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1、arXiv:hep-th/9510029v1 6 Oct 1995Virtual Black HolesS. W. HawkingDepartment of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeSilver StreetCambridge CB3 9EWUKAbstractOne would expect spacetime to have a foam-like structure on the Planckscale with a very high topology. If spacetime

2、 is simply connected (which isassumed in this paper), the non-trivial homology occurs in dimension two,and spacetime can be regarded as being essentially the topological sum ofS2 S2and K3 bubbles. Comparison with the instantons for pair creationof black holes shows that the S2 S2bubbles can be inter

3、preted as closedloops of virtual black holes. It is shown that scattering in such topologicalfluctuations leads to loss of quantum coherence, or in other words, to asuperscattering matrix $ that does not factorise into an S matrix and itsadjoint. This loss of quantum coherence is very small at low e

4、nergies foreverything except scalar fields, leading to the prediction that we may neverobserve the Higgs particle. Another possible observational consequence maybe that the angle of QCD is zero without having to invoke the problematicalexistence of a light axion. The picture of virtual black holes g

5、iven here alsosuggests that macroscopic black holes will evaporate down to the Planck sizeand then disappear in the sea of virtual black holes.11IntroductionIt was John Wheeler who first pointed out that quantum fluctuations inthe metric should be of order one at the Planck length. This would givesp

6、acetime a foam-like structure that looked smooth on scales large comparedto the Planck length. One might expect this spacetime foam to have a verycomplicated structure, with an involved topology. Indeed, whether spacetimehas a manifold structure on these scales is open to question. It might be afrac

7、tal. But manifolds are what we know how to deal with, whereas we haveno idea how to formulate physical laws on a fractal. In this this paper Ishall therefore consider how one might describe spacetime foam in terms ofmanifolds of high topology.I shall take the dimension of spacetime to be four.This m

8、ay soundrather conventional and restricted, but there seem to be severe problems ofinstability with Kaluza Klein theories.There is something rather specialabout four dimensional manifolds, so maybe that is why nature chose themfor spacetime. Even if there are extra hidden dimensions, I think one cou

9、ldgive a similar treatment and come to similar conclusions.There are at least two alternative pictures of spacetime foam, and I haveoscillated between them.One is the wormhole scenario 1, 2.Here theidea is that the path integral is dominated by Euclidean spacetimes withlarge nearly flat regions (par

10、ent universes) connected by wormholes or babyuniverses, though no good reason was ever given as to why this should be thecase. The idea was that one wouldnt notice the wormholes directly, but onlytheir indirect effects. These would change the apparent values of couplingconstants, like the charge on

11、an electron. There was an argument that theapparent value of the cosmological constant should be exactly zero. But thevalues of other coupling constants either were not determined by the theory,or were determined in such a complicated way that there was no hope ofcalculating them. Thus the wormhole

12、picture would have meant the end ofthe dream of finding a complete unified theory that would predict everything.A great attraction of the wormhole picture was that it seemed to providea mechanism for black holes to evaporate and disappear. One could imaginethat the particles that collapsed to form t

13、he black hole went offthrough awormhole to another universe or another region of our own universe. Sim-ilarly, all the particles that were radiated from the black hole during itsevaporation could have come from another universe, through the wormhole.2This explanation of how black holes could evapora

14、te and disappear seemsgood at a hand waving level, but it doesnt work quantitatively. In particu-lar, one cannot get the right relation between the size of the black hole andits entropy. The nearest one can get is to say that the entropy of a wormholeshould be the same as that of the radiation-fille

15、d Friedmann universe that isthe analytic continuation of the wormhole. However, this gives an entropyproportional to size to the three halves, rather than size squared, as for blackholes. Black hole thermodynamics is so beautiful and fits together so wellthat it cant just be an accident or a rough a

16、pproximation. So I began tolose faith in the wormhole picture as a description of spacetime foam.Instead, I went back to an earlier idea 3, which I will refer to as thequantum bubbles picture.Like the wormhole picture, this is formulatedin terms of Euclidean metrics.In the wormhole picture, one cons

17、ideredmetrics that were multiply connected by wormholes. Thus one concentratedon metrics with large values of the first Betti number, B1. This is equal to thenumber of generators of infinite order in the fundamental group. However, inthe quantum bubbles picture, one concentrates on spaces with large

