Holographic Thermodynamics Requires a Chemical Potential for Color
aa r X i v : . [ h e p - t h ] J a n Holographic Thermodynamics Requires a Chemical Potential for Color
Manus R. Visser
Department of Theoretical Physics, University of Geneva,24 quai Ernest-Ansermet, 1211 Gen`eve 4, Switzerland
The thermodynamic Euler equation for high-energy states of large- N gauge theories is derived from thedependence of the extensive quantities on the number of colors N . This Euler equation relates the energy ofthe state to the temperature, entropy, number of degrees of freedom and its chemical potential, but not to thevolume or pressure. In the context of the gauge/gravity duality we show that the Euler equation is dual to thegeneralized Smarr formula for black holes in the presence of a negative cosmological constant. We also matchthe fundamental variational equation of thermodynamics to the first law of black hole mechanics, when extendedto include variations of the cosmological constant and Newton’s constant. Introduction.
The thermodynamics of black holes remains oneof most important theoretical advancements in gravitationalphysics of the past half-century. In semiclassical general rel-ativity the energy E , entropy S and temperature T of a blackhole can be identified with its mass, horizon area A and sur-face gravity κ , respectively, [1–3] (setting k B = ~ = c = 1 ) E = M S = A G T = κ π . (1)For static, asymptotically flat black holes these thermody-namic quantities are simply related by the Smarr formula [4] M = d − d − κA πG in d + 1 spacetime dimensions. The dimen-sion dependent factors are a peculiar property of the Smarrformula, which typically do not appear in the Euler equationof thermodynamics relating all extensive and intensive quanti-ties. Furthermore, for black holes in the presence of a negativecosmological constant Λ there is an additional term in the gen-eralized Smarr formula M = d − d − κA πG − d − πG [5, 6]. Inan extended version of black hole thermodynamics the cosmo-logical constant is interpreted as the pressure P = − Λ / πG ,and is treated as a thermodynamic state variable in its ownright [6–9]. Its conjugate quantity Θ in the extended first lawof black holes is regarded as (minus) the thermodynamic vol-ume. An important, but slightly odd aspect of this interpre-tation is that the mass of the black hole is identified with theenthalpy instead of the internal energy of the system (see [10]for a recent review).From a holographic perspective the dimension dependentfactors and the Λ term in the generalized Smarr formula re-main somewhat elusive. In holography, or gauge/gravity du-ality, the thermodynamics of black holes in the ‘bulk’ space-time is equivalent to the thermodynamics of large- N strongly-coupled gauge theories living on the asymptotic boundary ofthe bulk spacetime [11–13]. In particular, the thermodynamicvariables (1) for black holes correspond with the energy E , en-tropy S and temperature T of thermal states in the boundarytheory [14, 15]. The generalized Smarr formula relating thesevariables in the gravity theory should be dual to the Euler rela-tion for the thermodynamic quantities in the field theory. Butthe pressure interpretation of the Λ term in the Smarr formuladoes not directly carry over to the field theory, since the bulkpressure P is not dual to the boundary pressure p and the bulkthermodynamic volume is not related to spatial volume V of the boundary [16].In several works [6, 16–20] it has been suggested that vary-ing Λ is related to varying the number of colors N , or the num-ber of degrees of freedom N , in the boundary field theory.For gauge theories arising from coincident D-branes, varying N corresponds to varying the number of branes. Further, inconformal field theories (CFTs) the number of degrees of free-dom is given by the central charge C , whose variation takesus from one CFT to another. In holographic CFTs dual to Ein-stein gravity the central charge corresponds to C ∼ L d − /G [21–23], where L is the curvature radius of the bulk geometry,related to the cosmological constant via Λ = − d ( d − / L ,and G is Newton’s constant in d + 1 dimensions. So varying C in the boundary CFT is dual to varying Λ and G in the bulktheory. In addition, it was argued in [20] that varying Λ in thebulk