---

BLACK HOLE TUNNELING ENTROPY

arXiv:gr-qc/9212010v1 15 Dec 1992ULB-TH 8/92TAUP 2017-92December 1992BLACK HOLE TUNNELING ENTROPYAND THE SPECTRUM OF GRAVITY⋆A. Casher and F. Englert†Service de Physique th´eorique, Universit´e Libre de Bruxelles, BelgiumandSchool of Physics and AstronomyRaymond and Beverly Sackler Faculty of Exact SciencesTel-Aviv University, Ramat-Aviv, 69978 Tel-Aviv, IsraelABSTRACTThe tunneling approach for entropy generation in quantum gravity is applied toblack holes.

The area entropy is recovered and shown to count only a tiny fractionof the black hole degeneracy. The latter stems from the extension of the wavefunction outside the barrier.

In fact the semi-classical analysis leads to infinitedegeneracy. Evaporating black holes leave then infinitely degenerate “planckons”remnants which can neither decay into, nor be formed from, ordinary matter in afinite time.

Quantum gravity opens up at the Planck scale into an infinite Hilbertspace which is expected to provide the ultraviolet cutoffrequired to render thetheory finite in the sector of large scale physics.⋆Supported in part by NATO CRG 890404† Postal address: Universit´e Libre de Bruxelles, Campus Plaine, C.P.225, Boulevard du Tri-omphe, B-1050 Bruxelles, Belgium.1

1. IntroductionTunneling in quantum gravity can generate entropy[1],[2].

To understand howsuch an apparent violation of unitarity may arise, let us first consider a classicalspace-time background geometry with compact Cauchy hypersurfaces. If quantumfluctuations of the background are taken into account, quantum gravity leaves no“external” time parameter to describe the evolution of matter configurations inthis background.

Indeed, the solutions of the Wheeler-De Witt equation[3]H|Ψ⟩= 0(1)where H is the Hamiltonian density of the interacting gravity-matter system cancontain no reference to such time when there are no contribution to the energyfrom surface terms at spatial infinity. This is due to the vanishing of the time dis-placement generator and even though the theory can be unambiguously formulatedonly at the semi-classical level, such a consequence of reparametrization invarianceshould have a more general range of validity.To parametrize evolution, one then needs a “clock” which could correlate mat-ter configurations to ordered sequences of spatial geometries.If quantum fluc-tuations of the metric field can be neglected, the field components gij at everypoint of space can always be parametrized by a classical time parameter, in accor-dance with the classical equations of motion.

This classical time, which is in facta function of the gij, can be used to describe the evolution of matter and consti-tutes thus such a dynamical “clock” correlating matter to the gravitational field[4].This description is available in the semi-classical limit of (1) where the classicalbackground evolving in time is represented by a coherent superposition of W.K.B.“forward” waves formed from eigenstates of (1). When quantum metric field fluc-tuations are taken into account, “backward” waves, which can be interpreted asflowing backwards in time, are unavoidably generated from (1) and the operationalsignificance of the metric clock gets lost outside the domain of validity of the semi-classical approximation.

Nevertheless, in domains of metric field configurations2

where both forward and backward waves are present but where quantum fluctua-tions are sufficiently small, interferences with such “time reversed” semi-classicalsolutions will in general be negligible‡. Projecting then out the backward wavesrestores the operational significance of the metric clock but the evolution markedby the correlation time is no more unitary: information has been lost in projectingthese backward waves stemming from regions where quantum fluctuations of theclock are significant.

This is only an apparent violation of unitarity which wouldbe disposed of if the full content of the theory would be kept, perhaps eventuallyby reinterpreting backward waves in terms of the creation of “universe” quantathrough a further quantization of the wavefunction (1).This apparent violation of unitarity is particularly marked if the gravitationalclock experiences the strong quantum fluctuations arising from a tunneling process.This can be illustrated from the simple analogy, represented in Fig.1, offered by anonrelativistic closed system of total fixed energy E where a particle in one spacedimension plays the role of a clock for surrounding matter and tunnels through alarge potential barrier. Outside the barrier, the clock is well approximated by semi-classical waves, but if on the left of the turning point one would take only forwardwaves, one would inevitably have on the right of the other turning point bothforward and backward waves with large amplitudes compared with the originalones.

The ratio between the squares of the forward amplitudes on the right and onthe left of the barrier for a component of the clock wave with given clock energyEc is the inverse transmission coefficient N0(Ec) through the barrier and providesa measure of the apparent violation of unitarity.In reference [2], it was shown that for a class of de Sitter type of space-times, which admit compact Cauchy hypersurfaces, tunneling could occur betweena “wormhole” and an expanding universe. It was proven that, for sufficiently smallcosmological constant, ln N0 played the role of a true thermodynamic entropy forthe metric clock transferable to matter in reversible processes.

Explicit evaluation‡ For a recent discussion of related problems see reference [5].3

gave for this tunneling entropy ln N0 = A/4 where A is the area of the event hori-zon. Thus one recovered in this way the horizon thermodynamics of Gibbons andHawking[6].The tunneling entropy ln N0 is in last analysis an effect of quantum fluctuationsin quantum gravity.

Therefore, despite the fact that no violation of unitarity wouldappear in a complete description including backward waves, this entropy shouldbe expressible in terms of density of states of matter and gravity. Tunneling offersan interesting perspective in this direction because it enlarges the semi-classicalwave function of space-time to include in its description the other side of thebarrier.

