In a recent paper 1 we proposed that the quantum states of black hole responsible for the BekensteinHawking entropy are given by Bell states of Fermi quanta in the interior of black hole. In this paper we include the effect of tunneling on these entangled states and show that partial tunneling of these Bell states of Fermi quanta give rises to the Page curve of Hawking radiation. We also show that the entirety of information initially stored in the black hole is returned to the outside via the Hawking radiation.
['High Energy Physics - Theory', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Phenomenology']
Tunneling Page Curve and Black Hole Information
{'Tunneling, Page Curve and Black Hole Information': 'Chong-Sun Chu 2 , 3 , 4 ∗ , Rong-Xin Miao 1 †  \n1 School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai 519082, China ‡  \n- 2 Center of Theory and Computation, National Tsing-Hua University, Hsinchu 30013, Taiwan\n- 3 Department of Physics, National Tsing-Hua University, Hsinchu 30013, Taiwan  \n4 National Center for Theoretical Sciences, Taipei 10617, Taiwan', 'Abstract': 'In a recent paper [1], we proposed that the quantum states of black hole responsible for the Bekenstein-Hawking entropy are given by Bell states of Fermi quanta in the interior of black hole. In this paper, we include the effect of tunneling on these entangled states and show that partial tunneling of these Bell states of Fermi quanta give rises to the Page curve of Hawking radiation. We also show that the entirety of information initially stored in the black hole is returned to the outside via the Hawking radiation.', '1 Introduction': "There are mounting theoretical evidences that a black hole obeys the first law of thermodynamics with an entropy  \nS BH = A 4 G (1.1)  \nand a temperature T H = 1 / (8 πGM ) due to a thermal Hawking radiation [2]. Despite remarkable progress that has been made in microstate counting [3], it is still not known what is the nature of the gravitational degrees of freedom that are being counted by (1.1).  \nAnother outstanding problem of the black hole is the information problem [4]. Consider a black hole form from a pure state. Assuming that the Hawking radiation is thermal, then the entanglement entropy outside the black hole increases monotonically for the entire course of life of the black hole. This Hawking curve violates the fundamental unitarity principle of quantum mechanics: the fine-grained entanglement entropy should not exceed the coarse-grained black hole entropy. On the other hand, Page argued that [5, 6] if the black hole evolution process is unitary, then the total system of black hole and radiation must go back to a pure state at the end of the evaporation. As a result, unitarity of quantum black hole requires the following properties for the Hawking radiation: 1. Page curve: the entanglement entropy of Hawking radiation should initially rise until the so called Page time t P when it starts to drop down to zero as the black hole completely vanishes. 2. Recovery of information: at the end of the black hole life, one must be able to recover from the Hawking radiation all the information of the initial pure state. It was Page's remarkable insight that the Page curve behavior of the Hawking radiation, due to its accessibility to external observers, can be especially useful as a decisive criteria for unitary theory of quantum gravity.  \nRemarkable progress has been made lately on this front where a fully quantum form [7] of the Ryu-Takayanagi entropy formula [8, 9] was employed to compute the entanglement entropy of the Hawking radiation and it was shown to obey the Page curve [10-12]. Central to this analysis is the emergence of island, regions of spacetime that are completely disconnected and spacelike separated from the region of the Hawking radiation. From the semi-classical point of view, island originated from the wormhole saddle in the replica path integral for the entanglement entropy [13]. It is interesting to note that the island is located just beneath the horizon for evaporating Schwarzschild black hole [10-12].  \nThat the Page curve is obtained gives confidence that the AdS/CFT correspondence and the generalized entropy formula constitutes a credible quantum gravity framework to study black holes. Yet, it still leaves many questions unanswered. For example, it is believed that the Bekenstein-Hawking entropy (1.1) is coarse-grained in nature [14, 15]. From this perspective, what role does island play in coarse graining the initial pure state? How does it help in returning the information to the environment? The main motivations of this work have been to understand better the origin of the Page curve and the return of information in terms of explicit spacetime quantum mechanical processes.  \nIn an earlier paper [16], it was proposed that the quantum states of black hole responsible for the Bekenstein-Hawking entropy are given by a thin shell of entangled quanta located at the region just underneath the horizon. It was argued that the configuration can be stabilized by a new kind of degeneracy pressure. However many questions were left unanswered. For example what is the nature of these quanta? how does the degeneracy pressure arises? In a recent paper [1], a model of the black hole spacetime as a system of Fermi quanta was proposed. Due to the Pauli exclusion principle, the system exhibits a quantum degeneracy pressure which is able to withstand the collapse of spacetime due to gravity. The horizon radius of the black hole is reproduced as a result of the equivalence principle. Moreover the Bekenstein-Hawking entropy is explained in terms of the coarse graining of the entangled configurations of the Fermi quanta. According to the proposal, the fermionic quanta observes a constant density of states, from which it follows that there can be no more than V/l 3 P quantum states in any volume V . This maximal capacity of states implies the existence of a minimal volume in space and suggests that any spacetime singularity as predicted in general relativity must be resolved in the quantum gravity.  \nWe note that these fermionic states in our model are characterized by energy and are not localized in location as assumed in [16]. Nevertheless the analysis performed in [16] can be easily adapted to the current consideration to include the effect of tunneling on the dynamical evolution of the black hole Fermi gas system. In section 2, we review our quantum mechanical model of black hole model as a gas of Fermi quanta. In section 3, we show that partial tunneling of the Bell states give rises to the Page curve of Hawking radiation. We also show that at the end of life of the black hole, the entirety of information initially stored in the black hole is returned to the outside via the Hawking radiation. Discussion and conclusion are in section 4.", '2 Black hole as a system of Fermi quanta': 'General relativity predicts that any object that has collapsed beyond a certain point would form a black hole, inside which there is a singularity. For a spherical black hole with horizon radius R , the spacetime is given by some classical metric solution of the Einstein equation. And the singularity at the origin represents a place where matter is compressed infinitely and the classical description of spacetime breaks down. In [1], a quantum mechanical model the interior of the black hole is proposed. According to the model, the spacetime inside the black hole is quantized whose fundamental quanta are fermionic. The system of Fermi gas has a Fermi energy µ that is inversely proportional to the horizon size R and the density of states is constant given by  \nµ = 1 2 πR , g ( E ) = 3 πV G . (2.1)  \nThis gives the total number N S = ∫ dEg ( E ) of fermionic states in the system  \nN S = 2 πR 2 G , (2.2)  \nwhich show an area dependence, and the total quantum energy of the Fermi sea is  \nU = R 2 G . (2.3)  \nWhen the black hole is charged or with a cosmological constant, there will be additional contribution to the internal energy of the Fermi system. It was shown in [1] that either by the equivalence principle, or equivalently, by the requirement of the balancing of matter and gravitational pressure, the black hole horizon radius is found to be reproduced remarkably. Moreover, it was shown that if there is a double bosonic degeneracy for each energy level, then the fermionic degrees of freedom at each energy level can be entangled in 4 different ways, which then give rises to the ground state degeneracy G = 4 N L . Here N L is the number of energy levels. It is related to the number of states N S as N S = 4 N L . The counting of these microstates gives the entropy S = log G = N S / 2 , which reproduces precisely the Bekenstein-Hawking entropy. A base 2 logarithm has been employed. It is remarkable that in our model, the holographic nature of the black hole is a direct result of the area dependence of the number of fermionic energy states of the black hole interior.', '3.1 Hawking radiation and unitarity violation': "Black hole information problem refers to the apparent violation of unitarity of black hole in the semiclassical analysis of it's evolution. There are at least two ways to see it, both related to  \nthe Hawking radiation. Consider a black hole that forms in flat space from a pure state and decays under Hawking radiation. If unitarity of quantum