18、 valuesof the second Betti number, B2. The spaces are generally taken to be simplyconnected, on the grounds that any multiple connectedness is not an essentialproperty of the local geometry, and can be removed by going to a coveringspace. This makes B1zero. By Poincare duality, the third Betti numbe

19、r, B3,is also zero. On this view, the essential topology of spacetime is containedin the second homology group, H2. The second Betti number, B2, is thenumber of two spheres in the space that cannot be deformed into each otheror shrunk to zero. It is also the number of harmonic two forms, or Maxwellf

20、ields, that can exist on the space. These harmonic forms can be dividedinto B2+self dual two forms and B2anti self dual forms. Then the Eulernumber and signature are given by = B2+ B2+ 2 =11282Zd4xgRR, = B2+ B2=1962Zd4xgRRif the spacetime manifold is compact. If it is non compact, = B2+B2+1and the v

21、olume integrals acquire surface terms.Barring some pure mathematical details, it seems that the topology ofsimply connected four manifolds can be essentially represented by glueing3Euler NumberSignatureS2 S240CP231CP23-1K32416K324-16Table 1:The Euler number and signature for the basic bubbles.togeth

22、er three elementary units, which I shall call bubbles. The three ele-mentary units are S2 S2, CP2and K3. The latter two have orientationreversed versions,CP2andK3. Thus there are five building blocks for simplyconnected four manifolds. Their values of the Euler number and signatureare shown in the t

23、able. To glue two manifolds together, one removes a smallball from each manifold and identifies the boundaries of the two balls. Thisgives the topological and differential structure of the combined manifold, butthey can have any metric.If spacetime has a spin structure, which seems a physically reas

24、onablerequirement, there cant be any CP2orCP2bubbles. Thus spacetime hasto be made up just of S2S2, K3 andK3 bubbles. K3 andK3 bubbles willcontribute to anomalies and helicity changing processes. However, their con-tribution to the path integral will be suppressed because of the fermion zeromodes th

25、ey contain, by the Atiyah-Singer index theorem. I shall thereforeconcentrate my attention on the S2 S2bubbles.When I first thought about S2S2bubbles in the late 70s, I felt that theyought to represent virtual black holes that would appear and disappear inthe vacuum as a result of quantum fluctuation

26、s. However, I was never able tosee how this correspondence would work. That was one reason I temporarilyswitched to the wormhole picture of spacetime foam. However, I now realizethat my mistake was to try to picture a single black hole appearing anddisappearing. Instead, I should have been thinking

27、of black holes appearingand disappearing in pairs, like other virtual particles. Equivalently, one canthink of a single black hole which is moving on a closed loop. If you deformthe loop into an oval, the bottom part corresponds to the appearance of apair of black holes and the top, to their coming

28、together and disappearing.In the case of ordinary particles like the electron, the virtual loops that4occur in empty space can be made into real solutions by applying an externalelectric field. There is a solution in Euclidean space with an electron mov-ing on a circle in a uniform electric field. I

29、f one analytically continues thissolution from the positive definite Euclidean space to Lorentzian Minkowskispace, one obtains an electron and positron accelerating away from eachother, pulled apart by the electric field. If you cut the Euclidean solution inhalf along = 0 and join it to the upper ha

30、lf of the Lorentzian solution, youget a picture of the pair creation of electron-positron pairs in an electric field.The electron and positron are really the same particle. It tunnels throughEuclidean space and emerges as a pair of real particles in Minkowski space.There is a corresponding solution

31、that represents the pair creation ofcharged black holes in an external electric or magnetic field. It was discoveredin 1976 by Ernst 4 and has recently been generalised to include a dilaton5 and two gauge fields 6. The Ernst solution represents two charged blackholes accelerating away from each othe

32、r in a spacetime that is asymptotic tothe Melvin universe. This is the solution of the Einstein-Maxwell equationsthat represents a uniform electric or magnetic field. Thus the Ernst solutionis the black hole analogue of the electron-positron pair accelerating awayfrom each other in Minkowski space.

