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VOLUME 75, NUMBER 7 PHYSICAL REVIEW LETTERS 14 AUGUsT 1995Thermodynamics of Spacetime: The Einstein Equation of State

Ted Jacobson*Department of Physics, University ofMaryland, College Park, Maryland 20742-411J(Received 23 May 1995)

The Einstein equation is derived from the proportionality of entropy and the horizon area togetherwith the fundamental relation BQ = T dS. The key idea is to demand that this relation hold for all thelocal Rindler causal horizons through each spacetime point, with BQ and T interpreted as the energyAux and Unruh temperature seen by an accelerated observer just inside the horizon. This requires thatgravitational lensing by matter energy distorts the causal structure of spacetime so that the Einsteinequation holds. Viewed in this way, the Einstein equation is an equation of state.PACS numbers: 04.70.Dy, 04.20.Cv, 04.62.+v

The four laws of black hole mechanics, which areanalogous to those of thermodynamics, were originallyderived from the classical Einstein equation [1]. Withthe discovery of the quantum Hawking radiation [2], itbecame clear that the analogy is, in fact, an identity.How did classical general relativity know that the horizonarea would turn out to be a form of entropy, and thatsurface gravity is a temperature? In this Letter I willanswer that question by turning the logic around andderiving the Einstein equation from the proportionality ofentropy and the horizon area together with the fundamentalrelation BQ = T dS connecting heat Q, entropy S, andtemperature T. Viewed in this way, the Einstein equationis an equation of state. It is born in the thermodynamiclimit as a relation between thermodynamic variables, andits validity is seen to depend on the existence of localequilibrium conditions. This perspective suggests thatit may be no more appropriate to quantize the Einsteinequation than it would be to quantize the wave equationfor sound in air.The basic idea can be illustrated by thermodynamics ofa simple homogeneous system. If one knows the entropyS(E,V) as a function of energy and volume, one can de-duce the equation of state from BQ = T dS. The firstlaw of thermodynamics yields BQ = dE + pdV, whiledifferentiation yields the identity dS = (BS/BE) dE +(BS/BV) dV. One thus infers that T ' = BS/BE andthat p = T BS/BV The l. atter equation is the equa-tion of state, and yields useful information if the func-tion 5 is known. For example, for weakly interact-ing molecules at low density, a simple counting argu-ment yields S = ln(No. accessible states) ~ lnV + f(E)for some function f(E). In this case, BS/BV ~ V ', soone infers that pV ~ T, which is the equation of state ofan ideal gas.In thermodynamics, heat is energy that fiows betweendegrees of freedom that are not macroscopically observ-able. In spacetime dynamics, we shall define heat as en-ergy that Aows across a causal horizon. It can be felt viathe gravitational field it generates, but its particular formor nature is unobservable from outside the horizon. For

the purposes of this definition it is not necessary that thehorizon be a black hole event horizon. It can be simplythe boundary of the past of any set 6 (for "observer").This set of horizon is a null hypersurface (not necessarilysmooth) and, assuming cosmic censorship, it is composedof generators which are null geodesic segments emanat-ing backwards in time from the set 6. We can considera kind of local gravitational thermodynamics associatedwith such causal horizons, where the "system" is the de-grees of freedom beyond the horizon. The outside worldis separated from the system not by a diathermic wall, butby a causality barrier.That causal horizons should be associated with entropyis suggested by the observation that they hide information[3]. In fact, the overwhelming majority of the informa-tion that is hidden resides in correlations between vacuumfiuctuations just inside and outside of the horizon [4].Because of the infinite number of short wavelength fielddegrees of freedom near the horizon, the associated "en-tanglement entropy" is divergent in continuum quantumfield theory. If, on the other hand, there is a fundamen-tal cutoff length l then the entanglement entropy is finiteand proportional to the horizon area in units of las longas the radius of curvature of spacetime is much longerthan l, . (Subleading dependence on curvature and otherfields induces subleading terms in the gravitational fieldequation. ) We shall thus assume for most of this Letterthat the entropy is proportional to the horizon area. Notethat the area is an extensive quantity for a horizon, as oneexpects for entropy [5].As we will see, consistency with thermodynamicsrequires that l, must be of order the Planck length(10 cm). Even at the horizon of a stellar mass blackhole, the radius of curvature is 10 times this cutoff scale.Only near the big bang or a black hole singularity or inthe final stages of evaporation of a primordial black holewould such a vast separation of scales fail to exist. Ouranalysis relies heavily on this circumstance.So far we have argued that energy Aux across a causalhorizon is a kind of heat flow, and that entropy of thesystem beyond is proportional to the area of that horizon.