Unfortunately, for the space-times considered above, the other side is awormhole, that is in the classical limit simply a point on the Euclidean section ofthe original manifold. This makes it illusory to describe in semi-classical terms theconfigurations of the wormhole side of the barrier⋆.In the present paper, we shall show that for black hole geometries, the areaentropy is also interpretable in terms of tunneling.

Now, Cauchy hypersurfaces arenot compact and evolution can be described by the Minkowskian time availablefrom surface terms at spacelike infinity. However for spherically symmetric solu-tions, as long as no mass is brought into the system from infinity, the Minkowskiantime appears as an irrelevant unmeasurable phase in the quantum state of thesystem and the above argument can be repeated.

Tunneling amplitudes betweenspherically symmetric configurations with same total mass M can still be searchedfor and the related tunneling entropy is still expected to be ln N0 and equal toA/4. This will indeed turn out to be the case but the crucial new feature whichwill emerge is that the black hole is not connected by tunneling to a wormhole (infact, there are no such tunneling amplitudes), but to a large scale matter-gravityconfiguration describable in classical terms.

These configurations are analogousto macroscopic collapsing states frozen just outside the Schwartzshild radius. We⋆One could deform the Euclidean section to open the point into a Planckian size universebut this would not change the conclusion.4

shall call these configurations “achronons” because they are, for the outside ob-server deprived of any time dependent properties as a consequence of an infinitetime dilation. Achronons of given mass will be shown to be quantum mechanicallyinfinitely degenerate.

Hence black holes in quantum gravity have also infinite de-generacy as their wave function is connected by tunneling to the achronon side ofthe barrier. This means of course that the number of states exp A/4 counted bythe tunneling entropy is only a finite number of “surface” states out of an infiniteset of internal states which cannot belong to the same finite Hilbert space as thematter surrounding the black hole in a finite volume.

This mismatch will entirelymodify the black hole evaporation process at its last stage. In fact the evaporationmust stop when the black hole reaches the Planck scale, leaving a stable “planckon”remnant which can neither be created out of, nor decay into, ordinary matter in afinite external time.

Such objects were introduced previously[7] to avoid the viola-tion of unitarity which would arise from a complete black hole evaporation†. Theyfollow here directly from the tunneling structure of the black hole-achronon wavefunction.To strengthen these tentative conclusions, one should improve the present anal-ysis in two respects.

First, we have been restricted by the semi-classical treatmentof quantum gravity and we can only surmise that the crucial element which cameout of it, namely the infinite degeneracy of the black hole wave function, will sur-vive the full quantum description. Second, although the semi-classical treatmentdefines unambiguously the achronon from the tunneling, we have not realised sucha configuration in a genuine field theoretic way.

We have only examplified its fea-tures by a phenomenological model, which although consistent, is too schematic tobe directly physically relevant. Hopefully a more complete and realistic illustra-tion of the achronon will secure the explicit construction of the black hole-achrononwave function.Notwithstanding these limitations, the present approach gives strong support† For a comprehensive review on recent attempts to solve the black hole unitarity puzzle, seereference [8].5

to the planckon hypothesis. It would of course be of great interest to find at leastsome indirect evidence in favour of their existence.

This is not totally impossibleas planckons may have interesting cosmological and astrophysical consequences ifthey where present in the early universe as might be the case if primordial blackholes played an important role in cosmogenesis[9]. At a more fundamental level,they would provide, as a consequence of unitarity, a natural cut-offat the Plancksize for the ultraviolet spectrum of Hilbert space of states describing large scalephysics and are therefore expected to render quantum gravity expressible as a finitetheory.In the presentation of the paper, rather than deducing achronon configurationsfrom the analysis of black hole tunneling amplitudes, we found it more convenientto motivate the latter by first introducing achronons as classical solutions of generalrelativity.

This is exemplified in section 2 in a simple shell model. In section 3,we show how achronons surrounding a black hole can screen their temperature tozero or to a finite quantity if the black hole has, classically, a vanishingly smallSchwartzschild radius.

These properties are then used in section 4 to prove thateternal black hole are related by tunneling to achronon configurations. The inversetransmission coefficient is computed in the semi-classical limit and its relation toentropy is proven.

Contact is made between the black hole tunneling entropy andthe Gibbons-Hawking thermodynamics. In section 5, the infinite degeneracy ofquantum black holes of given mass is established.The nature of the planckonremnants follows then from the evolution of the potential barrier during the blackhole decay.

Their properties are reviewed and their bearing on the spectrum andthe scope of quantum gravity is discussed. Mathematical details are relegated tothe Appendix.6

2. The AchrononOur basic action in four dimensional Minkowski space-time will beS = Sgrav + Smatter(2)where Sgrav has the conventional form (G = 1) :Sgrav = −116πZ √−gR d4x(3)and Smatter contains sufficiently many free parameters to allow for the stress tensorsconsidered below.

