John Preskill- Quantum Hair

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Physica Scripta. Vol. T36, 258-264, 1991.

Quantum Hair*

John Preskill

California Institute of Technology, Pasadena, CA 91 125, USA

Abstract

A black hole can carry “qu antu m hair” that , while undetectable classically,can influence quantum physics outside the event horizon. The effect ofquantum hair on black hole thermodynamics is computed here, in a par-ticular field-theoretic model.

to be exhaustive, but it will demonstrate th at q uan tum haircan exist, and that it can have dramatic and computableeffects on the thermodynamic behaviour of a black hole. We

Wi l l also ga in Some insight in to th e isSue Of the final State Of

an evaporating black hole.

1. Introduction 2. Quantum hair and the Aharonov-Bohm effect

The remarkable black hole uniqueness theoremsf assert thata stationary black hole can be completely characterized by itsmass, angular momentum, and electric charge. Since a black

hole carries (almost) no “hair,” an observer outside the eventhorizon is (almost) completely ignorant of the black hole’sinternal state. This absence of hair has therefore been cited asthe explanation of the intrinsic black hole entropy that wasconjectured by Bekenstein [2] an d calculated by Haw king [3] .

However, it is important to remember that the “no-hair

Krauss an d Wilczek [5] pointed out that quantum hair may

arise in a gauge theory as a consequence of the Higgs mechan-ism. To illustrate this phenom enon, we consider a theory with

gauge group U (l ) that contains both a charge-N scalar fieldq and a charge- 1 scalar field 4; under a gauge transformationparametrized by o(x), these fields transform as

o: + e’”+,

q + eiNoI . (2.1)theorems” are statements abou t the solutions to classical fieldequations. These theore ms d o not exclude the possibility tha tblack holes can carry additional attributes that influence

Suppose that (in unitary gauge) q condenses in the vacuum

state, and + does not,

quantum physics outside the horizon, but cannot bedescribed in classical language . Such “qua ntu m h air,” w ere itto exist, would require us to re-examine o ur ideas abo ut blackhole thermodynamics, and about quantum mechanics in a

black hole background.In this talk, I will describe a possible variety of quantum

hair that has recently attracted attention, and will exploresome implications concerning black hole physics.

Th e classical no-hair theorem s invite us to propose a m oregeneral no-hair principle - that when an event horizon

forms, any feature of a black hole that ca n be radiated away

will be radiated away [4]. Properties that cannot be radiatedaway are those that can be detected, at least in principle, atinfinite range. With this motivation, the term “h air” might beused, in a broader context than black hole physics, to meanan attribute of a localized object that can be measured at

arbitrarily long range. Hair, in this sense, can be either clas-

sical or quantum-mechanical. Classical hair, like electriccharge, is associated w ith a long-range classical field. Qu an-tum hair is associated with a long-range Aharonov-Bohmphenomenon that has n o classical analog. The qua ntum hairtha t will be considered here is of this Aharonov-Bohm type.

Of course, it is quite possible that, in order to give acomplete description of quantum mechanics on a black holebackground, it will be necessary to ascribe to the black holemany additional properties th at have nothing to d o with theAharonov-Bohm effect. In other words, a black hole mighthave many varieties of quantum hair that are outside thescope of the analysis in this talk. This analysis is not intended

-Invited talk at Nobel Symposium no. 79: The Birth and Early Evolution

t See [I ] for references.

Physica Scripta T36

of our Universe, Graftlv allen, Sweden, June 11-16, 1990.

( q ) = ‘U # 0, (4) = 0. (2.2)

Then the Higgs mechanism occurs; the photon acquires a

mass p - N ev and the classical electric fieid of a charge

decays at long range like

E - e- i iR, pR + 1 . (2.3)

However, the local U( 1) symmetry is not completelybroken. There is a surviving manifest Z, local symmetry

under which the scalar fields transform according to

+ q , + -+ eZniklN, k = 0 , 1 , 2 . . . , N - 1 . (2.4)

The physical content of the surviving Z, symmetry is that,

because the condensate that screens the classical electric field

of a charge is a condensate of c h a r g e 4 q particles, the con-densate is not capable of screening the charge modulo N .

