Calculation of Hawking Radiation in Local Field Theory
aa r X i v : . [ h e p - t h ] F e b Prepared for submission to JHEP
KEK-TH-2297
Calculation of Hawking Radiation in LocalField Theory
Shotaro Shiba Funai a and Hirotaka Sugawara ba Physics and Biology Unit, Okinawa Institute of Science and Technology (OIST),1919-1 Tancha Onna-son, Kunigami-gun, Okinawa 904-0495, Japan b High Energy Accelerator Research Organization (KEK),1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
E-mail: [email protected] , [email protected] Abstract:
Hawking radiation [1, 2] of the blackhole [3] is calculated based on theprinciple of local field theory. In our approach, the radiation is a unitary process,therefore no information loss will be recorded. In fact, observers in different regionsof the space communicate using the Hawking radiation, when the systems in thedifferent regions are entangled with each other. The entanglement entropy of theblackhole is also calculated in the local field theory. We found that the entanglemententropy of the systems separated by the blackhole horizon is closely connected tothe Hawking radiation in our approach. Both Hawking radiation and entanglemententropy of the four-dimensional blackholes are ultraviolet divergent quantity, butthe equation relating the two quantities is free of divergences and is given simply by S EE = √ A R H , where S EE is the entanglement entropy, A is the area of the horizon,and R H is the Hawking radiation. ontents The information loss occurs due to the transition from unitary process to thermody-namic or stochastic process. In this paper, we show that it will not happen in thecase of blackhole. This means that the blackhole solution corresponds to a pure butan entangled state.Consider a system with Hamiltonian H which is the sum of two parts H A , H B : H = H A + H B (1.1)– 1 –here A and B are different parts of space or composed of different particles. If H A and H B commute with each other, i.e., [ H A , H B ] = 0, the two parts are independentand the energy eigenstate of the whole system can be written as | ψ i = | ψ A i| ψ B i (1.2)where | ψ A i and | ψ B i are energy eigenstates of H A and H B , respectively. However, if H A and H B do not commute with each other, we have | ψ i = | ψ A i| ψ B i + | ψ ′ A i| ψ ′ B i + · · · (1.3)where | ψ ′ A i and | ψ ′ B i are other eigenstates. The state | ψ i is an entangled state inthis case. To describe such a situation, we have to take into account that both H A and H B are time-dependent, then we find that there is certain energy flow betweenthe systems A and B .The blackhole is such a pure but entangled state as will be shown in this paper: A in this case corresponds to inside of the blackhole horizon and B is the outside. H A and H B do not commute due to the existence of the boundary. There must bean energy flow through the boundary, which is in the form of Hawking radiation.We calculate this energy flow based on the local field theory and obtain the resultwhich is similar to the Planck formula but not quite: This result does not allow theconventional interpretation of the blackhole system to be a mixed state characterizedby certain temperature.This paper is organized as follows. In Sec. 2, we explain our formulation of thelocal field theory which is characterized by Schwinger commutation relation amongthe energy momentum tensor components. This gives a formula for the energy flowthrough the boundary of the two regions A and B . The Schwinger commutationrelation cannot be applied to the quantum spin-2 particle. Then in Sec. 3, we treatthe gravity as classical so that we can safely use it. In Sec. 4, we calculate theenergy flow just outside of the horizon using the Rindler coordinate, which is free ofsingularity at the horizon. We also calculate the entanglement entropy S EE of theblackhole and show that it is related to the Hawking radiation R H as S EE = 1 √ A R H (1.4)where A is the area of the horizon. Both Hawking radiation and the entanglemententropy are divergent quantity, but their ratio is finite.– 2 – Formulation
