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This content was downloaded from IP address 136.56.14.111 on 02/02/2021 at 20:10 JHEP09(2005)037 hep092005037.pdf June 17, 2005 September 5, 2005 September 14, 2005 trum in the Parikh- Received: gether with the possible Accepted: Published: evaporation. New Zealand energy of the black hole-emitted e terms arise from modifications 7599-3255, U.S.A. nomy University of Athens http://jhep.sissa.it/archive/papers/jhep092005037 /j Published by Institute of Physics Publishing for SISSA Black Holes. Planck-scale corrections to the black-hole radiation spec [email protected] [email protected] [email protected] University of North Carolina,E-mail: Chapel Hill, North Carolina 2 School of Mathematics, StatisticsVictoria and University Computer of Science Wellington,E-mail: PO Box 600, Wellington, Nuclear and Particle PhysicsGR-15771, Section, Athens, Physics Greece Department, E-mail: Institute of Field Physics, Department of Physics and Astro SISSA 2005 c ° Wilczek tunneling framework are calculated.in The the correctiv expression of the surfaceparticle gravity in system. terms of the The mass- consequences form for the of fate the of new black holes spectrum in is theKeywords: discussed late stages to of Hawking radiation as tunnelinghorizon through the quantum A. Joseph M. Medved Elias C. Vagenas Abstract: Michele Arzano JHEP09(2005)037 1 3 10 principles of Quan- ffects 8 schild geometry 2 tion of whether or not heorem [1] and the fact vity theory in regimes far hus violating the unitarity [4 – 7]). Among these, the hole information paradox” ried by a physical system mulated theory of Quantum is characterized by just one recovered. This would allow, ich Quantum Mechanics and ge via the Hawking radiation ection the authors find is such cts were studied in [8, 9] and f evaporation the usual picture e to gravitational back-reaction y conservation during the emis- o non-thermal corrections to the h different energies appears [12]. rikh and Wilczek [10] showed how – 1 – Many proposals have been made in the attempt to save the basic 1. Introduction It is widely believedwould that provide a a key final ingredient wordGravity. in on the search the The for so-called paradox athat “black yet relies to Hawking be on radiation for the exhibitsparameter: validity a of thermal the the spectrum, blackfalling “no i.e. toward hole a it hair” temperature black-hole t singularityfor [2]. then example, has a no pure Information way quantum toof car state be evolution to evolve in into Quantum athis mixed Mechanics violation one, [3]. occurs t willGeneral The deeply Relativity answer characterize would to emerge thefrom the as way the limits in ques Planck of wh scale. the Quantum Gra tum Mechanics in theidea presence that of information black lost holesseems behind (for particularly the reviews interesting. horizon see might Infor re-emer the fact emission in process the willshould late lose be stages its taken o validity and intolater effects account. in du [10 – These 12] (also “self-gravitation”the see, effe inclusion e.g., [13 of – 15]). back-reactionsion In effects, of particular which a Pa ensures particle via energ tunnelingblack-hole through radiation the spectrum. horizon, However leads thethat t form no of statistical the correlation corr between quanta emitted wit Contents 1. Introduction 2. Motion of a self-gravitating spherical3. shell in The a KKW Schwarz model in4. a nutshell Quantum tunneling and non-thermal5. spectrum The Parikh-Wilczek tunneling picture6. revisited Tunneling in the presence of7. near-horizon Planck-scale Conclusion e 4 6 JHEP09(2005)037 ll (2.2) (2.1) , . 2 Ω luated at the shell d -Wilczek tunneling schild geometry 2 ) ical spherical shell in a combined with Planck- ty corrections, no such t, r [16 – 20]. Moreover, the ek in terms of quantum ( ormalism with a specific the geodesic motion of a ormed energy-momentum on-thermal spectrum and h both back-reaction and lanck-scale corrections are R adiation. In section V we scuss a possible role of the g black hole. This is a novel + boundary terms ar-horizon emission process. wever, that (given a certain emarks (section VII). + e logarithmic corrective term 2 ssion spectrum. In section III ational back-reaction and the of different contexts (see [22] sical” treatment of [10 – 12]. ric system in ADM form ectrum. ) 2 r ] geometry were calculated. We elations between the probability on of a particle approaches zero ck scale departures from Lorentz n from the direct count of black ˆ o see [24]) it