IFUP-TH 6/94 January 1994 gr-qc/9401027 4 9 Black Holes as Quantum Membranes 9 1 n a Michele Maggiore J 5 I.N.F.N. and Dipartimento di Fisica dell’Universit`a, 2 piazza Torricelli 2, I-56100 Pisa, Italy. 1 v 7 2 0 Abstract. We propose a quantum description of black holes. 1 0 The degrees of freedom to be quantized are identified with the 4 microscopic degrees of freedom of the horizon, and their dynam- 9 ics is governed by the action of the relatistic bosonic membrane / c in D = 4. We find that a consistent and plausible description q - emerges, both at the classical and at the quantum level. We r g present results for the level structure of black holes. We find a : v “principal series” of levels, corresponding to quantization of the i X area of the horizon. From each level of this principal series starts r a quasi-continuum of levels due to excitations of the membrane. a We discuss the statistical origin of the black hole entropy and the relation with Hawking radiation and with the information loss problem. The limits of validity of the membrane approach turn out to coincide with the known limits of validity of the thermo- dynamical description of black holes. 1 Introduction In the search of a quantum theory of gravitation black holes might play a role similar to the hydrogen atom in quantum mechanics: they might be the system in which important conceptual problems and possibly even contra- dictions arise in a sharp form, and from which we can start to understand what are the “rules of the game” when both quantum mechanics and general relativity play an important role. An important problem arising in this con- text is the information loss paradox raised by Hawking [1], which apparently suggests that the evolution of states in a black hole background violates such a basic principle as unitarity. A closely related issue is the origin of the black hole entropy: the relation between the entropy of a black hole and the area of the horizon, S = A/4 in Planck units, suggests the existence of a large number of microstates (on the order of exp 4πM2 for a Schwarzschild ∼ { } black hole of mass M), of which there is no trace in the classical description. Another, less discussed, but possibly very important issue is the relation be- tween black holes and elementary particles; ’t Hooft [2] has proposed that the two concepts should merge as we cross the Planck mass (see also [3]). The hearth of the problem is that we do not know what is a black hole at the quantum level. Indeed, it is not even clear what are the degrees of freedom to be quantized. Various authors, from early work of Bekenstein [4] to more recent works of ’t Hooft [2] and Susskind, Thorlacius and Uglum [5] have suggested that the black hole is a normal quantum system with discrete energy levels. But it is far from clear what is the “Schroedinger equation” for a black hole, if any, or whether it can be described by a wave function. A first important hint for understanding what are the degrees of freedom to be quantized comes from the membrane paridigm [6], in which the inter- action of a classical black hole with the external environment, as seen by fiducial observers1, is entirely described in terms of a fictitious membrane lo- cated close to the horizon and endowed with physical properties like electric conductivity, temperature and viscosity. At the quantum level, the description of the black hole horizon as a quantum surface with approximately one degree of freedom per Planck unit of area has been advocated by ’t Hooft [2, 7], and it is strongly suggested by 1Wewillhereafterrefertoobserversexternaltotheblackholeandstaticwithrespectto itas“fiducial”observers. Theprecisedefinitionoffiducialobserversforbothnon-rotating and rotating black holes is given in ref. [6]. 1 the proportionality between the entropy and the area of the horizon. A possible objection to such a description is that, although from the point of view of a fiducial observer the black hole behaves as if there were a membrane on the horizon, such a membrane certainly does not exist for the free falling observer. Then, it is not clear what is the meaning of assigning actual microscopic degrees of freedom to the membrane. Furthermore, in order to have a useful description it turned out [6] to be necessary to move the membrane a bit outside the horizon, defining the so-called stretched horizon. This allows to get rid of the details of the infalling matter and regularizes the infinite red-shift factor between the horizon and infinity. The amount of stretching is however quite arbitrary, so that the position of the membrane is determined by our “regularization” procedure, rather than by physics. Theanswertothefirstobjectionfollowsfromtherecentworks[5,8,9,10], where a principle of “black hole complementarity” is proposed. The principle states that the points of view of the free falling and fiducial observers are complementary, in a sense close to Bohr complementarity: even if an ideal “superobserver”, who can compare the measurements of both the fiducial and the free falling observers, would conclude that their observations are actually