We consider four-dimensional Lorentz-invariant local field theory. Schwinger provedin 1963 [4] that in such a case the energy-momentum tensor Θ µν generally satisfies[Θ ( x ) , Θ ( y )] = − i (Θ i ( x ) + Θ i ( y )) ∂ i δ ( ~x − ~y ) (2.1)where the index i = 1 , , ~x, ~y are spatial vectors. The case with the quantumspin-2 field must be excluded from this formula, and the case of classical gravitationalbackground [3] will be discussed in the following sections.If we divide the space into two regions A and B with the functions f A and f B : f A = ( A and on the boundary of A and B , f B = ( B , (2.2)meaning that we assume the boundary belongs to A rather than B . Then we definethe Hamiltonians H A , H B as[ H A , H B ] := (cid:20)Z A dx f A ( x )Θ ( x ) , Z B dy f B ( y )Θ ( y ) (cid:21) = i Z A ∩ B dy ( f B ∂ i f A − f A ∂ i f B ) Θ i ( y )= − i Z bdy ds Θ n ( s ) . (2.3)The final form is obtained since A ∩ B collapses to the boundary of A and B .This formula simply means that we have energy-momentum flow between the regions A and B . Here n is the direction normal to the boundary (to the outside directionof A ) and ds is the surface area taken to be the normal vector to the surface.This formula also means that Hamiltonians H A , H B for the regions A and B donot generally commute unless the common region A ∩ B vanishes, i.e., A ∩ B = ∅ . A system described by Hamiltonian H = Z A ∪ B Θ ( x ) dx = Z A f A ( x )Θ ( x ) dx + Z B f B ( x )Θ ( x ) dx = H A + H B (2.4)cannot be described as the two independent systems described by H A and H B , respec-tively, because H A and H B do not commute as shown above. The time dependence– 3 –f H A and H B can be calculated as i∂ t H A = [ H, H A ] = [ H B , H A ] = i Z bdy Θ n ( s ) ds ,i∂ t H B = [ H, H B ] = [ H A , H B ] = − i Z bdy Θ n ( s ) ds . (2.5)Therefore, we obtain H A ( t ) = H A (0) + Z t dt Z bdy Θ n ( s ) ds ,H B ( t ) = H B (0) − Z t dt Z bdy Θ n ( s ) ds . (2.6) The next question is if we can separate out the boundary energy from H A in such away that [ ˜ H A , ˜ H B ] = 0 , [ ˜ H bdy , ˜ H A ] = − [ ˜ H bdy , ˜ H B ] = i Z bdy Θ n ( s ) ds . (2.7)We may write˜ H A = Z dx (1 − ˜ f A ( x ))Θ ( x ) , ˜ H B = Z dx (1 − ˜ f B ( x ))Θ ( x ) , (2.8)where the functions ˜ f A , ˜ f B are defined to be ˜ f A ( x ) = 1 in B < ˜ f A ( x ) < A and close to the boundary with B ˜ f A ( x ) = 0 in A and away from the boundary (2.9)and ˜ f B ( x ) = 1 in A < ˜ f B ( x ) < B and close to the boundary with A ˜ f B ( x ) = 0 in B and away from the boundary . (2.10)Then we have H = ˜ H A + ˜ H B + ˜ H bdy (2.11)– 4 –ith ˜ H bdy = Z A dx ˜ f A ( x )Θ ( x ) + Z B dx ˜ f B ( x )Θ ( x ) =: Z dx ζ ( x )Θ ( x ) . (2.12)where ζ ( x ) = 1 at the boundary and falls off rapidly away from the boundary.Note that this ˜ H bdy obviously has nothing to do with the conjectured AdS/CFTcorrespondence [5] where a specific gravity solution in the bulk corresponds to aspecific field theory at the boundary. We are merely taking the limit of an arbitraryfield configuration in local field theory to the boundary, excluding the possibilitythat spacetime itself has a boundary: we have always both inside and outside of theboundary.In the case of rotationally symmetric spacetime, for example,˜ f A ( r ) = θ ( r − ( R − ǫ )) , ˜ f B ( r ) = θ (( R + ǫ ) − r ) (2.13)where θ ( x ) is Heaviside step function, then the boundary is at r = R , and ǫ ≪ R .This means the region of A is inside of the boundary, while B is outside. We thenhave [ ˜ H A , ˜ H B ] = 0 (2.14)and [ ˜ H A , ˜ H