is showed how this dt N ) + ˙ t, r ˆ r ( ( r 2 ) N r ˆ L + ( − dr [ 2 2 – 2 – ) ) t ˆ N t, r ( ( L q + dt 2 Z dt 2 m ) t, r ( R− t g ). The above action can be written in hamiltonian form where a N ˆ r − − ˆ t, √ ( = x 4 µν 2 d g ds Z = π µν 1 g is the shell radius and the other quantities under “ˆ” are eva 16 r = It might have been hoped that, when back-reaction effects are Unfortunately, even with the inclusion of the Quantum Gravi In the next sections we briefly review how the motion of a class In this letter we proceed a step further and modify the Parikh S picture using aThe typical type Planck-scale of modification modification we forappearing consider in the is the ne directly Bekenstein-Hawking related entropy-areahole to relatio micro-states th in Stringlogarithmic Theory correction and has Loop beenfor Quantum substantiated a Gravity in list of acorrective relevant variety term citations). can For bedispersion instance, relation also in which derived emerges [23] in from various (als symmetry. models a of Plan general form of def scale corrections, one would observe theof appearance emission of of corr different modes in the black-hole radiationcorrelations sp are conclusively inconstraint to evidence. be discussed One below)when does its the find, energy probability becomes of ho of emissi feature the order of of our the analysis mass that of was the emittin not present inSchwarzschild the geometry “semiclas is modified inway this the is presence used of to gravit we compute corrections review to how the these black-holetunneling emi effects through were the described blackQuantum by hole Parikh Gravity horizon. and correction In Wilcz toits section the consequences IV entropy-area we for law di thepropose for a escape the modified of n version informationintroduced of via and the Hawking obtain tunneling r picture aQuantum in black Gravity which hole effects P emission areexample spectrum (section present. VI), in which whic We is then followed by illustrate some the concluding2. r f Motion of a self-gravitating spherical shellWe in summarize a here Schwarz thespherical results shell of due [8] tostart where by self-gravitation the writing in the corrections a metric to for Schwarzschild a general spherically symmet where ˆ through ˆ The action for the black hole plus the emitted shell system is JHEP09(2005)037 (2.5) (3.1) (2.4) (2.3) . Details E + , 2 M , Ω = d dr ) 2 r r + ( njugate momenta are − + M p e of the hamiltonian. In 2 the total mass/energy of ugate to the shell radius. n order to eliminate the system and allows energy ] (0) ey consider the Bogoliubov ergo a large red-shift when the mass of the hole is kept f . − dt r sion is dominated by modes r nto account one finds [9] has only one effective degree ) ) orrections of the type consid- ). This choice of the gauge R tional degrees of freedom to he case of a massless particle the observer crossing the hori- hell. In the approach followed ena being non-singular at the details see [25]). The effective r es which extremize this action E E lement (Painleve’ coordinates) zon. The (semiclassical) WKB Im we have + + = − , e E M M R ´ , the radial position of the shell, and ; ; = + r r r | ( ( 0 r r M kk = 1 N N β − | ˙ ˆ L r − + c ) p dr E ³ – 3 – ; + dt + [ dr 2 ) ] r Z M ( ; dt + r = ) p ( t E and the shell energy S (0) N f + + r r M R = M ; Im dt dr r , connecting the positive and negative frequency modes of an − 0 ( e t kk N = β [ | − 0 kk = and α 2 | 0 ds kk α is the momentum canonically conjugate to ˆ c p is the total mass of the shell-hole system which plays the rol Once the constraints are solved and the expression for the co = 0) and fixes the gauge appropriately ( + m of the lengthy derivation canare be the found null in geodesics [8]. of The the trajectori metric approximation is then used to expresszon. the mode Such solutions an for approximationthat is have valid very since small the wavelengthspropagating black close away hole to from emis the it. horizon Once and this und approximation is taken i asymptotic observer and one freely falling through the hori the canonically conjugate momenta appear.of Since freedom the system thedependence idea of is the to actionThis from solve all remaining the the degree constraints momenta ofthe but of freedom system the can and the one it be conj theory isin expressed obviously i [8], related in the to terms total the (ADM) of positionfixed. mass of is This the allowed time s to dependence varyconservation accommodates with to time the while hold dynamics at of any the time of the process. substituted into theobtain action an one effective integrates action.