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Black hole entropy from graviton entanglement Eugenio Bianchi Perimeter Institute for Theoretical Physics 31 Caroline St.N, Waterloo ON, N2J 2Y5, Canada Wearguethattheentropyofablackholeisduetotheentanglementofmatterfieldsandgravitons across the horizon. While the entanglement entropy of the vacuum is divergent because of UV correlations, we show that low-energy perturbations of the vacuum result in a finite change in the entanglemententropy. Thechangeisproportionaltotheenergyfluxthroughthehorizon,andequals thechangeinareaoftheeventhorizon dividedby4timesNewton’sconstant–independentlyfrom the number and type of matter fields. The phenomenon is local in nature and applies both to black hole horizons and to cosmological horizons, thus providing a microscopic derivation of the Bekenstein-Hawking area law. The physical mechanism presented relies on the universal coupling of gravitons to the energy-momentumtensor, i.e. on theequivalenceprinciple. 3 1 0 Blackholesareperhapsthemostperfectlythermalob- Vacuum entanglement. The vacuum state of a 2 jects in the universe: there is convincing theoretical evi- quantum field has correlations at space-like separations. n dencethattheyarehot,theyemitthermalradiation,and This entanglement implies that, although the vacuum is a they are endowed with a thermodynamic entropy [1, 2]. a pure state, the reduced density matrix associated to a J Ablackhole canbe perturbedawayfromits equilibrium regionofspaceismixedandthereisanentanglementen- 7 state, for instance by letting some matter fall into it. tropy S associated to it [3]. Such entropy is divergent ent ] When the system reaches equilibrium again, the area A and in 4 space-time dimensions scales as c of the black hole horizon has grown by an amount δA. q - The increase in the entropy of the black hole δSBH is Sent =c0A0Λ2 +c1logΛ + c2, (2) r given by the Bekenstein-Hawking formula,1 g wherec andc arenumericalconstants,A istheareaof [ κc3 δA 0 1 0 2 δSBH = ~ 4G. (1) the surface bounding the region, Λ is a high-energy cut- off, andc is the finite partofthe entanglemententropy. v 2 Explainingthemicroscopicoriginofblackhole’sthermal The entanglement entropy of field theoretical modes 2 2 nature is a long-standing problem in theoretical physics. acrossthe horizonhas beenextensively studied asa pro- 5 In this letter we prove that low-energy perturbations posed explanation of black hole entropy [3]. As orig- 0 of the ground state of matter fields and gravitons on a inally formulated however, the proposal suffered from 1. black-hole background result in a variation of the hori- three problems: (i) the entropy Sent in eq.(2) is diver- 1 zon entanglement entropy δSent that is finite, univer- gent,(ii)toreproducethe1/4prefactorofeq.(1)thehigh 2 sal, and reproduces the Bekenstein-Hawking area law, energycut-offΛhastobetunedatthePlanckscale,(iii) 1 : δSent = δSBH. More specifically we consider a small suchtuning depends onallthe physicsfromlow-energies v patch that approximates Minkowski space at the hori- up to the Planck scale, in particular on the number of Xi zon,andstudy the regionboundedby aRindler horizon. species of matter fields we unfreeze going to higher and We show that an energy flux through the horizon corre- higher energies [7]. Here we present a resolutionof these r a sponds to a variation of the entanglement entropy. This puzzles byconsideringlow-energyprocessesin whichthe quantity is finite and is shown to be independent of the entanglement entropy changes. In such processes only number of matter species because of the universal cou- the low-energy part of the entanglement entropy is per- pling of gravitons to the energy-momentum. We argue turbed, and the change δS is shown to be insensitive ent that this result establishes entanglement as the micro- to the UV behavior of the entanglement entropy. scopic origin of black hole entropy [3]. The phenomenon Perturbative quantum gravity. In perturbative is local in nature [4], it holds for all non-extremal black quantum gravity [8], the graviton field h is defined as µν holes [1], and extends to the case of