mechanics is not violated in the black hole formation process, then the black hole interior and exterior together is in a pure state. Due to our ignorance of the interior, this give rises to a number S BH of coarse-grained states with entropy (1.1). By definition, the knowledge of these states together with the knowledge of the exterior constitutes a pure state. Eventually, the black hole is exhausted by the Hawking radiation. According to Hawking's original calculation, Hawking radiation is thermal and a mixed state. This is, however, in direct contradiction with unitarity since a pure state cannot evolve into a mixed state. Instead if unitarity is preserved, then the final state of the Hawking radiation cannot be purely thermal, but it must encompass the information contained in the initial set of coarse-grained states so that, together with the outside information, a pure state can be reconstructed.  \nA related problem is the behavior of the entropy of the Hawking radiation. An easy way to see this is to consider the quantum mechanical picture of Parikh and Wilczek [17], where Hawking radiation is obtained from the tunneling of the virtual particles created out of the vacuum state of the black hole. For an outgoing particle with energy ω created just inside the horizon, they found that it can travels cross the horizon and results in a nonvanishing imaginary part for the particle action: Im S = 4 πω ( M -ω 2 ) . This corresponds to a tunneling process with the tunneling rate  \nΓ = Γ 0 e -2Im S = Γ 0 e -8 πω ( M -ω 2 ) , (3.1)  \nwhere Γ 0 is a prefactor which can be computed from a more detailed knowledge of the dynamics. The leading exponential ω dependence in Γ registers a Boltzmann thermal distribution with the Hawking temperature T . The higher order correction ω 2 term is a back reaction term, showing that the spectrum is slightly deviated from the thermal one. We note that such a result for the tunneling rate is more or less guaranteed by consistency due to the fact that Hawking temperature is an universal quantum effect of the horizon, which can be understood either in terms of the removal of conical singularity of geometry or as Unruh effect of acceleration. In addition, pair creation outside the horizon was also considered. In this so called anti-particle channel, the anti-particle created outside the horizon follows a time reversed ingoing geodesic crosses the horizon and also makes contribution to the Hawking radiation. Note that while the original analysis of [17] is for massless particles, one can show that the same result (3.1) holds for massive particles as well.  \nAlthough this picture of tunneling gives a clear physical picture of the origin of Hawking radiation and its temperature, it also leads to the Hawking curve of the entropy of Hawking radiation. To see this, consider a virtual pair created in the interior side of the horizon (particle channel). Since the pair was created entangled, an entanglement between the black hole and the Hawking radiation is created when the particle tunnels through and the anti-particle got absorbed by the black hole. The rate of increase of the entanglement entropy of the Hawking radiation is given by α ( ω ) A , where ω is the energy of the particle and α ( ω ) is the creation rate per unit area of virtual pairs on the interior side of horizon times the tunneling probability. Similarly, there is a contribution β ( ω ) A from the anti-particle channel, where β ( ω ) is the creation  \nrate of virtual pairs on the exterior side of horizon times the tunneling probability. As α, β are positive, the entanglement entropy of Hawking radiation increases monotonically. This Hawking curve violates the unitarity of quantum mechanics and is a fundamental problem for the quantum information of black hole.  \nIt has been suggested that some form of nonlocality is needed to resolve the black hole information problem (see [18] for a review). As quantum nonlocality is best captured by entanglement, it seems natural to consider the coarse-grained degrees of freedom to be rich in entanglement content. This is exactly what happens in our black hole model. In this section, we include the dynamical effect of tunneling and show how tunneling can affect Fermi gas of quanta and lead to the Page curve and the recovery of the full entanglement content of the coarse-grained states in late time correlation of the Hawking radiation.", '3.2 Tunneling and entanglement swapping of Bell pairs in the Fermi model': "Our model of black hole differs from the quantum vacuum model of black hole in a major way. Instead of a vacuum filled with virtual pairs of particle anti-particle [17], the black hole in our picture is filled