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Quantum Mechanics and Black Holes in Four-Dimensional

arXiv:hep-th/9112062v1 20 Dec 1991CERN-TH.6351/91ACT-55CTP-TAMU-100/91Quantum Mechanics and Black Holes in Four-DimensionalString TheoryJohn Ellis and N.E. MavromatosTheory Division, CERN, CH-1211, Geneva 23, SwitzerlandandD.V.

NanopoulosCenter for Theoretical Physics, Dept. of Physics,Texas A & M University, College Station, TX 77843-4242, USAandAstroparticle Physics GroupHouston Advanced Research Center (HARC),The Woodlands, TX 77381, USAandTheory Division, CERN, CH-1211, Geneva 23, SwitzerlandAbstractIn previous papers we have shown how strings in a two-dimensional target space reconcilequantum mechanics with general relativity, thanks to an infinite set of conserved quantumnumbers, “W-hair”, associated with topological soliton-like states.

In this paper we extendthese arguments to four dimensions, by considering explicitly the case of string blackholes with radial symmetry. The key infinite-dimensional W-symmetry is associated withtheSU(1,1)U(1)coset structure of the dilaton-graviton sector that is a model-independentfeature of spherically symmetric four-dimensional strings.

Arguments are also given thatthe enormous number of string discrete (topological) states account for the maintenanceof quantum coherence during the (non-thermal) stringy evaporation process, as well asquenching the large Hawking-Bekenstein entropy associated with the black hole. Definingthe latter as the measure of the loss of information for an observer at infinity, who -ignoring the higher string quantum numbers - keeps track only of the classical mass,angularmomentum and charge of the black hole, one recovers the familiar a quadratic dependenceon the black-hole mass by simple counting arguments on the asymptotic density of stringstates in a linear-dilaton background.CERN-TH.6351/91ACT-55CTP-TAMU-100/91

December 1991

1Introduction and SummaryString theory offers the possibility of resolving once and for all many of the deep-est problems in quantum gravity, such as finiteness, the reconciliation of quantummechanics with general relativity, the vanishing of the cosmological constant, theorigin and flatness of the Universe, and even the satisfactory definition of the grav-itational path integral. Varying amounts of progress have been made in presentingthe actual solutions to these fundamental problems.

For example, it has been shownthat all n-loop string amplitudes calculated in a fixed flat background are finite [1],but the enumeration of classical non-perturbative backgrounds and understanding ofthe string path integral are far from complete. Some of these fundamental problemshave been addressed from the point of view of the effective field theory of light stringstates, such as the existence of hair that might help us understand whether stringblack holes respect quantum coherence [2], and the discovery of “no-scale” models[3] that ensure the vanishing of the cosmological constant in some approximation.However, the resolutions of all these problems presumably require non-perturbativestring theory techniques.Such techniques have recently been succesfully developed and applied to two-dimensional string quantum gravity, leading first to the non-perturbative solutionof matrix models [4] and more recently to the construction of two-dimensional stringblack holes [5, 6, 7].

Earlier constructions exact cosmological solutions of subcriticalstring theory as conformal Wess-Zumino models had also been known [8]. One couldhope that these non-perturbative techniques have advanced sufficiently for the res-olutions of some of the above-mentioned fundamental problems to be within reach.Indeed, we have argued in a recent series of papers [9, 10, 11] that string theoryreconciles quantum mechanics and two-dimensional gravity.

We have identified aninfinite set of exactly-conserved gauge quantum numbers, “W-hair”, which are asso-ciated with the topological solitons 1 that form the final stages of two-dimensionalblack hole evaporation [9, 5, 6]. As a result of this W-symmetry, the two-dimensionalphase space volume of the matrix model is conserved under time evolution [14, 10],excluding the introduction of modifications of the conventional S-matrix or Hamil-tonian evolution of the density matrix [15] in two dimensions.

Furthermore, we havedemonstrated explicitly that the evaporation of a two-dimensional black hole is apurely quantum-mechanical higher-genus effect that does not introduce mixed states[11].In this paper we extend these arguments to argue that four-dimensional stringblack holes do not lead to mixed states, and hence that string reconciles quantummechanics with general relativity also in four dimensions. The central problem ofquantum coherence was seen originally for spherically-symmetric four-dimensional1These discrete states were discovered in the context of matrix models by Gross and Klebanov[12], and in the continuum Liouville theory by Polyakov [13].

