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1、arXiv:gr-qc/9409013v2 22 Sep 1994NI-94-012DAMTP/R 94-26UCSBTH-94-25gr-qc/9409013Entropy, Area, and Black Hole PairsS. W. Hawking,1Gary T. Horowitz,1and Simon F. Ross21Isaac Newton Institute for Mathematical SciencesUniversity of Cambridge, 20 Clarkson Rd., Cambridge CB3 0EH2Department of Applied Mat

2、hematics and Theoretical PhysicsUniversity of Cambridge, Silver St., Cambridge CB3 9EWInternet: S.F.Rossamtp.cam.ac.ukAbstractWe clarify the relation between gravitational entropy and the area of horizons. Wefirst show that the entropy of an extreme Reissner-Nordstr om black hole is zero, despitethe

3、 fact that its horizon has nonzero area. Next, we consider the pair creation of extremaland nonextremal black holes. It is shown that the action which governs the rate of this paircreation is directly related to the area of the acceleration horizon and (in the nonextremalcase) the area of the black

4、hole event horizon. This provides a simple explanation of theresult that the rate of pair creation of non-extreme black holes is enhanced by preciselythe black hole entropy. Finally, we discuss black hole annihilation, and argue that Planckscale remnants are not sufficient to preserve unitarity in q

5、uantum gravity.September 1994Current address: Department of Applied Mathematics and Theoretical Physics, Silver St.,Cambridge CB3 9EW. Internet: swh1amtp.cam.ac.ukCurrent address: Physics Department, University of California, Santa Barbara, CA. 93111.Internet: garycosmic.physics.ucsb.edu1. Introduct

6、ionThe discovery of black hole radiation 1 confirmed earlier indications 2 of a closelink between thermodynamics and black hole physics. Various arguments were given thata black hole has an entropy which is one quarter of the area of its event horizon in Planckunits. However, despite extensive discu

7、ssion, a proper understanding of this entropy is stilllacking. In particular there is no direct connection between this entropy and the numberof internal states of a black hole.We will re-examine the connection between gravitational entropy and horizon area intwo different contexts. We first conside

8、r charged black holes and show that while non-extreme configurations satisfy the usual relation S = Abh/4, extreme Reissner-Nordstr omblack holes do not. They always have zero entropy even though their event horizon hasnonzero area. The entropy changes discontinuously when the extremal limit is reac

9、hed. Wewill see that this is a result of the fact that the horizon is infinitely far away for extremalholes which results in a change in the topology of the Euclidean solution.The second context is quantum pair creation of black holes. It has been known forsome time that one can create pairs of oppo

10、sitely charged GUT monopoles in a strongbackground magnetic field 3. The rate for this process can be calculated in an instantonapproximation and is given by eIwhere I is the Euclidean action of the instanton. Formonopoles with mass m and charge q, in a background field B one finds (to leading order

11、in qB) that I = m2/qB.It has recently been argued that charged black holes cansimilarly be pair created in a strong magnetic field 4,5,6. An appropriate instanton hasbeen found and its action computed. The instanton is obtained by starting with a solutionto the Einstein-Maxwell equations found by Er

12、nst 7, which describes oppositely chargedblack holes uniformly accelerated in a background magnetic field.This solution has aboost symmetry which becomes null on an acceleration horizon as well as the black holeevent horizon, but is time-like in between. One can thus analytically continue to obtaint

13、he Euclidean instanton. It turns out that regularity of the instanton requires that theblack holes are either extremal or slightly nonextremal. In the nonextremal case, the twoblack hole event horizons are identified to form a wormhole in space. It was shown in 6that the action for the instanton cre

14、ating extremal black holes is identical to that creatinggravitating monopoles 8 (for small qB) while the action for non-extreme black holes isless by precisely the entropy of one black hole Abh/4. This implies that the pair creationrate for non-extremal black holes is enhanced over that of extremal

15、black holes by a factorof eAbh/4, which may be interpreted as saying that non-extreme black holes have eAbh/4internal states and are produced in correlated pairs, while the extreme black holes have aunique internal state. This was not understood at the time, but is in perfect agreementwith our resul

16、t that the entropy of extreme black holes is zero.To better understand the rate of pair creation, we relate the instanton action to anenergy associated with boosts, and surface terms at the horizons. While the usual energyis unchanged in the pair creation process, the boost energy need not be. In fa