contradictory, still no real contradiction arises because such super- observers cannot berealized inNature. Moreprecisely, theanalysis ofseveral Gedanken experiments [9] indicates that any attempt to compare the results of the measurements of a free falling observer crossing the horizon and of a fiducial observer turns out to involve assumptions on physics beyond the Planck scale. Similar considerations have been made in ref. [11], where it is argued that the commutators of the operators measuring Hawking radiation with operators inside the horizon grow uncontrollably large and prevent si- multaneous measurements of the relative observables. This suggestion has been elaborated in the very recent paper [12]. This removes the first objection to the membrane approach. The mem- brane actually exists for the fiducial observer, who can legitimately assign it microphysical degrees of freedom and try to study the dynamics of this microscopic structure. The membrane is not detected by the free falling observer who is crossing the horizon, but this does not lead to any logical contradiction. The second objection, concerning the arbitrariness in the position of the stretched horizon, canbeanswered takingintoaccount thegeneralizeduncer- 2 tainty principle in quantum gravity, which assigns to the horizon a physical thickness because of quantum fluctuations [13]. The principle of black hole complementarity has deep physical conse- quences which imply a radical change in our view of space-time. In particu- lar, as discussed by Susskind [10], it implies that even the notion of invariant event cannot be anymore relied upon. It seems to us that, if the ideas sug- gested by ’t Hooft and by Susskind and collaboratorswill prove to be correct, theywillrepresent a majorconceptualadvanceinourunderstanding ofquan- tum gravity. If we accept the point of view of black hole complementarity, the mem- brane is the natural candidate for the location of the microstates of the black hole and the degrees of freedom of the membrane are interpreted as the variables to be quantized. In this paper, in order to test these ideas, we carry them a bit further and propose a specific dynamical framework for the membrane. More precisely, we wish to explore the consequences of the following assumption: from the point of view of a fiducial observer, a black hole is described by a membrane whose dynamics is governed by an action principle. The action is taken to be that of the closed relativistic bosonic membrane in D = 4 dimensions, which is proportional to the world-volume swept. In sect. 2 we will show that this description makes sense at the classical level. While it is known that no stable classical solution of this action exists in flat space, we will find that in the external, fixed, background of a black hole (or, more in general, in metrics with a horizon, as in Rindler space) there is a stable classical solution which lies just on the horizon and other classical solutions, whose motion is confined to the region outside the horizon and which approach the horizon in a logarithmically divergent time. This result, independently of the applications to black hole physics, might be of interest in itself in the study of relativistic membranes. We willthen proceed toa quantum description (sect. 3). At this stage, we will have to resort to an approximation in which the only degree of freedom retained is the radius of the membrane, which is taken to have spherical sym- metry. Some aspects of black hole physics, like the value of the entropy, are irretrievably lost in this approximation, since we are drastically reducing the number of degrees of freedom; however, many interesting features survive, and therefore it is a useful starting point, at least for qualitative understand- ing. The greatadvantageofthe approximationisthat theproblem isreduced 3 to an ordinary problem of quantum mechanics, rather than of quantum field theory. We will find that an interesting and plausible quantum structure emerges, both for Schwarzschild and more general Reissner-Nordstrom black holes. In the Rindler case, we will find that the dynamics of the membrane is governed by Liouville quantum mechanics. We will discuss the limits of va- lidity ofthemembrane approach(sect. 3.3)and, remarkably, we willfindthat they coincide with the limits of validity of the thermodynamical description of black holes found in ref. [14]. Using these results, in sect. 4 we will be able to propose an energy level structure for black holes. We will be mainly concerned with static properties, in particular with the energy level structure; however, we will also make some comments on the transition amplitudes between these levels, and their relation with Hawking radiation (sect. 5). It should bestressed at this point that, while themathematical formalism that we use is well-defined, the physical interpretation of our results, at the present stage, is only tentative. This is unavoidable in a situation in which we are trying to unravel what the “rules of the game” are, and what physical concepts are appropriate for a quantum description of black