But if the classical electric field of any charge decays

according to eq. (2.3), what does it mean to say that thecharge modulo N is unscreened? Th at is, how can th e charge

modulo N be detected at long range? The key point is that theHiggs phase with manifest Z , local symmetry supports a

“cosmic string.” The core of this string traps a quantity ofmagnetic flux @, , /N, where (Do= 27c/e is the flux quantumassociated with a particle of charge 1. (Hence e is the gaugecoupling.) Thus, when a charge-1 4 particle winds aroun d thestring, it acquires the nontrivial Aharono v-Bohm phase [6]

exp ( i e 4, A - dx ) = e2”IN

This Aharonov-Bohm phase induces a long-range inter-

action between the string and the + particle. For example,although the string has a thickness of order p - ’ , a + particle

incident on the string with energy E 4 scatters with a



Quantum Hair 259

cross-section that is independent of the strin g thickness [6 , 71.

Similarly, a closed loop of string tha t winds aro un d an objectwith nonzero Z, charge acquires a nontrivial Aharonov-Bohm phase, and so the charge scatters an incident string

Even though the photon has acquired a mass due to theHiggs mechanism, and classical electric fields are screened,

the Aharonov-Bohm interaction is of infinite range. Thisinteraction enables us, in principle, to me asure the ZN hargeof an object while maintaining an arbitrarily large separationfrom the object. Thus, Z charge is a type of qua ntum hair [ 5 ] .

That such infinite-range phenomena can occur in theHiggs phase, even though there is a mass gap, has a numberof interesting consequences. For one, the existence (or not)of a long-range Aharonov -Bohm interaction provides acriterion for distinguishing different phases of a gauge theory- n the model just described, we have seen that two differentHiggs phases, with manifest and spontaneously broken Z,symmetry, can be distinguished in this manner. This obser-vation is readily generalized to other models [8].

Furthe rmo re, fascinating new phenomena occur in modelssuch that the unbroken gauge symmetry in the Higgs phaseis nonabelian. T here can be infinite-range processes in w hichcharges are transferred between point particles and stringloops, by means of the nonabelian Aharonov-B ohm effect

But our main interest here is in the implications of Z,quantum hair for black hole physics, to which we now turn.

loop.

~ ~ 9 1 .

3. The no-hair theorem and the H iggs mechanism

In a U(l) gauge theory in the Coulomb phase (in which thephoton is massless), an electrically charged particle has aninfinite-range electric field. Gauss’s law ensures t ha t his long-range electric field cannot be extinguished when the particledisappears behind the event horizon of a black hole. Thus welearn that a black hole must be able to carry electric charge;electric charge is a type of classical hair.

Now suppose tha t the Higgs mechanism occurs, so that thephoto n acquires a mass p . The electric field of a charge decaysat long range like E - e - p R ,where R is the distance from thecharge. But if a charged particle d rop s into a black hole, theevent horizon refuses to sup po rt a static electric field, and theexponentially decaying long-range field can not survive [lo].The no-hair principle (“whatever can be radiated is radiat-ed”) requires the electric flux to leak away from the horizon,either by propagating through the horizon or by escaping toinfinity. If the photon Compton wavelength p - ’ is muchlarger than the black hole radius RBH ,hen the characteristictime scale (in terms of Schwarzschild time t ) for this leakageprocess is of order p-’.* The analysis leading to the aboveconclusion is purely classical, and it is not affected if theHiggs mechanism leaves unbroken a local Z symmetry.Once the pho ton acquires a mass, regardless of any survivingmanifest discrete symmetry, a stationary black hole that isnonsingular at the event horizon must have a vanishing clas-sical electric field outside the horizon.

But this classical analysis cannot exclude the possibilitythat a black hole may carry quantum hair. Indeed, we know

that a particle with nonzero Z , charge (electric chargemodulo N ) has an infinite-range Aharonov-Bohm inter-action with a cosmic string. This long-range interaction can-not be extinguished when the particle disappears behind theevent horizon of a black hole. T hus we learn that a black holemust be able to carry ZN charge;+ Z charge is a type ofquantum hair.

It is essential to recognize tha t the Aharonov-Bohm inter-action remains undim inished even a s the classical electric fielddecays. This is already evident when we note that the phaseacquired by a loop of string that winds around a Z, chargeis nontrivial when the trajectory of the string stays arbitrarilyfar from the charge, even though the electric field encoun-tered by the string is exponentially small. Likewise, the phaseacquired by a loop of string a fixed distance from a chargepersists even after the charg e crosses the black hole horizon,and the electric field leaks away.

Aside from the Z, quantum hair described here, anotherexotic type of black hole hair has attrac ted attention recently- axionic hair [11]. Rathe r th an digress here o n this topic, I

have relegated a compa rison of Z, hair an d axionic hair to anappendix.