( over Furthermore the one gravita specializes to t 3. The KKW model in aIn nutshell [8] and [9], Keski-Vakkuri, Krausered and Wilczek above showed can how affect c thecoefficients, emission spectrum of a black hole. Th for which M terms of the black hole mass where corresponds to a particularwhich set is of particularly coordinates useful forhorizon to the and study line having across e euclidean horizon constantaction phenom time for slices a (for massless more gravitating spherical shell is then JHEP09(2005)037 (3.6) (3.2) (3.3) (3.5) (3.4) antum , one can E ergy modes + , to the typical k M ω = . H e form if one describes ) , one can evaluate (3.3). . is is done by considering E ) how energy conservation is dS that the amplitude in terms + E ) . adratic in π for the black hole entropy in M )] + 2 ( M e argument makes it possible to κ is seen as a tunneling through a ; dE in the KKW model, it is assumed M . r he WKB approximation in the analysis M ( ( ( 2 = r | S κ 0 2 N instead one has | . k − kk 0 ω k ackets are arbitrarily blue-shifted close to the 2 ) − dM ¯ ¯ ¯ ¯ dE kk − k ω 0 , we obtain the usual emission amplitude /α 0 ) 0 0 ω Z kk E /α kk M kk 0 β − α π is located outside the horizon. From Hamil- β + ¯ ¯ ¯ ¯ kk − f β one instead finds M r | ' − | – 4 – ( M 0 = ; ) 1 S r kk k ( dr β ω t ) ( r )= N ρ exp [ ( k k − ω ω ' , with the hamiltonian given by ( p ± ) ρ 0 ∂p k ∂E (0) f Z ω so that one can treat the wave-packets near the horizon as = 1. For r − ( r 0 = 1 ρ Z )= kk is small compared to r α k ( Im ω ± p ∂H/∂p are the trajectories and momenta of positive and negative en = is comparable with the mass of the black hole and at most one qu k ± r reads p ω 2 | 0 using (2.5) and kk ± p ± /α r 0 kk This limit is valid for the same reason that allowed us to use t β | 1 barrier set by the energyavoid of the the machinery particles of itself. Bogoliubovnaturally coefficients This preserved and simpl during also the shows emission process [12]. effective particles. The emission of each of these particles Boltzmann factor of the emission probability. 4. Quantum tunneling and non-thermal spectrum The results of [8]the and emission [9] of canthe a geometrical be particle optics recast as limit in a an tunneling elegant process and [10, simpl 11]. Th ton’s equation ˙ Substituting the standard Bekenstein-Hawkingthe expression previous equation leads to a non-thermal correction, qu It can be shown [9] that governed by the Hawking temperature. For large propagating in and out of the horizon and In order to calculate the amplitudethat for particle the production emission in theof low energy regime is uncorrelated so can be emitted of the previous section,horizon. i.e. the fact that the emitted wave p Using the first lawFor of the first black-hole case thermodynamics when where express Instead when JHEP09(2005)037 (4.4) (4.8) (4.3) (4.6) (4.1) (4.2) (4.7) (4.5) . One ticle is E is valid, 2 − M πM = + is the same as the M 4 = 4 S element (2.4) obtained A/ , dr . 0 e high energy regime. To = tion of the imaginary part ˙ r pt fixed while the hole mass is now itself. . dH , )] + . BH ∂H/∂p E ical model. The advantage of ng along a radial null geodesic of the black hole horizon with a en that in order to get (4.7) we S M M − + rection we get now is present at ¶¶ = ( dr . . r M 0 M r M ) M ˙ 2 E r BH Z 2 E dE dr , S r r fin − . E − r p in − r ± 0 r 1 S ) Z Z fin M = µ r in E r fin r 2( 2 Im r in Z − r N − e r Z = Im – 5 – M πME − ( ∼ 8 used in the KKW analysis. The same coefficient Im dr one easily obtains = Im 1 r Γ − 0 − r BH p ± S . So according to what one would expect from energy 1 ; µ kk S E given by the constraint equations [8] = fin ± β = r in Im r ˙ r − r , so the imaginary part of the action reads S exp = 0 Z N exp [ t E M ∼ Im N ∼ − Γ = and Γ = Im t M out S N r = 0 Im H and are just inside and outside the barrier through which the par M fin r we use once again Hamilton’s equation, ˙ = S in r and in r The authors consider now an explicit expression for the line characterizes the emission probabilityconsidering the in particle-tunneling the picture fieldall is energy theoret that regimes the even cor calculate though Im it becomes dominant only in th then has the following expression for a spherical shell movi tunneling. We canone now for see the the Bogoliubov coefficient key point: the expression for Im where conservation, the tunneling barrier ischange set in by the the radius shrinking set by the energy of the