holes. Thus, our results can only be considered a possible starting point for further investigations. At the appropriate points we will discuss the limits of validity of our computations. However, we want to stress now that the membrane approach is not, at our present level of understanding, a fundamental description of black holes. We rather think to it as an effective description, valid in a domain to be determined below. In some sense, the situation is similar to the large distance effects in QCD, where the quark-antiquark potential at scales of order of 1 fm is described, at the effective level, in terms of a string. Of course, this string is a useful description only on a given scale of distances: if we look more closely, there is no string, but just quarks and gluons. This means that we should not worry of the fact that the quantum theory of a purely bosonic membrane can be defined consistently only (and at most) in D = 27 dimensions [15] (as bosonic strings require D = 26), nor of the fact that, unlike string theory, a membrane theory is not renormalizable by power counting. For the same reason, the membrane tension which will be introduced is not a fundamental constant, but, as the string tension in QCD, it is in principle derivable from the underlying theory. 4 In thispaper we will onlystudy black holeswith zero angularmomentum. Although we believe that there is no fundamental difficulty in generalizing our approach to Kerr black holes, in this paper we will make use of the technical simplifications due to spherical symmetry. 2 Classical membrane dynamics The action of the relativistic bosonic membrane is given by S = d3ξ det(g ∂ xµ∂ xν), (1) µν i j −T − Z q where is the membrane tension, and has dimensions of (mass)3. The mem- T brane world-volume is parametrized by ξi = (τ,σ ,σ ), that is xµ = xµ(ξ). 1 2 The indices i,j take values 0,1,2 and ∂ = ∂/∂ξi. The metric of the target i space is g , with µ,ν = 0,1,2,3 and the signature is Minkowskian. We use µν Planck units, h¯ = c = G = 1. The action (1) is the natural generalization of the relativistic action of a spinless pointlike particle, which is proportional to its world-line, andof a bosonic string, which is proportionalto the world-area swept. This action or modifications of it have been considered in various physical problems. It was first proposed by Dirac [16] in 1962 as a possible model for an extended electron. Later it has been studied both as a generalization of bosonic strings andwith motivations coming frombag modelsof hadrons [17, 18]. In recent years it has been used to investigate the bubble nucleation process in the early Universe in inflationary cosmology [19, 20], while its supersymmetric extension (“supermembrane”) received some attention as a generalization of superstrings [21, 22]. For pointlike particles and strings, the square root can be eliminated introducing an einbein or a world-sheet metric, respectively; similarly, here one can introduce a world-volume metric h ,i,j = 0,1,2. At the classical ij level, the action given in eq. (1) is equivalent [23] to the action S = T d3ξ√ h hijg ∂ xµ∂ xν 1 , (2) µν i j −2 − − Z h i whereh = deth . Two differences withthestringcasearenoteworthy. First, ij the appearance of a cosmological term in eq. (2). Second, the fact that the metric h cannot be gauged away even at the classical level. ij 5 The action (2) is invariant under general coordinate transformations of the world-volume. The equations of motion obtained with variation with respect to h are ij h = g ∂ xµ∂ xν . (3) ij µν i j Thus h is the metric induced by g on the surface of the membrane. Vari- ij µν ation with respect to xµ gives 1 ∂ √ hhijg ∂ xν √ hhijg ∂ xρ∂ xσ = 0. (4) i µν j ρσ,µ i j − − 2 − (cid:16) (cid:17) In order to find solutions of the equations of motion with spherical symmetry one proceeds as follows [17]. One introduces “polar fields”, i.e. a set of three fields r(ξ),θ(ξ),φ(ξ) (where ξ is a shorthand for (τ,σ ,σ )) defined by 1 2 xµ = (x0,rsinθcosφ,rsinθsinφ,rcosθ). (5) Next one fixes the gauge x0(ξ) = τ, θ(ξ) = σ , φ(ξ) = σ . (6) 1 2 This gauge fixing is useful also at the quantum level, since in this gauge there is no Faddeev-Popov ghost [24]. The parameters σ ,σ satisfy 1 2 0 σ π, 0 σ 2π. (7) 1 2 ≤ ≤ ≤ ≤ One now makes the spherical ansatz, r(τ,σ ,σ ) = r(τ). This reduces the 1 2 problem to a single degree of freedom r(τ). Of course, at the classical level this simply means that we are looking for special solutions of the equations of motion, but the solutions we get are exact. At the quantum level, instead, restrictiontomembranesofthisformisadrasticapproximation, analogousto the minisuperspace approximation used in quantum cosmology. However, in the case of black holes, the no-hair theorem ensures that at the classical level the horizon has spherical symmetry, so we expect that the approximation might be useful even in the quantum theory. In any application, an especially important issue is the existence of stable classical solutions. The classical equations of motion of the action (2) have been studied in flat target space, g = η , in refs. [17, 25]. In this