4. Challenges posed by black hole radiance

Having established tha t a black hole can carry a conserved Z N

charge, even though no massless gauge field couples to thecharge, we now wish to consider how the Z N harge influencesthe Hawking radiation emitted by the black hole. Our hopeis that some of the mysterious aspects of black hole radiance

can be illuminated by this investigation.Specifically, the semiclassical theory of black hole rad iance

developed by Hawking [3] raises (at least) three fundamental

issues, none of which can be satisfactorily addressed withinthe context of the leading semiclassical appro xima tion. T heseissues, enumerated below, are all closely related but arelogically distinct.

4.1. The loss of quantum coherence

In Hawking’s semiclassical theory, the radiation emitted byan evaporating black hole is described by a density matrixthat is exactly thermal [12]. Since the outgo ing rad iation is ina mixed state, one might argue (as Hawking [I31 does) thatscattering of quan ta off a black hole cannot be described inthe usual S-matrix language - an incoming pure state canevolve into an outgoing mixed state. Similarly, a black hole

that forms from collapse in a state that is initially pure willeventually evaporate and yield a mixed final state. A blackhole, then, appears to provide a means of destroyingquantum-mechanical phase information. If so, quantum mech-anics as currently formulated is not applicable to processesinvolving black holes, and must be replaced by some moregeneral formalism.

This remarkable suggestion is much too radical to beaccepted lightly; we are obligated to subject it to the closestscrutiny. In particular, the argument for a loss of quantumcoherence presumes that the thermal character of the out-going density matrix computed by Hawking is an intrinsic

property of the final state, and not a mere artifact of the

approximations that are made in the leading semiclassical

* I thank Kip Thorne fo r a helpful discussion a bou t this. As Krauss and Wilczek [ 5 ]noted.

Physica Scripta T36



260 John Preskill

theory. The calculations performed to date do not, I think,exclude the possibility that the outgoing radiation, whenexamined with sufficient care, actually carries all of the com-plex phase information contained in the initial state fromwhich the black hole formed. But if quantum coherence isindeed preserved in black hole processes, it seems to benecessary for black holes to be able to carry non-classical

varieties of hair.’ Only then is it reasonable t o claim that theradiation emitted early on leaves an imp rint on th e black holethat can influence the radiation emitted later, thus establish-ing quantum-mechanical correlations between radiationemitted at different times.

4.2. The origin of black hole entropy

A remarkable consequence of Hawking’s semiclassical theoryof black hole radiance is that a black hole has an enormousintrinsic entropy (as Bekenstein [2] had anticipated). Effortsto attach a sensible physical interpretation to this entropyhave been partially successful,+but I think that the ultimateorigin of the black hole entropy remains quite obscure. In

general, the statistical-mechanical entropy of a system arisesbecause the system has access to m any m icroscopic internalstates as it undergoes thermal fluctuations. Unless black holesare fundamentally different than other systems, then, itshould be possible to relate the black hole entropy to a largenumber of accessible black hole microstates. Because a clas-sical black hole has n o h air, the intrinsic entropy suggests thata black hole has many non-classical degrees of freedom.

The black hole entropy, then, like the apparent loss ofquan tum coherence, hints at the existence of “quan tum hair”on black holes. It does not seem likely tha t all of the entropy

SBH = k A B H , (4.1)

where AB Hs the area of the event horizon in Planck units, canbe attribute d to hair of the Aharonov-Bohm type that we willconsider here. Rath er, eq. (4.1) tempts one to conjecture thatthe quantum-mechan ical internal state of a black hole can bedescribed in terms of a mem brane stretched over the horizontha t is approximately on e Planck length deep and is charac-terized by about one bit of information per Planck volume.Such a picture has been advocated by ’t Hooft [14].

4.3. T he jna l s ta te

An evaporating black hole loses mass. Eventually, as themass approaches the Planck mass Mp,Hawking’s semiclas-sical theory breaks down, for quantum fluctuations in the

background geometry can no longer be neglected. At thispoint, we are unable to predict what will happen without adeeper understanding of quantum gravity than we currentlypossess.

The black hole may evaporate completely, leaving behindno trace other than the emitted radiation. It is also conceiv-able that the ev aporatio n halts, leaving behind a stable blackhole remnant. This remnant would be a new type of “elemen-tary” particle, presumably with a mass comparable to Mp

Whether a black hole evaporates completely may haveimportant implications for quantum cosmology - Hawking

has argued that if a black hole can form and subsequentlydisappear, then we have stron g evidence tha t fluctuations inthe topology of spacetime must be included in quantumgravity [19].