1260 0031-9007/95/75(7) /1260(4) $06.00 1995 The American Physical Society


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VOLUME 75, NUMBER 7 PH YS ICAL REVIEW LETTERS 14 AUGUsT 1995It remains to identify the temperature of the system intowhich the heat is Bowing. Recall that the origin ofthe large entropy is the vacuum fluctuations of quantumfields. According to the Unruh effect [8], those samevacuum fluctuations have a thermal character when seenfrom the perspective of a uniformly accelerated observer.We shall thus take the temperature of the system to bethe Unruh temperature associated with such an observerhovering just inside the horizon. For consistency, thesame observer should be used to measure the energy fluxthat defines the heat How. Different accelerated observerswill obtain different results. In the limit that the accel-erated worldline approaches the horizon the accelerationdiverges, so the Unruh temperature and energy Aux di-verge; however, their ratio approaches a finite limit. It isin this limit we analyze the thermodynamics, in order tomake the arguments as local as possible.Up to this point we have been thinking of the systemas defined by any causal horizon. However, in general,such a system is not in "equilibrium ' because the horizon

is expanding, contracting, or shearing. Since we wish toapply equilibrium thermodynamics, the system is furtherspecified as follows. The equivalence principle is invokedto view a small neighborhood of each spacetime point pas a piece of flat spacetime. Through p we consider asmall spacelike 2-surface element 2 whose past directednull normal congruence to one side (which we call the"inside') has vanishing expansion and shear at p. Itis always possible to choose 'P through p so that theexpansion and shear vanish in a first order neighborhoodof p. We call the past horizon of such a I' the "localRindler horizon of 2," and we think of it as defininga systemhe part of spacetime beyond the Rindlerhorizonhat is instantaneously stationary (in "localequilibrium" ) at p. Through any spacetime point thereare local Rindler horizons in all null directions.The fundamental principle at play in our analysis is this:The equilibrium thermodynamic relation 6Q = T dS, asinterpreted here in terms of energy Aux and area of localRindler horizons, can be satisfied only if gravitationallensing by matter energy distorts the causal structure ofspacetime in just such a way that'the Einstein equationholds. We turn now to a demonstration of this claim.First, to sharpen the above definitions of temperatureand heat, note that in a small neighborhood of any space-like 2-surface element 2 one has an approximately flatregion of spacetime with the usual Poincare symmetries.In particular, there is an approximate Killing field ggenerating boosts orthogonal to 2 and vanishing at 2.According to the Unruh effect [8], the Minkowski vac-uum state of quantum fieldsr any state at very shortdistancess a thermal state with respect to the boostHamiltonian at temperature T = Rlr/2', where lr is theacceleration of the Killing orbit on which the norm ofA" is unity (and we employ units with the speed of lightequal to unity). The heat flow is to be defined by the

Tab A'(In keeping with the thermodynamic limit, we assume thequantum fluctuations in T,b are negligible. ) The integralis over a pencil of generators of the inside past horizonof 2 . If k' is the tangent vector to the horizongenerators for an affine parameter A that vanishes at 2and is negative to the past of P, then A' = rAk' anddX' = k' dA dW, where dA. is the area element on across section of the horizon. Thus the heat fIux can alsobe written as