does not only correspond to varying C (or N ), but also tovarying the volume V of the spatial boundary geometry. Thisis because the bulk curvature radius L is equal to the boundarycurvature radius for a particular boundary metric. We show,however, that for a different boundary metric varying Λ onlycorresponds to varying C (and E ) in the boundary theory, andnot to varying V . Overall, by building on (and refining) theholographic dictionary in [20], we propose a precise boundarydescription of extended black hole thermodynamics.In this paper we argue that the dual field theory descriptionof black hole thermodynamics requires a chemical potential µ for the central charge (see also [17, 18]). From the large- N scaling properties of the field theory we derive the holographicEuler equation E = T S + ν i B i + µC, (2)and show that it is holographically dual to the generalizedSmarr formula. Here ν i are additional chemical potentialsfor the conserved quantities B i (such as charge and angu-lar momentum). As expected, the dimension dependent fac-tors do not feature in (2), and the Λ term is incorporatedin µ . Moreover, E is the standard energy of the field the-ory and not the enthalpy. Strikingly though, there is no pV term in the large- N Euler equation. We explain why this isconsistent with the fundamental equation of thermodynamics, dE = T dS − pdV + ν i dB i + µdC , in which both V and C arevaried. Finally, we match this boundary variational equationwith the extended first law of black holes. Thermodynamics of large- N theories. We first derive the Eu-ler equation from the scaling properties of gauge theories atfinite temperature in the large- N ’t Hooft limit [24], N → ∞ for fixed coupling λ ≡ g N . Large- N SU ( N ) gauge theorieson compact spaces, with fields in the adjoint representation,exhibit a separation between low-energy states with energy of O ( N ) , and high-energy states for which the energy scales as E ∼ N [15, 25, 26]. This is because the low-energy excita-tions consist of color singlets, whose energy is independent of N , and at high energies all the N adjoint degrees of freedomcontribute on the same footing. The low-energy states are ina confined phase and are characterized by a thermal entropythat grows with energy, whereas the high-energy states are ina deconfined phase and behave as a gas of free particles (atnonzero λ there may also exist an intermediate phase [27]).Other gauge theories at finite temperature display a similar(de)confinement phase transition, but the energies in the de-confined phase scale with a different power of N , e.g. as N for an exotic theory in d = 6 with (0 , supersymmetry [28].In conformal theories the central charge C counts the num-ber of field degrees of freedom. For SU ( N ) gauge theo-ries with conformal symmetry the central charge scales as C ∼ N at large N , so high-energy states satisfy E ∼ C .Since holographic CFTs are the main examples of large- N theories we have in mind, we denote the number of degrees offreedom simply as C for all large- N theories.High-energy states in large- N theories obey interestinglarge- N scaling laws and are dual to black holes in holo-graphic field theories. By definition the internal energy ofthese equilibrium states depends on extensive quantities, suchas entropy S , volume V and conserved quantities B i , and onthe (intensive) central charge C , i.e. E = E ( S, V, B i , C ) . Formally, we can vary the energy with respect to each of thesequantities, while holding the others fixed. This leads to Gibbs’fundamental equation of thermodynamics dE = T dS − pdV + ν i dB i + µdC, (3)where the temperature T , pressure p , chemical potentials ν i ,and the chemical potential µ conjugate to C are defined as T ≡ (cid:18) ∂E∂S (cid:19) V,B i ,C , p ≡ − (cid:18) ∂E∂V (cid:19) S,B i ,C ,ν i ≡ (cid:18) ∂E∂B i (cid:19) S,V,C , µ ≡ (cid:18) ∂E∂C (cid:19) S,V,B i . (4)The variation of C in (3) moves one away from the originalfield theory content to a theory with a different number of de-grees of freedom. On the other hand, for variations whichonly compare different thermodynamic states within the sametheory, the variable C is kept fixed. Hence, depending on theensemble, the central charge could be varied or fixed in thefundamental equation of thermodynamics. However, we ob-serve next that the central charge necessarily has to appear inthe large- N Euler relation.The entropy and conserved quantities scale with the centralcharge for high-energy states,
