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Particle Creation by Loop Black Holes Emanuele Alesci Universita¨t Erlangen, Institut fu¨r Theoretische Physik III, Lehrstuhl fu¨r Quantengravitation, Staudtstrasse 7, D-91058 Erlangen, EU Leonardo Modesto Perimeter Institute for Theoretical Physics, 31 Caroline St.N., Waterloo, ON N2L 2Y5, Canada We study the black hole particle production in a regular spacetime metric obtained in a minisu- perspace approach to loop quantum gravity. In different previous papers the static solution was obtained and shown to be singularity-free and self-dual. In this paper expanding a previous study oftheblackholedynamicswerepeattheHawkinganalysiswhichleadstoathermalfluxofparticles at the future infinity. The evaporation time is infinite and the unitarity is recovered due to the 1 regularity of thespacetime and to thecharacteristic behaviorof thesurface gravity. 1 0 2 I. INTRODUCTION ist, this scenariocannotbe correct. Since the singularity plays a central role for the causalspacetime diagram,its n a Blackholesareoneofthe mostfascinatingpredictions absence in the presence of quantum gravitational effects J has consequences for the entire global structure [4], and of Einsten’s gravitational theory. Today we know that 0 its removal is essential for resolving the black hole infor- they are not just a mathematically possible solution of 3 mation loss problem [5, 6]. To understand the dynamics generalrelativity,but part ofNature. Since more thana ofthegravitationalandmatterfields,itisthennecessary decade now,we havegoodevidence that ourMilky Way, ] to have a concrete model. c as other galaxies, hosts many stellar black holes as well q as a supermassive black hole in its center. It is thus promising that a resolution of the big bang - as well as the black hole singularites [8–10] has been r From the perspective of quantum gravity, black holes g achieved in a simplified version of loop quantum gravity are of interest because of the infinite curvature towards [ (LQG) [38], known as loop quantum cosmology (LQC) their center which signals (probably) a breakdown of 1 General Relativity. It is an area where quantum grav- [7] . The regular static black hole metric was recently v ity effects are strong, and it is generally expected that derived in [11][12], and studied more closely in [13]. The 2 polymeric approach to solve the black hole singularity these prevent the formation of the singularity. 9 problem was also applied in [14]. An alternative reso- 7 In the mid-seventies Steven Hawking [1] and Jacob lution of the black hole singularity was obtained in an 5 Bekenstein [2] showed that since classicalblack holes ra- effective, noncommutative approach to quantum gravity . diateparticles,theirmassslowlydecreaseandfinallydis- 1 [15] and in asymptotically safe quantum gravity [16]. In 0 appear,leavinga paradox. Supposing thatmatterwhich others works [6][17], a 2-dimensional model was used to 1 enterstheblackholewasinapurequantumstateatearly study the evaporation process in the absence of a sin- 1 time, then the existence of an horizon transforms such gularity. Recently also regular spinning loop black holes : matter into the mix thermal state of Hawking radiation v were obtained in [18]. i at late time for an external observer, and the complete X evaporation of the singularity destroys the correlations Here, we will use a 4-dimensional model based on the r between the radiation and the information “swallowed” static solution derived in [11] and generalize it to a dy- a by the black hole that would allow to reconstruct the namical case which then allows us to examine the causal original pure quantum state. The role of the singularity structure. This generalization holds to good accuracy in is crucial in this paradox. Suppose there is an entan- all realistic scenarios. This approach should be under- gledpurestateatearlytime, andapartoftheentangled stoodnotasanexactsolutiontoaproblemthatrequires system ends inside the black hole while the other part knowledge of a full theory of quantum gravity, but as a remains outside the event horizon, the result is a mixed plausible modelbasedonpreliminarystudies thatallows state after the partial trace is taken over the interior of ustoinvestigatethegeneralfeaturesofsuchregularblack the black hole. Now, since everything that falls inside hole solutions with quantum gravitycorrectionsinspired by LQG. The main goal of this paper is to reproduce the black hole reaches the singularity (r = 0) in finite proper time, the part of the Hilbert space that is traced the Hawking calculation of particle creation in a partic- ular regularblack hole backgroundand to show that the over disappears and never appears againwith the disap- pearence of the black hole. In the standard semiclassical wholeprocess,collapseandcompleteevaporation,isuni- tary. In this analysis we will make extensively use of the treatment this is the scenario,since the black hole emits particles in the form of Hawking radiation and the hori- Fabbri and Navarro-Salasbook [20]. zonradiusdecreasesandapproachesthesingularityuntil Non-singular black holes were considered already by both, singularity and horizon, vanish in the endpoint of Bardeen in the late 60s and have a long history [19, 21– evaporation [3]. However, if the singularity does not ex- 30, 32–36]. In this paper we extend the procedure in 2 [30] [31] analyzing the particle production and the uni- the fixed graph decided above. The diff-constraint tary problem as result of the complete evaporation. The isthenidenticallyzerobecauseofhomogeneityand paper is organized as follows. We start in the section II the Gauss constraint is zero for the Kantowski- by recalling the regular static metric we will use in the Sachs spacetime. whole article. In section III we generalize it to a col- lapse scenario as already done in [31]. In section IV we ii We solve the Hamilton equation of motion for the holonomic Hamiltonian system imposing the derive and solve the scalar field equations of motion in this regular black hole background and in section V we Hamiltonian constraint to be zero. use these solutions to reproduce the Hawking analysis iii The third step consists to extend the solution to leading to a thermal spectrum. In section VI we sum- the whole spacetime; this step is mathematically marize the complete dynamics for the LBH collapse and correct but, since we found the solution in the ho- evaporation. WededicatesectionVII toreviewthe basic mogeneous region, we can not be sure of the cor- aspects of the unitarity and information loss problems rect Hamiltonian constraint polymerization in the associated to classical black holes and finally in section full spacetime. However we believe such polymer- VIIIweshowhowinthisquantumcorrectedmodelthese ization exists. problemsaresolved. The signatureofthe metric usedin this paper is ( ,+,+,+) and we use the natural unit This quantum gravitationally corrected Schwarzschild convention ~=−c=GN =1. metric can be expressed in the form dr2 II. THE REGULAR BLACK HOLE SPACETIME ds2 = G(r)dt2+ +H(r)dΩ2 , − F(r) (r r )(r r )(r+r )2 + − ∗ LQG is a candidate theory of quantum gravity. It is G(r)= − − , r4+a2 obtainedfromthe canonicalquantizationof the Einstein o equationswrittenintermsoftheAshtekarvariables[37], (r r+)(r r−)r4 F(r)= − − , that is in terms of an su(2) 3-dimensional connection A (r+r )2(r4+a2) ∗ o and a triad E. The result [39] is that the basis states of a2 LQGareclosedgraphsmadeofedgesandnodeslabelled H(r)=r2+ o . (2) r2 byirreducibleSU(2)representationsandintertwinersre- spectively. Physically,theedgesrepresentquantaofarea with dΩ2 = dθ2 +sin2θdφ2. Here, r = 2m and r = + − with area 8πγlP2 j(j+1), where j is the representation 2mP2 are the two horizons,and r∗ =√r+r− =2mP. P labeloftheedge(anhalf-integer),l isthePlancklength, p P is the polymeric function P =(√1+ǫ2 1)/(√1+ǫ2+ andγ isaparameteroforder1calledtheImmirziparam- − 1), withǫ 1being the productofthe Immirziparame- eter. Verticesofthegraphrepresentquantaof3-volume. ≪ ter (γ) andthe polymeric parameter (δ). With this, it is The area is quantized and the smallest possible quanta also P 1, such that r and r are very close to r =0. − ∗ correspond to an area ≪ The area a is equal to A /8π, A being the mini- o min min mum area gap of LQG. Note that in the above metric, r A =4π√3γl2 . (1) min P is only asymptotically the usual radial coordinate since H(r) is not just r2. This choice of coordinates however Toobtainthesimplifiedblackholemodel,thefollowing has the advantage of easily revealing the properties of assumptions were made. First, the number of variables this metric as we will see. The ADM mass is the mass was reduced by assuming spherical symmetry. Second, inferred by an observer at flat asymptotic infinity; it is insteadofallpossibleclosedgraphs,aregularlatticewith determined solely by the metric at asymptotic infinity. edge-lengths δ and δ was used. The dynamical so- b c ≈ The parameter m in the solution is related to the mass lution inside the homogeneous region (that is, inside the M by M =m(1+P)2. horizon,where spaceis homogeneousbut notstatic)was If one now makes the coordinate transformation R = then obtained. An analytic continuation to the region a /r with the rescaling t˜= tr2/a , and simultaneously outside the horizon shows that one can reduce the two o ∗ o substitutes R = a /r , R = a /r one finds that the free parameters by identifying the minimum area in the ± o ∓ ∗ o ∗ metricinthenewcoordinateshasthesameformasinthe solutionwith the minimum areaof LQG. The remaining oldcoordinatesandthus exhibits averycompelling type unknown constant of the model, δ , is the dimensionless b of self-duality with dual radius r =√a . Looking at the polymeric parameter. This determines, with A , the o min angularpartofthe metric, onesees thatthis dualradius strengthofdeviationsfromtheclassicaltheoryandmust corresponds to a minimal possible surface element. It is be constrained by experiment. then also clear that in the limit r 0, corresponding The procedure to obtain the metric is in short the