However their physical significance,especially for black hole physics, seems not to have been recognized by these authors.

black holes [16, 17], and the study of rotating four-dimensional black holes has notaltered the dilemma, so we emphasize here spherically-symmetric four-dimensionalblack holes 2. These are described by dimensional reduction of four-dimensionalstring theories.

The dimensionally-reduced theory is expressible as a SU(1,1)U(1)cosetconformal field theory of the two dimensional string black hole, and thus is an exactsolution. The radially-dependent dilaton field enters just as in the two-dimensionalmodel, whose massless “tachyon” represents model-dependent matter fields in fourdimensions.

There is infinite-dimensional W-hair associated with this SU(1,1)U(1)cosetmodel3, that conserves the 2-dimensional radial s-wave phase space volume element,and thereby prevents the appearance of non-quantum-mechanical terms in the S-matrix description of s-wave scattering. Just as in two dimensions, the quantum-mechanical evaporation of the four-dimensional black hole is a higher-genus effectthat does not involve mixed states.

Nevertheless, by simple counting argumentson the multiplicity of string states, one recovers the Hawking-Bekenstein formula[17, 19] for the apparent entropy of a black hole (S ∝M2) if one restricts oneselfto the classical conserved charges of a black hole (mass, angular momentum andelectric charge) and disregards the infinite string set of conserved quantities.2Non-local stringy gauge symmetriesThe central obstruction to reconciling quantum mechanics with general relativity,and thereby avoiding the evolution of pure states into mixed states, is the apparentloss of information across the horizon surrounding, e.g., a conventional black hole.This can be expressed mathematically via the formula [17, 19]S = 14kBA(1)for the entropy S associated with a horizon of area A (kB denotes Boltzmann’s con-stant). Using the usual relation between the mass and horizon area of a spherically-symmetric black hole, we findS = 1¯hkBM2(2)The entropy (1,2) is just one aspect of the black hole thermodynamics induced byquantum effects in local field theories.

There is also an apparent temperatureT =¯h8πM(3)2The extension of our arguments to rotating black holes is technically more complicated, butpresumably does not raise any fundamental issues of principle, since string has infinitely manygauge symmetries. See the discussion of the Kerr solution in section 4.3The one-dimensional axion “hair” found in the effective field theory approach [18] is just a singlestrand of this infinite-dimensional W-hair, and cannot by itself reconcile black hole dynamics withquantum mechanics.

associated with a spherically-symmetric black hole. There is an alternative statisticaldefinition of the entropy of a black hole :S = −kBlnNH(4)where NH denotes the total number of quantum-mechanical distinct ways that ablack hole, of given mass, angular momentum, and charge could have been made.The number NH can be viewed as counting the number of possible independentmicroscopic states of the black hole atmosphere.

The problem is that in any localfield theory there is only a finite number of conserved gauge quantum numbers, sothe entropy (1) or (2) can only be accommodated by a mixed state. Also it is clearthat a thermal state is necessarily mixed.

Recently attacks have been made on thisproblem using effective field theories derived from the string, which contain a finitenumber of additional conserved quantum numbers, associated, e.g., with the axion[18] or with discrete gauge symmetries [2]. However, these still do not touch thecore of the problem presented by the entropy (1) or (2), which seems to require aninfinite number of exactly conserved quantities, if the black hole state is not to bemixed.However, a stringy black hole has an infinite set of hair associated with the infinityof gauge symmetries that characterise any string theory [20, 21].

In this case the largeentropy (1) defined by Hawking is avoided by the following argument 4. Classically,mass, angular momentum and charge are the only type of observable hair that ablack hole can have, and hence the necessity of a mixed state to account for the largeentropy.

In string theories the entropy is zero, since quantum mechanics is valid andpure states never mix, due to the arguments in [9], and the evaporation scenarioof [11] is in vacuo and does not involve mixed states. It is the information carriedby the infinite string hair that makes the difference from the previous calculationof the entropy (1,2).

To get the latter, one considers only the classical charges ofthe black hole and treats the (infinity of) string gauge charges (associated with thethe rest of the excited string states-which in the effective two-dimensional case aretopological) as unobservable, using them just to count the number of quantum-mechanically distinct ways that a black hole of given mass, electric charge, andangular momentum is made. In this way, eq.

(4) accounts for the large Hawking-Bekenstein entropy, as we shall show explicitly in section 3.It is instructive to review briefly at this point target-space gauge symmetries incritical strings. The first approach to such symmetries was that of refs.