17、ct, we will seethat it is changed in the pair creation of nongravitating GUT monopoles. Remarkably, itturns out that it is unchanged when gravity is included. This allows us to derive a simple1formula for the instanton action. For the pair creation of nonextremal black holes we findI = 14(A + Abh),(

18、1.1)where A is the difference between the area of the acceleration horizon when the blackholes are present and when they are absent, and Abhis the area of the black hole horizon.For the pair creation of extremal black holes (or gravitating monopoles) the second termis absent so the rate is entirely

19、determined by the area of the acceleration horizon,I = 14A.(1.2)This clearly shows the origin of the fact that nonextremal black holes are pair created ata higher rate given by the entropy of one black hole.The calculation of each side of (1.2) is rather subtle. The area of the accelerationhorizon i

20、s infinite in both the background magnetic field and the Ernst solution.Tocompute the finite change in area we first compute the area in the Ernst solution out to alarge circle. We then subtract offthe area in the background magnetic field solution out toa circle which is chosen to have the same pro

21、per length and the same value ofHA (whereA is the vector potential). Similarly, the instanton action is finite only after we subtract offthe infinite contribution coming from the background magnetic field. In 5, the calculationwas done by computing the finite change in the action when the black hole

22、 charge is varied,and then integrating from zero charge to the desired q. In 6, the action was calculatedinside a large sphere and the background contribution was subtracted using a coordinatematching condition. Both methods yield the same result. But given the importance ofthe action for the pair c

23、reation rate, one would like to have a direct calculation of it bymatching the intrinsic geometry on a boundary near infinity as has been done for otherblack hole instantons. We will present such a derivation here and show that the result isin agreement with the earlier approaches. Combining this wi

24、th our calculation of A, weexplicitly confirm the relations (1.1) and (1.2).Perhaps the most important application of gravitational entropy is to the black holeinformation puzzle. Following the discovery of black hole radiation, it was argued thatinformation and quantum coherence can be lost in quan

25、tum gravity. This seemed to bean inevitable consequence of the semiclassical calculations which showed that black holesemit thermal radiation and slowly evaporate. However, many people find it difficult toaccept the idea of nonunitary evolution. They have suggested that either the informationthrown

26、into a black hole comes out in detailed correlations not seen in the semiclassicalapproximation, or that the endpoint of the evaporation is a Planck scale remnant whichstores the missing information. In the latter case, the curvature outside the horizon wouldbe so large that semiclassical arguments

27、would no longer be valid. However, considerationof black hole pair creation suggests another quantum gravitational process involving blackholes, in which information seems to be lost yet the curvature outside the horizons alwaysremains small.The basic observation is that if black holes can be pair c

28、reated, then it must be possiblefor them to annihilate. In fact, the same instanton which describes black hole pair creation2can also be interpreted as describing black hole annihilation. Once one accepts the ideathat black holes can annihilate, one can construct an argument for information loss asf

29、ollows. Imagine pair creating two magnetically charged (nonextremal) black holes whichmove far apart into regions of space without a background magnetic field. One could thentreat each black hole independently and throw an arbitrarily large amount of matter andinformation into them. The holes would

30、then radiate and return to their original mass. Onecould then bring the two holes back together again and try to annihilate them. Of course,there is always the possibility that they will collide and form a black hole with no magneticcharge and about double the horizon area. This black hole could eva

31、porate in the usualway down to Planck scale curvatures. However, there is a probability of about eAbh/4times the monopole annihilation probability that the black holes will simply annihilate,their energy being given offas electromagnetic or gravitational radiation. One can choosethe magnetic field a

32、nd the value of the magnetic charge in such a way that the curvature iseverywhere small. Thus the semiclassical approximation should remain valid. This impliesthat even if small black hole remnants exist, they are not sufficient to preserve unitarity.This discussion applies to nonextremal black hole

33、s. Since extremal black holes have zeroentropy, they behave differently, as we will explain.In the next section we discuss the entropy of a single static black hole and show thatan extreme Reissner-Nordstr om black hole has zero entropy. Section 3 contains a review ofthe Ernst instanton which descri

34、bes pair creation of extremal and nonextremal black holes.In section 4 we discuss the boost energy and show that while it is changed for pair creationin flat space, it is unchanged for pair creation in general relativity. Section 5 contains aderivation of the relations (1.1) and (1.2) and the detail