fol- → to R , the solution does not have a singularity, but lowing. →∞ insteadhasanotherasymptoticallyflatSchwarzschildre- i WedefinetheHamiltonianconstraintreplacingthe gion. The dual mass observed from the dual observer in homogeneousconnectionwiththeholonomiesalong r 0 is m =a (1+P)2/4mP2. d o ≈ 3 radius as radial coordinate. This is possible introducing the new radial coordinate x = √H which varies in the range x [√2a ,+ [ o ∈ ∞ 0 a2 r= r− r− r=0 x=rr2+ r2o , x∈[√2ao,+∞[. (3) B′ B The metric assumes the following form r=0 r− r− r=0 ds2 = G(r(x))dt2+ 1 dr 2dx2+x2dΩ2, (4) − F(r(x)) dx (cid:18) (cid:19) I+′ r+ r+ I+ where G, F are implicit functions of x. It is usual to de- fineg toberelatedtothemassinsideasphereofradius xx A′ A xandtheg componentofthemetrictobeproportional tt to a dirty factor e2Φ(x), since g = 1/g , tt xx 6 − I−′ r+ r+ I− −1 2m(x) g = 1 , xx − x (cid:18) (cid:19) 2m(x) g = G= 1 e2Φ(x) (5) tt − − − x (cid:18) (cid:19) and the dirty factor is FIG. 1: Penrose diagram of the regular static black hole so- lution with two asymptotically flat regions. Both horizons, (r(x)+r )4(r4(x)+a2) located at r+ and r−, are marked in blue and red respec- e2Φ(x) = (r4(∗x) a2)2 o . (6) tively. − o In the radial x coordinate the self-dual metric reads 2m(x) dx2 The causal diagram for this metric, shown in Fig 1, ds2 = e2Φ(x) 1 dt2+ +x2dΩ2. thenhastwohorizonsandtwopairsofasymptoticallyflat − − x 1 2m(x) (cid:18) (cid:19) − x regions, A,A′ and B,B′, as opposed to one such pair in thestandardcase. Intheregionenclosedbythehorizons, The metric in this form satisfies the following Einstein’s space- and timelikeness is interchanged. The horizon at equations r isafuturehorizonforobserversintheasymptotically + dm flat A,A′ region and a past horizon for observers inside =4πρx2, dx the two horizons. Similarly, the r horizon is a future − horizon for observers inside the two horizons but a past 1 dG 2 m(x)+4πPxx3 = , horizon for observes in B,B′. If one computes the time G dx x(x 2m(x)) (cid:0) − (cid:1) it takes for a particle to reach r = 0, one finds that it dP 1 dG 2 x = (ρ+P )+ (P P ), (7) takes infinitely long [13]. The diagram shown in Fig.1 x ⊥ x dx −G dx x − can be analytically continued on the dotted horizons at the bottom and top. whereenergydensityandpressuresaredefinedbyTµ = ν ThemetricinEq. (2)isasolutionofaquantumgravi- Gµν/8π = Diag( ρ,Px,P⊥,P⊥) using the coordinates − tationallycorrectedsetofequationsthat,intheǫ,a 0 (t,x,θ,φ) and the metric (4). The first equationein (7) o → limit,reproduceEinstein’sfieldequations. However,due showsthatm(x)isthemassinsideashellofradiusxand to these quantum corrections, the above metric is no fromthefirstequationin(5)wecanextractthefunction longer a vacuum-solution to Einstein’s field equations. m(x) (r =r(x)), Instead, if one computes the Einstein-tensor and sets it equal to a source term Gµν = 8πTµν, one obtains an m(x)= 1 a2o +r2 1 r4−a2o 2(r−r−)(r−r+) , effectivequantumgravitationalstress-energy-tensorTµν. 2rr2 − (cid:0) (a2o+(cid:1)r4)2(r+r∗)2 ! TheexactexpressionsforthecompoenentsofT aresome- whatunsightlyandcanbefoundinthe appendixof[e31]. where r = r(x) is defined implicitly by the relation (3). For our purposes it is here sufficient to notee that the The mass m(x) above tends to the ADM mass for r → entries are not positive definite and violate the positive + andtothedualADMmassforr 0(orx + ), ∞ → → ∞ energy condition which is one of the assumptions for the m(1+P)2 for x + , r >√a , singularity theorems. o m(x) → ∞ (8) We can write the metric (2) introducing the physical →( ao4(m1+PP2)2 for x→+∞, r <√ao. 4 Theexactrelationbetweenthephysicalradialcoordinate ThismetricreducestotheVaidyasolutionsatlargera- x and r is dius, or for ǫ 0,a 0. However,in the usual Vaidya o → → solutions,theingoingradiationcreatesacentralsingular- r= x22 − √x42−4a2o for r <√ao, (9) ictoyr.reBcutitona,stwhee sceeentheerrree,mwaitinhstrheeguqluaarn.tum gravitational  qx22 + √x42−4a2o for r >√ao. Wenotethattheingoingenergyfluxhastwozeros,one q atr =r∗(v)andoneatr =√ao,andisnegativebetween  these. What happens is that the quantum gravitational III. VAIDYA COLLAPSE correction works against the ingoing flux by making a negativecontributionuntiltheeffectivefluxhasdropped to zero at whatever is larger, the horizon’s geometric We will proceedby combiningthe static metric witha radiallyingoing null-dust, such that we obtaina dynam- mean r∗ or the location of the dual radius r = √ao. The flux then remains dominated by the quantum grav- ical space-time for a black hole formed from such dust. itational effects, avoiding a collapse, until it has passed In the present model this process, usually described by r and the dual radius where it quickly approacheswhat the Vaidya metric [40], will have corrections, negligible ∗ looks like an outgoing energy flux to the observer in the in the asymptotic region, but crucial to avoid the for- second asymptotic region. mation of a singularity in the strong-curvature region. The metric constructed this way in the following is not a strict solution of the LQC minusuperspace equations IV. A SCALAR FIELD ON THE LBH in the vacuum. In other words, keeping in mind what BACKGROUND we said in the previous section, it is a solution of the Einstein equations with an effective energy tensor which depends also on the time variation of the mass. The wave-equationfor a masslessscalar field in a gen- We use the radial coordinate r of the previous section eral spherically symmetric curved space-time reads andwestartbymakingacoordinatetransformationand 1 rewritethestaticspace-timeintermsoftheingoingnull- ∂ gµν√ g∂ Φ =0, (13) µ ν coordinate v. It is defined by the relation dv = dt + √ g − − dr/ F(r)G(r),whichcanbesolvedtoobtainanexplicit (cid:0) (cid:1) where Φ Φ(r,θ,φ,t). Inserting the metric of the self- expression for v. The metric then takes the form ≡ p dualblack hole weobtainthe followingdifferentialequa- tion G(r) ds2 = G(r)dv2+2 drdv+H(r)dΩ2 . (10) − sF(r) H(r) 2∂2Φ G(r)F′(r)Φ′ (14) ∂t2 − (cid:18) (cid:19) Nowweallowthemassminthestaticsolutiontodepend ∂2Φ ∂Φ ∂2Φ on the advanced time, m m(v). Thereby, we will 2G(r) +cotθ +csc2θ → − ∂θ2 ∂θ ∂φ2 assume the mass is zero before an initial value va and (cid:18) (cid:19) that the mass stops increasing at vb. We can then, as F(r) H(r)G′(r)Φ′+2G(r) H′(r)Φ′+H(r)Φ′′ =0, − before, use the Einstein equations G = 8πT to obtain h (cid:16) (cid:17)i the effective quantum gravitational stress-energy tensor where a dash indicates a partial derivative with respect T. Tv and Tr do not change when m(v) ies no longer to r. Making use of spherical symmetry and time- v r constant. The transversepressure Tθ =Tφ however has translation invariance, we write the scalar field as θ φ aen adeditionaleterm Φ(r,θ,φ,t):=T(t)ϕ(r)Y(θ,φ) . (15) e e Pr2m′(v) Tθ(m(v))=Tθ(m) , (11) θ θ − 2π(r+2m(v)P)4 omitting theindexesl,minthesphericalharmonicfunc- tions Y (θ,φ). The standard method of separation of lm whereme′ =dm/dv.eBecauseoftheingoingradiation,the variables allows us to split Eq.(15) in three equations, stress-energy-tensornow also has an additional non-zero one depending on the r coordinate, one on the t coor- component, Tr, which describes radially ingoing energy dinate and the remaining one depending on the angular v flux variables θ,φ, e Grv = 2(1+P()a2r2o4+(rr44−)2a(r2o)+(rr−∗(vr∗))(3v))m′(v) . (12) √HGF ∂∂r H√GF ∂ϕ∂(rr) = Gl(lH+1) −ω2 ϕ(r), (cid:18) (cid:19) (cid:18) (cid:19) ∂2 ∂ ∂2 Noticethatalsointhedynamicalcase,trappinghorizons +cotθ +csc2θ Y(θ,φ)= K2Y(θ,φ), stilloccurwheregrr =F(r,v)vanishes[42,43],sowecan ∂θ2 ∂θ ∂φ2 − (cid:18) (cid:19) continuetousethenotationfromthestaticcasejustthat ∂2 r (v) and r (v) are now functions of v. T(t)= ω2T(t), (16) ± ∗ ∂t2 − 5 where K2 = l(l + 1). To further simplify this expres- V. PARTICLE CREATION sion we rewrite it by use of the tortoise coordinate r∗ implicitly defined by We now work out the particle production in the back- groundgeometrydescribingtheformationofaLBH.The dr∗ 1 := . (17) spacetimeassociatedtothegravitationalcollapsetoform dr √GF ablackholeisnoteverywherestationaryandthenweex- pectparticlecreation. Theparticlecreationphenomenon Integration yields the new radial tortoise coordinate is due tothe nonstationarygravitationalcollapse. How- a2 (r +r ) ever the spacetime will be stationary at late time and r∗ =r o +a2 − + log(r) (18) thenparticlecreationisjustatransientphenomenonand − rr r o r2r2 − + − + we can forget all the details related to the gravitational a2o+r−4 log r r + a2o+r+4 log r r . collapse. We introduce the Hawking effect in the sim- −r2(r r ) | − −| r2(r r ) | − +| plest possible scenario a la Vaidya explained in section −(cid:0) +− −(cid:1) +(cid:0) +− −(cid:1) (III). Thebasic toolstoevaluate the particleproduction Further introducing the new radial field ϕ(r) := aretheBogulibovtransformationsneededtoconnectthe ψ(r)/√H the radial equation (16) simplifies to positive frequency modes of the field between the initial and final stationary regions . The field can in fact be ∂2 expanded in the initial stationary region as +ω2 V(r(r∗)) ψ(r)=0, (19) ∂r∗2 − (cid:20) (cid:21) Φ= ainf +ain†f∗ (23) GK2 1 GF ∂ GF ∂H ω ω ω ω V(r)= + . ω H 2 H ∂r H ∂r X r " r !# (the useofdiscretemodesfromthecontinousonecanbe Inserting the metric of the self-dual black hole we finally implemented smearingwith suitable wavepackets[1, 20, obtain 45]) or in the final one as (r r )(r r ) V(r)= −(r4−+a2−)4 + × Φ= aoωutpω+aoωut†p∗ω (24) o ω X r2 a4 r K2 2 r+r +r +2K2rr +K2r2 o − − + ∗ ∗ where ain,aout are the ladder operators verifying the ω ω +h 2a(cid:16)2r4(cid:0) (cid:0)K(cid:0)2+5 r(cid:1)2+2K2rr +(cid:1)K2r2 5r(r +r(cid:1)) usualcommutationrelationsandfω andpω arethe solu- o ∗ ∗ − − + tions of (19) in the initial and final regions respectively. +5r r (cid:16)(cid:0)+r8 K(cid:1)2(r+r )2+r(r +r ) 2r r . We definetheinFockspacewiththe naturaltimev at − + ∗ − + − + − I−. The positive frequency modes which are solution of (cid:17) (cid:0) (cid:1)(cid:17)i (19) at infinity (I−) are The potential V(r) is zero at r = r and r as for the + − classical Reissner-Nordstro¨m black hole. We therefore e−iω(r∗+t) e−iωv can follow the same analysis as for this case, approxi- f (r,v)= = , (25) ω mating V(r(r∗)) near the horizons via 4π√ω√H 4π√ω√H V(r∗) e2κ+r∗ , for r r or r∗ , where v =t+r∗ and they obey the scalar product + ∝ → →−∞ VV((rr∗∗))∝e0−2κ−r∗,, ffoorr rr →0r−ororr r∗+→+.