2005 黑洞中的信息损失-精品文档资料整理.pdf
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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
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