AT bk k dAdA. . (2)Assume now that the entropy is proportional to thehorizon area, so the entropy variation associated with apiece of the horizon satisfies dS = zlzz%, where BA.is the area variation of a cross section of a pencilof generators of A. The dimensional constant zj isundetermined by anything we have said so far (althoughgiven a microscopic theory of spacetime structure one

FIG. 1. Spacetime diagram showing the heat flux BQ acrossthe local Rindler horizon A of a 2-surface element 2 . Eachpoint in the diagram represents a two dimensional spacelikesurface. The hyperbola is a uniformly accelerated worldline,and A' is the approximate boost Killing vector on A.

boost-energy current of the matter, T,by", where T,b isthe matter energy-momentum tensor. Since the tempera-ture and heat Bow scale the same way under a constantrescaling of g, this scale ambiguity will not prevent usfrom inferring the equation of state.Consider now any local Rindler horizon through aspacetime point p (see Fig. I). Let A" be an approximatelocal boost Killing field generating this horizon, with thedirection of ~' chosen to be future pointing to the insidepast of 2 . We assume that all the heat flow across thehorizon is (boost) energy carried by matter. This heatflux to the past of 2 is given by


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VOLUME 75, NUMBER 7 PHYS ICAL REVIEW LETTERS 14 AUGUs T 1995may someday be able to compute g in terms of afundamental length scale). The area variation is given by

OdAd&,where 0 is the expansion of the horizon generators.The content of BQ = T dS is essentially to requirethat the presence of the energy flux is associated witha focusing of the horizon generators. At 2 the localRindler horizon has vanishing expansion, so the focusingto the past of 2 must bring an expansion to zero at justthe right rate so that the area increase of a portion of thehorizon will be proportional to the energy flux across it.This requirement imposes a condition on the curvature ofspacetime as follows.The equation of geodesic deviation applied to the nullgeodesic congruence generating the horizon yields theRaychaudhuri equation

d0 1dA 2= o R,bk'k, (4)

where a. = o o. b is the square of the shear and R bis the Ricci tensor. We have chosen the local Rindlerhorizon to be instantaneously stationary at 1, so that 0and cr vanish at 1 . Therefore, the Oz and a. terms arehigher order contributions that can be neglected comparedwith the last term when integrating to find 0 near 2 . Thisintegration yields 0 = AR bk'k" for sufficiently smallA. Substituting this into the equation for BA. we find

AR bk k dAdA. .With the help of (2) and (5) we can now see that 6Q =T dS = (h~/2')gBA can be valid only if T,bk'k(hg/2')R, bk'k for all null k', which implies that

(27r/hrj)T, b = Rb + fg, b for some function f. Localconservation of energy and momentum implies that T,bis divergence free and, therefore, using the contractedBianchi identity, that f = R/2 + A for some constantA. We thus deduce that the Einstein equation holds:1 2mRab Rgab + Agab Tab (6)RgThe constant of proportionality g between the entropy andthe area determines Newton's constant as G = (4hg)which identifies the length g '/ as twice the Plancklength (hG) ' . The undetermined cosmological constantA remains as enigmatic as ever.

Changing the assumed entropy functional would changethe implied gravitational field equations. For instance, ifthe entropy density is given by a polynomial in the Ricciscalar nn + niR + . , then BQ = T dS will implyfield equations arising from a Lagrangian polynomial inthe Ricci scalar [9]. It is an interesting question what"integrability" conditions must an entropy density satisfyin order for 6Q = T dS to hold for all local Rindlerhorizons. It seems likely that the requirement is that theentropy density arises from the variation of a generally

