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ON BLACK HOLES IN STRING THEORY⋆

arXiv:hep-th/9111052v1 25 Nov 1991ON BLACK HOLES IN STRING THEORY⋆Edward Witten†School of Natural SciencesInstitute for Advanced StudyOlden LanePrinceton, NJ 08540In its most elementary form, the Schwarzschild solution isds2 = −1 −2GMrdt2 +1 −2GMr−1dr2 + r2dΩ2. (1)One quickly sees that the space-time described by this metric is not geodesicallycomplete and hence that there must be more to the story.

Indeed, analytic contin-uation of the Schwarzschild solution, as described for instance in [1], transforms itto a formds2 = −du dv1 −uvew(u,v) + r(u, v)2dΩ2,(2)from which the global properties can be understood.Here r(u, v) and w(u, v)are certain functions whose details are not so material. The important propertiescome from the singularities at uv = 1.

There are two branches of this singularity,as u and v may be both negative or both positive. The “physical region” is theregion uv < 1 between the two singularities.

The branch of the singularity withu, v < 0 is the “white hole” singularity; it is to the past of the physical region,and can emit but not absorb matter. The branch of the singularity with u, v > 0is the “black hole” singularity; being to the future of the physical region, it can⋆Lecture at Strings ’91, Stonybrook, June 1991.† Research supported in part by NSF Grant PHY86-20266.1

absorb but not emit. The white hole violates the predictability of the classicaltheory, as classically one cannot predict what it will emit.

Relativists have dealtwith the white hole by the cosmic censorship hypothesis, according to which whiteholes and more general “naked singularities” never form from acceptable initialdata. (For instance, spherical stellar collapse from standard initial data gives aspacetime that coincides with the Schwarzschild solution only in the exterior ofthe collapsing star; the exterior contains the black hole singularity but not thewhite hole singularity.) Actually, there is not very much evidence for the cosmiccensorship conjecture, and it may well be false.

It is not clear that we should wishfor the truth of the conjecture. If cosmic censorship is false, then in principle wewould have the chance to observe new laws of physics that must take over near thewould-be naked singularity.

This might be more useful for physics than extendingthe scope of classical general relativity by proving cosmic censorship.It is usually assumed that the new physical laws associated with a possiblebreakdown of cosmic censorship would involve quantum gravity, but this may bewrong, particularly in view of the fact that the length scale of string physics isprobably a little larger than the Planck length. If indeed cosmic censorship is falsein general relativity but its analog is true in string theory, then it may well bethat it is classical string theory that becomes manifest near the would-be nakedsingularity.As for black holes, at the classical level they cause no breakdown in predictabil-ity for the outside observer.

Quantum mechanically, though, the idea that blackholes exist and white holes do not is paradoxical as the white hole is the CPTconjugate of the black hole. In any event, Stephen Hawking rendered the classicalpicture obsolete with his discovery of black hole radiation, showing that an isolatedhole will radiate until (after Hawking’s approximations break down) it reaches aquantum ground state or disappears completely.

A key aspect of the problem ofquantum mechanics of black holes is to describe this endpoint. More broadly, onewould like to describe the S-matrix for matter interacting with the black hole if (asI suspect) there is one.

If (as argued by Hawking) such an S-matrix does not exist,2

one would like to get a precise account of the nature of the obstruction and to learnhow to calculate in whatever framework (such as the density matrices advocatedby Hawking) replaces that of S-matrices.I think most physicists expect that Hawking radiation leads to complete disap-pearance of neutral black holes. For charged black holes the situation is likely to bequite different, under appropriate conditions, since a charged hole might be lighterthan any collection of “ordinary particles” of the same total charge.