[20, 21],who showed that there exists in string theory an infinite set of generalized Wardidentities inter-relating states of different spin and mass. The lowest such identity4Notice that a local field theory has necessarily a finite number of conserved charges, hencethe thermal evaporation scenario for the black holes seems the only consistent one, with all theinevitable consequences on the loss of quantum-mechanical coherence.

is that expressing general coordinate invariance:qµ < V Gµν(q)ΠNi=1V T(ki) >=NXi=1kiν < V T(ki + q)Πj̸=iV T(kj) >(5)and another involves states of rank four, three, and two [21]kµA(µ|ν|ρ)σ −iBνρσ =NXi=1kiνGρσ(ki + k, {kj; j ̸= i}) + perms(ν, ρ, σ)(6)where µ|ν|ρ ≡µνρ + ρνµ.In on-shell cases the sums on the rhs of the aboveequations (5,6) vanish on the basis of the cancelled propagator argument [22].Forour purposes we shall only deal with on-shell modes, in which case one avoids theusual ambiguities of extending these identities offstring-shell [21]. Ref.

[23] gave aconformal field theory analysis of such gauge symmetries, showing that there wasone associated with every (1, 0) or (0, 1) operator, of which string theories havean infinite number. Contained within this infinite set of gauge symmetries is theparticular W∞+1 symmetry located in studies of two-dimensional string gravity,whose associated “ground ring” algebraic structure of (1, 0) and (0, 1) operators hasbeen discussed in ref.

[14].A simple counting argument indicates that the number of such (1, 0) and (0, 1)operators in a four-dimensional string theory is comparable to the entropy (2) ofa massive black hole, and hence might be adequate to accommodate this entropywithout the necessity of a mixed state. This is based on the fact that the numberof gauge symmetries is at least in correspondence with the number of string levels.For example, a subclass of (1, 0) operators corresponding to string level 2N assumesthe generic form [23]ZdσΨ(∂X)N(∂X)N−1(7)where σ is a space-like world sheet parameter, and Ψ is a (2N −1)-index tensorspace-time field that is symmetric on the first N and last N −1 indices, as well asdivergence-free on each index.

Clearly this is an infinite set of operators in any stringtheory. The actual symmetries are bigger [23].

On the basis of general arguments,one could expect that each gauge stringy symmetry would lead to a conserved charge,which could participate in characterising the black hole.In two dimensions, the existence of an infinite set of conserved quantum numberswas first demonstrated in matrix models [12]. We subsequently pointed out thatthey should also appear as gauge symmetries of two-dimensional black holes [9],associated with the massive topological discrete discrete states that are known toexist in continuum Liouville models of two-dimensional gravity [13].

Indeed, thegauge nature of these symmetries was subsequently demonstrated in ref. [24].

Weargued [9] that this infinite set of conserved charges, “W-hair”, should be sufficient to

maintain quantum coherence for two-dimensional black holes, consistently with theknown existence of an S-matrix for two-dimensional matrix models. This quantum-mechanical behaviour was subsequently given an elegant geometrical interpretationin terms of an infinite phase-space area-preserving symmetry [10], which originatesfrom the ground ring of (1, 0) or (0, 1) world-sheet operators mentioned earlier [14].In confirmation of this point, it was shown subsequently [11] that the quantumevaporation of a two-dimensional black hole was related to the imaginary part of aformally divergent higher-genus string amplitude, associated with an integral overlarge tori.This evaporation did not have a finite-temperature interpetation, and didnot lead to a mixed state.

For the purposes of the later discussion we note that thisevaporation mechanism did not seem specific to two dimensions, and appeared tobe generalizable to four-dimensional black holes.3Spherically-Symmetric Four-dimensional BlackHolesWe note first that the original arguments given by Hawking referred to spherically-symmetric black holes originated by the spherically-symmetric collapse of macro-scopic matter [17]. Spherically-symmetric solutions to gravity theories in arbitrarydimensions have been classified in a wide class of theories [25].