In describing classical solutions, it should be kept in mind thatthey have to be interpreted as semi-classical solutions of (1). In particular, whenusing shells of infinite energy density, eventual smearing out by quantum spreadshould be understood.Spherically symmetric solutions in general relativity are entirely determinedin terms of the energy density function σ = T 00 and the radial pressure functionp1 = −T 11 in the coordinate systemds2 = g00(r) dt2 −g11(r) dr2 −r2(dθ2 + sin2 θ dφ2)(4)where we have restricted ourselves to tindependent configurations.The metrictensor is given byr2[g11(r2) −1] −r1[g11(r1) −1] = −2r2Zr1dM(r)dM(r) = 4πσr2dr(5)andg00(r2)g11(r2)g00(r1)g11(r1) = expr2Zr18πr(σ + p1)g11 dr.(6)For asymptotically flat solutions one chooses g00 = 1 at ∞; in absence of blackhole horizon, g11 > 0 and the solution considered is everywhere static.

The other7

components of the (diagonal) energy-momentum tensor pθ = −T θθ and pφ = −T φφare determined by the Bianchi identities and can be expressed in terms of p1 andσ:pθ = pφ = 14(σ + p1)8πr2p1 + 2M(r)/r1 −2M(r)/r+ 12rp′1 + p1. (7)Let us now consider a static spherically symmetric distribution of matter sur-rounded by an extended shell comprised between two radii ra and rb.

We defineˆσ ≡rbZraσg1/211 dr,ˆpθ ≡rbZrapθg1/211 dr,ˆp1 ≡rbZrap1g1/211 dr.(8)Assuming p1 = 0, one may perform the thin shell limit rb →ra = R in theseintegrals by using dM(r) = 4πσR2dr. From (5) and (7) one then gets4πRˆσ = (1 −2m−/R)12 −(1 −2m/R)12(9)8πRˆpθ =1 −m/R(1 −2m/R)12 −1 −m−/R(1 −2m−/R)12(10)ˆp1 = 0(11)where m and m−are the values of M(r) respectively at rb and ra and ms =m −m−is thus the mass of the shell.

Equations (9) and (10) are the standardresult[10]. As the radius R approaches 2m, these solutions become physically mean-ingless when ˆpθ becomes greater than ˆσ; this violates indeed the “dominant energycondition”[11], implying the existence of observers for which the momentum flow ofthe classical matter becomes spacelike.

In fact, the shell is mechanically unstableeven before this condition is violated[12].The divergence of ˆpθ when R →2m appears in (10) because of the vanishingdenominator in (7). Equation (10) depends however crucially on the radial pressurebeing zero inside the shell.

Relaxing this condition we see indeed that a finite valueof p1 multiplies in (7) the energy density σ which becomes infinite in the thin shell8

limit. It is in fact possible to avoid all singularities of the stress tensor as R →2mby requiring p1 inside the shell to satisfy, before performing the thin shell limit,4πr2p1 + M(r)r= 0.

(12)Inserting the solution of (12) back in (7), we get for the trace of the energy mo-mentum tensor T µµ the equation of stateT µµ = 2σ(13)which means that the source of the “Newtonian” force T 00 −1/2δ00T µµ due to the anyinner part of the shell on the remainder vanishes. An alternate way to discover thesolution (12) is precisely to impose the trace condition (13) in equation (7): the so-lution of this differential equation with p1(ra) fixed (and equal to −M(ra)/4π(ra)3)is equation (12).This solution is unsatisfactory if the (extended) shell sits in an arbitrary back-ground because of the finite discontinuity of the radial pressure across the shellboundaries which would lead to singularities in pθ.

We may ensure continuity ofthe radial pressure by immersing the shell in suitable left and right backgrounds.To avoid reintroducing stress divergences when rb approaches 2M(rb) these shouldsatisfy (σ + p1) = 0 at the shell boundaries. One can now perform the thin shelllimit.

The finite discontinuity of p1(r) at r = R leads toˆpθ = −ˆσ2 ,ˆp1 = 0(14)instead of (10),(11) and ˆσ is still given by (9). The dominant energy condition issatisfied everywhere, as is the “weak energy condition”[11] ensuring positivity of theenergy density for any observer.

Provided the background is smooth enough in theneighbourhood of the shell, no stress divergences will appear when it approachesthe Schwartzshild radius.9

Such static thin shells sitting outside the Schwartzschild radius but infinitesi-mally close to it will be referred to as “limiting shells”. The mass ms of the limitingshell plus the mass m−of the inner matter contribution is equal to the black holemass whose horizon would be at the limiting radius r = 2m.

The striking featureof the region bounded by the limiting shell is that it gives rise to an infinitely largetime dilation in the global Schwartzshild time. Indeedg00(r) = (1 −2M(r)r) exp−∞Zr8πr′(σ + p1)g11 dr′(15)and performing the explicit integration over the shell, we get in the region 0 ≤r <2mg00(r) = (1 −2M(r)r) R −2mR −2m−exp−R∞Zr8πr′(σ + p1)g11 dr′.

(16)Here the radius R of the shell is taken at R = 2m + ǫ where ǫ is a positiveinfinitesimal and the symbol R means that the integral is carried over the regularmatter contribution only. Clearly, g00(r) = O(ǫ) for 0 ≤r < 2m, t arbitrary.This domain of space-time is characterized by a Killing vector which is light-likein the limit ǫ →0.

When the space-time geometry presents a 4-domain endowedwith such a limiting light-like Killing vector, we shall call the domain an achronon.All spherically symmetric achronon configurations will exhibit an infinite dilationof the Schwartzshild time t with respect to the outside world, or equivalently,massless modes emitted by the achronon are infinitely redschifted. Classically, theachronon has the “frozen” appearance of a collapse at infinite Schwartzshild time.The difference is that it is also frozen in space-time.

