《2005 黑洞中的信息损失-精品文档资料整理.pdf》由会员分享，可在线阅读，更多相关《2005 黑洞中的信息损失-精品文档资料整理.pdf（5页珍藏版）》请在taowenge.com淘文阁网|工程机械CAD图纸|机械工程制图|CAD装配图下载|SolidWorks_CaTia_CAD_UG_PROE_设计图分享下载上搜索。
1、arXiv:hep-th/0507171v2 15 Sep 2005Information Loss in Black HolesS.W.HawkingDAMTP, Center for Mathematical Sciences, university of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UKThe question of whether information is lost in black holes is investigated using Euclidean pathintegrals. The formation
2、 and evaporation of black holes is regarded as a scattering problem withall measurements being made at infinity. This seems to be well formulated only in asymptoticallyAdS spacetimes. The path integral over metrics with trivial topology is unitary and informationpreserving. On the other hand, the pa
3、th integral over metrics with non-trivial topologies leads tocorrelation functions that decay to zero. Thus at late times only the unitary information preservingpath integrals over trivial topologies will contribute. Elementary quantum gravity interactions donot lose information or quantum coherence
4、.PACS numbers: 04.70.DyI.INTRODUCTIONThe black hole information paradox started in 1967 when Werner Israel showed that the Schwarzschild metric wasthe only static vacuum black hole solution 1. This was then generalized to the no hair theorem, the only stationaryrotating black hole solutions of the E
5、instein Maxwell equations are the Kerr Newman metrics 9. The no hair theoremimplied that all information about the collapsing body was lost from the outside region apart from three conservedquantities: the mass, the angular momentum, and the electric charge.This loss of information wasnt a problem i
6、n the classical theory. A classical black hole would last for ever andthe information could be thought of as preserved inside it, but just not very accessible.However, the situationchanged when I discovered that quantum effects would cause a black hole to radiate at a steady rate 2. At leastin the a
7、pproximation I was using the radiation from the black hole would be completely thermal and would carryno information3. So what would happen to all that information locked inside a black hole that evaporated awayand disappeared completely? It seemed the only way the information could come out would b
8、e if the radiation wasnot exactly thermal but had subtle correlations. No one has found a mechanism to produce correlations but mostphysicists believe one must exist. If information were lost in black holes, pure quantum states would decay into mixedstates and quantum gravity wouldnt be unitary.I fi
9、rst raised the question of information loss in 75 and the argument continued for years without any resolutioneither way. Finally, it was claimed that the issue was settled in favor of conservation of information by ADS-CFT.ADS-CFT is a conjectured duality between string theory in anti de Sitter spac
10、e and a conformal field theory on theboundary of anti de Sitter space at infinity ? . Since the conformal field theory is manifestly unitary the argumentis that string theory must be information preserving. Any information that falls in a black hole in anti de Sitter spacemust come out again. But it
11、 still wasnt clear how information could get out of a black hole. It is this question, I willaddress in this paper.II.EUCLIDEAN QUANTUM GRAVITYBlack hole formation and evaporation can be thought of as a scattering process. One sends in particles and radiationfrom infinity and measures what comes bac
12、k out to infinity. All measurements are made at infinity, where fields areweak and one never probes the strong field region in the middle. So one cant be sure a black hole forms, no matterhow certain it might be in classical theory. I shall show that this possibility allows information to be preserv
13、ed andto be returned to infinity.I adopt the Euclidean approach 5, the only sane way to do quantum gravity nonperturbatively. One might thinkone should calculate the time evolution of the initial state by doing a path integral over all positive definite metricsthat go between two surfaces that are a
14、 distance T apart at infinity. One would then Wick rotate the time intervalT to the Lorentzian.Electronic address: .2The trouble with this is that the quantum state for the gravitational field on an initial or final space-like surface isdescribed by a wave function which is a functional of the geome
15、tries of space-like surfaces and the matter fieldshij,t(1)where hijis the three metric of the surface, stands for the matter fields and t is the time at infinity. However thereis no gauge invariant way in which one can specify the time position of the surface in the interior. This means onecan not g
16、ive the initial wave function without already knowing the entire time evolution.One can measure the weak gravitational fields on a time like tube around the system but not on the caps at topand bottom which go through the interior of the system where the fields may be strong. One way of getting rid