S, B i ∼ C , reflecting the con- tribution from all the degrees of freedom. Thus, the energyfunction obeys the following scaling relation E ( αS, V, αB i , αC ) = αE ( S, V, B i , C ) , (5)with α a dimensionless scaling parameter. This means that inthe deconfined phase of large- N theories on compact spacesthe energy is not an extensive function. Differentiating withrespect to α and putting α = 1 leads to the Euler equation E = T S + ν i B i + µC. (6)Notice that pressure and volume do not appear in this Eulerequation, since the volume does not generically scale with C .It does scale with C in the infinite-volume limit of CFTs,i.e. pV = − µC as V → ∞ (see Appendix A). By varyingthis relation and employing the fundamental variational equa-tion (3), we find a slightly unusual Gibbs-Duhem equation SdT + pdV + B i dν i + Cdµ. (7)The variation of volume (instead of pressure) features in thisequation, since the Euler relation does not involve a pV term.Furthermore, in the grand canonical ensemble the thermo-dynamic potential or free energy is defined as W ≡ E − T S − ν i B i = µC. (8)It follows from the fundamental equation (3) that the grandcanonical free energy is stationary at fixed ( T, V, ν i , C ) . Theproportionality of free energy with C (or N ) is a signatureof deconfinement; in contrast, the free energy of the confinedphase is of order one [29]. In [20] the relation W ∼ N wasalready identified as the holographic origin of the generalizedSmarr formula, and it was coined the “holographic Smarr re-lation”. However, their matching between the large- N scal-ing relation (8) and the Smarr formula depends on a field the-ory pressure for compact spaces, whereas our correspondencehinges on the chemical potential.The Euler equation (6), or equivalently W = µC , onlyholds in a regime where /C corrections can be neglected.For generic CFTs on compact spaces this is the case in thehigh-temperature or large-volume regime T R ≫ , where R is the curvature radius. On the other hand, for holographic and d sparse CFTs the free energy already scales with the centralcharge at low temperatures T R ∼ O (1) (i.e. if ER ∼ C with C ≫ ). Further, the Euler equation is satisfied for anyvalue of λ , at weak and strong coupling, and for any large- N field theory, including conformal and confining theories, andtheories with unusual scaling behavior like Lifshitz theories.Each of these theories, though, satisfies a different equation ofstate, which is not encoded in the large- N Euler equation [20].For instance, the equation of state for conformal theories is E = ( d − pV , and for Lifshitz scale invariant theories withdynamical scaling exponent z it is given by zE = ( d − pV (see Appendix A). The fact that the Euler relation applies toboth conformal and Lifshitz theories, means that it not onlyholds for relativistic, but also for non-relativistic theories. Euler equation for two-dimensional CFTs.
Examples of large- N field theories are d modular invariant CFTs with large cen-tral charge c . The microcanonical entropy for these theories isgiven by the Cardy formula (setting c L = c R = c ) [30] S ( E L , E R , c ) = 2 π r c E L + 2 π r c E R , (9)with E L,R the left- and right-moving energies. On a circle oflength V = 2 πR , the total energy and angular momentum are,respectively, E = ( E L + E R ) /R and J = E L − E R . TheCardy formula holds for CFTs with a sparse light spectrum inthe regime C → ∞ with ER ≥ C [31], where we normal-ized the central charge (conjugate to µ ) as C = c/ . If weview the entropy (9) as the function S = S ( E, V, J, C ) , thenthe fundamental equation (3), with ν i dB i = Ω dJ , followsby taking partial derivatives of the entropy function. Conse-quently, the products of thermodynamic quantities are T S = 4 R p E L E R , pV = E, Ω J = E − R p E L E R , µC = − R p E L E R , (10)where Ω is the angular potential. They satisfy the relation E = T S + Ω J + µC. (11)Hence, the large- N Euler equation indeed holds for d CFTs. In fact, the Euler relation splits up into two separateequations, E = Ω J − µC and T S = − µC . Holographic black hole thermodynamics.