emitted particle If one integrates (4.6)must first have over the energies it is easily se corresponds to the KKW result If we consider the emission of a spherical shell we have Using (4.2) and integrating first over from the expressions of In the model proposed inis [10] allowed the total to mass vary. of This the system means is that ke the mass parameter In the WKB approximationof the the tunneling particle’s probability action is a func which, provided the usual Bekenstein-Hawking formula The hamiltonian is JHEP09(2005)037 (5.2) (5.1) 2 [27]. In / 1 − = 1 to keep track and two quanta = c 2 α )) (4.9) = E 2 4 ~ + E ) = 0. One concludes 1 2 = E k )Γ( , E ses that would allow the 1 o verify [12] that for the ermal correction obtained = 1 ropy is directly related to phase space factor which rivations of the black hole E E rom a quantum mechanical E t that takes into account the ; . st notice that the probability bilities. It is zero when the gravitational collapse but one 2 sly fixed at y he emitted radiation. Consider ! in-Hawking linear relation give E rward way in which information 2 p ning the square of the amplitude ring Theory and Loop Quantum for a thermal emission spectrum log(Γ( A + L ction effects are considered in transient . 1 − à treatments (see [22] for a list); for instance, ) E es of emission of quanta with different ( S )) O 2 χ opy-area law in ordinary QFT [21]. + E 2 p + A dependence on the area L 1 3 E ln = exp (∆ 2 – 6 – α in fin + S is a negative coefficient whose exact value was once S e e 2 p α , in the presence of back-reaction effects, put in the L A ∼ 4 E = 1 units of the previous sections to ) = log(Γ( Γ 2 = G , E = 1 QG c S E = ; 2 ~ E = + . The function k 2 1 E E ( χ and 1 E An interesting aspect to analyze is whether or not the non-th This observation calls for an immediate generalization. De As proposed in [12] correlations might appearA when similar logarithmic back-rea term has also emerged in various other We now switch from 2 3 4 that back-reaction effects alone docan not provide emerge a from straightfo theinformation horizon. of a There pure quantum mightwould state well to have be be to recovered resort other after to proces its other mechanisms. presence of Quantum Gravity-induced corrections.of Let emission us of fir a shell with energy 5. The Parikh-Wilczek tunneling picture revisited In this section we discuss a modification of the above argumen form (4.8) is highly suggestive. It is what one would expect f calculation of a transition rateof where, up the to process, a factor contai measures the statisticalprobabilities correlation are between independent the (orlike “uncorrelated”) two the as proba e.g. one ofnon-thermal correction a due radiating to black back-reaction effects body. Using (4.7) it is easy t of energies of the Planck-scale suppressed terms. phases of the black hole emission. the calculation of one-loop quantum corrections to the entr leads to statistical correlations between the probabiliti energies. This would allow for informationfor to example be the encoded probability in of t emission of a quantum of energ In the case of Loop Quantum Gravity an object of debate (see e.g. [26]) but has since been rigorou In other wordsdepends the on emission the probability initialthe and is number final proportional of entropy micro-states of to available the to a system. the system The itself. ent entropy-area relation by direct micro-stateGravity counting [16 – in 20] St besides reproducinga leading the order familiar correction Bekenste with a logarithmic JHEP09(2005)037 − ) 1 (5.3) (5.5) (5.6) (5.4) (5.7) (5.8) E − . ). And so . ¸ 2 M 2 ¸ ( E . 2 E S − E − 1 1 ¶¶ − 1 E E . 1 M M E E ¸ ion probabilities for − 2 − E , then 1 − 2 E − − M M E of three or more quanta hermal deviation found ( M 1 − M S + M · µ · 1 . M − ntum Gravity framework in ln p to any perturbative order gous to (5.1) with the usual E ) · ln α )) elations after all. 2 after just a few iterations, one ce, it does not appear that the tion that information leaks out ns, or level calculation. By which we elds α ng on the ratio of the energy of ln E from the black hole. One might M .4), we have, for a first emission 2 ( + 2 and α , πGME + 2 E 8 2 − ¶ QG − 2 + 2 1 ¶ E S ) µ E , E 1 ¶ ) = 0 1 2 − 2 M E + 1 E ) 2 M − 2 1 exp E , E 2 − E 1 E E α 2 M − − E replaced by (5.2), i.e. ( M − ; ¶ 1 S – 7 – 2 p 2 2( 1 M µ E L E M ( µ 6= 4 1 − ) ) + − 2 gives us QG 1 1 1 A/ 1 S E µ 2 E E µ 2 = ( E + − E χ πGME 1 ) exp ( 8 1 )= BH E M ( ( − E S ∼ depends on the number of field species appearing in