cosmological hori- the perturbation of the space-time metric g about the µν zons [5], thus explaining the universality of the formula flat Minkowski backgroundη , µν (1). The physicalmechanismpresentedis a perturbative version of the one recently discovered in the context of g =η +√32πGh . (3) µν µν µν loopquantumgravitywheretheentanglemententropyis finite because of non-perturbative effects [6]. AtleadingorderinaperturbativeexpansioninNewton’s constant G, the action for gravity coupled to a scalar field is givenby the Fierz-Pauliactionfor free gravitons, 1 Weworkwithunits κ=c=~=1and keep Newton’s constant plus the action of a free scalar field φ, plus a universal Gexplicit. Themetricsignatureis( +++). coupling of the graviton to the total energy-momentum − 2 tensor. Schematically, Entanglement and thermality. The Minkowski vacuum state Ω of the interacting quantum field the- I =Igrav[hµν]+Imatt[φ]+√8πGZ d4xhµνTµν, (4) ory described|byi the action (4) is Poincaré invariant. In particular it is invariant under boosts in the x direc- where T is the energy-momentum of matter fields and µν tionandthereforeit appearsstationaryto the uniformly gravitons. Varying the action with respect to the gravi- accelerated observers discussed above. Moreover, being ton field, we find the equation of motion the ground state, it is stable under dynamical pertur- 1 bations. These two properties suggest that accelerated (cid:3)h = √8πG(T η Tσ), (5) µν − µν − 2 µν σ observers see the Minkowski vacuum as a thermal equi- where (cid:3) = ∂ ∂µ and we have adopted the harmonic librium state. That this is in fact the case is the core µ gauge ∂νh = 1∂ hν . This equation will be crucial of the Unruh effect [9]. The physical mechanism behind µν 2 µ ν to the following. itsthermality isthe entanglementwithmodesacrossthe The perturbed Rindler horizon. A uniformly ac- Rindler horizon. This factis manifestwhenthe interact- celeratedobserverinMinkowskispacecannotreceivesig- ing Minkowski vacuum of matter fields and gravitons is nalsfrombehindtheRindlerhorizon. Letxµ =(t,x,y,z) written in the form [2] beCartesiancoordinatesadaptedtoaninertialobserver, 1 and (η,ρ,y,z) be Rindler coordinates, where η is a di- Ω = Dϕ Dϕ ϕ e−πK ϕ ϕ ϕ , (8) mensionless “angular” coordinate, i.e. t = ρsinhη and | i √Z Z L R h L| | Ri| Li⊗| Ri x=ρcoshη. The Minkowski line element is given by where ϕ stands for the matter field and the graviton L ds2 = ρ2dη2+dρ2+dy2+dz2. (6) (φ,hµν) both restricted to x<0, that is to the left half- − space, and similarly ϕ for the right half-space3. The Anobserveratρ=ℓhasauniformaccelerationa=ℓ−1, R hermitian operator K is the Rindler Hamiltonian and is causally connected only with the portion x>t of Minkowski space. Its boundary is the Rindler horizon. K = T χµdΣν. (9) Let (v,y,z), with v = t+x > 0, be coordinates on the Z µν future RindlerhorizonH. We callA the areaofasmall v patch v of the space-like surface (y,z) at given v. where χµ = ∂xµ is the boost Killing vector, ξµ = 1∂xµ B ∂η ρ ∂η In the presence of space-time curvature there is thesamevectornormalizedto 1,anddΣµ =ξµdρdydz − geodesicdeviation. Considerthesmallperturbationshµν is the volume element. The operator K is the generator of the metric away from flat space, and a beam of light ofboosts,i.e. translationsinη. As acceleratedobservers rays in H that crosses the plane v =0 drawing on it the cannotprobethe regionx<0, alltheir observationscan surface B0 of area A0. We call kµ = ∂∂xvµ the tangent be described using the reduced density matrix vectors to such light rays. At v > 0 they will draw a new patch with a different area. Because ofthe grav- e−2πK itational peBrvturbation, the beam is focused. The event ρ0 =TrL|ΩihΩ| = Z , (10) horizon, or perturbed Rindler horizon,is defined by light where the trace is on left modes ϕ of the matter field rays that are finely balanced between falling out of sight L andthegraviton. ThisdensitymatrixrepresentsaGibbs of the accelerated observer and escaping to infinity [2]. state [11], it is thermal with respect to boost evolution The area A of the event horizon is defined as the area H with associated geometric4 temperature T = 