The three issues outlined above are all of fundamentalinterest. They all also have in common that they cannot beadequately addressed within the leading semiclassical theory

of black hole radiance, and so they provide us with powerfulmotivation for improving on that theory.

5. Black holes with ZN uantum hair

Ou r desire for a better grasp of black hole physics beyond theleading semiclassical approximation leads us back to theblack hole with Z, charge described in S ection 3 . The blackhole with Z, charge provides a model in which dramaticcorrections t o the leading semiclassical theory are expected.Furthermore, to a limited extent, these corrections can bestudied in a systematic and well-controlled approxim ation.

To see that dram atic corrections should be expected, let us

suppose that N 9 1, so that it is possible for the Z, chargeQ o be a large number. Supp ose further that the only elemen-tary particles with n onzero Z charge have charge 1 and massm , where m is small compared to the Planck mass Mp.

By assembling Q charge-1 particles, with N / 2 > Q 4 1,one can create a black hole with charge Q nd mass MBH

Qm; if N is sufficiently large, we can also arrange thatMBH9 Mp,so tha t semiclassical theory can safely be appliedto this black hole. The black hole then loses mass by radiatinggravitons, say, but does not radiate away its Z, charge.Eventually, MBHecomes less than, perhaps much less than,Q m , yet is still comfortably above Mp.

At this point, complete evaporation of the black hole to

elementary q uant a is no longer possible; it is kinematicallyforbidden. Th e Z, charge is exactly conserved, and there is noavailable decay chan nel with Z, charge Q and a sufficientlysmall mass. Inevitably, then, the evaporation process muststop, leaving behind a stable remnant.$ We may regard theremnant as an exotic “nucleus,” a highly relativistic boundstate of Q elementary charged particles.

However, as noted in Section 3, the Z, quantum hair hasno effect whatsoever o n th e classical geometry of a stationaryblack hole. The leading semiclassical theory is just free quan-tum field theory on the background geometry of the blackhole, and so it too is unaffected by the 2 hair. Indeed, the 2,hair has no effect to any finite order in the semiclassical

expansion in h. To understand the mechanism by which theZ, hair inhibits the evaporation of a black hole, we mustinvestigate physics that is non-perturbative in h.

6. A remnant stabilized by classical hair

Before we continue the discussion abou t 2, quan tum hair, itwill be enlightening to consider an example of a black hole

remnant that is stabilized by classical hair - an electricallycharged black hole.§ We can concoct a model in w hich the

* This point has been especially stressed by ’t Hooft [14]. See [15-171 fo r

+ I am thinking in particular of the “thermal atmosphere” picture developedother critiques of the proposal in [13].

in [18].

Physica Scripta T36

It is a logical possibility that the charge -Q object will be able to decay int o

black holes of lower charge and mass, but the existence of at least onespecies of exotic stable remnant is guaranteed.

4 In the case where the pho ton is massless.



Quantum Hair 261

complete evaporation of a charged black hole is kinematic-ally forbidden. But when the hair is classical, the mechanismby which evaporation stops can be discussed in terms offamiliar semiclassical theory.

In the Coulomb phase of electrodynamics, electric chargeon a black hole has two effects, both of which can be des-cribed using classical language [20]. First, since the charged

black hole has an electric field outside the horizon, and theelectric field carries stress-energy, the e lectric charge modifiesthe black hole geometry. Specifically, if two black holes havethe same mass, the one with more charge has a smallersurface gravity K , and so a smaller Hawking temperatureTBH= 4271. Second, a charged black hole has a nonzeroelectrostatic potential at the event horizon; if the black holeis positively charge d, say, then m ore work is required to carrya positive charge th an a negative charge fro m spatial infinityto the horizon. In black hole thermodynamics, the electro-static potential plays the role of a chemical potential. Apositively charged black hole canno t be in equilibrium w ith aneutral plasma, because it prefers to emit positive charge an d

to a bsorb negative charge.As a charged black hole evaporates, it loses mass. If we

suppose that its charge Q e remains fixed as it evaporates,*

then the charge to mass ratio Qe/MBH steadily increases. Thisratio asymptotically ap proa che s the critical value 1 (in Planckunits), at which the surface gravity IC and Hawking tem-perature T B H vanish.+ Heuristically, a black hole of givenmass M B H has a maximal charge because, when the Coulombenergy of order (@?)’/& stored in the electric field outsidethe horizon becomes comparable t o MBH , here is no mass left

over at the center to support the horizon. (In fact, as thecharge to mass ratio increases toward one, the Cauchy hor-izon interior to the event horizon asymptotically approaches

the event horizon. In this instance, then, the cosmic cen-sorship hypothesis [21] can b e identified with the third law ofthermodynamics. A third law of black hole dynamics hasbeen proved, under suitable assumptions, by Israel [22]).

