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Black Holes Entropy and the Semiclassical Approximation

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Black Holes Entropy and the Semiclassical Approximation View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server hep-th/9404135 BLACK HOLES ENTROPY AND THE SEMICLASSICAL APPROXIMATION SAMIR D. MATHUR Center for Theoretical Physics Massachussetts Institute of Technology, Cambridge, MA 02139, USA ABSTRACT We compute the entropy of the Hawking radiation for an evap orating black hole, in 1+1 dimensions and in 3+1 dimensions. We give a cri- terion for semiclassicality of the evap oration pro cess. It app ears that there might b e a large entropyofentanglementbetween the infalling matter and the radiation when quantum gravity is considered. Invited talk given at the International Collo quium on Mo dern Quantum Field Theory I I at TIFR (Bombay) January 1994. 1. Intro duction Recently there has b een a renewed interest in the black hole evap oration prob- lem, and the asso ciated problem of information loss. One reason is the construction 1;2 of 1+1 dimensional mo dels where evap orating holes can b e easily studied . An- 3 other reason is a spate of work on the prop osal of 't Ho oft that the black hole evap oration pro cess may not b e semiclassical. This idea is based on the fact that + 1 although the Hawking radiation at I is low frequency ( M ) it originates in very M high frequency vacuum mo des at I , the latter frequency b eing e times the planck frequency.(Mis the mass of the black hole in planck units.) The need to consider sup er-planckian frequencies suggests that the black hole evap oration pro cess involves quantum gravity, and cannot b e approximated bya semiclassical calculation of ` eld theory on curved space'. Susskind et. al. have argued that the information of the infalling matter is transferred at the horizon 4 to the Hawking radiation, thus avoiding information loss . The paradox then is: How do es this information transfer o ccur when seen from the frame of an infalling observer, who probably sees nothing sp ecial at the horizon? For this issue Susskind suggests a breakdown of Lorentz symmetry at large b o osts, and a principle of `com- plementarity' whichsays that one can observe either the state outside the horizon or the state inside, but somehow it should make no sense to talk of b oth states at 5 at the same time . The strongest supp ort for such a conjecture of complementarity comes from the work of ref. 6. The study of quantum gravity plus matter in 1+1 dimensions reveals large commutators b etween op erators representing infalling matter and op erators + of the radiating elds at I . This result emphasises the role of quantum gravityin the evap oration pro cess, and suggests a transfer of information from the infalling matter to the Hawking radiation. Can we conclude that black hole evap oration is not a semiclassical pro cess? There seems to b e no universal agreement on this p oint. One reason seems to b e that there is no suciently explicit identi cation of what go es wrong with the semiclassical reasoning. A second reason is that di erent quantities are computed by prop onents of di erent views on the information question; this obscures the precise role of quantum gravity in the problem. A third, more minor, reason is the puzzlementover the role of certain b oundary conditions assumed in ref. 6. In this talk we prop ose an approach that should eliminate the ab ove diculties. Issues of information and entropy are b est examined through the `state' of quantum elds on a spacelikehyp ersurface, rather than through correlation functions of the quantum theory. Consider a hyp ersurface suchas shown in gure 1. This is a `1-geometry' in 1+1 spacetime, and a `3-geometry' in 3+1 spacetime. The matter on this slice we separate into three classes. At the extreme left wehave the infalling matter that makes the black hole; we call this comp onent `A'. Next wehave the quantum radiation that falls into the singuarity (`B'). At the extreme rightwehave + quantum radiation (`C') which escap es to I as Hawking radiation. + ∞ σ− + cr σ− = 0 III Ω = C Σ − ∞ + ∞ B Σ′ A II y+ I − cr y + =0 Ω = − ∞ Figure 1. The semiclassical geometry of a black hole in the RST mo del. In the rst part of the talk we examine the semiclassical approach. Matter `A' is classical, with mass M , and the metric is determined by M and <T > from the radiation. On this given 2-dimensional spacetime geometry the quantum eld `B' and `C' are examined, and