The large- N Eu-ler equation applies in particular to strongly-coupled large- N CFTs with a semiclassical, gravitational dual description. Wenow investigate the gravity dual of the Euler equation.The best-established example of holography, the Anti-deSitter/Conformal Field Theory (AdS/CFT) correspondence,states that the partition function of the CFT and of the grav-itational theory in asymptotically AdS spacetime are equal Z CFT = Z AdS [12, 13]. For field theories at finite temperaturethe thermal partition function is related to the free energy via W = − T ln Z CFT . On the other hand, the gravitational par-tition function is given by the Euclidean path integral, whichin the saddle-point approximation is computed by the on-shellEuclidean action, I E = − ln Z AdS [32]. Since thermal statesin the CFT are dual to black holes in AdS, the on-shell ac-tion should be evaluated on the classical black hole saddle.Here, we consider rotating, charged black hole solutions [5]to the Einstein-Maxwell action with a negative cosmologicalconstant, i.e. I E = − πG R d d +1 x √ g ( R − − F ) . In thegrand canonical ensemble (at fixed T and Φ ) the free energyof the holographic field theory corresponds to [33–35] W = T I E = M − κA πG − Ω J − Φ Q. (12)The final equality follows from evaluating the action – includ-ing the Gibbons-Hawking boundary term [32] and a back-ground subtraction term – on the black hole solution with an-gular momentum J and electric charge Q . The corresponding chemical potentials are the angular velocity Ω and the electricpotential Φ of the horizon. We note it is straightforward togeneralize this equation to black holes with multiple electriccharges and angular momenta [36–38].The thermodynamic Euler equation for these black holesfollows from (12) by inserting the holographic dictionary (1)for energy, entropy and temperature, and the dictionary for thecharge ˜ Q = QL and potential ˜Φ = Φ /L [20, 33], and usingthe relation (8) between free energy and chemical potential, E = T S + Ω J + ˜Φ ˜ Q + µC. (13)The thermodynamic variables in this equation are well-knownblack hole parameters, except for the chemical potential andcentral charge. What is their gravitational dual description?Essentially, µC is the on-shell Euclidean action (times T ). Inaddition, a different expression for the chemical potential isobtained from the generalized Smarr formula for AdS blackholes, which relates all the black hole parameters and is thusa gravitational reorganization of the Euler equation [5–8, 34,39] M = d − d − (cid:18) κA πG + Ω J (cid:19) + Φ Q − d − πG . (14)The Λ term is absent for asymptotically flat black holes, but isnecessary for the consistency of the Smarr formula of asymp-totically AdS black holes. The quantity Θ can be defined as R Σ bh | ξ | dV − R Σ AdS | ξ | dV [40], where a subtraction with re-spect to the pure AdS background is implemented to cancelthe divergence at infinity. In this definition the domain of in-tegration Σ BH extends from the horizon to infinity, while Σ AdS in the pure AdS integral extends across the entire spacetime.Further, ξ is the timelike Killing field ξ = ∂ t + Ω ∂ φ , which(in the black hole geometry) generates the event horizon, and | ξ | = √− ξ · ξ is its norm. In the literature [7, 8] (minus) Θ is often called the “thermodynamic volume”, since it is theconjugate quantity to Λ (the bulk pressure) in the first law ofblack hole mechanics, see equation (16). For our purposes,however, a geometric name is probably more suitable, suchas (background subtracted) “Killing volume”, because we areinterested in the field theory thermodynamics rather than thebulk thermodynamics.Comparing the Euler equation and the Smarr formula wesee that the chemical potential (times central charge) corre-sponds to three combinations of the black hole paramaters µC = M − κA πG − Ω J − Φ Q = 1 d − (cid:18) M − Φ Q − ΘΛ4 πG (cid:19) = 1 d − (cid:18) κA πG + Ω J − ΘΛ4 πG (cid:19) . (15)Note that the dimension dependent factors in the Smarr for-mula are absorbed in the chemical potential. Moreover, thechemical potential is not proportional to Θ , in contrast to pre-vious proposals [17, 41]. These references obtained a differentresult for the chemical potential, since they did not distinguishbetween the bulk duals to spatial volume and central charge.The expression above for the chemical potential shouldalso follow from its definition in (4), µ ≡ (cid:0) ∂E∂C (cid:1) S,V,J, ˜ Q . Wecheck this explicitly by rewriting the extended first law of AdSblack hole mechanics as a thermodynamic variational identity,where µ plays the role of the conjugate quantity to the centralcharge variation dC . For CFTs dual to Einstein gravity theholographic dictionary for the central charge depends on thecosmological constant and on Newton’s constant, hence thevariation of