case, µν µν no stable classical solution has been found. The membrane contracts under its self-attraction, until it collapses. While strings can stabilize themselves 6 rotating, this is not possible for a membrane in D = 4, since the centrifugal force will be zero on the rotation axis (stable rotating classical solutions exist in dimensions higher than four [26]). Alternatively, one can study the equations on spaces with non trivial topology, such as S2 RD−2, where × stable solutions exist [22, 24]. In the Dirac extended electron model, instead, the self attraction is bal- ancedby theelectrostatic repulsion, giving risetoa stablesolution. However, the model is unstable under quadrupole deformations from spherical symme- try [18]. The equations of motion have also been studied in the case of a self- gravitating membrane [19, 20, 27], in which the metric is generated by the membraneitself, sothatitistakentobeaReissner-Nordstrommetricoutside the membrane and a flat metric inside, with suitable junction conditions. We will instead be interested in the motion of a membrane in a background metric which is fixed from the beginning (it is generated by the matter which has collapsed to form the black hole) and in this section we only consider the situation in which the back-reaction of the membrane on the metric is neglegible. The limits of this approximation will be discussed in sect. 3.1. We will find that stable solutions of the action given in eq. (1) exist when the metric g is taken to be a metric with a horizon. Let us now analyze µν separately the Reissner-Nordstrom and Rindler cases. 2.1 Reissner-Nordstrom metric We consider a general Reissner-Nordstrom black hole with mass M and charge Q. We use Boyer-Lindquist coordinates, which are appropriate for a fiducial observer, and we only consider the metric outside the horizon. As we will see, this approach is consistent since, for the fiducial observer, a membrane initially outside the horizon always remains outside. The Reissner-Nordstrom metric is given by ds2 = αdt2 +α−1dr2 +r2(dθ2 +sin2θdφ2), (8) − 2M Q2 1 α = 1 + = (r r )(r r ), (9) − r r2 r2 − + − − and r = M √M2 Q2. The outer horizon is at r = r . The induced ± + ± − 7 metric is computed from eq. (3) and in the gauge x (τ,σ ,σ ) = τ is 0 1 2 (α2 r˙2)/α − − h = r2 , (10) ij   r2sin2θ     where r˙ = dr/dτ. Using the ansatz (5), the equation of motion (4) with µ = 0 gives immediately r˙2 ∂ r2α(α )−1/2 = 0. (11) 0 " − α # The equations with µ = 1,2,3 give a single second order differential equation rα′ rr¨+2(α2 r˙2)+ (α2 3r˙2) = 0, (12) − 2α − where α′ = dα/dr. Eq. (11) is the first integral of eq. (12), except that eq. (11) also admits the solution r˙ = r¨ = 0, r = arbitrary constant, while in eq. (12) this is a solution only if this arbitrary constant is equal to the horizon radius (so that α = 0). Then, the equation governing the radial motion is eq. (12), or eq. (11) with the solution r = r removed unless the 0 constant r is equal to r . Therefore, we have found that there is a stable 0 + static solution corresponding to a spherical membrane lying on the horizon. Integration of eq. (11) gives r˙2 = α2 C2r4α3, (13) − where C is an integration constant fixed by the initial conditions. (Note that timelike motions outside the horizon satisfy the condition α2 r˙2 > − 0). Eq. (13) is formally identical to the motion of a non-relativistic point particle with unit mass and zero energy in the potential U(r) = ( α2 + − C2r4α3)/2,r > r (see fig. 1). Depending on the sign of r˙(0), the membrane + can expand until it reaches a maximum value and then recontract, or it will contract immediately. In both cases, it reaches asymptotically r = r , in a + logarithmically divergent time (or in a linearly divergent time for extremal black holes, M = Q). This asymptotic approach to the horizon is just what we expect for any object, as seen by a fiducial observer. Notethat we never need to know the metricinside the“nominal” horizon, r = r . A membrane initially at r > r will always remain outside, and a + + 8 U(r) | r + r Figure 1: The effective potential discussed in the text. The classical motion of the spherical membrane is formally equivalent to the motion of a particle with E = 0 in this potential. fiducial observer can consistently describe the dynamics of the membrane without making any reference to the region r < r . Of course, this is not + surprising; it is just what happens for any object, pointlike or extended, from the point of view of a fiducial observer. Ifwesubstitutethesphericalansatz(5)intothemembraneaction, eq.(2), with the induced metric given by eq. (3) and integrate over the variables σ ,σ , we find the action 1 2 r˙2 1/2 S = 4π dτ r2 α (14) − T − α! Z The Euler-Lagrange equation of this action is eq. (12). Thus, eq. (14) is the effective action for the radial coordinate. It is interesting to study the stability of the solution that we have found under non-spherical deformations. Using again the gauge fixing (6) but keep- ing r(τ,σ ,σ ) generic, from eq. (4) with µ = 0 we immediately find the more 1 2 general equation r˙2(τ,θ,φ) = α2 C2(θ,φ)r4α3, (15) − 9