In particular, theso-called second-order formalism has been adopted for a description of gravity the-ories in arbitrary number of dimensions, involving in general higher powers of thecurvature tensor. The result is that, with the exception of some unphysical cases,all spherically-symmetric solutions are static [25] and some of them are known toexhibit singularities hidden by event horizons, and therefore are of black hole type.Since all such spherically-symmetric singularities can be regarded as in some sensetwo-dimensional, the angular variables being inessential, we analyze them using re-sults from string theory in two-dimensional space-time, which we now review briefly.Witten [6] showed that it is possible to describe the region of two-dimensionaltarget space-time around the singularity by an exact conformal field theory, whichis a coset SU(1,1)U(1)Wess-Zumino σ-model formulated on an arbitrary Riemann sur-face Σ.

In [11] we have argued that summation over Riemann surfaces of arbitrarytopology, as required by a consistent string formalism, produces modular infinitieswhich have to be regularised by analytic continuation, thereby leading to imaginarymass shifts of the black hole solution and hence instabilities. The latter will causethe black hole to evaporate, but such an evaporation, although quantum in origin,is different from the thermal scenario argued by Hawking [17] in conventional localfield theories of gravity.

The evaporation is necessitated by the fact that the stringblack hole solutions carry an infinite number of conserved quantum numbers, aris-ing from stringy gauge symmetries [20, 23, 9] mixing the various mass levels. In thecase of two-dimensional black holes these charges are known [24, 10] to form a W∞+1

extended conformal algebra. This symmetry is a subgroup of an area-preserving in-finite dimensional algebra generated by world-sheet currents of conformal spin (1, 0)or (0, 1) [14].

Due to this fact, we have shown in [10] that the symmetry is elevatedinto a target space one leading to the infinity of conserved charges mentioned before.The reason is that such world-sheet symmetries constitute a canonical deformationof the conformal field theory ( stringy σ-model) describing the world-sheet dynam-ics [23]. The latter are represented as induced transformations of the (target space)background fields of the σ-model.

The important feature, relevant for issues of quan-tum coherence, is that this symmetry preserves the phase-space area of the matrixmodel [14], which describes the interaction of c = 1 matter with the black hole.In [10] we pointed out that it is precisely this property of the infinite-dimensionalstring symmetry that ensures preservation of quantum coherence during the blackhole evaporation process. This was the feature that was believed to be violated ac-cording to the Hawking arguments [16] on the non-factorisability of the conventionalscattering matrix due to the presence of space-time singularities 5These arguments were originally formulated in two-dimensional string cases, andone can naively think that they have nothing to do with the real four-dimensionalcase.However we shall now argue that this is not the case, since spherically-symmetric solutions of four-dimensional gravity theories are effectively two-dimensionaltheories.

We conjecture that, in order to describe the spherically-symmetric singu-larities one can use the formalism of two-dimensional strings. This conjecture seemsalso to be in agreement with Witten’s point of view [27].

However, as we arguedalready in [9, 10] and we shall repeat below, it seems to us that full consistency ofgeneral relativity with quantum mechanics is achieved only upon inclusion of theentire spectrum of topological string states, which in two dimensions constitute theremnants of excited string states in higher-dimensional target spaces.To be systematic, we start from the observation that in a D-dimensional target-space string theory there is an infinity of discrete topological states, which are similarin nature to those of the two-dimensional case [13]. Indeed these states can be seenin the gauge conditions for a rank n tensor multiplet,Dµ1Aµ1µ2...µn = 0(8)where Dµ is a (curved space) covariant derivative.To illustrate our arguments,consider the simplified case of weak gravitational perturbations around flat space,5According to Hawking [16] these constituted an obstruction to the analytic continuation fromEuclidean to Minkowskian space, causing the non-factorisability property.Such modificationswould invalidate the CPT -theorem of quantum mechanics in its ordinary sense, as the later is in-compatible with a non-factorisable $ - matrix.

However if one abandons the concept of a superscat-tering operator, while keeping the density matrix formalism as fundamental, then CPT -invariancecan be preserved [26], perhaps at the cost of not having definite mixed states, a situation even lessdeterministic than that of Hawking [17]. In our case, the factorisability of the Hawking matrix$ is guaranteed due to symmetries of the theory and hence CPT -invariance holds in the strong(ordinary) sense.

with a linear dilaton field of the form Φ(X) = QµXµ. One finds the following Fouriertransform of (8),(p + Q)µ1 ˜A(k)µ1µ2....µn = 0(9)We then observe that there is a jump in the number of degrees of freedom at dis-crete momenta p = −Q.

Due to the complete uncertainty in space, such statesare delocalised, and can be considered as quasi-topological and non-propagatingsoliton-like states. In ordinary string theories, such states presumably carry a smallstatistical weight, due to the continuous spectrum of the various string modes.