the low S QG − Γ S α − )] = ) πGM M 1 2 ( 8 E E − πG exp (∆ − 8 1 ∼ ln[Γ( − )] = E ) 2 E − E , )] = Γ( + 1 2 M 1 ( E E S E ln[Γ( )+ Now consider the emission of a particle of energy We would now like to know whether or not, in our case, the emiss On the other hand, it is interesting to note that the emission ln[Γ( M ( there still appears toin be the viability, on tunneling some process. level, for the no The previous expression written in explicit form reads Then a second emission of energy S expect that a derivationpresence of of the back-reaction emission would probabilityBekenstein-Hawking in lead entropy a to Qua an expression analo String Theory the signenergy of approximation [17]. Alternatively, a single emission of the same total energy yi The exponential inin this [10]. equation In shows thisthe case, the emitted however, same an quantum and additional type the factor of mass dependi of non-t the blackdifferent hole modes is are present. statisticallyof correlated energy [28]. Using (5 It is now easily verified that the vanishing of the correlatio could still induce amean non-zero that, correlation for even the for sequential the emission tree- of (e.g.) is still in effectshould at readily least be to this convincedof logarithmic that the order. this quantum-corrected entropy outcome In [cf,inclusion will fact, equation of persist such (5.2)]. correlations u Hen can account for the mode corr JHEP09(2005)037 A ln (6.1) 2 1 me limit is , one might 0; whereas a α uppression term ffects α > alue for -Wilczek derivation, in le radiation spectrum seen n in all the derivations of ome is actually quite rea- ssically neutral and static uantum Gravity prediction at bath. In this event, the n of the energy-momentum ecessary to account for the ate from “near” the horizon zon looks, to a freely falling ace when ive term to the entropy-area ability can, nevertheless, be le is (by virtue of a negative tities.) Hence, the canonical corrections appears to ensure ia and Procaccini showed how n thermal equilibrium is most lengths can easily go below the aches that of the Hawking tem- tions in the charge and spin are nto an increase in the black hole rease is, at the very least, e to an emission spectrum of the flat space-time fore quite remarkable and, at the . der corrections may conspire to induce a 2 E ¢ 2 E 2 p ηL – 8 – 1+ ¡ ' [29]. To elaborate, the canonical ensemble allows for 2 p α 0 when the energy of the emitted quantum approaches the mass → ) E will, conversely, cause the probability to diverge as the sa α . 5 α 2 [27]. π 3 / 2 1 − = η = Let us also notice how the appearance of the Quantum Gravity s As we already observed in the previous sections, the black ho Considering that Loop Quantum Gravity predicts a negative v Let us, however, point out one possible loophole: higher-or 5 α 6. Tunneling in the presence of near-horizon Planck-scale e We give now anthe explicit presence of example Planck-scale of effects,type a which (5.4). indeed modification In gives of [23] ris onein the of the Parikh the context authors of (M.A.), Loop Amelino-Camel law Quantum of Gravity a the logarithmic type correct presentdispersion in relation (5.2) for can a be massless related particle to propagating a in modificatio with suppression that is stronger than this divergence. in (5.4) can cause Γ( (e.g., [30 – 32]) and almostproperly certainly accounted larger once for the [29].black fluctua hole (The would point stillcorrection being experience is that more fluctuations than even in sufficient aof to these compensate cla quan for the Loop Q from an observer at infinity iswhere dominated they by have modes arbitrarily that high propag Planck frequencies length and their [33, wave 34].the Hawking It radiation turns is out that the then quantum that state a near key the assumptio hori of the black hole. However, this suppression can only take pl negative value of approached! have preferred the abovesonable, pattern inasmuch to as be an reversed.heat evaporating capacity) Schwarzschild But black highly our ho unstable outc resolved with when the in immersion isolation.system of will the equilibrate This black until inst the holeperature; temperature in at of a the which bath suitable point, re suitably he stability described ensues. by Since acanonical a canonical corrections system ensemble, to i the it black nowsame hole becomes entropy. time, n It reassuring is that there a the strictly inclusion non-negative of value thesethermal for fluctuations canonical in the horizonentropy. area, A which translates rigorous i calculation demonstrates that this inc JHEP09(2005)037 , we (6.4) (6.5) (6.6) (6.2) R = r replaced by dr . 