∞, (20) (fω,fω′)=−(fω∗,fω∗′)= → → → ∞ i dvHdΩ(f ∂ f∗ f∗ ∂ f )=δ(ω ω′) (26) − ω v ω′ − ω′ v ω − Acrucialquantity to study black hole evaporationis the ZI− surface gravity and (f ,f∗ ) = 0. We define also the out Fock space at ω ω′ I+ associated with the natural time parameter u. The 1 F (G′(r))2 κ2 = gµνg χρ χσ = , (21) positive frequency modes are ρσ µ ν −2 ∇ ∇ 4G where ′ denotes the derivative respectto the radialcoor- p (r,u)= e−iω(−r∗+t) = e−iωu , (27) ω dinare r and χµ = (1,0,0,0) is a timelike Killing vector 4π√ω√H 4π√ω√H in r >r and r<r but space-likein r <r <r . For + − − + where u= t r∗ and the outgoing modes obey the nor- the metric (2) we find the following values − malization condition 4m3P4(1 P2) 4m3(1 P2) κ− = 16m4P8−+a2o , κ+ = 16m4−+a2o , (22) (pω,pω′)=−(p∗ω,p∗ω′)= for the surface gravity on the inner and outer horizons. −i duHdΩ(pω∂up∗ω′ −p∗ω′∂upω)=δ(ω−ω′) (28) ZI+ 6 and (pω,p∗ω′)=0. The modes fω and pω are solutions of r=0 (19) atI− and I+ respectively; we canalso approximate H r2 in these the asymptotic regions. We decided abo≈utI+ asCauchysurfacebutitisnotproperlycorrect. r=0 We must include the future event horizon H+ to have a u complete Cauchy surface: I+ H+. The modes pω are r=r∗ not complete and we have to add those that cross the future horizon H+. HoweverSwe do not need them to na la evaluatethe particleproductionatI+ becausetheresult r− U+= ǫ − is insensitive to the ingoing modes. The next step is to γH pω caanldcuolauttegotihnegBboagsoisliusoblouvticoonesffifcienantsdrepla.tinAgsstuhme iinnggotinhge r=0 r+ ǫ γ I+ ω ω frequency to be discrete pω = Aωω′fω′ +Bωω′fω∗′, (29) ω′ X and Aωω′ =(pω,fω′) & Bωω′ =−(pω,fω∗′) (30) r=0 ǫ la na whichsatisfythefollowingmatrixrelationsagainassum- v0 ing discrete values for the frequency, Flatspace−time fω v=v0−ǫ v AA† BB† =1, I− − ABT BAT =0 (31) − and the matrix elements are FIG. 2: Penrose diagram for the Vaidya collapse and slowly A = i dvHdΩ(p ∂ f∗ f∗ ∂ p ), (32) evaporation. ThePenrosediagramforthewholecollapseand ωω′ − ZI− ω v ω− ω′ v ω evaporation process will be given in thenext figure. where, for mathematical convenience we have chosen I− astheCauchysurfacetocalculatethescalarproductthat consider a geometric optic approximation in which the isinsensitivetothischoice. TheBogulibovcoefficientsA massless particle world-line is a null ray, γ, of constant and B can also be used to expand one of the two sets of u and we trace this ray backwardsin time from I+ until creationandannihilationoperatorsintermsoftheother, I− (seeFig.2). ThelateritreachesI+,thecloseritmust approachH+. Therayγ isoneoftherayswhoselimitas aiωn = Aω′ωaoωu′t+Bω∗′ωaoωu′t† (33) t + is a null generator γH of H+. To specify γ we → ∞ Xω′ canthenuseits affinedistancefromγH alonganingoing nullgeodesicthroughH+. Thiscanbeeaslyfoundusing the Kruscal-Type coordinates in the region outside the aout = A∗ ain B∗ ain† (34) ω ωω′ ω′ − ωω′ ω′ horizon r+ =2m. These are defined by [11] ω′ X 1 1 If any of the B are non zero the particle content of U+ = e−κ+u , V+ = eκ+v, (36) ω,ω′ −κ κ the vacuum state at I− (which we indicate with the ket + + in ) respect to the Fock space at I+ is non trivial, where κ is the surface gravity calculated in (22). The | i + affineparameterintheingoingnullgeodesicclosertoH+ hin|NˆωI+|ini= |Bω,ω′|2. (35) is U+ =−ǫ and so using the first of (36) we find ω′ X 1 where NˆωI+ is the particle number operator at frequency u=−κ+ logǫ+const. (37) ω atI+. Incontrast,ifallthe coefficientB are equal ω,ω′ tozero,the firstofthe relations(31)reducesto AA† =1 substituting the previousexpressioninthe solution(27), andthenthepositivefrequencymodebasisf andp are we can see that it oscillates rapidly at later times t and ω ω relatedby a unitary transformationandthe annihilation this justifies the geometric optics approximation. We operators (33) and (34) define the same vacuum. need to match p with the solution of the Klein-Gordon ω To evaluate the products (30) we need to know the equation near I−. In the geometric optic approximation behavior of the modes p at I−. For this propose we we just parallel-transport the vector na, tangent to the ω 7 nullgeodeticwhichisingoingatH+,andla whichis the 0.10 null generator of H+, back to I− along the continuation ofγ . Wecallv thepointwherethecontinuationmeets H 0 I− thenthecontinuationoftherayγ,alongtheoutgoing 0.08 null geodesic, meets I− at v =v ǫ so 0 − 0.06 eiκω+ log(v0−v) p = for v <v , Lm ω 4π√ω√H 0 HT 0.04 p =0 for v >v , (38) ω 0 where v is the latter time at which the field can reach 0 0.02 infinity without entering in the black hole. Next step is the calculation of the Bogolubov matrix A. Introducing (38) and (25) in (32) we find 0.00 0 1 2 3 4 5 6 m (iω)−iω/κ+ ω A = Γ 1+i , ω,ω′ 2π√ωω′ (cid:18) κ+(cid:19) FIG. 3: Plot of the temperature versus the loop black hole B = iA , (39) mass. The dashed line represents the classical temperature ω,ω′ ω,−ω′ − T =1/8πm. wherewedefinev =0. TheBogoliubovcoefficientA 0 ω,ω′ is the fourier transform of a function that vanishes for π2/60,andA (m)=4π[(2m)2+a2/(2m)2]isthesurface v >v thanitisanalyticinthelowerhalfofthecomplex H o 0 ω′ plane. It has a logarithmic branch point in ω′ = 0 areaofthehorizon. Insertingthetemperature,weobtain then the branch cut extends into the upper half plane. 16m10α(1 P2)4 Therefore, we have the following relation between the L(m)= − . (44) π3(a2+16m4)3 coefficients A and B o πω The mass loss of the black hole is given by the following |Aω,ω′|=eκ+|Bω,ω′|. (40) equation for the black hole mass Finally introducing (40) in the first relation(31) we find dm = L(m) (45) dv − δ =(AA†) (BB†) ω,ω′ ω,ω′ ω,ω′ − = eπ(ω+ω′)/κ+ −1 (BB†)ω,ω′. (41) atinodnwme(vc)a.nTinhteegrerasuteltitosfitnhviserisnetetgoraotbitoaninwtihtheimniatsiaslfuconnc-- h i dition m(v =v )=m is 0 0 Taking ω = ω′ the number of particles (35) in the ωth mode at I+ is (5a6+432a4m4+34560a2m8 61440m12)π3 ∆v = o o o − 720m9(1 P2)4α hin|NˆωI+|ini=(BB†)ω,ω = e2πω/κ1+ 1. (42) (5a6o+432a4om40+34560a2om−80−61440m102)π3 . (46) − − 720m9(1 P2)4α 0 − The result (42) coincides with the Planck distribution of thermal radiation for bosons at the temperature T = In the limit m 0 this expression becomes ∆v κ /2π. BH a6π3/(144m9(1 →P2)4α), and one thus concludes tha≈t + o − Evaporation time. Theevaporationproceedsthrough the black hole needs an infinite amount of time to com- the Hawking emission at r , and the black hole’s pletely evaporate. In the complete evaporation process, + Bekenstein-Hawking temperature, given in terms of the we neglected the backreaction for any value of the mass surface gravity κ by TBH =κ/2π, yields [13] because at v ≈ +∞, when m . mP, dm/dv << m and then, contrary to the classical case, such approximation (2m)3(1 P2) is valid also in the final stages of evaporation. T (m)= − . (43) BH 4π[(2m)4+a2] o This temperature coincides with the Hawking temper- VI. COLLAPSE AND EVAPORATION ature in the limit of large masses but goes to zero for m 0 (Fig.3). Wearenowreadytocombinetheblackholeformation → The luminosity canbe estimated by use of the Stefan- and evaporation. As in section III, we divide space-time Boltzmann law L(m) = αA (m)T4 (m), where (for a intoregionsofadvancedtime. Westartwithemptyspace H BH single massless field with two degrees of freedom) α = before v then the mass increase from v to v since the a a b 8 gravitationalcollapse. For astrophysical black holes this i+ evaporationwillproceedveryslowly,andmremainscon- stanttogoodaccuracyatm ,butatsomelatertime, v , Hawkingradiationbecomesr0elevantandmdecreasesunc- r=0 I+ tilitreacheszeroagaininaninfinitytimeaswehaveseen u in the previous section. We thus have the partition < v < v < v < with −∞ a b c ∞ r=0 uc v ( ,va) : m(v)=0, √ao ∀ ∈ −∞ v (v ,v ) : m′(v)>0, a b ∀v∈ (v ,v ) : m(v)=m , r− i0 b c 0 ∀ ∈ v (vc,+ ) : m′(v)<0, r=0 r∗ ∀ ∈ ∞ for v + : m(v) 0. (47) r+ → ∞ → The mass would immediately start to drop without in- vc coming energy flux and thus v =v , but stretching this a b regionoutwillbemoreilluminatingto clearlydepictthe long time during which the hole is quasistable. v TodescribetheHawking-radiationwewillconsiderthe √ao vb I− r=0 creation of (massless) particles on the horizon such that va locally energy is conserved. We then have an ingoing radiation with negative energy balanced by outgoing ra- Flatspace−time diation of positive energy. Both fluxes originate at the horizonandhavethesamemassprofilewhichisgivenby the Hawking temperature. The area with ingoing neg- i− ative density is again described by an ingoing Vaidya solution, while the one with outgoing positive density is FIG. 4: Penrose diagram for the formation and evaporation described by an outgoing Vaidya solution. of theregular black holemetric. Thered anddark bluesolid The outgoing Vaidya solution has a mass-profile that linesdepictthetwotrappinghorizonsr−andr+. Thebrown, depends on the retarded time u instead of v and the dottedlineisthecurveofr=√aoandthebrown,longdashed massdecreasesinsteadofincreases. Theretardedtimeis oneisr∗. Thelightbluearrowsrepresentpositiveenergyflux, the magenta arrows negative energy flux. defined by du=dt dr/ F(r)G(r). After a coordinate − transformation, the metric reads p v >v ,theinnerandoutertrappinghorizonsarepresent. G(r,u) a ds2 = G(r,u)du2 2 dudr+H(r)dΩ2 , (48) These horizonsjoinsmoothlyatr =0 inaninfinite time − − sF(r,u) and enclose a non-compact region of trapped surfaces. A black hole begins to form at v = v from null dust where F(r,u) and G(r,u) have the same form as in the a whichhascollapsedcompletelyatv =v toastaticstate static case (2) but with m replaced by a function m(u). b with mass m . It begins to evaporate at v =v , and the We fix the zero point of the retarded time u so that 0 c complete evaporation takes an infinite amount of time. r = r corresponds to u = v . Then