the central charge is dual to the variation of bothcoupling constants in the gravity theory [18, 20]. So we needto include the variations of both Λ and G in the bulk first law.The mass of rotating, charged AdS black holes can be re-garded as the function M ( A, J, Q, Λ , G ) . From a scaling ar-gument [4, 6] and from the generalized Smarr formula (14) itfollows that the extended first law for these black holes is dM = κ πG dA + Ω dJ + Φ dQ + Θ8 πG d Λ − ( M − Ω J − Φ Q ) dGG . (16)Usually, in extended black hole thermodynamics only thevariation of Λ is taken into account in the first law, but thevariation of Newton’s constant can be easily included by not-ing that M, J, Q ∼ G − [42]. Remarkably, the Λ and G varia-tions in (16) cannot be combined into one single term propor-tional to d (Λ /G ) , because there is a term remaining involvingthe variation of G . This seems to imply that the standard in-terpretation of the extended first law in terms of bulk pressure P = − Λ / πG is inconsistent, if Newton’s constant is allowedto vary. On the other hand, we can find a consistent bound-ary interpretation by expressing the right-hand side of the firstlaw in terms of variations of the entropy S = A/ G , electriccharge ˜ Q = QL , central charge C ∼ L d − /G and spatial vol-ume V ∼ L d − of the holographic field theory. To this endwe rewrite the extended first law above as dM = κ π d (cid:18) A G (cid:19) + Ω dJ + Φ L d ( QL ) − Md − dL d − L d − + (cid:18) M − κA πG − Ω J − Φ L QL (cid:19) d ( L d − /G ) L d − /G . (17)Here we used again the Smarr relation and d Λ / Λ = − dL/L ,and observed that ( d − dL/L − dG/G is equal to the fractionin the final term. It is crucial that the L and G variations ap-pear in this combination, otherwise the holographically dualfirst law would not involve a variation of the central charge.Consequently, from the holographic dictionary we deduce thatthe extended first law for AdS black holes is dual to the fun-damental equation in thermodynamics dE = T dS + Ω dJ + ˜Φ d ˜ Q − pdV + µdC. (18)By comparing (17) and (18) we see that the pressure p satisfiesthe CFT equation of state E = ( d − pV , and the chemicalpotential µ fulfils the Euler equation (13). This justifies ourdictionary for the chemical potential in (15). Note that weonly used the scaling properties V ∼ L d − and C ∼ L d − /G to arrive at (18), but did not need their proportionality con-stants, because the fractions dV /V and dC/C appear in thefirst law and hence the proportionality constants drop out.We found a precise match between the bulk and boundary“first laws”. Still there is one issue we need to address: thevariations of ˜ Q , V and C on the boundary are mixed in thebulk, because they all correspond to varying the AdS radius. Itwas pointed out in [20] how to disentangle these variations inthe bulk, but we want to add that V is more cleanly separatedfrom ˜ Q and C with a different choice of CFT metric. The de-pendence of ˜ Q and C on L is fixed by the AdS/CFT dictionary(and the definition of the gravitational action). However, therelation between V and L only holds for a particular choice ofCFT metric, ds = − dt + L d Ω k,d − , where the curvatureradius L is equal to the AdS radius (see Appendix C). Rescal-ing the CFT metric with the Weyl factor λ = R/L changesthe curvature radius into R and CFT time into Rt/L . Then,the holographic dictionary is replaced by [43] E = M LR , T = κ π LR , ˜Ω = Ω LR , ˜Φ = Φ
R . (19)For this choice of boundary metric, the volume V ∼ R d − is clearly distinct from the electric charge and central charge.Although both ˜ Q and C involve a variation of L , they can alsobe distinguished in the bulk: the central charge variation isdual to the combination of variations ( d − dL/L − dG/G ,and the electric charge variation can be identified as the vari-ation conjugate to ˜Φ . Importantly, when the dictionary in (19)holds, the bulk and boundary first law in (17) and (18) stillmatch and the chemical potential again satisfies the Euler re-lation. Thus, the volume, electric charge and central chargevariations in the CFT first law are in one-to-one correspon-dence with terms in the extended first law of AdS black holes.Finally, we comment on the criterion for chemical or diffu-sional stability to fluctuations of the central charge (if they arepossible, e.g. by coupling two CFTs with different C ’s)Chemical stability: (cid:18) ∂ W∂C (cid:19) T,V,ν i = (cid:18) ∂µ∂C (cid:19) T,V,ν i ≥ . (20)We have µ = W/C hence the derivative evaluates to − W/C ,which is negative if W > and positive if W < . So chem-ical stability changes precisely when the free energy switchessign. This provides a new perspective on the Hawking-Pagetransition [14] as a phase transition between chemically un-stable (thermal spacetime) and stable solutions (black holes). Discussion.