We see that the Hawking evaporation process ceases toope rate for a maximally charged black hole. Nevertheless, inthe real world, a maximally charged black hole would con-tinue to radiate both ch arge an d mass [20]. The reason is tha tthe electrostatic potential energy

U = eM , (6.1)

of a positron at the horizon of a positive critically chargedblack hole is enormous compared to the electron mass.

Hence, dielectric breakdown of the vacuum occurs outsidethe horizon. It is energetically favourable to produc e an e +e-pair, allowing the electron to be absorbed by the black holewhile the positron is ejected to infinity with a n ultrarelativisticvelocity. The black hole would discharge rapidly.

But since our interest here is in matters of principle, we arefree to contemplate a fictitious world with

Gm2 > e2 (6.2)

where m , e are the electron mass and charge. In this world,

* Loss of charge is negligible if all elementary charged particles are suf-

ficiently heavy; see below.

We assume, for simplicity, that the angular momentum JBHs zero.It is conceivable that such an inequality applies, even in the real world, to

the mass and charge of a magnetic monopole.

the electrostatic interaction between two electrons is weakerthan the gravitational interaction. And the electrostaticpotential energy of an electron at the horizon of a maximally

charged black hole is less than m , so that dielectric break-down of the vacuum cannot occur. The maximally chargedblack hole in this model with MBH= e Q M p s lighter tha n Q

electrons, an d so its decay t o elementary qu anta is kinematic-

ally forbidden.The only decay channel potentially available to the maxi-

mally charged black hole is a state containing black holes oflower mass and charge. At the classical level, all maximallycharged black holes have the same charge to mass ratio, sothat such decay modes would be just marginally allowed.However, at least when the black hole radius RBH is muchsmaller than th e electron C om pton w avelength m - ’ , expectthat renormalization effects (particularly charge renormaliz-ation) cause the charge to mass ratio of the maximally

charged black hole to increase with increasing M B H .$Thus,the particle spectrum of the model should contain a wholetower of stable black hole “solitons” of increasing charge.

It could be quite illuminating to consider in detail thequantum mechanics of scattering processes involving thesesolitons. On the one hand, it may be feasible, using standardmethods of soliton quantizatio n, to construct, ord er by order

in an expansion in h, an S-matrix for these processes.7 Onthe other hand, if, as Hawking insists, such processes inevi-tably involve a loss of quantum-mechanical phase infor-mation, then no such S-matrix has a right to exist.

7. A world-sheet instanton

In the model just considered, the complete eva poration of ablack hole can be kinematically forbidden, if the electric

charge on the black hole is large enoug h, and we were able tounderstand using semiclassical reasoning why the evapor-ation of a charged black hole eventually stops. In the modeldescribed in Section 5, the complete evaporation of a blackhole can be kinematically forbidden, if the Z, charge on theblack hole is large enough. Yet, to any finite order in theexpansion in h , the Z, charge has no effect on the evap oration

process. Sm all fluctuations of the qu an tum fields on th e blackhole background are completely insensitive to the Z, charge.

The only readily identifiable quantum fluctuations that do

depend on the value of the Z, charge are virtual cosmicstrings that wind around the event horizon of the black hole;a string that winds k times around an object with Z, charge

Q acquires the Aharonov-Bohm phase exp (27cikQlN). Weneed a better grasp of the physics of the cloud of virtualstrings surro und ing a black hole, if we hope to understand theeffect of Z, quantum hair on black hole radiance.

In fact, th e effects of interest, which are nonpe rturbative inA can be incorporated into a systematic small-A approx i-mation. The analysis is done most conveniently usingEuclidean path integral methods. And since we are studyingquan tum field theory o n the Schwarzschild backgro und, the-

Loosely speaking, it is the “bare” charge of the black hole, as measured at

the distance scale RE,, hat enters into the relation eQ = M satisfied by

a black hole with TBH 0. The renormalized charge, as measured far

away, is partially screened by vacuum polarization, and this charge screen-

ing becomes less effective as RE, ncreases.

ll This point was emphasized to be by David Gross.

Physica Scripta T36



262 John Preskill

appropriate arena is the Euclidean section of the Schwarzs-child geometry [23].