1 , [9]. of the cross-section ∞ of such light rays and given by geom 2π B The horizon is hot because of entanglement. ∞ A = A +√8πG ( (cid:3)h kµkν)vdvdydz, (7) Arealawfromentanglementperturbations. The H 0 Z0 ZBv− µν entanglement entropy of the Minkowski vacuum |Ωi is defined as the von Neumann entropy of its reduced den- at the leading order in √8πG. The effect is due to light sity matrix S (Ω ) = Tr(ρ logρ ). This expression deflectionbyagravitationalperturbationandcanbe de- ent | i − 0 0 is divergent due to the high-energy correlations present scribedgeometricallyintermsoftheRaychaudhuriequa- tion for the expansion of null geodesics2 [2]. 3 This property of the Minkowski vacuum can be derived usinga 2 In the harmonic gauge and at the linear order in the graviton pathintegralrepresentationforthegroundstateofaninteracting fiθeqe=ulda,∇ttiµohnkeµR∂∂voθicfc=niut√elln8gsπeoGordi(cid:3)sesRhicµµsννska=µtik−sνfi√.es8TπthhGeee(cid:3)lvihneµenaνtr,hiazoenrddizRothnaeyhceahxsapeuaxdnphsaiuonrn-i tqShuiteatonerrtyusmp[2a,fice9el][d.5s],Iotwnghaeenrseetrasatilimiczieblsalartcockr-othhsosel-ehHobaraircztkolegn-rHocouarnwrdeklia[n1tg0io]vnaasncdaupuopmneadoref. ssoioluntiθo(nv)of=th√e8RπaGycRhv∞au−dh(cid:3)uhriµνeqkuµaktνiodnvw′.ithThfiinsailsbtohuendaadrvyanccoend- 4 Tcehleeratteimonpearataunrde Ttagneogmentisvdeicmtoernsuioµnl=ess.a∂Axnµ/o∂bηsermveeraswuirtehsaacn- ditionθ(∞)=0. Theareachangeisrelatedtotheexpansionby energy density Tµνuµξν and a local temperature Tloc = a/2π, theequation∆A=R θdvdydz. See[4]foradetailedanalysis. theUnruhtemperature. 3 in the vacuum state as explained above. Notice however whereA istheoperatordefinedintermsofthegraviton H that in thermodynamics the physically relevantquantity fieldh andthe backgroundvalueA by eq.(7),andδA µν 0 is notthe entropyinitself, but howit changesin aphys- isthechangeinthe areaoftheeventhorizon. Theresult ical process. We consider now a perturbation of the vac- reproduces the variation of the Bekenstein-Hawking en- uumcorrespondingtoanenergyfluxthroughtheRindler tropy [1] of the Rindler horizon [4]. The entropy is due horizon, and compute the variation of the entanglement to the variation in the amount of entanglement for mat- entropy during the process. ter fields and gravitons, and has a universal expression Let beanexcitedstateofthesystem(4)ofinteract- in terms of the area change δA thanks to the universal |Ei ing gravitons and matter fields. It could be for instance coupling of gravitons to the energy-momentum tensor. the state of a particle crossing the Rindler horizon [12]. Black hole horizons and cosmological horizons. Wecallρ =Tr itsreduceddensitymatrix. Ifthe The result presented above for the Rindler horizon ap- 1 L |EihE| statehaslowenergy,thenρ canbeconsideredasasmall plies directly to the entropy of black hole horizons [1] 1 perturbation of ρ , i.e. ρ = ρ +δρ. The entanglement andofcosmologicalhorizons[5]. Theargumenthingeson 0 1 0 entropy S ( ) = Tr(ρ logρ ) has exactly the same the local nature of the results presented: we considered ent 1 1 |Ei − UV structure (2) as the vacuum state because only the a small surface and associated to it an entanglement 0 B low-energy modes of the two differ. In particular, the entropyperunitsurfacearea. Asdiscussedin[4],thisen- change in entropy tropy density governs near-horizon thermodynamic processes for all horizons. More specifically, in the case of a δS =S ( ) S (Ω ) (11) ent ent |Ei − ent | i non-extremal Kerr-Newman black hole, we can consider is finite and can easily be computed at the first order in a near-horizon co-rotating frame. The local geometry is the perturbation δρ, stationary and described by the Rindler metric7. Com- patibly with the equivalence principle, the vacuum state δSent = Tr(δρ logρ0)= 2πTr(Kδρ). (12) behaves locally as the Minkowski vacuum [15, 16], and − we canapply our analysisto this system: a perturbation The first equality follows from the identity Tr(δρ) = 0, of the vacuum corresponding to an energy-flux through the secondfrom the Gibbs form(10) of ρ . Moreover,as 0 the horizon results in an increase of the horizon entan- the boost energy is conserved, we can