there is a static + c c The observer at I+ sees particle emission set in at some region with total mass m for v >v , u<u . Note that 0 c c retardedtime u . The regionwith v >v isthen divided since the spacetime described here has neither a singu- c c into a static region for u < u , and the dynamic Vaidya laritynoraneventhorizon,wecanconsiderpaircreation c region for u > u , which is further subdivided into an to happen directly at the trapping horizon instead of at c ingoing and an outgoing part. a different timelike hypersurface outside the horizon, as done in [44]. We have in this way further partitioned As previously mentioned, the radially ingoing flux spacetime in regions, broken down by retarded time: (light blue arrows) in the collapse region is not positive everywhere due to the quantum gravitational contribu- u<u : m(u)=m , tion. It has a flipped sign in the area between r (black c 0 ∗ ∀ u>uc : m′(u)<0 . (49) short dashed curve) and r = √ao (brown dotted curve) ∀ which is grey shaded in the figure. Likewise, the ingoing Nowthatwehaveallpartstogether,letus explainthe negative flux during evaporation (magenta arrows) has complete dynamics as depicted in the resulting causal anothersuchregionwithflippedsign. Itisinthisregion, diagram Fig.4. between the two horizon’s geometric mean value r and ∗ In the region v <v we have a flat and empty region, the dual radius corresponding to the minimal area, that a described by a piece of Minkowski-space. For all times the quantumgravitationalcorrectionsnoticeablymodify 9 the classical and semi-classical case, first by preventing i+ theformationofasingularity,andthenbydecreasingthe blackhole’stemperaturetowardszeroinaninfinitytime. I+ Σ′ VII. UNITARITY PROBLEM f u r=0 InthissectionfollowingWaldbook[45],weresumethe main features of the loss of information and unitarity of Σ f classicalblack holes. The evolutionofa physicalsystem, Σ representedby a quantum state in in an initial Cauchy int Σ surface Σ in Minkowski space, is| giiven by a unitary op- out i0 i eratorthatmapsitintoafinalstate f inafinalCauchy | i surfaceΣ . Let’sseewhathappenswhenaclassicalblack f holeispresent: TheCauchysurfaceΣ cansplitintotwo f surfaces: Σ = Σ Σ of which the first lies inside Matter f int out the black hole and the second lies outside. We can also think aslimiting caseSΣ =Σ andΣ =Σ . Now r=0 int H+ out I+ if in = 0 we know from [45] that we can formally | i | iI− Σi v write ∞ I− U|0iI− = ( e−Nπκωi|Niiint⊗|Niiout), (50) i N=0 Y X with U given by a unitary transformation relating the vacuum ofthe initial Fock space to the N particle states of the final Fock space. The form of the right hand side isduetothepresenceofabifurcatedKillinghorizonand i− theconsequentdifferentnotionsoftimeontheinitialand final Cauchy surfaces. Note that the expression (50) is formal because the state on the right hand side fails to FIG. 5: Hawking scenario be normalizable if the condition Tr(B†B)< (51) ∞ particles propagating to infinity are strongly correlated is not fulfilled [45]. This condition implies that the total with particles that enter the black hole at early times. number of particles produced in the transition between The presence of a density matrix to describe the state of the initial and final state is finite. If it is not satisfield the field in the exterior region from the algebraic point theinitialandfinalFockspacesareunitaryinequivalent, ofviewsimplycorrespondstothefactthattherestricion neverthless the expression (50) can still be used to de- of the state of the field to the subalgebra associated to scribe approximated states (see [45]). Note in fact that the intD(Σout)(whereintmeansinteriorofasetandD in both the usual Unruh and Hawking effects, this is the is the domain of dependence, see [45]) leads to a mixed case. state and this is due to the fact that the ”‘domainof de- Now the presence of the horizon obliges the exter- terminacy”’ of int D(Σout) is not the entire spacetime. nal observer to trace on the internal degrees of freedom, The breakdown of the unitarity arises from the presence transforming the initial pure state in in a density ma- of a true singularity in the case of complete evaporation trixρ. Thisistheinformationloss. T| hiusitisthehorizon see Fig.5. that causes the presence of a mixed state with an asso- The disappearence of the black hole would completly ciatedtemperature,as aconsequenceofourignoranceof remove from the spacetime the correlations (that could the complete system. The state that the observer at I+ have preservedpurity) hiddenby the Horizon. As a con- sees will not only be mixed but also made of uncorre- sequence,onasurfaceΣ′ inthefinalspacetime,afterthe f lated radiation (stochastic thermal radiation). The im- evaporation, we would have really a mixed state also at portantpoint howeveris that the externalobserverloses a fundamental level. The conclusion is that not only we the correlation between the two regions due to the trac- have information loss in the process in ρ, but also | i → ing operation: but the correlations do exist!. Altought that this process is not Unitary. The problem is that the information is lost we can claim that the quantum now at late time, the entire algebraic state of the field unitary evolution is preserved at a fundamental level. If is mixed. In algebraic terms we can look at the whole we were able to look inside the horizonwe would see the process as an evolution of the state in a time labeled by correlationsandthestatewouldcontinuetobepure. The the Cauchy surfaces. In the figure (5) we are looking 10 at the evolution from the surface Σ to the surface Σ′. Now form 0, B 0 andthen the initialstate evolves f f → → Now the domain of determinacy of the hypersurface Σ in itself f is the entire spacetime (see [45]). On the other hand int D(Σ′) includes the entire future of the spacetime in = out . (55) f | i | i (after the evaporation), but its domain of determinacy We conclude the unitarity is restored in an infinite is not the entire spacetime, since it does not include the amount of time. Since B = 0 the positive frequency blackholeregion. Theevolutioncorrespondsthento the mode basis f and p are related by the unitary transfor- restrictionofapurestateω fromtheWeylalgebraofthe ω ω mation A as immediate consequence of the first relation entire spacetime to the subalgebra associated to the re- gionintD(Σ′)i.eevolutionfromapuretoamixedstate. in (31). Note also that due to this behaviour the out f state,unlikeintheusualtreatment,canberelatedto| thei Theentireproblemcanbesummarizedsayingthatweare initial vacuum state by an exact unitary transformation evolving from a Cauchy surface to a surface that fails to because the necessary condition (51) is satisfied. beaCauchysurfaceforthewholespacetime. Inthenext To test the radiation emitted is thermal, for non zero section we will see in which sense these problems can be black hole mass, we can calculate the probabilities of cured in our model. emitting different numbers of particles. For instance, we can calculate [20] VIII. UNITARITY RESTORED inNˆI+NˆI+ in = e−2πω/κ+(1+e−2πω/κ+), (56) We have shown in this paper that LBHs evaporate h | ω ω | i (1 e−2πω/κ+)2 − emitting particle in thermodynamical equilibrium at the which agrees with a thermal distribution. In a similar temperature T =κ /2π. The black hole evaporation BH + way we can find all the higher moments that coincide time is infinite (46). As we have seen in the previous with the thermal probability section, the black hole paradox presents essentialy two features: the loss of information, strictly linked to the P(N )=(1 e−2πω/κ+)e−2πNω/κ+. (57) existence of an horizon, and the not unitary evolution, ω − that is essentially related to the presence of a singular- to emit N particles in the mode ω. If we wait for an ity. The effective spacetime that we have described in infinity amount of time the mass goes to zero together the previous sections seems to be a good candidate to with the thermal emission probability solvebothproblems. Thespacetimeisinfactsingularity freeandtheBogoulibovcoefficientsevolvewiththemass lim P(N =0)=0 , lim P(N =0)=1 (58) ω ω allowinga unitarytranformationbetween the initialand m→0 6 m→0 final vacuum. Let’s see in detail the behaviour of these andtheparticleemissionstopsafteraninfiniteamountof coefficients: for small but non zero black hole mass we time. Using(57)theexpectationvalueoftheoutparticle have thermal radiation but when the mass goes to zero, number operator reads in an infinite amount of time, the unitarity is restored. Moreconcretelylookingatthesurfacegravityκ+,wesee +∞ the mass goes to zero (in an infinite time) together with inNˆI+ in = NP(N ). (59) the temperature. This is the crucial difference with the h | ω | i ω N=0 X classicalexplosivecasewherethetemperaturegoestoin- finity when the mass reduces to zero. From the Planck The emission probability yet meets the normalization to spectrumin(42)wededuce thatinaninfinite amountof one in the zero mass limit. time To check the thermal properties of the radiation pro- duced by the black hole we calculate lim(BB†) =0 ω, (52) ω,ω m→0 ∀ inNˆI+NˆI+ in = inNˆI+ in inNˆI+ in , because κ 0 for m 0. Since (BB†) is positive h | ω ω′ | i h | ω | ih | ω′ | i ω,ω → → semi-define, this vanish iff B = 0. Let us now recall this resultshowsthe absenceofcorrelationsbetweendif- anotherusefulrelationbetweenthe initialandfinalFock ferentmodesastypicalofthethermalradiation,thenthe spaces. Using the Bogoliubov transformation between quantum state at I+ is exactly described by a thermal creation and annihilation operators respectively at I− density matrix and I+, we can find the relation between the vacuum state in (at I−) and the vacuum state out (at I+), +∞ | i | i ρ= (1 e−2πω/κ+) e−2πωN/κ+ N N , (60) ω ω − | ih | in = outin e12aˆoωut†Vωω′aˆoωu′t† out , (53) Yω NX=0 | i h | i | i where N is the state at I+ with N particles in the where the matrix V is ω mode ω|. Wi e see that any measure at I+ is described V = B∗ A−1 . (54) by the density matrix ρ and the von Newmann entropy ωω′ − ωω′′ ω′′ω′