In gauge/gravity duality, black holes in the bulkcorrespond to thermal states in the boundary theory. We pro-posed a new dictionary between the bulk and boundary ther-modynamics, by introducing a chemical potential for the num-ber of colors in the gauge theory. The chemical potential isessential for the correspondence between the Euler equationfor large- N theories and the Smarr formula relating the blackhole parameters. Since the Euler relation determines the en-ergy as a function of other variables, it contains the essentialthermodynamic information about the field theory.Our field theory interpretation of the extended thermody-namics of black holes stands in contrast to the common grav-itational interpretation in terms of bulk pressure and volume.One notable difference is that the black hole mass is equiva-lent to the internal energy of the field theory, whereas in [6]it is identified with the enthalpy of the gravitational system.Moreover, we found that the extended first law of black holescannot be solely written in terms of the variation of bulk pres-sure P = − Λ / πG , if both Λ and G are allowed to vary,but can be consistently interpreted as a field theory first law.Thus, as the thermal field theory has a natural thermodynamicdescription, the boundary interpretation seems unavoidable.As for future work, we expect that the dictionary forthe chemical potential can be generalized to a multitudeof black holes in the presence of a cosmological constant,such as black holes in higher-curvature gravity, Lifshitz andhyperscaling violating solutions, black rings, and de Sitterblack holes. On the field theory side, an interesting problemis to extend the Euler equation beyond the large- N limit, byincluding /N corrections [20, 44]. Acknowledgments.
I would like to thank Jan de Boer,Pablo Bueno, Matthijs Hogervorst, Arunabha Saha and WatseSybesma for useful discussions, and Andreas Karch, Juan Pe-draza and Andrew Svesko for detailed comments on a draft.This work was supported by the Republic and canton ofGeneva and the Swiss National Science Foundation, throughProject Grants No. 200020-182513 and No. 51NF40-141869The Mathematics of Physics (SwissMAP).
Appendix A: Euler equation in flat space
In flat spacetime, static equilibrium states satisfy the stan-dard thermodynamic Euler equation, E = T S + ν i B i − pV, (A1)which is often formulated instead in terms of densities since V is infinite. Note that the energy is purely extensive in thisformula, since it satisfies E ( αS, αB i , αV ) = αE ( S, B i , V ) .This Euler relation applies in particular to conformal and Lif-shitz theories on the plane (see e.g. [45, 46]). It is not imme-diately clear why this equation is consistent with the large- N Euler equation, therefore in this appendix we explain the rela-tion between the two for Lifshitz scale invariant theories.Anisotropic scaling symmetry (cid:8) t, x i (cid:9) → (cid:8) ζ z t, ζx i (cid:9) withdynamical scaling exponent z implies that the product T R z isLifshitz scale invariant, where R is the curvature radius of thecompact space, such as a sphere. Therefore, for Lifshitz theo-ries with positive z the infinite-volume limit R → ∞ is effec-tively the same as T → ∞ , so on the plane these theories areessentially always in the high-temperature deconfining phase.In this limit, the energy scales as E ∼ T d − zz and entropyand conserved quantities as S, B i ∼ T d − z , so the scaling re-lation is E ( α d − z S, V, α d − z B i , C ) = α d − zz E ( S, V, B i , C ) . This imposes the condition ( d − z ) E = ( d − T S + ν i B i ) ,which in combination with the large- N Euler equation yields zE = − ( d − µC . We can now compare this to the Lifshitzequation of state zE = ( d − pV , which is a consequenceof the anisotropic scaling relation E ( S, α d − V, B i , C ) = α − z E ( S, V, B i , C ) . As a result, we find µC = − pV as V → ∞ , turning the large- N Euler equation into the standardone. The same argument works for conformal theories (by set-ting z = 1 ), hyperscaling violating theories and possibly otherlarge- N theories. Notably, the standard Euler equation onlyapplies in the infinite-volume limit of large- N theories. Thelarge- N Euler relation, on the other hand, also holds at finitetemperature on compact spaces for holographic field theoriesand d sparse CFTs (but not for generic CFTs). Appendix B: The extended first law of entanglement