The Euclidean section of th e Schwarzschild geometry hasthe topology of R2 x S 2 .Th e plane R2 s parametrized by theSchwarzchild radial coordinate r and the E uclidean Schwarzs-child time coordinate z.These turn out to be polar coordi-nates for the plane; z s actually an angular variable, periodic

with period P , where P - ’ = T B H is the Hawking temperature.Indeed, this periodicity in Euclidean time provides one (rath erformal) way of understanding the origin of black hole radi-ance [24]. Sitting on top of each point in the plane is atwo-sphere with radius r . The two-sphere with minimalradius r = R B , = 2MB, sits above the origin of the plane.

We can compute the free energy F of a black hole inequilibrium with a radiation b ath by using the Euclidean pathintegral method [23]. If the charge of the black hole is notspecified, the n F is given by

where SE enotes the Euclidean action, a nd the pa th integral

is restricted to geometries with topology R2 x S2 that areperiodic in zwith period PA. Hawking’s black hole thermo-dynamics can be recovered in the limit h -, 0 by expandingSE about its saddle point, the Euclidean Schwarzschildsolution.

If, however, we wish t o com pute the free energy F(/?,Q) inthe charge-Q sector of a theory with m anifest Z, local sym-metry, then the field configurations fall into distinct sectorsthat must be weighted by different phases. This is easiest tounderstand in the “thin-string’’ limit; that is, when thenatural width p-’ of the string is negligible compared to thesize R B H of the black hole. So , to begin with, let us considerthis limit.

The path integral includes a sum over the world sheet ofa (thin) cosmic string. But on a background with R2 x S2

topology the world sheets are classified by a w inding numberk tha t specifies how many times the world sheet is wrappedaround the two-sphere. Dependence of F on the charge Qarises because a world sheet with winding number k isweighted in the path integral by the Aharonov-Bohm phase

exp(2nikQ/N).’ The action of the string is just the stringtension Tstringimes the area of the world sheet. So it is evidentthat, among the k = 1 configurations, the one of lowestaction is such that the world sheet is stretched around thetwo-sphere with the minimal radius R B H . This configurationis the “world-sheet instanton” that dominates the Q-depen-dence of the free energy in the limit h -+ 0.

The presence of the string world sheet perturbs the back-ground geometry, but this effect is small if the tension Tstring

is small in Planck units. So the string wo rld sheet increases theaction by

Sstring = r t r i n g = 4nRiH Tstr ing, (7.2)

and the one-instanton contribution to the partition functionis of the form

~ X P-AB, T t r i n g l h l ,(e- F (B .Q ) ) -SSchwarznchild/h

one-instanton e

(7.3)-Formally, this phase is introduced into the path integral in order to projectthe sum over states, in the evaluation of the partition function tr(e-Pn),onto the states with charge Q.

Physica Scripta T36

where SSchwarzrchllds the gravitational action of the EuclideanSchwarzschild geometry. This contribution is suppressed

compared to the leading, Q-independent, term

(7.4)

by a factor tha t is exponentially sm all when h + 0, or whenthe surface area of the black hole is large compared to the

inverse string tension.By summing the contributions from one instanton an d one

anti-instanton, we find

pF(p , Q) - pF@, Q = 0) - [ l - cos (2nQ/lv>]e-ABHTs‘nng’h.

This expression is independen t of the volume of the radiationbath surrounding the black hole, and so should be regardedas a contribution to the intrinsic free energy of the black hole.From the thermodynamic identity

(e-BW$Q)) - Schwarzrhild/hzero-instanton e

(7.5)

aM(P,Q ) = -jj PW, Q>l, (7.6)

we may extract the leading dependence on Q of the black ho letemperature,

P(MBH, ) - P(MBH, = 0)

(7.7)- 6XMBHstm,- [I - cos (ZzQ/N)]e

Given two black holes with the same mass, the one withlarger 2, charge is cooler. Quantum hair inhibits the emissionof Hawking radiation, in a computable manner.

The Q-dependence of the temperature in eq. (7.7) isvery weak in the thin-string limit, A B H Tstnng 1. But theQ-dependent contribution can be appreciable when

(7.8)

- 1 ’ 2

MBH stnng

(in Planck units). If rtrings small in Planck units, then, thethermodynamic effects of quantum hair can become impor-tant while M B H is still large in Planck units, so that it isreasonable to continue to neglect the effects of quantumgravity. In order to analyze the Q-dependence when thethin-string approximation is no longer valid, it is helpful toreformulate the topological classification of the m atter fieldconfigurations in a m ore general way.