This is the reason why, incontradistinction to collapsing shells, stresses (14) were needed to build the purelystatic solution considered above. However, requiring exact staticity everywhere inspace-time is mathematically convenient but perhaps a too stringent and physicallyunnecessary constraint.

Thus our shell solution (even if extended to a finite width)10

and the concomitant restriction on the background, should be viewed as a simpleillustrative model. More elaborate achronon solutions will be discussed elsewhere.3.

Thermal Screening of a Black HoleUp to now, we have considered achronons in a trivial space-time topology butthey can also be introduced in the topology of an eternal black hole. An eter-nal black hole of mass m0 contains two asymptotically flat Schwartzshild patchesconnected by a throat.

It represents the maximal extension of the Schwartzschildsolution which is singular at r = 0 and is dynamical outside the patches as seenfrom the well known Kruskal representation (Fig.2). One can add in the patches astatic distribution of matter without changing the topology as long as outer hori-zons are avoided.

We shall consider such distributions and we shall limit ourselvesto matter configurations which are identical in both patches. Thus, achronons ofmass m −m0 surrounding a black hole of mass m0 are defined in this topology bytheir matter distribution in a static patch.Let us now consider such a achronon, possibly surrounded by static matter.Using the metric (4) in the static patches, M(r) is defined in general from (5) forr > 2m0 byM(r) = m0 +rZ2m04πσr′2 dr′.

(17)The Kruskal metric isds2 = dT 2 −dX2F ′2(ξ)−r2(ξ) (dθ2 + sin2 θ dφ2)(18)which is related to the static metric (4) within a patch byg1/211 dr = dξ,ξ = 0 at the horizon,(19)11

andF(ξ) =pX2 −T 2. (20)A Cauchy hypersurface Σc represented in Kruskal coordinates by T = 0 (Fig.2)connects the space-time with Minkowskian signature to a solution of the EuclideanEinstein equations.

The latter can be described by the metric (4) with t = −iteand is periodic in the Euclidean time te. The Euclidean period T −1 can then becomputed from the metric (4) in the vicinity of the black hole horizon r0:T = 14π[g00(r0) g11(r0)]−1/2dg00(r)dr|r=r0(21)and from (5) and (6) one getsT =18πm0exp−∞Z2m04πr′(σ + p1)g11 dr′.

(22)Comparing (22) with (15), one immediately sees that the inverse Euclidean periodof a black hole surrounded by an achronon vanishes.To illustrate this phenomenon consider an achronon of mass m −m0 boundedby a limiting shell of mass ms ≤m −m0.The limiting shell sits at a radiusR = 2m + ǫ and we may rewrite (22) asT =18πm0limǫ→0 exp−R+ǫZR−ǫ4πr′(σ + p1)g11 dr′exp−R−ǫZ2m04πr′(σ + p1)g11 dr′exp−∞ZR+ǫ4πr′(σ + p1)g11 dr′. (23)The first factor is easily evaluated in the limit ǫ →0 and (23) yieldsT =18πm0 R −2mR −2m− 12exp−R∞Z2m04πr′(σ + p1)g11 dr′(24)where m−= m −ms.

A glance at (24) shows that the limiting inverse Euclidean12

period when ǫ →0 is indeedTǫ→0 = 0(25)We know, from the work of Gibbons and Hawking[7] that T in (21) is the tem-perature at infinity of quantum matter in the background of the classical gravity-matter system considered and is its equilibrium temperature in the energy con-jugate to the static time t. In particular, when no matter surrounds the blackhole, T reduces to the usual black hole temperature 1/8πm0. We shall show in thefollowing section that T is also the equilibrium temperature, in the semi-classicallimit of quantum gravity, of the interacting gravity-matter system itself.Thus,(25) implies that the thermal effects of a black hole of mass m0 can be entirelyscreened by a achronon of mass m −m0, as expected from the infinite redshift dueto the achronon.A different situation can however arise if the black hole has, classically, avanishing small mass.Such an object, which we shall call a germ black hole,generates a non trivial topology.

As long as the surrounding matter does not forma achronon, the value of T tends to infinity when the mass m0 of the germ tendsto zero. But in the presence of an achronon the resulting Euclidean periodicity cantake any value, depending on the limiting process.

In particular, one may haveTǫ→0 = Tm(26)where Ts is the inverse Euclidean period of a black hole of mass m surrounded bythe same matter distribution as the corresponding achronon of mass m−m0. Thisis exemplified in (24) by letting R −2m go to 0 as Cm20 and tuning the constantC to satisfy (26).The possibility of constructing, in presence of a germ, an achronon with thesame behaviour in Euclidian time as a genuine black hole will be the key to thetunneling between achronons and black holes.13

4. The Black Hole Tunneling EntropyWe first review⋆and generalize to the present case the description of tunnelingin quantum gravity, obtained in reference [2] for geometries with de Sitter topology.Consider in general two spacelike hypersurfaces Σ1 and Σ2 which are turning pointsin superspace (or turning hypersurfaces) along which solutions of the Minkowskianclassical equations of motion for gravity and matter meet a classical solution oftheir Euclidean extension.