In this appendix we compare our chemical potential forAdS black holes to the chemical potential in the extendedfirst law for entanglement entropy of ball-shaped regions inthe CFT vacuum [18, 47]. This CFT first law takes the form d ¯ E = ¯ T dS ent + ¯ µdC, (B1)where ¯ E denotes the modular Hamiltonian expectation value, S ent is the vacuum entanglement entropy of the ball-shapedregion and C is the universal coefficient of the entanglemententropy (commonly denoted as a ∗ d ) [23, 48, 49]. The CFTfirst law is dual to the first law of static hyperbolic AdS blackholes which are isometric to pure AdS space [50–52], a spe-cial case of the black holes considered in the main text, with J = Q = 0 . The boundary first law follows from reformulat-ing our fundamental variational equation (18) in terms of di-mensionless quantities ¯ E = M L, ¯ T = κL/ π , and ¯ µ = µL .The volume variation drops out of the first law, since it is a di-mensionful quantity. In the vacuum ¯ E = 0 , hence the chemi-cal potential in (13) reduces to ¯ µ = − ¯ T S ent /C , which agreeswith the results in [18] (where the temperature was normal-ized as ¯ T = 1 ). Appendix C: The renormalized holographic Euler equation
In the main text the energy was defined with respect to theground state, so the vacuum energy was effectively set to zero(except in the two-dimensional example). However, CFTs ona curved background exhibit the Casimir effect, which impliesthat the ground state could have non-vanishing energy. InAdS/CFT the ground-state energy can be computed with themethod of holographic renormalization, by regularizing thegravitational action with local counterterms at the boundary[21, 53]. In this appendix we derive the renormalized holo-graphic Euler equation for static vacuum AdS black holes, andfind that the ground-state energy contributes a constant termto the chemical potential associated to the central charge.We consider static, vacuum asymptotically AdS black holeswith hyperbolic, planar and spherical horizons [54] ds = − f k ( r ) dt + dr f k ( r ) + r d Ω k,d − , (C1)where f k ( r ) = k + r L − πGM ( d − k,d − r d − . (C2)For k = 1 the unit metric d Ω k,d − is the metric on a unit S d − sphere, for k = 0 it is the dimensionless metric L P d − i =1 dx i on the plane R d − , and for k = − the unit metric on hy-perbolic space H d − is du + sinh ud Ω k =1 ,d − . The massparameter M is related to the horizon radius r + via M = ( d − k,d − r d − πG (cid:18) r L + k (cid:19) . (C3)According to the GKPW prescription in AdS/CFT [12, 13]the CFT metric is identified with the boundary metric of thedual asymptotically AdS spacetime up to a Weyl rescaling, i.e. g CFT = lim r →∞ λ ( x ) g AdS where λ ( x ) is a Weyl scale factor.As r → ∞ the boundary metric approaches ds = r L dt + L r dr + r d Ω k,d − . (C4)A common choice of Weyl factor is λ = L/r , so that the CFTmetric becomes − dt + L d Ω k,d − . The curvature radius ofthe spatial geometry is then equal to the AdS radius and thespatial volume is V = Ω k,d − L d − / ( d − . Moreover, theCFT time is the same as the global AdS time t , which impliesthat the CFT energy E can be identified (up to a constant) withthe ADM mass M , the conserved charge associated to time t translations.The temperature, entropy and energy of the black holes are T = d r + k ( d − L πL r + , S = Ω k,d − r d − G , (C5) E ren = ( d − k,d − L d − πG r d + L d + k r d − L d − + 2 ǫ k d − ! . The energy was derived from the renormalized boundarystress-energy tensor in [53] and from the on-shell Euclideangravitational action with counterterms in [55]. The resultingenergy, E ren = M + E k , differs from the mass paramater bya constant term, the Casimir energy of the dual field theory E k = Ω k,d − L d − πG ǫ k , (C6)with ǫ k = 0 for odd d and equal to [55] ǫ k = ( − k ) d/ ( d − d ! for even d. (C7) For instance, ǫ k = − k/ for d = 2 and ǫ k = 3 k / for d = 4 . The renormalized version of the Smarr formula (14) reads E ren = d − d − T S − d − ren Λ4 πG , (C8)with a new (counterterm subtracted) Killing volume Θ ren = − Ω k,d − d (cid:18) r d + − d − d − L d ǫ k (cid:19) . (C9)The holographic Euler equation still takes the form E ren = T S + µ ren C, (C10)since the Casimir energy is also proportional to the centralcharge, which we normalize here as C = Ω k,d − L d − / πG .But the chemical potential is not given by (15) anymore, sinceit receives a constant contribution from the vacuum energy µ ren = − r d − L d − (cid:18) r L − k (cid:19) + 2 L ǫ k . (C11)For d = 2 we find the chemical potential µ ren = − r /L .For planar black holes ( k = 0 ) or very large hyperbolic orspherical black holes (with r + ≫ L ), the Casimir energy iseffectively zero and hence there is no distinction between therenormalized energy and the vacuum-subtracted energy. 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