35、ed calculations of the accelerationhorizon area and instanton action which confirm them. Finally, section 6 contains furtherdiscussion of black hole annihilation and some concluding remarks.In Appendix A we consider the generalization of the Ernst instanton which includesan arbitrary coupling to a d

36、ilaton 9. We will extend the development of the precedingsections to this case, showing that the boost energy is still unchanged in this case, andcalculating the difference in area and the instanton action using appropriate boundaryconditions. The result for the instanton action is in complete agree

37、ment with 6.2. Extreme Black Holes Have Zero EntropyIn this section we consider the entropy of a single static black hole. The reason thatgravitational configurations can have nonzero entropy is that the Euclidean solutions canhave nontrivial topology 10.In other words, if we start with a static spa

38、cetime andidentify imaginary time with period , the manifold need not have topology S1 where is some three manifold. In fact, for non-extreme black holes, the topology is S2 R2.This means that the foliation one introduces to rewrite the action in Hamiltonian formmust meet at a two sphere Sh. The Euc

39、lidean Einstein-Maxwell action includes a surfaceterm,I =116ZM(R + F2) 18IMK,(2.1)where R is the scalar curvature, Fis the Maxwell field, and K is the trace of the extrinsiccurvature of the boundary. In fact, if the spacetime is noncompact, the action is defined3only relative to some background solu

40、tion (g0,F0). This background is usually taken to beflat space with zero field, but we shall consider more general asymptotic behavior. Whenone rewrites the action in Hamiltonian form, there is an extra contribution from the twosphere Sh. This arises since the surfaces of constant time meet at Shand

41、 the resultingcorner gives a delta-function contribution to K 10. Alternatively, one can calculate thecontribution from Shas follows 11 (see 12 for another approach). The total action canbe written as the sum of the action of a small tubular neighborhood of Shand everythingoutside. The action for th

42、e region outside reduces to the standard Hamiltonian form1,which for a static configuration yields the familiar result H. The action for the smallneighborhood of Shyields Abh/4 where Abhis the area of Sh. Thus the total Euclideanaction isI = H 14Abh.(2.2)The usual thermodynamic formula for the entro

43、py isS = ? 1?logZ,(2.3)where the partition function Z is given (formally) by the integral of eIover all Euclideanconfigurations which are periodic in imaginary time with period at infinity. The action forthe solution describing a nonextremal black hole is (2.2) so if we approximate logZ I,we obtain

44、the usual resultS =14Abh.(2.4)Recall that the Reissner-Nordstr om metric is given byds2= ?1 2Mr+Q2r2?dt2+?1 2Mr+Q2r2?1dr2+ r2d.(2.5)For non-extreme black holes Q2 M2, the above discussion applies. But the extremeReissner-Nordstr om solution is qualitatively different. When Q2= M2, the horizon r = Mi

45、s infinitely far away along spacelike directions. In the Euclidean solution, the horizon isinfinitely far away along all directions. This means that the Euclidean solution can beidentified with any period . So the action must be proportional to the period I . Itfollows from (2.3) that in the usual a

46、pproximation logZ I, the entropy is zero,Sextreme= 0.(2.6)This is consistent with the fact that gravitational entropy should be associated with non-trivial topology. The Euclidean extreme Reissner-Nordstr om solution (with periodicallyidentified) is topologically S1RS2. Since there is an S1factor, t

47、he surfaces introduced1The surface terms in the Hamiltonian can be obtained directly from the surface terms in theaction. For a detailed discussion which includes spacetimes which are not asymptotically flat (e.g.the Ernst solution) and horizons which are not compact (e.g. acceleration horizons) see

48、 13.4to rewrite the action in canonical form do not intersect. Thus there is no extra contributionfrom the horizon and the entropy is zero. Since the area of the event horizon of an extremeReissner-Nordstr om black hole is nonzero, we conclude that the entropy of a black hole isnot always equal to A

49、bh/4; (2.4) holds only for nonextremal black holes.The fact that the entropy changes discontinuously in the extremal limit implies thatone should regard non-extreme and extreme black holes as qualitatively different objects.One is already used to the idea that a non-extreme black hole cannot turn in

50、to an extremehole: the nearer the mass gets to the charge the lower the temperature and so the lowerthe rate of radiation of mass. Thus the mass will never exactly equal the charge. However,the idea that extreme and non-extreme black holes are distinct presumably also impliesthat extreme black holes