How-ever, in strings propagating in spherically-symmetric four-dimensional backgroundspace-times, these discrete states become relevant. Such backgrounds are effectivelytwo-dimensional, and therefore all the transverse modes of higher rank tensors canbe gauged away using Ward identities of the form (8), except for the topologicalmodes.

In a four-dimensional spherically-symmetric background formalism, theseare s-wave topological modes. For spherically-symmetric black holes, these modesconstitute the final stage of the evaporation [9, 5], and they are responsible for themaintenance of quantum coherence [9, 10].

For clarity we shall recapitulate the argu-ments of [9, 10, 11], emphasizing that now one is really dealing with four-dimensionalspace-time spherically-symmetric singularities.The analysis of [25] implies that in pure gravity all the classical spherically-symmetric solutions to the equations of motion obtained from higher-derivativegravitational actions with an arbitrary number of curvature tensors are static. Asimilar result occurs in the case of string-theoretic black holes at tree string-level.However, the arguments of [11] imply instabilities in spherically-symmetric blackhole solutions, since these are massive string states and as such should be able todecay to lighter states [28, 29].

This mechanism also exists for superstring theo-ries. Consider then a superstring theory, and a spherically-symmetric gravitationalbackground of black hole type.

The metric tensor will be given by an Ansatz of theform:ds2 = gαβdxαdxβ + eW (r,t)dΩ2(10)where W(r,t) is a non-singular function and xα, xβ denote r, t coordinates. Also,dΩ2 = dθ2 + sin2θdφ2 denotes the line element on a spherical surface that does notchange with time.

It can be shown that the standard Schwarzschild solution of thespherically-symmetric four-dimensional black hole [30] can be put in the above formby an appropriate transformation of variables. Consider the Schwarzschild solutionin Kruskal-Szekeres coordinates [30]ds2 = −32M3re−r2M dudv + r2dΩ2(11)Here r is a function of u, v, since it is given by( r2M −1)er2M = −uv(12)

Notice that despite the static character of the black hole solution, upon changingvariables, the two-dimensional metric components depend on both variables u, v.Changing variables toe−r4M u = u′e−r4M v = v′(13)and taking into account the Jacobian J of the transformation in the (positive-definite) area element dudv, we can put the two-dimensional metric in the formgbh(u′, v′) = eD(u′,v′)du′dv′1 −u′v′(14)where the scale factor is given by 16M2e−r′(u′,v′)2MJ(u′, v′), with r′ the function rexpressed in u′, v′ coordinates. This form of the metric is just a conformally-rescaledform of Witten’s two-dimensional black hole solution [6].

Since the latter is describedby an exact conformal field theory, so is the conformally-rescaled metric, which froma σ-model point of view simply expresses a sort of renormalisation scheme change6.From now on we shall work directly with the conformally-rescaled metric. The globalproperties (singularities) remain unchanged from the two-dimensional string case.In particular, according to the interpretation of Witten’s work [6] by Eguchi [31], a(conformal) Wess-Zumino coset model is suitable for the description of the region oftarget space-time around the singularity, where the conventional σ-model formalismbreaks down.To understand this point better, let us consider the gravitational sector of a four-dimensional supersymmetric string effective action It has the generic formSeff =Zd4x√GeΦ{ 1κ2R(4) −12(∇µφ)2 −e−2√2κφH2µνρ + ...}(15)where φ is a four-dimensional dilaton field,and Hµνρ = ∂[µBνρ] + ωL −ωY is thefield strength of an antisymmetric tensor field, which by a duality transformation,upon using the equations of motion, is equivalent to a pseudoscalar λ.

The dots... denote higher-derivative terms as well as gauge or other matter fields comingfrom compactification, in the case that one starts from a string theory in the crit-ical dimension. Their presence does not affect our discussion.

Upon dimensionallyreducing (15) to discuss the spherically-symmetric gravitational background, oneobserves that another dilaton (W(r, t)) is going to be generated by the angularpart of the Ansatz (10), as well as a two-dimensional cosmological constant term,even if the four-dimensional theory has zero cosmological constant. The reasonsare simple.

Since the metric is spherically-symmetric, there will be the radial partgαβ in (10) (depending on time in general), which yields a two-dimensional scalar6The function D(u, v) can be regarded also as a part of the two-dimensional dilaton in the givenrenormalisation scheme.

curvature term R(2). From the four-dimensional determinant one obtains exponen-tial W-dilaton factors accompanying the two-dimensional metric determinant √g,and from the derivatives with respect to r and t of the angular part one gets two-dimensional W-dilaton kinetic terms.