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Particle Creation by Loop Black Holes Emanuele Alesci Universita¨t Erlangen, Institut fu¨r Theoretische Physik III, Lehrstuhl fu¨r Quantengravitation, Staudtstrasse 7, D-91058 Erlangen, EU Leonardo Modesto Perimeter Institute for Theoretical Physics, 31 Caroline St.N., Waterloo, ON N2L 2Y5, Canada We study the black hole particle production in a regular spacetime metric obtained in a minisu- perspace approach to loop quantum gravity. In different previous papers the static solution was obtained and shown to be singularity-free and self-dual. In this paper expanding a previous study oftheblackholedynamicswerepeattheHawkinganalysiswhichleadstoathermalfluxofparticles at the future infinity. The evaporation time is infinite and the unitarity is recovered due to the 1 regularity of thespacetime and to thecharacteristic behaviorof thesurface gravity. 1 0 2 I. INTRODUCTION ist, this scenariocannotbe correct. Since the singularity plays a central role for the causalspacetime diagram,its n a Blackholesareoneofthe mostfascinatingpredictions absence in the presence of quantum gravitational effects J has consequences for the entire global structure [4], and of Einsten’s gravitational theory. Today we know that 0 its removal is essential for resolving the black hole infor- they are not just a mathematically possible solution of 3 mation loss problem [5, 6]. To understand the dynamics generalrelativity,but part ofNature. Since more thana ofthegravitationalandmatterfields,itisthennecessary decade now,we havegoodevidence that ourMilky Way, ] to have a concrete model. c as other galaxies, hosts many stellar black holes as well q as a supermassive black hole in its center. It is thus promising that a resolution of the big bang - as well as the black hole singularites [8–10] has been r From the perspective of quantum gravity, black holes g achieved in a simplified version of loop quantum gravity are of interest because of the infinite curvature towards [ (LQG) [38], known as loop quantum cosmology (LQC) their center which signals (probably) a breakdown of 1 General Relativity. It is an area where quantum grav- [7] . The regular static black hole metric was recently v ity effects are strong, and it is generally expected that derived in [11][12], and studied more closely in [13]. The 2 polymeric approach to solve the black hole singularity these prevent the formation of the singularity. 9 problem was also applied in [14]. An alternative reso- 7 In the mid-seventies Steven Hawking [1] and Jacob lution of the black hole singularity was obtained in an 5 Bekenstein [2] showed that since classicalblack holes ra- effective, noncommutative approach to quantum gravity . diateparticles,theirmassslowlydecreaseandfinallydis- 1 [15] and in asymptotically safe quantum gravity [16]. In 0 appear,leavinga paradox. Supposing thatmatterwhich others works [6][17], a 2-dimensional model was used to 1 enterstheblackholewasinapurequantumstateatearly study the evaporation process in the absence of a sin- 1 time, then the existence of an horizon transforms such gularity. Recently also regular spinning loop black holes : matter into the mix thermal state of Hawking radiation v were obtained in [18]. i at late time for an external observer, and the complete X evaporation of the singularity destroys the correlations Here, we will use a 4-dimensional model based on the r between the radiation and the information “swallowed” static solution derived in [11] and generalize it to a dy- a by the black hole that would allow to reconstruct the namical case which then allows us to examine the causal original pure quantum state. The role of the singularity structure. This generalization holds to good accuracy in is crucial in this paradox. Suppose there is an entan- all realistic scenarios. This approach should be under- gledpurestateatearlytime, andapartoftheentangled stoodnotasanexactsolutiontoaproblemthatrequires system ends inside the black hole while the other part knowledge of a full theory of quantum gravity, but as a remains outside the event horizon, the result is a mixed plausible modelbasedonpreliminarystudies thatallows state after the partial trace is taken over the interior of ustoinvestigatethegeneralfeaturesofsuchregularblack the black hole. Now, since everything that falls inside hole solutions with quantum gravitycorrectionsinspired by LQG. The main goal of this paper is to reproduce the black hole reaches the singularity (r = 0) in finite proper time, the part of the Hilbert space that is traced the Hawking calculation of particle creation in a partic- ular regularblack hole backgroundand to show that the over disappears and never appears againwith the disap- pearence of the black hole. In the standard semiclassical wholeprocess,collapseandcompleteevaporation,isuni- tary. In this analysis we will make extensively use of the treatment this is the scenario,since the black hole emits particles in the form of Hawking radiation and the hori- Fabbri and Navarro-Salasbook [20]. zonradiusdecreasesandapproachesthesingularityuntil Non-singular black holes were considered already by both, singularity and horizon, vanish in the endpoint of Bardeen in the late 60s and have a long history [19, 21– evaporation [3]. However, if the singularity does not ex- 30, 32–36]. In this paper we extend the procedure in 2 [30] [31] analyzing the particle production and the uni- the fixed graph decided above. The diff-constraint tary problem as result of the complete evaporation. The isthenidenticallyzerobecauseofhomogeneityand paper is organized as follows. We start in the section II the Gauss constraint is zero for the Kantowski- by recalling the regular static metric we will use in the Sachs spacetime. whole article. In section III we generalize it to a col- lapse scenario as already done in [31]. In section IV we ii We solve the Hamilton equation of motion for the holonomic Hamiltonian system imposing the derive and solve the scalar field equations of motion in this regular black hole background and in section V we Hamiltonian constraint to be zero. use these solutions to reproduce the Hawking analysis iii The third step consists to extend the solution to leading to a thermal spectrum. In section VI we sum- the whole spacetime; this step is mathematically marize the complete dynamics for the LBH collapse and correct but, since we found the solution in the ho- evaporation. WededicatesectionVII toreviewthe basic mogeneous region, we can not be sure of the cor- aspects of the unitarity and information loss problems rect Hamiltonian constraint polymerization in the associated to classical black holes and finally in section full spacetime. However we believe such polymer- VIIIweshowhowinthisquantumcorrectedmodelthese ization exists. problemsaresolved. The signatureofthe metric usedin this paper is ( ,+,+,+) and we use the natural unit This quantum gravitationally corrected Schwarzschild convention ~=−c=GN =1. metric can be expressed in the form dr2 II. THE REGULAR BLACK HOLE SPACETIME ds2 = G(r)dt2+ +H(r)dΩ2 , − F(r) (r r )(r r )(r+r )2 + − ∗ LQG is a candidate theory of quantum gravity. It is G(r)= − − , r4+a2 obtainedfromthe canonicalquantizationof the Einstein o equationswrittenintermsoftheAshtekarvariables[37], (r r+)(r r−)r4 F(r)= − − , that is in terms of an su(2) 3-dimensional connection A (r+r )2(r4+a2) ∗ o and a triad E. The result [39] is that the basis states of a2 LQGareclosedgraphsmadeofedgesandnodeslabelled H(r)=r2+ o . (2) r2 byirreducibleSU(2)representationsandintertwinersre- spectively. Physically,theedgesrepresentquantaofarea with dΩ2 = dθ2 +sin2θdφ2. Here, r = 2m and r = + − with area 8πγlP2 j(j+1), where j is the representation 2mP2 are the two horizons,and r∗ =√r+r− =2mP. P labeloftheedge(anhalf-integer),l isthePlancklength, p P is the polymeric function P =(√1+ǫ2 1)/(√1+ǫ2+ andγ isaparameteroforder1calledtheImmirziparam- − 1), withǫ 1being the productofthe Immirziparame- eter. Verticesofthegraphrepresentquantaof3-volume. ≪ ter (γ) andthe polymeric parameter (δ). With this, it is The area is quantized and the smallest possible quanta also P 1, such that r and r are very close to r =0. − ∗ correspond to an area ≪ The area a is equal to A /8π, A being the mini- o min min mum area gap of LQG. Note that in the above metric, r A =4π√3γl2 . (1) min P is only asymptotically the usual radial coordinate since H(r) is not just r2. This choice of coordinates however Toobtainthesimplifiedblackholemodel,thefollowing has the advantage of easily revealing the properties of assumptions were made. First, the number of variables this metric as we will see. The ADM mass is the mass was reduced by assuming spherical symmetry. Second, inferred by an observer at flat asymptotic infinity; it is insteadofallpossibleclosedgraphs,aregularlatticewith determined solely by the metric at asymptotic infinity. edge-lengths δ and δ was used. The dynamical so- b c ≈ The parameter m in the solution is related to the mass lution inside the homogeneous region (that is, inside the M by M =m(1+P)2. horizon,where spaceis homogeneousbut notstatic)was If one now makes the coordinate transformation R = then obtained. An analytic continuation to the region a /r with the rescaling t˜= tr2/a , and simultaneously outside the horizon shows that one can reduce the two o ∗ o substitutes R = a /r , R = a /r one finds that the free parameters by identifying the minimum area in the ± o ∓ ∗ o ∗ metricinthenewcoordinateshasthesameformasinthe solutionwith the minimum areaof LQG. The remaining oldcoordinatesandthus exhibits averycompelling type unknown constant of the model, δ , is the dimensionless b of self-duality with dual radius r =√a . Looking at the polymeric parameter. This determines, with A , the o min angularpartofthe metric, onesees thatthis dualradius strengthofdeviationsfromtheclassicaltheoryandmust corresponds to a minimal possible surface element. It is be constrained by experiment. then also clear that in the limit r 0, corresponding The procedure to obtain the metric is in short the fol- → to R , the solution