In this con nection, we recall tha t the abelian Higgs modeldescribed in S ection 2 contains instanto ns when it is form u-lated in flat two-dimensional spacetime. The in stanton is justthe magnetic vortex on RZ, ith magnetic flux 2nlNe trapped

in its core. This instanton survives on a four-dimensionalEuclidean background , ift he background has the topology ofR2 x S2. In the thin-string limit, our world-sheet instantonmay be re-interpreted as jus t such a vortex. The advantage ofthis interpretation is that the vortex-num ber classification ofthe field configurations can be extended to the regime inwhich the thin-string approximation no longer applies, andthe action of the vortex is no longer large compared to h.

By extending the analysis beyond th e domain of vilidity ofthe thin-string approxim ation, one can hope to attain a quan -titative understanding of how ZNqua ntum hair turns off theHawking evaporation of the black hole. Work on thisproblem is in progress [25].

An important lesson to be learned from the discussionabove is that the no-hair principle carries less force on theEuclidean section of the black hole geometry than on the



Quantum Hair 263

Lorentzian section. The Lorentzian black hole cannot sup-port a stationary electromagnetic field, if the photon mass pis nonvanishing. But the Euclidean section can support a

vortex with nonzero flux. In the limit p -, 0, the flux of thevortex spreads out, and the Euclidean vortex solutionsmooth ly approaches the Euclidean section of the electricallycharged black hole. Thus, the p - limit is much less singu-

lar on the Euclidean section tha n o n the Lorentzian section.Correspondingly, the p + 0 limit is less singular in q uantu mphysics than in classical physics.

8. Conclusions

The main point of this talk has been to demonstrate that ablack hole can havy Aharonov-Bohm “qu antu m hair” tha tinfluences its thermodynamic behavior. We have seen thatthis is the case, at least in a particular field-theoretic model.The existence of q uan tum hair exposes the limitations of theno-hair principle in the quantum domain.

However, I have no t directly addressed an im portan t ques-

tion - Do black holes in Nature have quantum hair?I don’t know. But it is appropriate to note, in this con-

nection, that our current ideas about physics at very shortdistances (particularly superstring theory) suggest that fun-damental ph ysics is governed by an eno rmo us group of localsymmetries.’ Nearly all of these symmetries suffer spon-taneous breakdown at o r near the Planck scale. One is stronglytempted to speculate [5] tha t this symm etry breakdown givesrise to many varieties of qu antu m hair.+

Of course, even if quantum hair of the Aharanov-Bohmtype does not exist in Nature, black holes may well carryqua ntu m hair in a more general sense; that is, in order to givea complete quantum-mechanical description of processes

involving black holes, it may be necessary to ascribe to theblack hole many non-classical degrees of freedom. Quantumhair in this sense seems unavoidable if (contrary to Hawk-ing’s bold hypothesis) quantum coherence is actually main-tained in processes involving black holes.

In any event, I expect that furth er investigation of modelsof qua nt um hair like that described in this talk will rewarduswith valuable insights into the quantum-mechanical behav-iour of black holes, an d hence into some of the deep my steriesof quantum gravity.

Acknowledgements

Scme of the work described here has been done in collaboration with SidneyColeman and Fran k Wilczek. I have also benefited from h elpful discussionswith Murray Gell-Mann, Jim Hughes, Lawrence Krauss, Malcolm Perry,Alex Ridgway, Patricia Schwarz, Andy Strominger, Kip Thom e, an d manyof the participants at the 1990Nobel Symposium, particularly David Gross,

Alan Gu th, Stephen Hawking, Valery Rubakov, and Gerard ’t Hooft. Thiswork supported in part by the U.S. Department of Energy under ContractNO . DEAC-03-8 1-ER40050.

Appendix: Quantum hair and a xionic hair

I described in Section 2 how quantum hair can arise if anontrivial local discrete symmetry survives when the Higgs

* See, for example, [ 2 6 ] .

t It is far from clear, though, tha t the effects of such qua ntum hair could bereliably studied using semiclassical method s.

mechanism occurs, and argued in Section 3 that a black holecan carry such quan tum hair. In this appendix, I compare theblack hole hair associated with a manifest local symmetry

and another exotic type of black hole hair that has beenproposed recently.

A theory in which a global U(l) symmetry is spon-taneously broken contains an exactly massless Goldstone

boson, the axion, and also a topological defect, the axionstring. An axionic charge operator can be defined, and anobject that carries this charge exhibits a nontrivial Aharon ov-Bohm interaction with an axion string [27].By means of thisAharon ov-Bo hm effect, axionic charge can in principle bedetected at long range.$ Thu s, axionic charge is a type of h airthat, according to the argument in Section 3, can be carriedby a black hole. Black holes with axionic hair were firstdescribed by Bowick et al. [l 11.