Σ1 and Σ2 are thus the boundaries of a region E ofEuclidean space-time defined by the Euclidean solution. If E can be continuouslyshrunk to zero one can span E by a continuous set of hypersurfaces τ = constantsuch that τ ≡τ1 on Σ1 and τ ≡τ2 on Σ2.

These τ = constant surfaces define acoordinate system which we shall call synchronous; the Euclidean metric in E canbe written in the formds2 = N2(τ, xk) dτ2 + gij(τ, xk) dxi dxj(27)where N(τ, xk) is a lapse function. The Euclidean action Se over E, from Σ1 toΣ2, is obtained by analytic continuation from the Minkowskian action (2) and canbe written asSe(Σ2, Σ1) =ZEΠij∂τgij d4x +ZEΠa∂τφa d4x −ZEddτ (gijΠij) d4x−18πZE∂k[(∂jN)gkjqg(3)] d4x.

(28)Here Πij and Πa are the Euclidean momenta conjugate to the gravitational fieldsgij and to the matter fields φa; g(3) is the three dimensional determinant.On the turning hypersurfaces Σ1 and Σ2, all field momenta (Πij, Πa) are zeroin the synchronous system and the third term in (28) vanishes. The last term in(28) also vanishes if the hypersurfaces Σ1 and Σ2 are compact (which was the case⋆For a more detailed discussion see reference [2].14

considered in reference [2]) but may receive contributions from infinity otherwise.We shall have to consider here the case where the two non compact turning hy-persurfaces merge at infinity so that the Euclidean action Se(Σ1, Σ2) does not getcontributions in E from the last term in (28). The classical Minkowskian solutionin the space-time M1 containing Σ1 can be represented quantum mechanically bya “forward wave” solution Ψ(gij, φa) of the Wheeler-de Witt equation (1) in thesemi-classical limit.

At Σ1, this wave function enters, in the WKB limit, the Eu-clidean region E and leaves it at Σ2 to penetrate a new Minkowskian space-timeM2. The tunneling of Ψ(gij, φa) through E engenders in addition to the “forwardwave” solution a time reversed “backward wave”.

The inverse transmission coeffi-cient N0 through the barrier measures the ratio of the norms of the forward wavesat Σ2 and Σ1. For large N0 one may write in the synchronous systemN0 = exp −[2(ZEΠij∂τgij d4x +ZEΠa∂τφa d4x)].

(29)As all surface terms in (28) vanish in this system, (29) can be rewritten in thecoordinate invariant formN0 = exp [2Se(Σ1, Σ2)]. (30)Consider now an eternal black hole of mass m surrounded by a sphericallysymmetric distributions of matter, the same in both static patches.

Compare thisclassical solution of general relativity to another one consisting of an achronon ofmass m−m0(ǫ) surrounded by the same matter distribution and screening a germblack hole of mass m0(ǫ) →0 to the same Euclidean period. Both solutions arethus characterized by the same total mass M, the same matter distribution of massM −m outside the radius 2m + η, η infinitesimal† and the same Euclidean periodT −1.

We shall identify M1 with the achronon solution and M2 with the black† For the shell model of section 2, one may take η = ǫ as g00 and R −2m are of the sameorder of magnitude. For sake of generality we do not impose this relation here.15

hole one. We label by ΣB.H.cand ΣAc respectively the turning hypersurfaces in theblack hole and in the achronon geometries.ΣB.H.cand ΣAc can be represented in Kruskal coordinates by hypersurfacesT = 0 and are depicted in Fig.3.They belong to Euclidean sections of thesesolutions EB.H.

and EA which can be described by Euclidean Kruskal metrics (17)with Euclidean time Te = iT or by static coordinates (4) with a periodic Euclideantime te = it; for both solutions the period has the same value T −1. It is clear, fromthe static coordinate description, that the two Euclidean space-time geometriesEB.H.

and EA coincide for r > 2m+η but, while the Euclidean black hole terminatesat r = 2m, the achronon solution has an extra “needle” in the region 0 < r < 2mwhose 4-volume is of order ǫ.We now identify, at finite η, Σ1 with ΣAc and consider instead of a secondturning hypersurface Σ2 a hypersurface Σ′B.H.cwhich lies in EB.H. and is such thatr > 2m + η everywhere on it.

Σ′B.H.cis then contained in the intersection of EB.H.and of EA. When η →0, Σ′B.H.ccan be taken arbitrarily close to ΣB.H.cand we shallprove in the Appendix that all gravitational momenta on Σ′B.H.cin a synchronoussystem vanish in this limit.

We may then identify Σ′B.H.cwith Σ2. The region E isthus contained in the needle 0 < r < 2m+η of EA.

Because of the Kruskal twofoldsymmetry ΣAc is mapped onto itself by a Euclidean time rotation of half a periodand thus E spans only half the needle 4-volume. From (28), we learn that theinverse transmission coefficient N0 is simply the exponential of the total Euclideanaction of the needle.

Although the limiting 4-volume of the needle vanishes, theaction is computable as the difference between the Euclidean action of the blackhole SB.H.eover EB.H. and of the achronon SAe over EA.

This difference is finite andwell defined by cutting offthe two spaces at an arbitrary radius rc greater than 2mas the two geometries and the two actions coincide for all r > rc. We thus writeN0 = exp[SB.H.e−SAe ].

(31)To evaluate these actions we take advantage of the covariance to express themin terms of the static coordinate system with gravitational and matter momenta16

everywhere vanishing. Thus only the last surface integral in (28) contributes nowto the action and can be expressed asSe =Zddr−r24T [g00g11]−1/2dg00drdr.