does not have a singularity, but lowing. →∞ insteadhasanotherasymptoticallyflatSchwarzschildre- i WedefinetheHamiltonianconstraintreplacingthe gion. The dual mass observed from the dual observer in homogeneousconnectionwiththeholonomiesalong r 0 is m =a (1+P)2/4mP2. d o ≈ 3 radius as radial coordinate. This is possible introducing the new radial coordinate x = √H which varies in the range x [√2a ,+ [ o ∈ ∞ 0 a2 r= r− r− r=0 x=rr2+ r2o , x∈[√2ao,+∞[. (3) B′ B The metric assumes the following form r=0 r− r− r=0 ds2 = G(r(x))dt2+ 1 dr 2dx2+x2dΩ2, (4) − F(r(x)) dx (cid:18) (cid:19) I+′ r+ r+ I+ where G, F are implicit functions of x. It is usual to de- fineg toberelatedtothemassinsideasphereofradius xx A′ A xandtheg componentofthemetrictobeproportional tt to a dirty factor e2Φ(x), since g = 1/g , tt xx 6 − I−′ r+ r+ I− −1 2m(x) g = 1 , xx − x (cid:18) (cid:19) 2m(x) g = G= 1 e2Φ(x) (5) tt − − − x (cid:18) (cid:19) and the dirty factor is FIG. 1: Penrose diagram of the regular static black hole so- lution with two asymptotically flat regions. Both horizons, (r(x)+r )4(r4(x)+a2) located at r+ and r−, are marked in blue and red respec- e2Φ(x) = (r4(∗x) a2)2 o . (6) tively. − o In the radial x coordinate the self-dual metric reads 2m(x) dx2 The causal diagram for this metric, shown in Fig 1, ds2 = e2Φ(x) 1 dt2+ +x2dΩ2. thenhastwohorizonsandtwopairsofasymptoticallyflat − − x 1 2m(x) (cid:18) (cid:19) − x regions, A,A′ and B,B′, as opposed to one such pair in thestandardcase. Intheregionenclosedbythehorizons, The metric in this form satisfies the following Einstein’s space- and timelikeness is interchanged. The horizon at equations r isafuturehorizonforobserversintheasymptotically + dm flat A,A′ region and a past horizon for observers inside =4πρx2, dx the two horizons. Similarly, the r horizon is a future − horizon for observers inside the two horizons but a past 1 dG 2 m(x)+4πPxx3 = , horizon for observes in B,B′. If one computes the time G dx x(x 2m(x)) (cid:0) − (cid:1) it takes for a particle to reach r = 0, one finds that it dP 1 dG 2 x = (ρ+P )+ (P P ), (7) takes infinitely long [13]. The diagram shown in Fig.1 x ⊥ x dx −G dx x − can be analytically continued on the dotted horizons at the bottom and top. whereenergydensityandpressuresaredefinedbyTµ = ν ThemetricinEq. (2)isasolutionofaquantumgravi- Gµν/8π = Diag( ρ,Px,P⊥,P⊥) using the coordinates − tationallycorrectedsetofequationsthat,intheǫ,a 0 (t,x,θ,φ) and the metric (4). The first equationein (7) o → limit,reproduceEinstein’sfieldequations. However,due showsthatm(x)isthemassinsideashellofradiusxand to these quantum corrections, the above metric is no fromthefirstequationin(5)wecanextractthefunction longer a vacuum-solution to Einstein’s field equations. m(x) (r =r(x)), Instead, if one computes the Einstein-tensor and sets it equal to a source term Gµν = 8πTµν, one obtains an m(x)= 1 a2o +r2 1 r4−a2o 2(r−r−)(r−r+) , effectivequantumgravitationalstress-energy-tensorTµν. 2rr2 − (cid:0) (a2o+(cid:1)r4)2(r+r∗)2 ! TheexactexpressionsforthecompoenentsofT aresome- whatunsightlyandcanbefoundinthe appendixof[e31]. where r = r(x) is defined implicitly by the relation (3). For our purposes it is here sufficient to notee that the The mass m(x) above tends to the ADM mass for r → entries are not positive definite and violate the positive + andtothedualADMmassforr 0(orx + ), ∞ → → ∞ energy condition which is one of the assumptions for the m(1+P)2 for x + , r >√a , singularity theorems. o m(x) → ∞ (8) We can write the metric (2) introducing the physical →( ao4(m1+PP2)2 for x→+∞, r <√ao. 4 Theexactrelationbetweenthephysicalradialcoordinate ThismetricreducestotheVaidyasolutionsatlargera- x and r is dius, or for ǫ 0,a 0. However,in the usual Vaidya o → → solutions,theingoingradiationcreatesacentralsingular- r= x22 − √x42−4a2o for r <√ao, (9) ictoyr.reBcutitona,stwhee sceeentheerrree,mwaitinhstrheeguqluaarn.tum gravitational  qx22 + √x42−4a2o for r >√ao. Wenotethattheingoingenergyfluxhastwozeros,one q atr =r∗(v)andoneatr =√ao,andisnegativebetween  these. What happens is that the quantum gravitational III. VAIDYA COLLAPSE correction works against the ingoing flux by making a negativecontributionuntiltheeffectivefluxhasdropped to zero at whatever is larger, the horizon’s geometric We will proceedby combiningthe static metric witha radiallyingoing null-dust, such that we obtaina dynam- mean r∗ or the location of the dual radius r = √ao. The flux then remains dominated by the quantum grav- ical space-time for a black hole formed from such dust. itational effects, avoiding a collapse, until it has passed In the present model this process, usually described by r and the dual radius where it quickly approacheswhat the Vaidya metric [40], will have corrections, negligible ∗ looks like an outgoing energy flux to the observer in the in the asymptotic region, but crucial to avoid the for- second asymptotic region. mation of a singularity in the strong-curvature region. The metric constructed this way in the following is not a strict solution of the LQC minusuperspace equations IV. A SCALAR FIELD ON THE LBH in the vacuum. In other words, keeping in mind what BACKGROUND we said in the previous section, it is a solution of the Einstein equations with an effective energy tensor which depends also on the time variation of the mass. The wave-equationfor a masslessscalar field in a gen- We use the radial coordinate r of the previous section eral spherically symmetric curved space-time reads andwestartbymakingacoordinatetransformationand 1 rewritethestaticspace-timeintermsoftheingoingnull- ∂ gµν√ g∂ Φ =0, (13) µ ν coordinate v. It is defined by the relation dv = dt + √ g − − dr/ F(r)G(r),whichcanbesolvedtoobtainanexplicit (cid:0) (cid:1) where Φ Φ(r,θ,φ,t). Inserting the metric of the self- expression for v. The metric then takes the form ≡ p dualblack hole weobtainthe followingdifferentialequa- tion G(r) ds2 = G(r)dv2+2 drdv+H(r)dΩ2 . (10) − sF(r) H(r) 2∂2Φ G(r)F′(r)Φ′ (14) ∂t2 − (cid:18) (cid:19) Nowweallowthemassminthestaticsolutiontodepend ∂2Φ ∂Φ ∂2Φ on the advanced time, m m(v). Thereby, we will 2G(r) +cotθ +csc2θ → − ∂θ2 ∂θ ∂φ2 assume the mass is zero before an initial value va and (cid:18) (cid:19) that the mass stops increasing at vb. We can then, as F(r) H(r)G′(r)Φ′+2G(r) H′(r)Φ′+H(r)Φ′′ =0, − before, use the Einstein equations G = 8πT to obtain h (cid:16) (cid:17)i the effective quantum gravitational stress-energy tensor where a dash indicates a partial derivative with respect T. Tv and Tr do not change when m(v) ies no longer to r. Making use of spherical symmetry and time- v r constant. The transversepressure Tθ =Tφ however has translation invariance, we write the scalar field as θ φ aen adeditionaleterm Φ(r,θ,φ,t):=T(t)ϕ(r)Y(θ,φ) . (15) e e Pr2m′(v) Tθ(m(v))=Tθ(m) , (11) θ θ − 2π(r+2m(v)P)4 omitting theindexesl,minthesphericalharmonicfunc- tions Y (θ,φ). The standard method of separation of lm whereme′ =dm/dv.eBecauseoftheingoingradiation,the variables allows us to split Eq.(15) in three equations, stress-energy-tensornow also has an additional non-zero one depending on the r coordinate, one on the t coor- component, Tr, which describes radially ingoing energy dinate and the remaining one depending on the angular v flux variables θ,φ, e Grv = 2(1+P()a2r2o4+(rr44−)2a(r2o)+(rr−∗(vr∗))(3v))m′(v) . (12) √HGF ∂∂r H√GF ∂ϕ∂(rr) = Gl(lH+1) −ω2 ϕ(r), (cid:18) (cid:19) (cid:18) (cid:19) ∂2 ∂ ∂2 Noticethatalsointhedynamicalcase,trappinghorizons +cotθ +csc2θ Y(θ,φ)= K2Y(θ,φ), stilloccurwheregrr =F(r,v)vanishes[42,43],sowecan ∂θ2 ∂θ ∂φ2 − (cid:18) (cid:19) continuetousethenotationfromthestaticcasejustthat ∂2 r (v) and r (v) are now functions of v. T(t)= ω2T(t), (16) ± ∗ ∂t2 − 5 where K2 = l(l + 1). To further simplify this expres- V. PARTICLE CREATION sion we rewrite it by use of the tortoise coordinate r∗ implicitly defined by We now work out the particle production in the back- groundgeometrydescribingtheformationofaLBH.The dr∗ 1 := . (17) spacetimeassociatedtothegravitationalcollapsetoform dr √GF ablackholeisnoteverywherestationaryandthenweex- pectparticlecreation. Theparticlecreationphenomenon Integration yields the new radial tortoise coordinate is due tothe nonstationarygravitationalcollapse. How- a2 (r +r ) ever the spacetime will be stationary at late time and r∗ =r o +a2 − + log(r) (18) thenparticlecreationisjustatransientphenomenonand − rr r o r2r2 − + − + we can forget all the details related to the gravitational a2o+r−4 log r r + a2o+r+4 log r r . collapse. We introduce the Hawking effect in the sim- −r2(r r ) | − −| r2(r r ) | − +| plest possible scenario a la Vaidya explained in section −(cid:0) +− −(cid:1) +(cid:0) +− −(cid:1) (III). Thebasic toolstoevaluate the particleproduction Further introducing the new radial field ϕ(r) := aretheBogulibovtransformationsneededtoconnectthe ψ(r)/√H the radial equation (16) simplifies to positive frequency modes of the field between the initial and final stationary regions . The field can in fact be ∂2 expanded in the initial stationary region as +ω2 V(r(r∗)) ψ(r)=0, (19) ∂r∗2 − (cid:20) (cid:21) Φ= ainf +ain†f∗ (23) GK2 1 GF ∂ GF ∂H ω ω ω ω V(r)= + . ω H 2 H ∂r H ∂r X r " r !# (the useofdiscretemodesfromthecontinousonecanbe Inserting the metric of the self-dual black hole we finally implemented smearingwith suitable wavepackets[1, 20, obtain 45]) or in the final one as (r r )(r r ) V(r)= −(r4−+a2−)4 + × Φ= aoωutpω+aoωut†p∗ω (24) o ω X r2 a4 r K2 2 r+r +r +2K2rr +K2r2 o − − + ∗ ∗ where ain,aout are the ladder operators verifying the ω ω +h 2a(cid:16)2r4(cid:0) (cid:0)K(cid:0)2+5 r(cid:1)2+2K2rr +(cid:1)K2r2 5r(r +r(cid:1)) usualcommutationrelationsandfω andpω arethe solu- o ∗ ∗ − − + tions of (19) in the initial and final regions respectively. +5r r (cid:16)(cid:0)+r8 K(cid:1)2(r+r )2+r(r +r ) 2r r . We definetheinFockspacewiththe naturaltimev at − + ∗ − + − + − I−. The positive frequency modes which are solution of (cid:17) (cid:0) (cid:1)(cid:17)i (19) at infinity (I−) are The potential V(r) is zero at r = r and r as for the + − classical Reissner-Nordstro¨m black hole. We therefore e−iω(r∗+t) e−iωv can follow the same analysis as for this case, approxi- f (r,v)= = , (25) ω mating V(r(r∗)) near the horizons via 4π√ω√H 4π√ω√H V(r∗) e2κ+r∗ , for r r or r∗ , where v =t+r∗ and they obey the scalar product + ∝ → →−∞ VV((rr∗∗))∝e0−2κ−r∗,, ffoorr rr →0r−ororr r∗+→+.