Suppose that the spontaneous breakdown of the globalU( 1) symmetry is driven by th e con densa tion of a scalar field

where

( P > = v (A .2 )

an d 8 is the axion field. Then th e axionic charge Q in a volumeR may be expressed as

where * denotes the Hodge dual. The Aharonov-Bohm phaseacquired by an axion string that w inds around an object Q is

exp (2742). 04.4)

Thus, it is only the non-integer pa rt of Q that can be detectedat long range by means of the Aharon ov-Bo hm effect, an donly the non-integer part of Q tha t classifies the axio nic hairon a black hole.

It is natural to suggest [l13 that axionic wormholes [28]play a role in the evaporation of a black hole with axioniccharge. However, the axionic charge that flows down awormhole must be an integer [27-291; wormhole processestherefore have no effect on the axionic hair. Since the non-integer part of Q can be detected at infinite range, it must be

exactly conserved, in spite of wormholes or other exotica[ 5 , 8, 301.

As Bowick et al. [I ll noted, it is convenient to describeaxionic charge using an alternative formalism, related to theabove by a duality transformation. We may introduce athree-form field strength H defined by

H/ 2n = v 2 de. (A.5)

This field strength can be expressed as the curl of a two-formpotential

I assume here that the axion mass is exactly zero. Effects of an axion massare discussed in Ref. [8].

Physica Scripta T36



264 John Preskill

and the axionic charge in a volume R can be written as

1 1Q = -j H = -1 B ,

271 n 27c =

where C s the boundary of R .

The advantage of this dual formalism is that it allows us

to discuss axionic hair using the language of classical field

theory. According to the identity eq. (A.7), an object thatcarries axionic charge mu st have a long-range B field. Indeed,Bowick et al . [l 11 fou nd t ha t a black hole with axionic chargecould be constructed as a solution to classical field equations.In this solution, the field strength H vanishes outside theevent horizon, but the potential B is nonvanishing, asrequired by eq. (A.7).

Even though it admits this “classical” description, how-ever, axionic hair should be regarded as a type of quantumhair. The B field is not a classical local observable; is not even

gauge invariant.* The long-range B field of a black hole canbe detected only by means of the Aharonov-Bohm inter-action of the black hole w ith an axion string. This interaction

is no more classical than is the Aharonov -Bohm interactionof amagnetic solenoid with an electrically charged particle, inmassless electrodynamics.

In fact, even the Z, quantum hair that results from theHiggs mechanism is amenable to this sort of ‘‘classical’’description [25, 311. A duality transformation similar toeq . (AS) can be applied to the abelian Higgs model ofSection 2. Then the three-form field strength is

H / 2 n = v 2 (d e + e A ) ( A 4

where 6 is the phase of the Higgs field, and the electric chargemay be expressed as in eq. (A.7).t In this dua l version of theHiggs model, the Aharonov -Bohm interaction of a charge

with a string arises from a term in the action of the model th atresembles a Chern-Simons term; the Aharonov-Bohm phase

is [25, 321

exp (& j A F ) ,

where F = dA is the electromagnetic field strength. To seethat eq . (A.9) is just the Aharono v-Bohm phase, consider ahistory such tha t the world sheet of a cosmic string lies on aclosed surface E, where the string carries magnetic fluxCD = 2n/e. Then, if we neglect the thickness of the string,eq. (A.9) reduces to

exp (i Sr B ) = exp (27ciQ), (A . 10)

where Q is the charge enclosed by the world sheet. In theHiggs model of Section 2 , this charge Q is an integer multipleof 1/N, and eq. (A.lO) is the Z, Aharonov-Bohm phase.

By means of this dual reformulation of the abelian Higgsmodel, the black hole with 2, quantum hair can be con-structed as a solution to “classical” field equations [25, 311. Iemphasize once again, however, that in spite of the existence

* The dua l formulatio n respects the local symmetry B + B + dA, where A

is a one-form. A gauge transformation that is nonsingular at the eventhorizon of a black hole can change the (un measurab le) integer part of Q,

but cannot change the (measurable) non-integer part.t In a departure from the notation in Section 2, I have taken the charge of

the Higgs field to be one.

Physica Scripta T36

of this “classical” description, the Z, charge is a type ofquantum hair that escapes detection in the classical limit.

And the above discussion demonstrates that axionic hairmay be regarded as a special case (the e --t 0 limit) of themore general quantum hair considered in this paper.

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