. Hawking, 'Particle Creation by Black Holes,' Commun. Math. Phys. 43 , 199-220 (1975) [Erratum: Commun. Math. Phys. 46 , 206 (1976)] doi:10.1007/BF02345020\n- [2] R. M. Wald, 'General Relativity,' Chicago Univ. Pr., 1984, doi:10.7208/chicago/9780226870373.001.0001\n- [3] G. W. Gibbons and S. W. 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Kim, 'The Hawking temperature of a dynamical black hole in de Sitter spacetime,' Grav. Cosmol. 20 ,  \ncomponents  \nu a = δ a 0 √ -g 00 . (A.1)  \nand the fluid quantities are time-independent,  \n∂ t ρ = u c ∇ c ρ ≡ ˙ ρ = 0 , (A.2)  \nΘ ≡ ∇ c u c = 1 √ -g ∂ c ( √ -g u c ) = ∂ t u 0 = 0 . (A.3)  \n˙ q a ≡ u b ∇ b q a = u 0 ( ∂ t q a +Γ a 0 j q j ) , (A.4)  \nOne also has h b a ˙ q b = 0 , i.e., ˙ q a reduces to its time component in the static coordinates. In fact, u a = ( u 0 , 0 , 0 , 0 ) and q a = ( 0 , q i ) with ∂ t q i = 0 . Then,  \nwhere the first term on the right-hand side vanishes because q i is time-independent and the second term vanishes because  \nΓ i 0 j = 1 2 g ik ( ∂ t g jk + ∂ j g 0 k -∂ k g j 0 ) = 0 . 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