17、of thedifficulties of caps would be to join the final surface back to the initial surface and integrate over all spatial geometriesof the join. If this was an identification under a Lorentzian time interval T at infinity, it would introduce closed timelike curves. But if the interval at infinity is
18、the Euclidean distance the path integral gives the partition function forgravity at temperature = 1.Z() =ZDgDeIg,= Tr(eH)(2)There is an infrared problem with this idea for asymptotically flat space. The partition function is infinite becausethe volume of space is infinite. This problem can be solved
19、 by adding a small negative cosmological constant whichmakes the effective volume of the space the order of 3/2. It will not affect the evaporation of a small black holebut it will change infinity to anti-de Sitter space and make the thermal partition function finite.It seems that asymptotically ant
20、i-de Sitter space is the only arena in which particle scattering in quantum gravityis well formulated. Particle scattering in asymptotically flat space would involve null infinity and Lorentzian metrics,but there are problems with non-zero mass fields, horizons and singularities. Because measurement
21、s can be madeonly at spatial infinity, one can never be sure if a black hole is present or not.III.THE PATH INTEGRALThe boundary at infinity has topology S1 S2. The path integral that gives the partition function is taken overmetrics of all topologies that fit inside this boundary. The simplest topo
22、logy is the trivial topology S1 D3whereD3is the three disk. The next simplest topology and the first non-trivial topology is S2 D2. This is the topologyof the Schwarzschild anti-de Sitter metric. There are other possible topologies that fit inside the boundary but thesetwo are the important cases, t
23、opologically trivial metrics and the black hole. The black hole is eternal: it can notbecome topologically trivial at late times.The trivial topology can be foliated by a family of surfaces of constant time. The path integral over all metricswith trivial topology can be treated canonically by time s
24、licing. The argument is the same as for the path integralfor ordinary quantum fields in flat space. One divides the time interval T into time steps t. In each time stepone makes a linear interpolation of the fields qiand their conjugate momenta between their values on succesive timesteps. This metho
25、d applies equally well to topologically trivial quantum gravity and shows that the time evolution(including gravity) will be generated by a Hamiltonian. This will give a unitary mapping between quantum states onsurfaces separated by a time interval T at infinity.This argument can not be applied to t
26、he non-trivial black hole topologies. They can not be foliated by a familyof surfaces of constant time because they dont have any spatial cross-sections that are a three cycle, modulo theboundary at infinity. Any global symmetry would lead to conserved global charges on such a three cycle. Thesewoul
27、d prevent correlation functions from decaying in topologically trivial metrics. Indeed, one can regard the unitaryHamiltonian evolution of a topologically trivial metric as a global conservation of information flowing through a threecycle under a global time translation. On the other hand, non-trivi
28、al black hole topologies wont have any conservedquantity that will prevent correlation functions from decaying. It is therefore very plausible that the path integralover a topologically non trivial metric gives correlation functions that decay to zero at late Lorentzian times. Thisis born out by exp
29、licit calculations. The correlation functions decay as more and more of the wave falls through thehorizon into the black hole.3IV.GIANT BLACK HOLESIn a thought provoking paper 6, Maldacena considered how the loss of information into black holes in AdS couldbe reconciled with the unitarity of the CFT
30、 on the boundary of AdS. He studied the canonical ensemble for AdS attemperature 1. This is given by the path integral over all metrics that fit inside the boundary S1 S2where theradius of the S1is times the radius of the S2. For there are three classical solutions that fit inside theboundary: perio
31、dically identified AdS, a small black hole and a giant black hole. If one normalizes AdS to have zeroaction, small black holes have positive action and giant black holes have very large negative action. They thereforedominate the canonical ensemble but the other solutions are important.Maldacena con
32、sidered two point correlation functions in the CFT on the boundary of AdS. The vacuum expectationvalue can be thought of as the response at y to disturbances at x corresponding to the insertion of theoperator O. It would be difficult to compute in a strongly coupled CFT but by AdS-CFT it is given by
33、 boundary toboundary Green functions on the AdS side which can be computed easily.The Green functions in the dominant giant black hole solution have the standard form for small separation betweenx and y but decay exponentially as y goes to late times and most of the effect of the disturbance at x fa