the entropyofentanglementbetween these comp onents 7 computed . The same redshift that yields the thermal nature of Hawking radiation also yields, in this calculation, the entropy of the Hawking radiation. Thus we see more transparently the origin of `black hole thermo dynamics'. (If evap oration were ignored in the black hole geometry, then the Hawking temp erature could b e calculated but the resulting entropywould b e in nite.) In the second part of the talk we prop ose a criterion for the validity of the semiclassical approximation in the evap oration pro cess. The state of the quantum gravity - matter system is describ ed byawavefunction in `extended sup erspace'. But sup erspace describ es D 1 dimensional spacelikehyp ersurfaces, not D dimen- sional spacetime. An approximate D dimensional spacetime emerges by a WKB approximation on the wavefunction, whichisgiven to satisfy the Wheeler-de Witt equation in extended sup erspace. Simple estimates suggest the the semiclassical approximation breaks down at hyp ersurfaces `cro oked enough' to capture b oth the Hawking radiation and the infalling matter. 2. Entropy in the Semiclassical Approximation 2 Consider rst the RST mo del of dilaton gravity coupled to massless scalar elds . The matter comp onent `A' de ned ab ove is classical. The entanglement of `B' and `C' is the entropy of pair creation: one memb er of the pair falls into the singularity + while the other escap es to I . + + Let b e the null Minkowski co-ordinate at I , and y the null Minkowski co- + ordinate at I .We wish to compute the entropy collected by an observer at I , who collects radiation upto some time < 0. (The radiation nishes at =0 1 b ecause all of the mass M has evap orated away.) + In our region of interest a null rayat can b e followed back from I , re ected o the `strong coupling b oundary' = and followed backto I to reach at some crit + + co-ordinate y . (y ) is given by + M y x e + 1 s = ln[ ] (1) M x + s M 4 M = 1 Here x = (1 e ) , is the cosmological constant (which sets the planck 2 s scale) and = N=12, N b eing the numb er of matter elds. + For y ! 0 (1) can b e approximated as 4M 1 + = ln(y ) (2) 2 The Bogoliub ov transformation given by (2) converts the vacuum at I to a thermal + state at I with temp erature T = . 2 8 To compute entropywe recall a result of Srednicki . Consider a free scalar eld on a 1-dimensional lattice, with lattice spacing a. Let this eld b e in the vacuum state. Select a region of length R of this lattice and trace over the eld degrees of freedom outside this region. This gives a reduced density matrix , from whichwe compute S = Trf ln g which is the entropyofentanglement of the selected region with the remainder of the lattice. This entropy is given by S = ln(R=a)+ln(R) (3) 1 2 for one scalar eld. (For N sp ecies the result must b e multiplied by N .) One nds =1=6. is an infrared cuto , and the co ecient is sensitive to the 1 2 choice of b oundary conditions. (For a detailed discussion of b oundary conditions and a heuristic derivation of the form (3), see ref. 7. For an alternative analytical derivation of see ref. 9.) 1 + To compute the entropy collected at I in 1 < < , consider the hyp ersur- 1 + face in g. 1. runs near I upto , then b ends down to avoid the singularity 1 and reaches = , remaining spacelike throughout. The instrument measuring crit the radiation is to b e switched o at , but the switching o pro cess cannot b e to o 1 sudden, otherwise it will generate extra radiation which will render the measure- ment inaccurate. It seems reasonable to switch o the radiation over a time of the order of the p erio d of the Hawking radiation; the exact choice will not matter. The entanglemententropywe seek corresp onds to separating the eld mo des on the left of the switcho p oint with the eld mo des on the right of this p oint, with the scale of cuto b eing . The left moving comp onents of the matter elds on are not excited, and with the ab ovechoice of give no signi cant contribution to S in (3). The rightmoving comp onents are not in a vacuum state, so we cannot directly apply (3). But wemay use any set of co-ordinates on to compute the entanglement of mo des. Through any p oint P on , followa null ray towards the past until it hits = ; re ect here to a null ray which is followed crit + + to I , reaching at some co-ordinate y . In terms of the co-ordinate y (P ) the right moving eld mo des on are in the vacuum state, but the cuto scale is squeezed (4 M =+ ) 1 to a e .
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