with a collection of entangled fermionic energy eigenstates. In the picture of [17], it is the motion of particles over the classically forbidden trajectories that give rises to the tunneling and the Hawking radiation. Our fermionic states are not localized in space and the mechanism for their tunneling has to be different. Nevetheless, we do expect that in quantum gravity, the Fermi quanta will also be able to tunnel and leak through the horizon. Although we do not currently have the technology of quantum gravity to determine the tunneling rate, we can expect the tunneling rate will take the generic form (3.1) 1 In any case, our following analysis and conclusion will be generic and will not depend on the detailed form of the tunneling rate.  \nIn addition to tunneling of the existing Fermi quanta, there will also be vacuum pair creation process as usual and this leads to an interesting effect of entanglement swapping on the Fermi quanta. Consider the particle channel, the anti-fermionic quanta ¯ p that is left behind can get annihilated by a quanta b 2 of one of the Bell pairs. As a result, the fermionic p becomes entangled with the other partner b 1 of the Bell pair, and entanglement is swapped from the Bell pair and the virtual pair to one between the Hawking radiation quanta and the remaining quanta inside the black hole. Similarly, entanglement swapping occurs in the anti-particle channel. See Figure 1. Both these processes will not only alter the entropy content of the black hole, but also introduce an entanglement entropy for the Hawking radiation due to the partially escaped entangled pairs.  \nTo study the dynamics of the entangled states, let us denote the energies of the quanta of  \nFigure 1: Tunneling of entangled quanta  \n<!-- image -->  \neach entangled pair by 0 ≤ ω ≤ M 0 , where M 0 is the original mass of the black hole. Let n 1 ( ω, t ) dω be the number of entangled pairs inside the black hole and have energies within the interval ( ω, ω + dω ) , n 2 ( ω, t ) dω be the number of entangled pairs that has one of the quanta inside the horizon and the other outside, and n 3 ( ω, t ) dω be the number of entangled pairs that are located entirely outside the horizon. See Figure 2. Taking into account of tunneling of the Bell states and quantum entanglement swapping, we have  \n∂n 1 ∂t = -2Γ n 1 -2 δA, (3.2)  \n∂n 2 ∂t = 2Γ n 1 -Γ n 2 +2 δA, (3.3)  \n∂n 3 ∂t = Γ n 2 , (3.4)  \nwhere Γ := Γ( ω, M ( t )) is the tunneling rate for a Fermi quanta of energy ω from a black hole of mass M ( t ) . α := α ( ω, M ( t )) (resp. β := β ( ω, M ( t )) ) is the production rate per unit area of the conventional vacuum created Hawking radiation in the anti-particle channel (resp. particle channel), and δ := α + β is the total production rate from both channels. Physically, the n i terms and the A terms on the RHS of (3.2) (resp. (3.3) or (3.4)) represent the effect of direct tunneling and quantum swapping on the entangled pairs that are entirely inside the black hole (resp. partially inside or entirely outside the black hole). The factors of 2 in (3.2), (3.3) is due to the fact that both ends of the entangled pair can participate in these processes. The total number of each types of entangled pairs is  \nN i ( t ) = ∫ D dωn i ( ω, t ) , i = 1 , 2 , 3 , (3.5)  \nwhere the integration region is over D := { 0 ≤ ω ≤ M ( t ) } . Physically, 2 N 1 represents the amount of coarse-grained entropy of the black hole at time t , N 2 represents the entanglement entropy of Hawking radiation, and N 3 represents the amount of entanglement information contained in the Hawking radiation. In our model, the mass of the black hole is given by  \nM ( t ) = ∫ D dω (2 n 1 + n 2 ) ω (3.6)  \nFigure 2: Entanglement swapping induced by virtual pairs  \n<!-- image -->  \nwhere the appropriate domains of integration is understood. In a consistent analysis, (3.6) will have to be considered together with (3.2) - (3.4). This is quite a complicated system to solve. In the following, we show that without assuming the form of Γ( ω, M ) and δ ( ω, M ) , and without solving the system explicitly, one can immediately show that the Hawking radiation obeys the Page curve and that the complete information of the black hole is returned.  \nLet us consider a black hole formed at t = 0 , implying the conditions n 2 = n 3 = 0 initially. Consider first (3.2). As the RHS of (3.2) is non-positive, n 1 will continue to decrease until it reaches zero, where also ∂n 1 /∂t = 0 . Denote this time as t 1 ,ω and define t 1 := Supp ω ∈ D ( t 1 ,ω ) . We have  \nN 1 = 0 for t ≥ t 1 . (3.7)  \nNext consider (3.3). Starting with the initial conditions n 2 = 0 and n 1 > 0 , n 2 is set to increase initially. As time progress, there will be a cross over time where the RHS of (3.3) vanishes and turns negative subsequently. Then n 2 will continue to decrease until it reaches zero at some time t 2 ,ω . Define t E := Supp ω ∈ D ( t 2 ,ω ) , we have  \nN 2 = 0 for t ≥ t E . (3.8)  \nFinally consider (3.4). It is clear that n 3 increases monotonically until n 2 reaches zero at t = t 2 ,ω , then it remains a constant. Therefore we have  \nN 3 = N 3 f a constant for t ≥ t E . (3.9)  \nIt is clear that t 1 is when the black hole's degrees of freedom become completely entangled with the environment and t E is the end time that the black hole dies.  \nNote that as N 2 starts zero at t = 0 and ends at zero again at t = t E , it must follow an inverted V-shape curve and reaches a maximum at some intermediate time t P . This is generic and we obtain the Page curve for the Hawking radiation, with t P being the Page time. By expressing N 2 as integrals over the domain D , it is easy to establish the conservation equation:  \nd dt ( N 1 + N 2 + N 3 ) = 0 . (3.10)  \nIt is remarkable that this leads immediately to  \nN 3 f = N 10 , (3.11)  \nmeaning that all the entanglement information originally stored in the Bell pairs are returned to the exterior observer via the Hawking radiation. Without the tunneling of Bell particles, our equations (3.2)-(3.4) will reduce to the equation ∂n 2 /∂t = δ 1 A as in the conventional tunneling model of Hawking radiation. There would then be the Hawking curve and it would not possible to return information via n 3 . In the supplementary material, we provide justification that t E is finite in our model. We also show the explicit time dependence of the n i 's for some typical values of the energies, which confirm the generic behavior discussed here. We also note that in our model, the Hawking radiation is not purely thermal and contains correlations that are necessarily highly nonlocal as they could arise from tunneling process that occurs at very different times over the life of the black hole.  \nAcknowledgments: We thank Pei-Ming Ho for useful discussion. C.S.C acknowledge support of this work by NCTS and the grant 110-2112-M-007-015-MY3 of the National Science and Technology Council of Taiwan. R.X.M thank support by NSFC grant (No. 11905297).", 'A Appendix': "In a semiclassical estimation, Hawking radiation reduces the mass of the black hole as [20]  \ndM dt = -Q 3 M 2 , (A.1)  \nwhere Q = 3 α/G 2 and α is some numerical constant. This gives  \nM ( t ) = M 0 [ 1 -Qt M 3 0 ] 1 / 3 (A.2)  \nand a finite lifetime of the black hole t E = M 3 0 /Q . In our model, this should come out from the consistent set of equations (3.2)-(3.4) and (3.6). Although the complete analysis is quite complicated, we can see that our model does roughly give (A.1) near the final stage of evaporation and so t E is finite. In fact, using (3.2) and (3.3), we have  \ndM dt = ∫ D -Γ(2 n 1 + n 2 ) ω, (A.3)  \nwhere we have ignored the much smaller term -2 δωA on the RHS since δ ≪ Γ as δ involves an additional vacuum creation rate. Near the final stage of evaporation, it is ∫ n 1 ∼ O (1) and we can use the mean value theorem of calculus to estimate that ∫ Γ ωn = Γ ω ∫ n ∼ 1 /M 2 , where we have used ω ∼ M and that Γ ∼ Γ 0 for small M . Here the prefactor of the tunneling rate has a dependence Γ 0 ∼ 1 /M 3 coming from the phase space volume. As a result, (A.3) is consistent with the semiclassical result (A.1) at time close to the end point and so t E is finite.  \nTo get a better feeling of the time evolution of the Bell pairs, let us consider the approximate mass function M ( t ) (A.2) with M 0 = 1 , Q = 0 . 05 so that t E = 20 . In Figure 3, we show the plots  \nFigure 3: Evolution of Bell states for the energies ω = (a) 0.001, (b) 0.05, The mass (A.2) with M 0 = 1 , Q = 0 . 05 is adopted for these plots.  \n<!-- image -->  \n̸  \nof n 1 , n 2 , n 3 for two typical set of energies ω given by (a): 0.001 and (b): 0.05. Since δ ≪ Γ , we have taken δ = 0 here for simplicity. Including δ = 0 won't affect the qualitative features of these plots. We see that the generic behaviors of the n i 's discussed above do get captured very well here, showing that (A.2) is a decent approximation. In general, a perturbative scheme can be devised to solve the system (3.2)-(3.4), (3.6) numerically.", 'References': "- [1] C.-S. Chu and R.-X. Miao, 'Fermi model of a quantum black hole,' Phys. Rev. D 110 no. 4, (2024) 046001, arXiv:2307.06164 [hep-th] .  \n[2] S. W. Hawking, 'Particle Creation by Black Holes,' Commun. Math. Phys. 43 (1975) 199-220. [Erratum: Commun.Math.Phys. 46, 206 (1976)].  \n- [3] A. Strominger and C. 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