(32)Using (21) and the fact that the integrand is the same at rc for SB.H.eand for SAe ,we getN0 = exp [4πm2 −4πm20(ǫ →0)](33)or, as m0 vanishes in the limit,N0 = exp A/4(34)where A = 16πm2 is the area of the event horizon of the black hole.We have thus learned that black holes are related by quantum tunneling toanother classical solution for gravity and matter, namely to an achronon⋆. Eachachronon is connected through a “potential” barrier to a corresponding black holeof mass m. Let us tentatively take boundary conditions in field space by assigningpure forward waves to achronons; the relative probability of finding a black holewith respect to an achronon is then N0, since in the classical limit interferencesbetween black holes propagating forward or backward in time must be negligible.Consider then two distinct achronons surrounded by matter distributions suchthat the total mass M is the same for both classical solutions but m and thesurrounding mass M −m need not be the same.Each achronon is related bytunneling to its corresponding black hole and the ratio of inverse transmissioncoefficients N0 between the two achronon-black hole configurations is, from (34),⋆One might have thought that a single point on EB.H.

would represent, in the classical limit,a wormhole-like turning hypersurface. This is not the case as there is no continuous mappingof ΣB.H.conto a point.

More generally, one can show that no lower manifold contained inEB.H. constitutes a turning hypersurface.17

equal to exp ∆A/4, where ∆A is the change in black hole area. From the differentialKilling identity of reference [13], or equivalently from the variation of the integratedconstraint equation over a static patch[2],we haveδM −T δA4 = δλHmatter(35)where λ labels the explicit dependence of the matter hamiltonian Hmatter deducedfrom the action (2) on all (non gravitational) “external” parameters.

As we areconsidering only contributions with fixed total mass M, it follows from (35) thatmatter configurations with neighbouring energies in a static patch of the black holewould be Boltzmann distributed at the global temperature T provided achrononswith different mass are taken to be equally probable. This is indeed a consistentassumption as all configurations describing achronons and surrounding matter withfixed total mass M have, from (35) with A = 0, the same total energy and may bedescribed by a microcanonical ensemble.

Thus the temperature of the static patchis indeed T . Therefore (35) also implies that δA/4 is the differential entropy ofthe black hole and that the latter is in thermal equilibrium with the surroundingmatter at the temperature T .

As the entropy must be an intrinsic property ofthe black hole, not only is equilibrium a consequence of the chosen boundaryconditions in field space but the converse is also true: the temperature obtaineddirectly from (35) with entropy identified as A/4 must agree at equilibrium with thethermal distribution generated from the field boundary conditions. This justifiesa posteriori the above choice of boundary conditions†.The tunneling approach to the horizon entropy and temperature[1],[2] appliedhere to the black hole differs from the analysis based on the Euclidean periodicityof Green’s functions[6] in two respects.

On the one hand, the present approachyields the thermal spectrum, and then the entropy, from the backreaction of thethermal matter on the gravitational field, in contradistinction to the Green’s func-tion approach. On the other hand however, the thermal matter considered here is† up to changes which would not alter the probability ratios in the large N0 limit18

taken in the classical limit while the Green’s function describes genuine quantumradiation. Both methods fall short of a fully consistent quantum treatment of thebackreaction problem but the interpretation of the horizon entropy from tunnelingwill permit us to uncover the quantum states building the black hole entropy; infact, we shall see that the number of states exp A/4 count only a minute fractionof the full black hole degeneracy.5.

From Achronons to PlanckonsThe entropy A/4 which can be exchanged reversibly from a black hole to ordi-nary matter was rederived in the preceding section from the existence of a “poten-tial barrier” between a black hole of mass m and an achronon of the same mass.This was done in the context of an eternal black hole admitting a Kruskal twofoldsymmetry with two achronons separated by an infinitesimal throat, each imbeddedin the surrounding geometry of a static space emerging from the eternal black holethroat. Within each space black hole-achronon states are in thermal equilibriumwith their surroundings.

We are therefore led to picture a black hole-achrononstate , in the semi-classical limit, as a quantum superposition of two coherent (nor-malized) states, |B.H.⟩and |A.⟩representing respectively a classical black hole anda classical achronon. The relative weight of the two states in thermal equilibriumis approximately, up to a phase, exp(−A/8) .

It follows from detailed balanceat equilibrium between radiated matter and the black hole that the same super-position should hold for a the black hole who would only emit (and not receive)thermal radiation at the equilibrium temperature. As a black hole formed fromcollapse indeed emits such a thermal flux, we infer that its state |C⟩should containan achronon component with the same weight as in thermal equilibrium.

We thuswrite|C⟩= |B.H.⟩+ exp(−A/8)|A.⟩. (36)To a single black hole configuration one may associate many distinct classicalachronon configurations.

In the shell model, for instance, there are infinitely many19

distinct classical matter configurations of the same total mass m. The argumentis however much more general and infinite quantum degeneracy of the achronon isa direct consequence of the infinite time dilation. Indeed, the Hamiltonian H is ofthe formH =Z √g00K(φa, gij, Πa, Πij) d3x(37)and all its eigenvalues are squashed towards zero by the Schwartzschild time di-lation factor √g00, thus generating an infinite number of orthogonal zero energymodes on top of the original achronon.