evaluate it in the glemententropybyδS =δA/4G,whereδAisthearea limit of large η in which the vector ξµ becomes light-like ent change of the event horizon of the black hole. The con- and parallel to kµ. In this limit, if no energy escapes to dition of applicability of our approximation is that the futurenullinfinity5,theboostenergybecomesanintegral perturbation is of low energy with respect to the scale over the horizon, of the UV cut-off and of short wavelength compared to K = T vkµdHν, (13) curvature scale at the black-hole horizon. Z µν Similarly in the case of de Sitter space, we can con- H sider a static patch. Static observers near the cosmolog- where kµ = ∂∂xvµ, dHµ = kµdvdydz, and we have used6 icalhorizonhave a largeaccelarationa=ℓ−1, where ℓ is χµ =Hvkµ, and dΣµ =HdHµ. As a result the proper distance from the horizon. The near-horizon geometry is again Rindler, independently of the positive δSent = 2π Tr Z Tµνkµkνvdvdydz δρ . (14) value of the cosmological constant. Near-horizon pro- (cid:0) H (cid:1) cesses can be considered [17] and the result (16) applies: Wecannowexpresstheenergy-momentumofmatterand the change in the entanglement entropy is δA/4G where gravitons in terms of the d’Alembertian of the graviton δA is the increase in area of the cosmologicalhorizon. field using the wave equation (5), Conclusions. In this letter we have shown that en- tanglementprovidesthemicroscopicexplanationofblack 2π δS = Tr (cid:3)h kµkνvdvdydz δρ . (15) holeentropy: causalhorizonshavethermalpropertiesbe- ent √8πG Z − µν (cid:0) H (cid:1) cause of entanglement, and the Bekenstein-Hawking for- As (cid:3)h controls the deflection of light rays and the ex- mula describes the variation in the amount of entangle- µν pansion of the area of the surface , we have ment during perturbations that have low-energies com- v B pared to the scale of a physical cut-off Λ. Such pertuba- 1 δA δS = 2π Tr(A δρ) = , (16) tions change only the infrared part of the entanglement ent H 8πG 4G entropy,leavingtheultravioletpartuntouched. Thevari- 5 Theenergyfluxatfuturenullinfinityvanishesforagenericstate because the only way for a wave not to fall across the Rindler 7 Near-horizonstationaryobservershaveaccelerationa=ℓ−1and horizonistotravelinthedirectionexactlyperpendiculartothe coincide with the ZAMOs of [14] as ℓ 0. Photons at the horizonasarguedin[13]. → local Unruhtemperature Tloc=a/2π thatmanage toescapeto 6 Weusethesymbol=HtodenoteequalityonthehorizonH. infinitywillbered-shiftedtotheHawkingtemperature[1] 4 ationδS isfiniteanduniversalbecauseofgravitational old is reached a new contribution to the entropy ap- ent backreaction: the universal coupling of gravitons to all pears. Similarly for other matter fields, so that we may matter allowsone to expressthe energyflux throughthe argue that at high energies there is an extra chemical- horizon in terms of the deflection of the light rays that potential contribution µ to the Bekenstein-Hawking en- define the horizon, therefore reproducing the geometric tropy δS = 2π( δA + µδN) where N is the number 8πG quantity δA/4G. The area A of the small patch ap- of unfrozen matter fields. An analogous phenomenon is 0 0 B pearing in eq.(2) is unchanged, what changes is the area being investigated in loop quantum gravity [23] and, to- A of the event horizon defined by eq.(7). The phe- getherwiththederivation[6]ofthehorizonentanglement H nomenon persists even in the absence of matter fields entropy,itmayleadto newphysicaleffects associatedto as gravitons couple to their own energy-momentum ten- the high-energy behavior of causal horizons. sor. The derivation of the universality of δS rests on ent Acknowledgements. Thanks to F. Cachazo, L. Frei- the equivalence principle, here formulated in a particle del, T. Jacobson, R. Myers, C. Rovelli, R. Sorkin, and physicist’s language [8]. A. Satz for discussions. This research is supported by Itisoftenarguedthatamicroscopicderivationofblack the Perimeter Institute for Theoretical Physics and by a hole entropy should consist in a counting of microstates Banting Fellowship through NSERC. [18, 19]. 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