∞, (20) (fω,fω′)=−(fω∗,fω∗′)= → → → ∞ i dvHdΩ(f ∂ f∗ f∗ ∂ f )=δ(ω ω′) (26) − ω v ω′ − ω′ v ω − Acrucialquantity to study black hole evaporationis the ZI− surface gravity and (f ,f∗ ) = 0. We define also the out Fock space at ω ω′ I+ associated with the natural time parameter u. The 1 F (G′(r))2 κ2 = gµνg χρ χσ = , (21) positive frequency modes are ρσ µ ν −2 ∇ ∇ 4G where ′ denotes the derivative respectto the radialcoor- p (r,u)= e−iω(−r∗+t) = e−iωu , (27) ω dinare r and χµ = (1,0,0,0) is a timelike Killing vector 4π√ω√H 4π√ω√H in r >r and r<r but space-likein r <r <r . For + − − + where u= t r∗ and the outgoing modes obey the nor- the metric (2) we find the following values − malization condition 4m3P4(1 P2) 4m3(1 P2) κ− = 16m4P8−+a2o , κ+ = 16m4−+a2o , (22) (pω,pω′)=−(p∗ω,p∗ω′)= for the surface gravity on the inner and outer horizons. −i duHdΩ(pω∂up∗ω′ −p∗ω′∂upω)=δ(ω−ω′) (28) ZI+ 6 and (pω,p∗ω′)=0. The modes fω and pω are solutions of r=0 (19) atI− and I+ respectively; we canalso approximate H r2 in these the asymptotic regions. We decided abo≈utI+ asCauchysurfacebutitisnotproperlycorrect. r=0 We must include the future event horizon H+ to have a u complete Cauchy surface: I+ H+. The modes pω are r=r∗ not complete and we have to add those that cross the future horizon H+. HoweverSwe do not need them to na la evaluatethe particleproductionatI+ becausetheresult r− U+= ǫ − is insensitive to the ingoing modes. The next step is to γH pω caanldcuolauttegotihnegBboagsoisliusoblouvticoonesffifcienantsdrepla.tinAgsstuhme iinnggotinhge r=0 r+ ǫ γ I+ ω ω frequency to be discrete pω = Aωω′fω′ +Bωω′fω∗′, (29) ω′ X and Aωω′ =(pω,fω′) & Bωω′ =−(pω,fω∗′) (30) r=0 ǫ la na whichsatisfythefollowingmatrixrelationsagainassum- v0 ing discrete values for the frequency, Flatspace−time fω v=v0−ǫ v AA† BB† =1, I− − ABT BAT =0 (31) − and the matrix elements are FIG. 2: Penrose diagram for the Vaidya collapse and slowly A = i dvHdΩ(p ∂ f∗ f∗ ∂ p ), (32) evaporation. ThePenrosediagramforthewholecollapseand ωω′ − ZI− ω v ω− ω′ v ω evaporation process will be given in thenext figure. where, for mathematical convenience we have chosen I− astheCauchysurfacetocalculatethescalarproductthat consider a geometric optic approximation in which the isinsensitivetothischoice. TheBogulibovcoefficientsA massless particle world-line is a null ray, γ, of constant and B can also be used to expand one of the two sets of u and we trace this ray backwardsin time from I+ until creationandannihilationoperatorsintermsoftheother, I− (seeFig.2). ThelateritreachesI+,thecloseritmust approachH+. Therayγ isoneoftherayswhoselimitas aiωn = Aω′ωaoωu′t+Bω∗′ωaoωu′t† (33) t + is a null generator γH of H+. To specify γ we → ∞ Xω′ canthenuseits affinedistancefromγH alonganingoing nullgeodesicthroughH+. Thiscanbeeaslyfoundusing the Kruscal-Type coordinates in the region outside the aout = A∗ ain B∗ ain† (34) ω ωω′ ω′ − ωω′ ω′ horizon r+ =2m. These are defined by [11] ω′ X 1 1 If any of the B are non zero the particle content of U+ = e−κ+u , V+ = eκ+v, (36) ω,ω′ −κ κ the vacuum state at I− (which we indicate with the ket + + in ) respect to the Fock space at I+ is non trivial, where κ is the surface gravity calculated in (22). The | i + affineparameterintheingoingnullgeodesicclosertoH+ hin|NˆωI+|ini= |Bω,ω′|2. (35) is U+ =−ǫ and so using the first of (36) we find ω′ X 1 where NˆωI+ is the particle number operator at frequency u=−κ+ logǫ+const. (37) ω atI+. Incontrast,ifallthe coefficientB are equal ω,ω′ tozero,the firstofthe relations(31)reducesto AA† =1 substituting the previousexpressioninthe solution(27), andthenthepositivefrequencymodebasisf andp are we can see that it oscillates rapidly at later times t and ω ω relatedby a unitary transformationandthe annihilation this justifies the geometric optics approximation. We operators (33) and (34) define the same vacuum. need to match p with the solution of the Klein-Gordon ω To evaluate the products (30) we need to know the equation near I−. In the geometric optic approximation behavior of the modes p at I−. For this propose we we just parallel-transport the vector na, tangent to the ω 7 nullgeodeticwhichisingoingatH+,andla whichis the 0.10 null generator of H+, back to I− along the continuation ofγ . Wecallv thepointwherethecontinuationmeets H 0 I− thenthecontinuationoftherayγ,alongtheoutgoing 0.08 null geodesic, meets I− at v =v ǫ so 0 − 0.06 eiκω+ log(v0−v) p = for v <v , Lm ω 4π√ω√H 0 HT 0.04 p =0 for v >v , (38) ω 0 where v is the latter time at which the field can reach 0 0.02 infinity without entering in the black hole. Next step is the calculation of the Bogolubov matrix A. Introducing (38) and (25) in (32) we find 0.00 0 1 2 3 4 5 6 m (iω)−iω/κ+ ω A = Γ 1+i , ω,ω′ 2π√ωω′ (cid:18) κ+(cid:19) FIG. 3: Plot of the temperature versus the loop black hole B = iA , (39) mass. The dashed line represents the classical temperature ω,ω′ ω,−ω′ − T =1/8πm. wherewedefinev =0. TheBogoliubovcoefficientA 0 ω,ω′ is the fourier transform of a function that vanishes for π2/60,andA (m)=4π[(2m)2+a2/(2m)2]isthesurface v >v thanitisanalyticinthelowerhalfofthecomplex H o 0 ω′ plane. It has a logarithmic branch point in ω′ = 0 areaofthehorizon. Insertingthetemperature,weobtain then the branch cut extends into the upper half plane. 16m10α(1 P2)4 Therefore, we have the following relation between the L(m)= − . (44) π3(a2+16m4)3 coefficients A and B o πω The mass loss of the black hole is given by the following |Aω,ω′|=eκ+|Bω,ω′|. (40) equation for the black hole mass Finally introducing (40) in the first relation(31) we find dm = L(m) (45) dv − δ =(AA†) (BB†) ω,ω′ ω,ω′ ω,ω′ − = eπ(ω+ω′)/κ+ −1 (BB†)ω,ω′. (41) atinodnwme(vc)a.nTinhteegrerasuteltitosfitnhviserisnetetgoraotbitoaninwtihtheimniatsiaslfuconnc-- h i dition m(v =v )=m is 0 0 Taking ω = ω′ the number of particles (35) in the ωth mode at I+ is (5a6+432a4m4+34560a2m8 61440m12)π3 ∆v = o o o − 720m9(1 P2)4α hin|NˆωI+|ini=(BB†)ω,ω = e2πω/κ1+ 1. (42) (5a6o+432a4om40+34560a2om−80−61440m102)π3 . (46) − − 720m9(1 P2)4α 0 − The result (42) coincides with the Planck distribution of thermal radiation for bosons at the temperature T = In the limit m 0 this expression becomes ∆v κ /2π. BH a6π3/(144m9(1 →P2)4α), and one thus concludes tha≈t + o − Evaporation time. Theevaporationproceedsthrough the black hole needs an infinite amount of time to com- the Hawking emission at r , and the black hole’s pletely evaporate. In the complete evaporation process, + Bekenstein-Hawking temperature, given in terms of the we neglected the backreaction for any value of the mass surface gravity κ by TBH =κ/2π, yields [13] because at v ≈ +∞, when m . mP, dm/dv << m and then, contrary to the classical case, such approximation (2m)3(1 P2) is valid also in the final stages of evaporation. T (m)= − . (43) BH 4π[(2m)4+a2] o This temperature coincides with the Hawking temper- VI. COLLAPSE AND EVAPORATION ature in the limit of large masses but goes to zero for m 0 (Fig.3). Wearenowreadytocombinetheblackholeformation → The luminosity canbe estimated by use of the Stefan- and evaporation. As in section III, we divide space-time Boltzmann law L(m) = αA (m)T4 (m), where (for a intoregionsofadvancedtime. Westartwithemptyspace H BH single massless field with two degrees of freedom) α = before v then the mass increase from v to v since the a a b 8 gravitationalcollapse. For astrophysical black holes this i+ evaporationwillproceedveryslowly,andmremainscon- stanttogoodaccuracyatm ,butatsomelatertime, v , Hawkingradiationbecomesr0elevantandmdecreasesunc- r=0 I+ tilitreacheszeroagaininaninfinitytimeaswehaveseen u in the previous section. We thus have the partition < v < v < v < with −∞ a b c ∞ r=0 uc v ( ,va) : m(v)=0, √ao ∀ ∈ −∞ v (v ,v ) : m′(v)>0, a b ∀v∈ (v ,v ) : m(v)=m , r− i0 b c 0 ∀ ∈ v (vc,+ ) : m′(v)<0, r=0 r∗ ∀ ∈ ∞ for v + : m(v) 0. (47) r+ → ∞ → The mass would immediately start to drop without in- vc coming energy flux and thus v =v , but stretching this a b regionoutwillbemoreilluminatingto clearlydepictthe long time during which the hole is quasistable. v TodescribetheHawking-radiationwewillconsiderthe √ao vb I− r=0 creation of (massless) particles on the horizon such that va locally energy is conserved. We then have an ingoing radiation with negative energy balanced by outgoing ra- Flatspace−time diation of positive energy. Both fluxes originate at the horizonandhavethesamemassprofilewhichisgivenby the Hawking temperature. The area with ingoing neg- i− ative density is again described by an ingoing Vaidya solution, while the one with outgoing positive density is FIG. 4: Penrose diagram for the formation and evaporation described by an outgoing Vaidya solution. of theregular black holemetric. Thered anddark bluesolid The outgoing Vaidya solution has a mass-profile that linesdepictthetwotrappinghorizonsr−andr+. Thebrown, depends on the retarded time u instead of v and the dottedlineisthecurveofr=√aoandthebrown,longdashed massdecreasesinsteadofincreases. Theretardedtimeis oneisr∗. Thelightbluearrowsrepresentpositiveenergyflux, the magenta arrows negative energy flux. defined by du=dt dr/ F(r)G(r). After a coordinate − transformation, the metric reads p v >v ,theinnerandoutertrappinghorizonsarepresent. G(r,u) a ds2 = G(r,u)du2 2 dudr+H(r)dΩ2 , (48) These horizonsjoinsmoothlyatr =0 inaninfinite time − − sF(r,u) and enclose a non-compact region of trapped surfaces. A black hole begins to form at v = v from null dust where F(r,u) and G(r,u) have the same form as in the a whichhascollapsedcompletelyatv =v toastaticstate static case (2) but with m replaced by a function m(u). b with mass m . It begins to evaporate at v =v , and the We fix the zero point of the retarded time u so that 0 c complete evaporation takes an infinite amount of time. r = r corresponds to u = v . Then there is a static + c c The