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1、The Information Paradox for Black Holes.S. W. Hawking,DAMTP,Centre for Mathematical Sciences,University of Cambridge,Wilberforce Road,Cambridge, CB3 0WAUK.ABSTRACTI propose that the information loss paradox can be resolved by considering thesupertranslation of the horizon caused by the ingoing parti

2、cles. Informationcan be recovered in principle, but it is lost for all practical purposes.Talk given on 28 August 2015 at “Hawking Radiation”, a conference held at KTH Royal Institute ofTechnology, Stockholm.arXiv:1509.01147v1 hep-th 3 Sep 2015Forty years ago I wrote a paper, “Breakdown of Predictab

3、ility in Gravitational Col-lapse” 1, in which I claimed there would be loss of predictability of the final state if theblack hole evaporated completely. This was because one could not measure the quantumstate of what fell into the black hole. The loss of information would have meant theoutgoing radi

4、ation is in a mixed state and the S-Matrix was non-unitary.Since the publication of that paper, the AdS/CFT correspondence has shown thereis no information loss. This is the information paradox: How does the information ofthe quantum state of the infalling particles re-emerge in the outgoing radiati

5、on? Thishas been an outstanding problem in theoretical physics for the last forty years. Despitea large number of papers (see reference 2,3 for a list), no satisfactory resolution hasbeen found. I now propose that the information is stored, not in the interior of the blackhole (as one might expect),

6、 but on its boundary, the event horizon. This is a form ofholography.The concept of supertranslations was introduced in 1962 by Bondi, Metzner and Sachs(BMS) 4,5, to describe the asymptotic isometries of an asymptotically flat spacetimein the presence of gravitational radiation. In other words the B

7、MS group describes thesymmetry on I+. For an asymptotically flat spacetime, a supertranslation shifts theretarded time u tou0= u + ,(1)where is a function of the coordinates on the 2-sphere. The supertranslation moves eachpoint of I+a distance to the future along the null geodesic generators of I+.

8、Notethat the usual time and space translations form a four parameter sub-group of the infinitedimensional supertranslations but they are not an invariant sub-group of the BMS group.Listening to a lecture by Strominger on the BMS group, 6, at the Mitchell Institutefor Fundamental Physics and Astronom

9、y workshop this April, I realized that stationaryblack hole horizons also have supertranslations. In this case, the advanced time v is shiftedby , that is,v0= v + .(2)The null geodesic generators of the horizon need not have a common past end point andthere is no canonical cross section of the horiz

10、on. The tangent vector l to the horizon istaken to be normalized such that it agrees with the Killing vectors, of time translationand rotation, on the horizon.Classically, a black hole is independent of its past history. I shall assume this is alsotrue in the quantum domain. How then can a black hol

11、e emit the information about theparticles that fell in? The answer I propose, as explained above, is that the informationis stored in a supertranslation associated with the shift of the horizon that the ingoingparticles caused.The supertranslations form a hologram of the ingoing particles. The varyi

12、ng shiftsalong each generator of the horizon leave an imprint on the outgoing particles in a chaoticbut deterministic manner.There is no loss of information.Note that although thediscussion in this paper focuses on the asymptotically flat case, this proposal also worksfor black holes on arbitrary ba

13、ckgrounds, e.g., in the presence of a nonzero cosmologicalconstant.Polchinski recently used a shock wave approximation to calculate the shift on a gen-erator of the horizon caused by an ingoing wave packet 7. Even though the calculation2may require some corrections, this shows in principle that the

14、ingoing particles determinea supertranslation of the black hole horizon. This in turn, will determine varying delays inthe emission of wave packets. The information about the ingoing particles is returned, butin a highly scrambled, chaotic and useless form. This resolves the information paradox.For

15、all practical purposes, however, the information is lost.Unlike the suggestion of t Hooft, 8- 9, that relies on a cut-offof high energy modesnear the horizon, the resolution of the information loss paradox I proposed here is basedon symmetries, namely supertranslation invariance of the S-matrix betw

16、een the ingoingand outgoing particles scattered offthe horizon, which by construction is unitary.A full treatment of the issues presented here will appear in a future publication withM. J. Perry and A. Strominger, 10.References1 S. Hawking, “Breakdown of Predictability in Gravitational Collapse”, Ph

17、ys. Rev. D14 (1976) 2460.2 A. Almheiri, D. Marolf, J. Polchinski, J. Sully, “Black Holes: Complementarity orFirewalls?”, JHEP 1302 (2013) 062, arXiv:1207.3123 hep-th.3 A. Almheiri, D. Marolf, J. Polchinski, D. Stanford, J. Sully, “An Apologia for Fire-walls”, JHEP 1309 (2013) 018, arXiv:1304.6483 he

18、p-th.4 H. Bondi, M. G. van der Burg, A. W. Metzner, “Gravitational Waves in GeneralRelativity. 7. Waves from Axi-symmetric Isolated Systems”, Proc. Roy. Soc. Lond.A 269 (1962) 21.5 R. K. Sachs, “Gravitational Waves in General Relativity. 8. Waves in AsymptoticallyFlat Space-time”, Proc. Roy. Soc. Lo

19、nd. A 270 (1962) 103.6 A. Strominger and A. Zhiboedov, “Gravitational Memory, BMS Supertranslationsand Soft Theorems”, arXiv:1411.5745 hep-th.7 J. Polchinski, “Chaos in the Black Hole S-matrix”, arXiv:1505.08108 hep-th.8 G. t Hooft, “Black Holes, Hawking Radiation, and the Information Paradox”, Nucl.Phys. B (Proc. Suppl.) 43 (1995) 1.9 G. t Hooft, “The Scattering Matrix Appraoch for the Quantum Black Hole,”Int. J.Mod. Phys. A11 (1996) 4623, arXiv:9607022 gr-qc.10 S. W. Hawking, M. J. Perry and A. Strominger, in preparation.3