Let us callthe electric and magnetic charges of the hole q and m. A charged black hole has aclassical ground state with mass M =pq2 + m2 in Planck units; this ground stateis simply the extreme Reissner-Nordstrom solution of the classical field equations.One would imagine that for suitable values of the charges the classical ground stateis some sort of approximation to the quantum ground state.In this respect, the case of a magnetically charged black hole is especiallyinteresting:(1) This case may be realized in nature if MGUT ∼MPlanck, since then the ’tHooft-Polyakov monopole is a black hole. In particular it is at least conceivable thatthe dark matter in our galactic halo could consist of magnetically charged blackholes.

This hypothesis is subject to experimental test; indeed recent experimentsare relatively close to the sensitivity required to exclude it [2]. Any type of discoveryof galactic halo particles would give particle physics a big boost, of course, butmagnetic black holes would be particularly exciting.

(2) For e << 1, the magnetic black hole is much heavier than the Planck massand much larger than the Planck radius. Quantum gravity should therefore, in asuitable sense, have only a weak effect on the structure of the magnetic black hole,and such objects might well be accessible to human understanding, maybe evenrelatively soon.

In particular, the deviation of the mass of a magnetic black holefrom the classical value is likely to be a quantum gravity or string theory effectof order e2/¯hc. It would certainly be an unusually interesting thing to measure ifmagnetic black holes were ever discovered.3

(3) For MGUT << MPlanck, Callan and Rubakov have already solved a problemthat is a protoype of what we want to do for MGUT ∼MPlanck. They consideredthe case of a Dirac point monopole interacting with charged fermions.

This classi-cal system does not quite make sense, since there are certain classical s wave modesthat go in but do not come out, or go out but did not come in; in more technicalterms, to give a self-adjoint extension of the Hamiltonian requires additional in-formation not present in the classical theory. The problem has a striking analogywith the Schwarzschild space-time where again the problem is that there are modes(emitted from the past singularity) that go out but did not come in, and there areother modes (absorbed by the future singularity) that go in but do not go out.

Insolving the MGUT << MPlanck problem, Callan and Rubakov showed that a twodimensional s-wave effective field theory gives a good description of monopoles inthat regime. It seems natural to hope that this is still true if MGUT ∼MPlanck.That would mean that some of the important qualitative aspects of black holephysics could be described by an effective two dimensional field theory.In the case studied by Callan and Rubakov, the relevant two dimensional fieldtheory is weakly coupled, but still requires careful analysis and exhibits strikingphenomena, precisely because a naive form of the weak coupling limit would giveback the pathologies of the classical system.

It is reasonable to hope that theirwork is a prototype for at least some aspects of the black hole problem.What is the status of black holes in string theory? One might have believedthat, because of the excellent short distance behavior, there would be no classicalsingularities in string theory.

But it has been found recently that the very metricds2 = −du dv/(1 −uv) that is the essence of the black hole gives a conformal fieldtheory. In fact, there is a conformal field theory with a two dimensional targetspace, parametrized by two variables u and v, and a world-sheet actionI =Zd2σ∂αu∂αv1 −uv+ R(2)Φ(u, v),(3)where Φ(u, v) = ln(1 −uv) + constant is the dilaton field.

The Euclidean version4

of this solution was first found by Elitzur, Forge, and Rabinovici [3]. Differentforms of the solution were rediscovered by several authors from various points ofview [4,5,6] (and implicitly discussed in current algebra by Bars; see his lectureat this conference and references therein) before I rediscovered the solution as anSL(2, IR)/U(1) coset model and interpreted it as a black hole [7].It is no accident that this system involves a target space-time so similar tothe Schwarzschild solution.