observer at I+ sees particle emission set in at some region with total mass m for v >v , u<u . Note that 0 c c retardedtime u . The regionwith v >v isthen divided since the spacetime described here has neither a singu- c c into a static region for u < u , and the dynamic Vaidya laritynoraneventhorizon,wecanconsiderpaircreation c region for u > u , which is further subdivided into an to happen directly at the trapping horizon instead of at c ingoing and an outgoing part. a different timelike hypersurface outside the horizon, as done in [44]. We have in this way further partitioned As previously mentioned, the radially ingoing flux spacetime in regions, broken down by retarded time: (light blue arrows) in the collapse region is not positive everywhere due to the quantum gravitational contribu- u<u : m(u)=m , tion. It has a flipped sign in the area between r (black c 0 ∗ ∀ u>uc : m′(u)<0 . (49) short dashed curve) and r = √ao (brown dotted curve) ∀ which is grey shaded in the figure. Likewise, the ingoing Nowthatwehaveallpartstogether,letus explainthe negative flux during evaporation (magenta arrows) has complete dynamics as depicted in the resulting causal anothersuchregionwithflippedsign. Itisinthisregion, diagram Fig.4. between the two horizon’s geometric mean value r and ∗ In the region v <v we have a flat and empty region, the dual radius corresponding to the minimal area, that a described by a piece of Minkowski-space. For all times the quantumgravitationalcorrectionsnoticeablymodify 9 the classical and semi-classical case, first by preventing i+ theformationofasingularity,andthenbydecreasingthe blackhole’stemperaturetowardszeroinaninfinitytime. I+ Σ′ VII. UNITARITY PROBLEM f u r=0 InthissectionfollowingWaldbook[45],weresumethe main features of the loss of information and unitarity of Σ f classicalblack holes. The evolutionofa physicalsystem, Σ representedby a quantum state in in an initial Cauchy int Σ surface Σ in Minkowski space, is| giiven by a unitary op- out i0 i eratorthatmapsitintoafinalstate f inafinalCauchy | i surfaceΣ . Let’sseewhathappenswhenaclassicalblack f holeispresent: TheCauchysurfaceΣ cansplitintotwo f surfaces: Σ = Σ Σ of which the first lies inside Matter f int out the black hole and the second lies outside. We can also think aslimiting caseSΣ =Σ andΣ =Σ . Now r=0 int H+ out I+ if in = 0 we know from [45] that we can formally | i | iI− Σi v write ∞ I− U|0iI− = ( e−Nπκωi|Niiint⊗|Niiout), (50) i N=0 Y X with U given by a unitary transformation relating the vacuum ofthe initial Fock space to the N particle states of the final Fock space. The form of the right hand side isduetothepresenceofabifurcatedKillinghorizonand i− theconsequentdifferentnotionsoftimeontheinitialand final Cauchy surfaces. Note that the expression (50) is formal because the state on the right hand side fails to FIG. 5: Hawking scenario be normalizable if the condition Tr(B†B)< (51) ∞ particles propagating to infinity are strongly correlated is not fulfilled [45]. This condition implies that the total with particles that enter the black hole at early times. number of particles produced in the transition between The presence of a density matrix to describe the state of the initial and final state is finite. If it is not satisfield the field in the exterior region from the algebraic point theinitialandfinalFockspacesareunitaryinequivalent, ofviewsimplycorrespondstothefactthattherestricion neverthless the expression (50) can still be used to de- of the state of the field to the subalgebra associated to scribe approximated states (see [45]). Note in fact that the intD(Σout)(whereintmeansinteriorofasetandD in both the usual Unruh and Hawking effects, this is the is the domain of dependence, see [45]) leads to a mixed case. state and this is due to the fact that the ”‘domainof de- Now the presence of the horizon obliges the exter- terminacy”’ of int D(Σout) is not the entire spacetime. nal observer to trace on the internal degrees of freedom, The breakdown of the unitarity arises from the presence transforming the initial pure state in in a density ma- of a true singularity in the case of complete evaporation trixρ. Thisistheinformationloss. T| hiusitisthehorizon see Fig.5. that causes the presence of a mixed state with an asso- The disappearence of the black hole would completly ciatedtemperature,as aconsequenceofourignoranceof remove from the spacetime the correlations (that could the complete system. The state that the observer at I+ have preservedpurity) hiddenby the Horizon. As a con- sees will not only be mixed but also made of uncorre- sequence,onasurfaceΣ′ inthefinalspacetime,afterthe f lated radiation (stochastic thermal radiation). The im- evaporation, we would have really a mixed state also at portantpoint howeveris that the externalobserverloses a fundamental level. The conclusion is that not only we the correlation between the two regions due to the trac- have information loss in the process in ρ, but also | i → ing operation: but the correlations do exist!. Altought that this process is not Unitary. The problem is that the information is lost we can claim that the quantum now at late time, the entire algebraic state of the field unitary evolution is preserved at a fundamental level. If is mixed. In algebraic terms we can look at the whole we were able to look inside the horizonwe would see the process as an evolution of the state in a time labeled by correlationsandthestatewouldcontinuetobepure. The the Cauchy surfaces. In the figure (5) we are looking 10 at the evolution from the surface Σ to the surface Σ′. Now form 0, B 0 andthen the initialstate evolves f f → → Now the domain of determinacy of the hypersurface Σ in itself f is the entire spacetime (see [45]). On the other hand int D(Σ′) includes the entire future of the spacetime in = out . (55) f | i | i (after the evaporation), but its domain of determinacy We conclude the unitarity is restored in an infinite is not the entire spacetime, since it does not include the amount of time. Since B = 0 the positive frequency blackholeregion. Theevolutioncorrespondsthento the mode basis f and p are related by the unitary transfor- restrictionofapurestateω fromtheWeylalgebraofthe ω ω mation A as immediate consequence of the first relation entire spacetime to the subalgebra associated to the re- gionintD(Σ′)i.eevolutionfromapuretoamixedstate. in (31). Note also that due to this behaviour the out f state,unlikeintheusualtreatment,canberelatedto| thei Theentireproblemcanbesummarizedsayingthatweare initial vacuum state by an exact unitary transformation evolving from a Cauchy surface to a surface that fails to because the necessary condition (51) is satisfied. beaCauchysurfaceforthewholespacetime. Inthenext To test the radiation emitted is thermal, for non zero section we will see in which sense these problems can be black hole mass, we can calculate the probabilities of cured in our model. emitting different numbers of particles. For instance, we can calculate [20] VIII. UNITARITY RESTORED inNˆI+NˆI+ in = e−2πω/κ+(1+e−2πω/κ+), (56) We have shown in this paper that LBHs evaporate h | ω ω | i (1 e−2πω/κ+)2 − emitting particle in thermodynamical equilibrium at the which agrees with a thermal distribution. In a similar temperature T =κ /2π. The black hole evaporation BH + way we can find all the higher moments that coincide time is infinite (46). As we have seen in the previous with the thermal probability section, the black hole paradox presents essentialy two features: the loss of information, strictly linked to the P(N )=(1 e−2πω/κ+)e−2πNω/κ+. (57) existence of an horizon, and the not unitary evolution, ω − that is essentially related to the presence of a singular- to emit N particles in the mode ω. If we wait for an ity. The effective spacetime that we have described in infinity amount of time the mass goes to zero together the previous sections seems to be a good candidate to with the thermal emission probability solvebothproblems. Thespacetimeisinfactsingularity freeandtheBogoulibovcoefficientsevolvewiththemass lim P(N =0)=0 , lim P(N =0)=1 (58) ω ω allowinga unitarytranformationbetween the initialand m→0 6 m→0 final vacuum. Let’s see in detail the behaviour of these andtheparticleemissionstopsafteraninfiniteamountof coefficients: for small but non zero black hole mass we time. Using(57)theexpectationvalueoftheoutparticle have thermal radiation but when the mass goes to zero, number operator reads in an infinite amount of time, the unitarity is restored. Moreconcretelylookingatthesurfacegravityκ+,wesee +∞ the mass goes to zero (in an infinite time) together with inNˆI+ in = NP(N ). (59) the temperature. This is the crucial difference with the h | ω | i ω N=0 X classicalexplosivecasewherethetemperaturegoestoin- finity when the mass reduces to zero. From the Planck The emission probability yet meets the normalization to spectrumin(42)wededuce thatinaninfinite amountof one in the zero mass limit. time To check the thermal properties of the radiation pro- duced by the black hole we calculate lim(BB†) =0 ω, (52) ω,ω m→0 ∀ inNˆI+NˆI+ in = inNˆI+ in inNˆI+ in , because κ 0 for m 0. Since (BB†) is positive h | ω ω′ | i h | ω | ih | ω′ | i ω,ω → → semi-define, this vanish iff B = 0. Let us now recall this resultshowsthe absenceofcorrelationsbetweendif- anotherusefulrelationbetweenthe initialandfinalFock ferentmodesastypicalofthethermalradiation,thenthe spaces. Using the Bogoliubov transformation between quantum state at I+ is exactly described by a thermal creation and annihilation operators respectively at I− density matrix and I+, we can find the relation between the vacuum state in (at I−) and the vacuum state out (at I+), +∞ | i | i ρ= (1 e−2πω/κ+) e−2πωN/κ+ N N , (60) ω ω − | ih | in = outin e12aˆoωut†Vωω′aˆoωu′t† out , (53) Yω NX=0 | i h | i | i where N is the state at I+ with N particles in the where the matrix V is ω mode ω|. Wi e see that any measure at I+ is described V = B∗ A−1 . (54) by the density matrix ρ and the von Newmann entropy ωω′ − ωω′′ ω′′ω′

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