0 BH ˙ r S dE )] (6.7) E ) (6.3) neling framework of E 2 0 ) ompute the tunneling previous sections with − Z R fin M r or break [37, 38] Lorentz in lanck scale corrections of − dr . ( symmetry should hold up .8) with r . ) r 0 ave, in fact, been proposed Z ) gral is being evaluated, one esic of the spherical shell in QG 2 (( E ics related to (6.1). As shown . Now we proceed to evaluate Quantum Gravity corrections ) S le then that the motion of our Im O E roduce via the first law of black R − . − − ) − − 0 ) 0 M )+ = r ( E M M 0 κ M dE (( ( for the pole on the real axis − ( ) dr = O 0 κ 6 dE R QG ) ˙ M r S H ( − [ R dH )+ κ 1 2 r ) is the horizon surface gravity. Taking into E − ( H E 0 , the Quantum Gravity corrected entropy-area r = 0 M E − Z ( ( – 9 – Z 0 κ dS Z π fin ' ) dS M r in ( ) ) π − r fin M κ 2 ( r E in Z κ r ) M = − Z ; ) R M r S = ( can be approximated by M ( 0 ( − r Im = Im r QG Im r N QG − S ( dM S dr − r Z = ' p ) = ˙ 1 2 r S 0 fin M − r in ; r dE Im r = Z ( t S N , using the Feynman prescription r Im = Im S Im . Here we provide an example of such an analysis within the tun Once again we consider the emission of a spherical shell and c One would expect that an analysis analogous to the ones of the The pole is moved in the lower half plane as in [10]. 6 QG We can then write S amplitude (4.3) through (4.4) Parikh and Wilczek. where R is the Schwarzschild radius and account self-gravitation effects, ˙ Integrating over The surface gravity appearingcoming in from the Planck-scale above modifications integral ofexplicitly carries near in horizon [23] phys these modificationshole are thermodynamics, such that they rep law (5.2). Using the first law, (6.6) becomes observer, like the Minkowskito vacuum. extremely In short scales otherparticle or words very tunneling Lorentz large through boosts. thethe It horizon type is (6.1). might plausib beas These affected low-energy type by Quantum of Gravity P modifiedsymmetry effects, (see dispersion which also relations [39]). deform h [35, 36] opportune modifications should lead to a result of the form (4 where we used the fact that for the hamiltonian the integral without usingterms an of explicit its form energy.has for In the fact, null near geod the horizon, where our inte get JHEP09(2005)037 (6.8) (1972) B 433 (1976) , D 5 D 14 ¶¶ ersity Research (1975) 199. Nucl. Phys. M E ecent results , 2 43 Phys. Rev. − l funds). , Phys. Rev. 1 partment of Physics of ]. on physics provides an d Royal Society) and by , µ king radiation. Although . e and possible Planck-scale cts combined might provide ssions and valuable sugges- tum evolution in Quantum direction of study could have December 2004. Research for e work of E.C.V. is supported πGME erent aspects of Quantum Gravity 8 − µ . gr-qc/9305012 hep-th/9209058 exp Commun. Math. Phys. , α , 2 ¶ E M – 10 – (1993) 4779 [ − 1 hep-th/9305040 µ , Quantum mechanics, common sense and the black hole D 48 )= S ]. Selfinteraction correction to black hole radiance 2Im − Phys. Rev. , Nonexistence of baryon number for static black holes . exp ( Particle creation by black holes Breakdown of predictability in gravitational collapse ∼ The information problem in black hole evaporation: old and r ) gr-qc/9408003 Black hole information Do black holes destroy information? E Γ( 1239. 2460. information paradox hep-th/0501132 (1995) 403 [ [1] J.D. Bekenstein, [2] S.W. Hawking, [3] S.W. Hawking, [4] J. Preskill, [5] D.N. Page, [6] U.H. Danielsson and M. Schiffer, [7] J.G. Russo, [8] P. Kraus and F. Wilczek, which leads to a probability of emission which is analogous to (5.4). 7. Conclusion We have discussed how Quantum Gravity anda back-reaction effe way for information tothis be matter recovered still from remains a unsettled,important black we implications hole would via argue for Haw thatGravity. the our fate Our of formalexcellent the arena treatment for unitarity studying gives the of interplay an ofas quan seemingly idea the diff number of of how microscopicmodifications degrees near of of space-time horiz freedom symmetries. of a black hol Acknowledgments M.A. would like to thanktions Giovanni and Amelino-Camelia Jack for Ng discu the for University useful of Rome comments. “LaA.J.M.M. Sapienza” M.A. is for also supported hospitality thanks by during the thethe Marsden University De Fund Research (c/o Fund the (c/o Newby Victoria Zealan EPEAEK University). 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