The general s wave ansatz of four dimensional generalrelativityds2 = gαβdxαdxβ + eΦdΩ2(4)involves a two dimensional metric tensor g and a two dimensional scalar field Φ.These are analogous to the metric and dilaton field of the low energy limit ofstring theory, and indeed the two dimensional action that one gets by dimensionalreduction of four dimensional general relativity with the ansatz (4) is very similarto the familiar lowest order actionI =Zd2x√geΦ R + (∇Φ)2 + 8(5)of the graviton-dilaton system in string theory.The analogy is even better ifone considers the dimensional reduction of general relativity in the presence of aspherically symmetric magnetic field, so as to get a system in which the black holehas a classical ground state.If one writes the black hole sigma model (3) in Schwarzschild-like coordinates,and traces what the Hawking radiation ought to mean in this situation [7], one sees,at least heuristically, what the endpoint of the Hawking radiation ought to be: itshould be the standard flat space-time of the usual c = 1 model, with a lineardilaton field. In making an analogy of the c = 1 model with four dimensionalblack holes, this makes it clear how we should think about the linear dilaton field:it is the “field” of an “object” sitting at r = −∞(where the tachyon potentialcorresponding to the world-sheet Liouville interaction is large).The standard5

c = 1 flat space-time should therefore be thought of as an analog, not of fourdimensional Minkowski space, but of the extreme Reissner-Nordstrom solution infour dimensions. This enables us to understand intuitively the otherwise annoyingabsence of Poincar´e invariance in the model.The field theory of the s-wave sector of general relativity is an interestingtwo dimensional field theory, which for many years has looked like a temptingmodel of the quantum physics of black holes – particularly in view of the analogywith the Callan-Rubakov effect that I explained before.

Of course to get a realmodel of black hole physics, one must couple this system to “matter.” The s-wave component of a neutral scalar field would be a satisfactory form of mattermathematically, but there is a perhaps more “physical” candidate – in the fieldof a monopole of minimal Dirac charge, the usual quarks and leptons have s-wavecomponents (which are important in the Callan-Rubakov effect). The combinedsystem of spherically symmetric geometry and s-wave fermions is weakly coupledif e << 1.

Despite the weak coupling, the s-wave field theory of magnetic blackholes has defied understanding (at least at my hands).String theory with c = 1 or D = 2 is superficially a very similar system, withthe massless “tachyon” playing the role of the bosonized s-wave fermions. It isbelieved to be exactly soluble via matrix models.

The already computed S-matrixof this model can probably be understood as a long-sought example of an S-matrixfor matter interacting quantum mechanically with a black hole. If we could get agood understanding of the space-time interpretation of the matrix model results,we could probably sharpen our understanding of black holes.The most striking puzzle in this area is probably the apparent absence inthe matrix model of an analog of the expected back-reaction of matter on thegravitational field.

In other words, if D = 2 string theory really does describetarget space gravity, one would expect to see in the theory a suitable analog of theback-reaction of matter on gravity. In the matrix model, all that we presently seethat might even loosely be regarded as “gravity” is the one body Hamiltonian of the6

free fermions (which is the “background” in which they are moving). But the freefermions have no back reaction on their own Hamiltonian.

Perhaps some degreesof freedom (analogous to the dilaton?) have been suppressed or integrated out inthe matrix model description.

A proper understanding of this point will probablyhelp make it clear how much can be learned about real black holes from D = 2string theory.REFERENCES1. C. W. Misner, K. S. Thorne, and J.

A. Wheeler, Gravitation (W. H. Freemanand Co., 1973), especially pp. 831-2.2.

S. Orito et. al., “Search For Supermassive Relics With a 2000m2 Array OfPlastic Track Detectors,” Phys.

Rev. Lett.

66 (1991) 1951.3. S. Elitzur, A.

Forge, and E. Rabinovici, “Some Global Aspects of StringCompactifications,” preprint RI-141(90).4. K. Bardakci, M. Crescimanno, and E. Rabinovici, “Parafermions From CosetModels,” LBL preprint (1990).5.

M. Rocek, K. Schoutens, and A. Sevrin, “Off-Shell WZW Models In ExtendedSuperspace,” IASSNS-HEP-91/14.6. G. Mandal, A. Sengupta, and S. Wadia, “Classical Solutions Of 2-Dimensional String Theory,” IASSYS-HEP/91/10.7.

E. Witten, “String Theory And Black Holes,” Phys. Rev.

D44 (1991) 314.7

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