In the classical limit, the phase space ofzero-energy solutions becomes infinite and the Killing identity (35) with A = 0confirms that the modes give no contribution to the achronon mass m. The sameconclusions can be arrived at by considering the wave equation instead of thecanonical Hamiltonian. For example, the scalar wave equation is1√g∂µ√ggµν∂νΦ = 0.

(38)Thus the frequency of any mode is proportional to √g00 and vanishes in the limitg00 →0. By imposing boundary conditions at the surface, φ may be expandedin creation and annihilation operators for the above modes, thus realizing thedegenerate spectrum.The infinity of zero energy modes around any background implies an infinite de-generacy of achronons of given mass and thus an infinity of distinct quantum blackhole states of the same mass differing by the achronon component of their wavefunction.

This infinite degeneracy of the quantum black hole provides the reser-voir from which are taken the finite number of “surface” quantum states exp A/4counted by the area entropy A/4 transferable reversibly to outside matter.Except for providing a rational for the large but finite testable entropy of theblack hole, achronons do not modify the behaviour of large macroscopic blackholes. However when their mass is reduced by evaporation and approaches thePlanck mass the barrier disappears and quantum superposition completely mixes20

the two components. Of course, this means that both the description in terms ofsemiclassical configurations and of tunneling disappears.

What remains howeveras a consequence of unitarity, is the infinity of distinct orthogonal quantum statesavailable which have no counterpart in the finite number of decayed states. Thequantum black hole has become a planckon[7], that is a planckian mass object withinfinite degeneracy.

Causality and unitarity prevent the decay (and the production)in a finite time of such object[7], and the argument applies to the “parent achronon”as well. Indeed, if a state |Ai⟩of finite size and mass m decays, or is produced,within a finite time τ in an approximately flat space-time, the total number ofpossible final states is limited by the number N of orthogonal states with totalmass m in a volume τ3.

From unitarity, the degeneracy ν(m) of the states |Ai⟩isat most N . Thus if ν(m) →∞, the time τ tends to infinity.

Thus, achronons andin particular planckons can neither decay nor be formed in a finite time.As recalled in the introduction, planckons are a solution to the unitarity puzzle⋆arising from black hole evaporation and may have played a crucial role in seedingour universe and its large scale structure. At a more fundamental level they havefar reaching implications on the spectrum of quantum gravity.

The opening atthe Planck size of an infinite number of states, an unavoidable consequence of theexistence of planckons, may appear as a horrendous complication which could makequantum gravity definitely unmanageable but hopefully the converse may be true.Indeed planckons should make quantum gravity ultraviolet finite.The Hilbertspace of physical states available to macroscopic observer must be orthogonal tothe infinite set of states describing planckian bound states. Their wave functionat planckian scales where planckon configurations are concentrated are thereforeexpected to be vanishingly small.

In this way, planckons would provide the requiredshort distance cut-offfor a consistent field theoretic description of quantum gravitywithin our universe while leaving the largest part of its information content hiddenat the Planck scale.⋆see also reference[14].21

An operational formulation of quantum gravity applicable within our universeand based on conventional four dimensional gravity, may thus well be within reach.But it is nevertheless tempting to dwell upon the further significance of the pic-ture that emerges. The sudden widening of the spectrum of physical states at thePlanck scale strongly suggests that the relative scarcity of states which describelarge distance physics (as compared to the Planck size) is due to the fact that theexsistence of observables whose correlations survive at macroscopic range is con-tingent on the notions of scale and metric.

The appearance of these these conceptsin the organization of long long distance physics , at the cost of relegating most ofthe information to the Planck scale , would imply that scale should be absent froma fundamental description of the physical world. Consistency may then ultimatelyrequire a unified theory, of which string theory is perhaps a precursor, which byeliminating the gravitational scale from the basic formulation would render obsoletethe use of a standard of length or of time.AcknowledgementsWe are very grateful to R. Balbinot, R. Brout,J.

Katz, J. Orloff, R. Parentaniand Ph. Spindel for most enjoyable, stimulating and clarifying discussions.22

APPENDIXIn computing the tunneling amplitude from the achronon turning hypersurfaceΣAc to the black hole hypersurface ΣB.H.c, we have, in section 4, replaced ΣB.H.cby a hypersurface Σ′B.H.cwhich lay in the intersection of the Euclidean sectionsof the achronon and of the black hole solutions EA and EB.H. but could be takenarbitrarily close to ΣB.H.cin the limit η →0.

In this way the existence of a classicalEuclidean motion from ΣAc to Σ′B.H.cwas self evident and the computation of thetunneling Eq (31) was straightforward by identifying in the limit Σ′B.H.cto ΣB.H.c.It must be shown however that the limit is smooth enough so that the momentathat flows between ΣAc and Σ′B.H.cin a synchronous system vanishes indeed on thelatter hypersurface when η →0. This is proven below.Let us take, in the vicinity of the black hole throat a Euclidean Kruskal coor-dinate systemds2e = dT 2e + dX2F ′2(ξ)+ r2(ξ) (dθ2 + sin2 θ dφ2)(A.1)whereF(ξ) =qX2 + T 2e .