33、Like the electron-positron solution, theErnst solution can be analytically continued to a Euclidean solution. Onehas to adjust the parameters of the solution, like the mass and charge ofthe black holes, so that the temperatures of the black hole and accelerationhorizons are the same. This allows one

34、 to remove the conical singularitiesand obtain a complete Euclidean solution of the Einstein-Maxwell equations.The topology of this solution is S2 S2minus a point which has been sentto infinity.The Ernst solution and its dilaton generalisations represent pair creationof real black holes in a backgro

35、und field, as was first pointed out by Gibbons7. There has been quite a lot of work recently on this kind of pair creation.However, in this paper I shall be less concerned with real processes like paircreation, which can occur only when there is an external field to provide theenergy, than with virt

36、ual processes that should occur even in the vacuumor ground state. The analogy between pair creation of ordinary particlesand the Ernst solution indicates that the topology S2 S2minus a pointcorresponds to a black hole loop in a spacetime that is asymptotic to R4.But S2 S2minus a point is the topolo

37、gical sum of the compact bubbleS2S2with the non compact space R4. Thus one can interpret the S2S25bubbles in spacetime foam as virtual black hole loops. These black holes neednot carry electric or magnetic charges, and will not in general be solutionsof the field equations. But they will occur as qu

38、antum fluctuations, even inthe vacuum state.If virtual black holes occur as vacuum fluctuations, one might expect thatparticles could fall into them and re-emerge as different particles, possiblywith loss of quantum coherence. I have been suggesting that this processshould occur for some time, but I

39、 wasnt sure how to show it. In fact Page,Pope and I did a calculation in 1979 of scattering in an S2 S2bubble, butwe didnt know how to interpret it 8. I feel now, however, that I understandbetter what is going on.The usual semi-classical approximation involves perturbations about asolution of the Eu

40、clidean field equations. One could consider particle scat-tering in the Ernst solution. This would correspond to particles falling intothe black holes pair created by an electric or magnetic field. The energy ofthe particles would then have to be radiated again before the pair came backtogether agai

41、n at the top of the loop and annihilated each other. However,such calculations are unphysical in two ways. First, the Ernst solution is notasymptotically flat, because it tends to a uniform electric or magnetic fieldat infinity. One might imagine that the solution describes a local region offield in

42、 an asymptotically flat spacetime, but the field would not normallyextend far enough to make the black hole loop real. This would mean thatthe field would have to curve the universe significantly. Second, even if onehad such a strong and far reaching field, it would presumably decay becauseof the pa

43、ir creation of real black holes.Instead, the physically interesting problem is when a number of particleswith less than the Planck energy collide in a small region that contains a vir-tual black hole loop. One might try and find a Euclidean solution to describethis process. There are reasons to beli

44、eve that such solutions exist, but itwould be very difficult to find them exactly, and such effort wouldnt reallybe appropriate, because one would expect the saddle point approximation tobreak down at the Planck length. Instead, I shall take the view that S2S2bubbles occur as quantum fluctuations an

45、d that the low energy particles thatscatter offthem have little effect on them. This means one should considerall positive definite metrics on S2S2, calculate the low energy scattering inthem, and add up the results, weighted with exp(I) where I is the actionof the bubble metric. If one were able to

46、 do this completely, one would have6calculated the full scattering amplitude, with all quantum corrections. How-ever, we neither know how to do the sum, nor how to calculate the particlescattering in any but rather simple metrics.Instead, I shall take the view that the scattering will depend on thes

47、pin of the field and the scale of the metric on the bubble, but will not be sosensitive to other details of the metric. In section 3 I shall therefore consider aparticular simple metric on S2S2in which one can solve the wave equations.I show that scattering in this metric leads to a superscattering

48、operator thatdoes not factorise. Hence there is loss of quantum coherence. In section 4, Iconsider scattering on more general S2S2metrics, and again find that the $operator doesnt factorise. The magnitude of the loss of quantum coherenceand its possible observational consequences are discussed in se

49、ction 5. Section6 examines the implications for the evaporation of macroscopic black holes,and section 7 summarises the conclusions of the paper.2The superscattering operatorIn this section I shall briefly describe the describe the results of reference 9on the superscattering operator $ which maps i

50、nitial density matrices to finaldensity matrices,A+B= $ADBCCD.The idea is to define n point expectation values for a field by a path integralover asymptotically Euclidean metrics,GE(x1,.,xn) =nYj=1 iJ(xj)!ZJ|J=0,ZJ =ZdeI,J.Because of the diffeomorphism gauge freedom, the expectation values havean in