34、lls through thehorizon of the black hole. This looks very like information loss into the black hole. On the CFT side it corresponds toscreening of the correlation function whereby the memory of the disturbance at x is washed out by repeated scattering.However the CFT is unitary, so theoretically it
35、must be possible to compute its evolution exactly and detect thedisturbance at late times from the many point correlation function. All Green functions in the black hole metricswill decay exponentially to zero but Maldacena realized that the Green functions in periodically identified AdS dontdecay a
36、nd have the right order of magnitude to be compatible with unitarity. In this paper I have gone further andshown that the path integral over topologically trivial metrics like periodically identified AdS is unitary.So in the end everyone was right in a way. Information is lost in topologically non-t
37、rivial metrics like black holes.This corresponds to dissipation in which one loses sight of the exact state. On the other hand, information aboutthe exact state is preserved in topologically trivial metrics. The confusion and paradox arose because people thoughtclassically in terms of a single topol
38、ogy for spacetime. It was either R4or a black hole. But the Feynman sum overhistories allows it to be both at once. One can not tell which topology contributed to the observation, any more thanone can tell which slit the electron went through in the two slits experiment. All that observation at infi
39、nity candetermine is that there is a unitary mapping from initial states to final and that information is not lost.V.SMALL BLACK HOLESGiant black holes are stable and wont evaporate away. However, small black holes are unstable and behave likeblack holes in asymptotically flat space if M 127. Howeve
40、r, in the approach I am using, one can not justset up a small black hole, and watch it evaporate. All one can do, is to consider correlation functions of operatorsat infinity. One can apply a large number of operators at infinity, weighted with time functions, that in the classicallimit would create
41、 a spherical ingoing wave from infinity, that in the classical theory would form a small black hole.This would presumably then evaporate away.For years, I tried to think of a Euclidean geometry that could represent the formation and evaporation of a singleblack hole, but without success. I now reali
42、ze there is no such geometry, only the eternal black hole, and pair creationof black holes, followed by their annihilation. The pair creation case is instructive. The Euclidean geometry canbe regarded as a black hole moving on a closed loop, as one would expect. However, the corresponding Lorentzian
43、geometry, represents two black holes that come in from infinity in the infinite past, and accelerate away from each otherfor ever. The moral of this is that one should not take the Lorentzian analytic continuation of a Euclidean geometryliterally as a guide to what an observer would see. Similarly,
44、the formation and evaporation of a small black hole, andthe subsequent formation of small black holes from the thermal radiation, should be represented by a superpositionof trivial metrics and eternal black holes. The probability of observing a small black hole, at a given time, is givenby the diffe
45、rence between the actions. A similar discussion of correlation functions on the boundary shows that thetopologically trivial metrics make black hole formation and evaporation unitary and information preserving. One canrestrict to small black holes by integrating the path integral over along a contou
46、r parallel to the imaginary axiswith the factor eE0. This projects out the states with energy E0.Z(E0) =Z+iidZ()eE0(3)For E0 12most of these states will correspond to thermal radiation in AdS which acts like a confiningbox of volume 32. However, there will be thermal fluctuations which occasionally
47、will be large enough to cause4gravitational collapse to form a small black hole. This black hole will evaporate back to thermal AdS. If one nowconsiders correlation functions on the boundary of AdS, one again finds that there is apparent information loss inthe small black hole solution but in fact i
48、nformation is preserved by topologically trivial geometries. Another way ofseeing that information is preserved in the formation and evaporation of small black holes is that the entropy in thebox does not increase steadily with time as it would if information were lost each time a small black hole f
49、ormed andevaporated.VI.CONCLUSIONSIn this paper, I have argued that quantum gravity is unitary and information is preserved in black hole formationand evaporation. I assume the evolution is given by a Euclidean path integral over metrics of all topologies. Theintegral over topologically trivial metr
50、ics can be done by dividing the time interval into thin slices and using a linearinterpolation to the metric in each slice. The integral over each slice will be unitary and so the whole path integralwill be unitary.On the other hand, the path integral over topologically non trivial metrics will lose