The angular part of the metric (10), withconstant two-dimensional curvature, yields a cosmological constant part7. Thus theeffective description of the theory is given by the following two-dimensional effectiveaction (to lowest order in derivatives)4πZd2xeW√g(R(2) −(∇W)2 −2 + (∇T)2 + ...)(16)We are interested in the extra charges that the black-hole can have in a string effec-tive model.

In the case of spherical geometry, this implies a spherically-symmetricAnsatz for the matter fields in (15).This leads to scalar s-mode structures forthe antisymmetric tensor (axion) and higher-dimensional dilaton or other matterfields’ s-wave modes which are collectively represented as a two-dimensional string“tachyonic” mode, T 8.Having expressed the theory as a two-dimensional effective string model, one canapply the whole machinery of two-dimensional strings to study the dynamics of theevaporation of the black holes and determine the final stage. It should be stressedthat, from the general analysis mentioned in the beginning [25], the static characterof the physically-interesting solutions to string-inspired gravitational theories impliesprobably that the only way that these black holes evaporate is the one suggestedin [11], i.e.

through string quantum corrections, requiring a formulation in higherworld-sheet genera and summation over them. Independent arguments to supportthis claim will be given below.

At present, we note that such decay is non-thermal,and hence maintains quantum coherence, as guaranteed by the W∞-symmetry asso-ciated with the discrete topological s-wave modes. The s-wave matter phase-space,which would be the problematic one from the point of view of quantum coherence,due to modes going into and not coming out or vice versa, is two-dimensional inthe spherically-symmetric case and hence the arguments of ref.

[10] apply. Thesymmetry of the effective two-dimensional target space is phase-space volume (areain two dimensions) preserving, and hence Liouville’s theorem for the time evolutionof the density matrix remains valid.

Thus, there is no modification of the evolutionequation of the density matrix in the presence of a spherically symmetric black hole[15], and the factorisation of Hawking’s superscattering operator holds.7 The dΩ2 represents the metric of a 2-sphere of unit radius, with scalar curvature 2. Integrationover the angular variables yields an extra factor of 4π.8We should stress that in our formalism the “dilaton” of the two-dimensional string modeloccurs necessarily in the spherical Ansatz for four-dimensional gravity, and therefore one does nothave to start from a higher-dimensional string model and compactify.

All such compactificationmodes that occur in traditional superstring-inspired models [32] appear in our two-dimensionaleffective model as matter “tachyon” T -fields.

A further comment concerning the thermodynamical relations (1), (2) of the blackhole solutions is in order. We noted in [11] that the quantum instabilities associatedwith modular infinities, which cause the evaporation of two-dimensional black holes(and hence of spherically-symmetric four-dimensional configurations as well), arenon-thermal in origin.

Arguments have been given [11] for the thermal stability ofthese objects on the basis of compactified c = 1 matrix models, believed to representa stringy regularisation of two-dimensional strings (and, in view of the picture inthis article, of four-dimensional strings propagating in spherically-symmetric back-grounds). Here we would like to give an independent argument in support of theabsence of thermal evaporation, at least in the conventional sense, by showing thatthe available string states account for the quadratic mass dependence of the blackhole entropy (2).The argument is based on the fact that the black hole is a particular string stateof mass M. The Hawking entropy is viewed as the number of ways N(M) one canconstruct a state of this mass (ignoring the associated string W quantum numbers,which, in view of our previous arguments, would make the exact string entropyvanish).

In this picture N(M) may be considered the same as the multiplicity ofstring states of mass level M. This entropy is measured by an observer at spatialinfinity, where the string propagates in a flat background with a linear dilaton field,QµXµ. In the two-dimensional black hole case the black hole mass is determined bya constant shift 2a in the dilaton which is non-trivial.

The precise relation is [6, 5]√α′M = Qe2a(17)A choice of a selects a particular black-hole configuration (i.e. a vacuum for thestring).

If we rescale the Regge slope by e−2a, then in units of the rescaled slope theblack hole mass is given by Q. The situation is then analogous to that of ref.