(A.2)Here ξ is defined by (19) so that the Euclidean static coordinate system can bewritten asds2e = g00(r) dt2e + dξ2 + r2(dθ2 + sin2 θ dφ2),(A.3)where2πT F(ξ)F ′(ξ)= g1200(ξ)(A.4)Consider a hypersurface Σ(T 0e ) which coincide with the Te = T 0e hypersurfacein an interval |X| < Xmax where Xmax is determined by the solution ofF ′−1(ξ) = g1/200 (ξ). (A.5)Thus at |Xmax| the lapse functions of static and Kruskal systems are equal.

We23

then complete the hypersurface Σ(T 0e ) by matching at |Xmax| hypersurfaces ofconstant te.For every T 0e > 0, we have r > 2m and hence one can always find a positive ηsuch that the hypersurface Σ(T 0e ) lays in the intersection of EA and EB.H.. We maythus choose Σ′B.H.cto coincide with Σ(T 0e ). We must then show that the momentumflow through Σ(T 0e ) vanishes in the limit T 0e →0.

More precisely, definingI ≡ZΣ(T 0e )Πijgij d3x,(A.6)we must prove thatlimT 0e →0 I = 0(A.7)as I is indeed the surface integral in (28) which has to vanish in order to validatethe tunneling result (30).The integral I receives only contributions from the region −Xmax < X < Xmaxand usingΠij =pg(3)32πN [gimgjn −gijgmn]∂τgmn(A.8)where N is the lapse function, one hasI = −XmaxZ0F ′(ξ) ∂∂Te r2(ξ)F ′(ξ)dX(A.9)Using (A.4) and (A.5), one gets|I| = T 0e12πTZT 0e1pF 2e −(T 0e )2 ∂∂ξ r2(ξ)F ′(ξ)dF < AT 0e12πTZT 0e1pF 2 −(T 0e )2 dF(A.10)24

where A is a positive number. Therefore, as T 0e →0,|I| < −AT 0e ln T 0e(A.11)and Eq (A.7) follows.25

References[1] F. Englert,“From Quantum Correlations to Time and Entropy” in “The Gar-dener of Eden” Physicalia Magazine (special issue in honour of R. Brout’sbirthday), (1990) Belgium, Ed. by P. Nicoletopoulos and J.

Orloff. [2] A. Casher and F.Englert, Class.

Quantum Grav. 9 (1992) 2231.

[3] B. De Witt, Phys.

Rev. 160 (1967) 113.

[4] T. Banks, Nucl. Phys.

249 (1985) 332,R. Brout, Foundations of Physics 17 (1987) 603,R.

Brout, G. Horwitz and D. Weil, Phys. Lett.

B192 (1987) 318. [5] W. Unruh and W.H.

Zurek, Phys. Rev.

D40 (1989) 1064,J.J. Halliwell, Phys.

Rev. D39 (1989) 2912.

[6] G. Gibbons and S. Hawking, Phys. Rev.

D15 (1977) 2738; 2752. [7] Y. Aharonov, A. Casher and S. Nussinov, Phys.

Lett. B191 (1987) 51.

[8] J.A. Harvey and A. Strominger, “Quantum Aspects of Black Holes”, PreprintEFI-92-41; hep-th/9209055.

[9] A. Casher and F. Englert, Phys. Lett.

B104 (1981) 117. [10] J. Frauendiener, C. Hoenselaers and W. Conrad, Class.

Quantum Grav. 7(1990) 585.

[11] S.W. Hawking and G.F.R.

Ellis, “The Large Scale of Space-Time”, (1973),(Cambridge University Press, Cambridge, England). [12] P.R.

Brady, J Louko and E. Poisson, Phys. Rev.

D44 (1991) 1891. [13] J.M.

Bardeen, B. Carter and S.W. Hawking, Comm.

Math. Phys.

31 (1973)161. [14] T.Banks,M.O’Loughlin and A.Strominger , RU-92-40 hep-th/9211030.26

FIGURE CAPTIONS.Figure 1. Tunneling of a nonrelativistic “clock”.The energy of the clock Ec is represented by the dashed line.

On the left of theturning point a the clock is well represented by a forward wave only depicted hereby a single arrow. On the right of the turning point b the amplification of theforward wave and the large concomitant backward wave are indicated.Figure 2.

The Kruskal representation of a black hole, eventually surrounded bystatic matter.The dashed straight line is the Euclidean axis Te. The dashed circle is the analyticcontinuation in Euclidean time of the solid hyperbolae representing trajectoriesr =constant in the static patches I and III.

These are separated from the dynam-ical regions II and IV by the horizons r = 2m0 where lay the past and futuresingularities r = 0 depicted by the dashed hyperbolae. The Schwartzschild time trun on opposite directions on the two hyperbolae r =constant and the Euclideantime te spans the period T −∞on the analytically continued circle.Figure 3.

Black hole-achronon tunneling.The figure represents the Euclidean sections of an achronon and of its correspondingblack hole. Each point is a 2-sphere and the circles span the Euclidean time te.The achronon geometry is depicted by thick lines and the black hole geometry bythin lines in the region where it differs from the first.

The picture is not on scaleas there are no 3-dimensional Euclidean imbedding of these surfaces.The curve a is the turning hypersurface ΣAc . The curve b is the turning hypersurfaceΣB.H.cThe curve c is the hypersurface Σ′B.H.cwhich lay in the intersection of EB.H.and EA and tends to ΣB.H.cin the limit η →0.27

---

출처: arXiv:9212.010 • 원문 보기