covariant action just as it does for black hole entropy.Then the implied field equations will be those arising fromthat same action.Our thermodynamic derivation of the Einstein equationof state presumed the existence of local equilibrium con-ditions in that the relation BQ = T dS applies only tovariations between nearby states of local thermodynamicequilibrium. For instance, in free expansion of a gas, en-tropy increase is not associated with any heat flow, andthis relation is not valid. Moreover, local temperature andentropy are not even well defined away from equilibrium.In the case of gravity, we chose our systems to be dehnedby local Rindler horizons, which are instantaneously sta-tionary, in order to have systems in local equilibrium. Ata deeper level, we also assumed the usual form of shortdistance vacuum fluctuations in quantum fields when wemotivated the proportionality of entropy and horizon areaand the use of the Unruh acceleration temperature. View-ing the usual vacuum as a zero temperature thermal state[11], this also amounts to a sort of local equilibrium as-sumption. This deeper assumption is probably valid onlyin some extremely good approximation. We speculatethat out of equilibrium vacuum fluctuations would entailan ill-defined spacetime metric.Given local equilibrium conditions, we have in the Ein-stein equation a system of local partial differential equa-tions that is time reversal invariant and whose solutionsinclude propagating waves. One might think of these asanalogous to sound in a gas propagating as an adiabaticcompression wave. Such a wave is a traveling distur-bance of local density, which propagates via a myriadof incoherent collisions. Since the sound field is only astatistically defined observable on the fundamental phasespace of the multiparticle system, it should not be canon-ically quantized as if it were a fundamental field, eventhough there is no question that the individual moleculesare quantum mechanical. By analogy, the viewpoint de-veloped here suggests that it may not be correct to canoni-cally quantize the Einstein equations, even if they describea phenomenon that is ultimately quantum mechanical.For sufficiently high sound frequency or intensity oneknows that the local equilibrium condition breaks down,entropy increases, and sound no longer propagates ina time reversal invariant manner. Similarly, one mightexpect that sufficiently high frequency or large amplitudedisturbances of the gravitational field would no longerbe described by the Einstein equation, not because somequantum operator nature of the metric would becomerelevant, but because the local equilibrium conditionwould fail. It is my hope that, by following this line ofinquiry, we shall eventually reach an understanding of thenature of "nonequilibrium spacetime. "I am grateful to S. Corley, J.C. Dell, R.C. Myers,J.Z. Simon, and L. Smolin for helpful comments on thepresentation in earlier drafts of this letter. This work wassupported in part by NSF Grant PHY94-13253.

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VOLUME 75, NUMBER 7 PH YS ICAL REVIEW LETTERS 14 AU@Use 1995*Electronic address: jacobsonumdhep. umd. edu[1] J.M. Bardeen, B. Carter, and S.W. Hawking, Commun.Math. Phys. 31, 161 (1973).[2] S.W. Hawking, Commun. Math. Phys. 43, 199 (1975).[3] J.D. Bekenstein, Phys. Rev. D 7, 2333 (1973).[4] L. Bombelli, R.K. Koul, J. Lee, and R.D. Sorkin, Phys.Rev. D 34, 373 (1986).[5] Another argument that might be advanced in support ofthe proportionality of entropy and area comes from theholographic hypothesis [6,7], i.e., the idea that the stateof the part of the universe inside a spatial region can befully specified on the boundary of that region. However,currently the primary support for this hypothesis comesfrom black hole thermodynamics itself. Since we aretrying to account for the occurrence of thermodynamiclikelaws for classical black holes, it would, therefore, becircular to invoke this argument.

[6] G. 't Hooft, Utrecht Report No. THU-93/26, gr-qc/9310026 (to be published).[7] L. Susskind, Stanford Report No. SU-ITP-94-33, hep-th/9409089 (to be published).[8] W.G. Unruh, Phys. Rev. D 14, 870 (1976).[9] If the entropy density on the horizon takes the forme)' dA. , with p a function of the Ricci scalar, onefinds that BQ = T dS implies (2m/hrI). T,b = Rb,V,Vbp + fg, b for some function f [10]. Imposing thelocal energy conservation equation V'T, & = 0 then allowsone to solve uniquely for f up to a constant.[10] T. Jacobson, G. Kang, and R.C. Myers, McGill ReportNo. 94-95, Maryland Report No. UMDGR-95-047, gr-qc/9503020 (to be published).[11] D.W. Sciama, P. Candelas, and D. Deutsch, Adv. Phys.30, 327 (1981).

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