[8],where for large mass levels the multiplicities are given asymptotically by 9N(M) = lima,M→∞e2π√α′√2+Q2e2aM = e2πα′M2(18)where α′ is the redefined Regge slope, depending on the particular black hole back-ground. There is no loss of generality if we express the black hole mass in unitsof this.

The entropy is determined by taking the logarithm −kBlnN(M), therebyleading to quadratic mass dependence of the form (2), as argued by Hawking andBekenstein [17, 19] using classical thermodynamics and information theory. It shouldbe noticed that as the mass decreases there will be a point where formula (18) willbe no longer applicable, and thus the situation is analogous to that of Hawking [17]where quantum effects become important for small black hole masses and classicalthermodynamics arguments cannot be used.9We should stress that the same arguments of [8] apply here, despite the Wick-rotated Q relativeto their case.

So far we have dealt with static black holes.Let us now consider the abovethermodynamical relations in connection with the evaporation mechanism describedin [11]. The latter is based on the fact that any massive string state decays to lighterstates in such a way that the stringy gauge symmetries associated with the relevantmass levels remain intact.

The formal origin of such an instability appears as amodular divergence of the two-point function of the state in question. Regularisationof the infinity by means of, say, analytic continuation yields an imaginary part I[28, 29].

The latter, in view of the validity of the S-matrix formalism due to theW-hair property of the string black holes, implies due to the optical theorem a decaywith a decay rate Γ = −2I. The dimensionality of space-time plays a crucial rˆole indetermining the life-time.

To see this, let us consider a simple field theory example,which however captures all the essential features of the string case. Consider [29]two fields Φ and φ of masses M and m respectively, with M > 2m, and a “stringinspired” 12λΦφ2 interaction.

The imaginary part of the one-loop two-point functionof the state M can be computed in terms of M, m [29]Γ ≡1MdMdt =λ2πMD−5(16π)D−12 Γ( D−12 )(1 −4m2M2 )D−32(19)The above computation necessarily goes off-mass shell for the propagating light par-ticle in the loop. In the two-dimensional case one observes that the decay rate dMdtis proportional to M−2.

This property, when transcribed into the case of black holedecay in vacuo yields the entropy relation (2), with coefficient dependent on thedetails of the process 10. In string theory one has an infinity of states propagating inthe loop.

However, in the two-dimensional effective string theory the only propagat-ing states are those of the massless “tachyons” [13]. The rest of the string states aretopological.

As a crude estimate therefore of the corresponding decay rate we take(19) with m = 0, d = 2 and M the black hole mass. Although detailed computationsmust be done to estimate the magnitude of the proportionality coefficients, we be-lieve the heuristic argument we gave above is sufficient to demonstrate the essentialphysics of black hole evaporation.

Although the evaporation is not thermal, one cangive a thermodynamic interpretation `a la Hawking [17], if one restricts oneself to theentropy defined in the classical black hole case (see the discussion in section 2 andin the previous paragraph). Then, the thermodynamic relation (2) follows from thesimple fact that the decay in vacuo occurs with a rate proportional to M−2.

Upontime-averaging during the decay from an initial black hole mass M down to masszero, we observe that the average Hawking entropy still obeys a quadratic-mass law.This is a feature only of the inverse squared-mass behaviour of the decay rate. Asa side remark, we would like to point out that in local field theories the relation (2)can be also explained in a thermal way by considering the evaporation of the blackhole in equilibrium with a heat bath of temperature approaching the Hawking onefrom below [33].

In such a scenario, the crucial point for getting a large statistical10The proportionality coefficient depends on detailed string computations, to which we hope toreturn in the near future.

entropy is the zero-point energy of summed up excited states which disappear if weallow the black hole to evaporate down to a final mass zero. This is to be comparedwith the situation in the present case, where however the evaporation/decay takesplace in vacuo.

The topological states, which are responsible for the maintenanceof quantum coherence during the evaporation process, also “disappear”’ as externalstates in the limiting zero-mass case represented by matrix model. However, theirpresence is essential in yielding the enormous statistical entropy for the stringy blackhole.

In the formal sense, their presence guarantees the correct value of the propor-tionality coefficient in (2) to match in order of magnitude that of Hawking’s originalcomputation, based on classical black body radiation [17].4Non-Spherically-Symmetric Black HolesWhat about spherically non-symmetric singularities ? Such objects are known assolutions of Einstein’s equations, the Kerr rotating black holes for example 11.

Inthe context of the present formalism, rotating objects with event horizons can beconstructed by appropriate tensoring of Wess-Zumino models in two dimensions[35]. The original papers on the possibility of the loss of quantum coherence haveconcentrated on the spherically-symmetric case, which we have argued does not haveany such problems in the string case.

What happens in spherically-non-symmetriccases is not yet fully understood. However, we now note that even in the case ofKerr black holes, one can argue that in physically-interesting cases the final stage ofthe evaporation excites s-wave topological modes that are similar to the spherically-symmetric case.Consider the Kerr metric [36]ds2Kerr = r2 + A2cos2θ −2Mrr2 + A2cos2θdt2 −r2 + A2cos2θr2 + A2 −2Mrdr2 −(r2 + A2cos2θ)dθ2 −[(r2 + A2)2 −A2sin2θ(r2 + A2 −2Mr)]sin2θr2 + A2cos2θdφ2 −4MArsin2θr2 + A2cos2θdtdφ (20)where M is the mass of the black hole, and A = αM is the angular momentum.

Forphysically-interesting cases [36, 34] 0 < α2 < M2. The region M2 < α2 correspondsto very rapid rotation of the body, which probably does not occur for real physicalbodies, as they would fly apart before rotating so rapidly.

The final stage of the11For classical Einstein gravity coupled to Maxwell’s electromagnetism there are uniquenesstheorems for rotating black holes [34]. The situation is less clear for quantum theories, especiallythe ones obtained as a low energy limit of string theories, where higher-derivative modifications ofEinstein-Maxwell’s equations occur.

For our purposes we shall concentrate on the Kerr solution,which is the only one studied extensively so far. We believe that this is sufficient to demonstrateour arguments.

evaporation, M →0, therefore, would correspond to α →0 as well. Expandingin powers of M and keeping only the leading order it is straightforward to seefrom (20) that the limit is again one with a topological Q-graviton as in the two-dimensional string case (upon redefining r →eQρ, where Q is 2√2, as required bythe conformal field theory interpretation of the spherically-symmetric case [5, 9]).Perturbations around the Kerr solution by the other stringy modes will presumablylead to an excitation of the rest of the topological s-wave modes in the final stageof the evaporation, which is similar to that in the spherically-symmetric case.This oversimplified argument suggests that W-symmetry, or rather a generalisedform of it, appropriate for spherically-non-symmetric matter, generated by topologi-cal (discrete) stringy modes, characterises in general singularities in four-dimensionaltarget space-times.

The formal reason is that the latter are described by topologicaltheories which are in some sense characterised by an infinite-dimensional extendedconformal symmetry. We expect that the generalisation (to non-symmetric spaces)of the area-preserving W-symmetry characterising the two-dimensional symmetriccase will be a phase-space volume preserving algebra.

In the same way that W-symmetry characterises SU(1,1)U(1)coset Wess-Zumino models [37], one might find thatthe appropriate extension of these theories to describe spherically-non-symmetricmatter would be phase-space volume-preserving groups. This, however, is still aspeculation, but one expects that the quantum coherence problem can be solved ingeneral due to the enormous stringy symmetries, without reference to any particulargeometry for the singularity.5ConclusionsWe have argued that spherically-symmetric four-dimensional objects with phys-ical curvature singularities and event horizons (black holes) can be described byeffective two-dimensional string theories.

Such models have necessarily a dilatonfield, and can be represented as coset SU(1,1)U(1)Wess-Zumino models, which are knownto possess W-symmetries that preserve the two-dimensional s-wave phase-space un-der time evolution, and thus maintain quantum coherence during the (non-thermal)stringy evaporation process of a spherically-symmetric four-dimensional black hole.The latter expresses decay of the massive stringy black hole state due to instabil-ities induced by the string propagation in summed up world-sheet topologies. Wehave argued that such an evaporation is consistent with a large Hawking entropyassociated with the black hole, the latter being viewed as the entropy the blackhole appears to have if one does not take into account the infinity of (observable)quantum charges associated with topological excited string states.

The quadraticblack-hole-mass dependence of this entropy, conjectured by Bekenstein and Hawking[19, 17], can be obtained from simple formulas [8] yielding the asymptotic density ofstring states in a linear dilaton background, which resembles the asymptotic formof the black-hole space-time.

AcknowledgementsOne of us (J.E.) thanks the University of Miami Physics Department for itshospitality while this work was being completed.

The work of D.V.N. is partiallysupported by DOE grant DE-FG05-91-ER-40633.

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