bdy ] = − i Z bdy ds Θ r (cid:12)(cid:12)(cid:12) r = R − ǫ , [ ˜ H B , ˜ H bdy ] = i Z bdy ds Θ r (cid:12)(cid:12)(cid:12) r = R + ǫ . (2.15) In the ordinary quantum theory, if the ground state is a non-entangled pure state | i , it satisfies h | [ H A , H B ] | i = h | i Z bdy Θ n ( s ) ds | i = 0 . (2.16)This allows us to ignore the non-commutativity of H A and H B at the boundary whenthey act on | i , making it possible to write | i = | i A ⊗ | i B . (2.17)Notable exceptions are the following:1. In two-dimensional conformal theory where the commutator [Θ ( x ) , Θ ( y )]– 5 –as anomaly [6], its vacuum expectation value does not vanish. This leads toan entangled vacuum state.2. When we deal with the quantum theory with the classical blackhole backgroundground state | Ω i , we can show that h Ω | i Z bdy Θ n ( s ) ds | Ω i 6 = 0 . (2.18)In fact, this provides the amount of radiation from a hidden region A or B ,i.e., Hawking radiation. We will discuss the case of blackholes in detail below. We apply the above formulation to the neutral scalar system with gravity back-ground, especially with the blackhole background [3]. Here we are interested in thesystem defined by the Lagrangian L = Z d x L ; L = −√− g (cid:0) g µν ∂ µ φ∂ ν φ + M φ (cid:1) . (3.1)Then the energy-momentum tensor is obtained asΘ µν = 1 √− g ∂ L ∂g µν = − ∂ µ φ∂ ν φ + 12 g µν (cid:0) ∂ ρ φ∂ ρ φ + M φ (cid:1) . (3.2)Assuming g µν is diagonal, we obtainΘ i ( s ) = − ∂ φ∂ i φ (3.3)and Θ = −
12 ( ∂ φ ) + 12 g (cid:0) ∂ i φ∂ i φ + M φ (cid:1) (3.4)where the index i runs only spatial directions 1 , ,
3. In principle, the classical self-consistent gravity is defined as R µν − g µν R = 8 πG h Θ µν i + (other sources) , (3.5)but here we treat g µν as given or just as a solution to the Einstein equation withoutscalar fields [3]. Therefore, the ground state (vacuum state without scalar particles)is described by g µν (e.g., a blackhole) with certain zero point energy.– 6 –e now consider the region A ( B ) to be the inside (outside) of three-dimensionalspace with spherical symmetry, respectively. In this case, we have ds = g ( r, t ) dt + g rr ( r, t ) dr + r ( dθ + sin θdϕ ) (3.6)where √− g = √− g g rr r sin θ . We calculate the commutation relation [Θ ( x ) , Θ ( y )] directly using[ P φ , φ ] = − g [ ∂ φ, φ ] = iδ ( ~x − ~y ) (3.7)where the conjugate momentum P φ := √− g ∂ L ∂ ( ∂ φ ) . Then, from Eq. (3.4), we obtain[Θ ( x ) , Θ ( y )] = − ig ( x ) g ( y ) (cid:0) g ij Θ j ( x ) + g ij Θ j ( y ) (cid:1) ∂ i δ ( ~x − ~y ) . (3.8)This is the same as the case of no gravity, i.e., Eq. (2.1), if we replace Θ i ( x ) in theright hand side with g ij Θ j ( x ) and except for the factor g ( x ) g ( y ). Taking intoaccount these factors, the Hamiltonian is defined as H = Z d x √− g g Θ ( ~x ) . (3.9)Then we have, as in the case without gravity background (2.3),[ H A , H B ] = (cid:20)Z d x √− g f A g Θ ( ~x ) , Z d y √− g f B g Θ ( ~y ) (cid:21) = i Z A ∩ B d x √− g ( f B ∂ i f A − f A ∂ i f B ) g ij Θ j ( ~x ) , (3.10)where f A , f B are defined as in Eq. (2.2), meaning that f A = 1 and f B = 0 at theboundary of A and B . Since A ∩ B is just the boundary of A and B in our case, itbecomes [ H A , H B ] = − i Z bdy ds √− g g rr Θ r = − i Z bdy ds r − g g rr r sin θ Θ r . (3.11)We define ˜ H A , ˜ H B as in Eq. (2.8):˜ H A := Z d x √− g g (1 − ˜ f A ( ~x ))Θ ( ~x ) , ˜ H B := Z d x √− g g (1 − ˜ f B ( ~x ))Θ ( ~x ) . (3.12)– 7 –hen we obtain [ ˜ H A , ˜ H B ] = 0 (3.13)and [ ˜ H A , ˜ H bdy ] = − i Z bdy ds r − g g rr r sin θ Θ r (cid:12)(cid:12)(cid:12) r = R − ǫ (3.14)where R, ǫ are defined in Eq. (2.13), and ˜ H bdy := H − ˜ H A − ˜ H B is written as˜ H bdy =: Z d x √− g ζ ( ~x ) g Θ ( ~x ) . (3.15)In the case of Schwarzschild solution, we have g = − (cid:16) − r s r (cid:17) , g rr = 11 − r s r , (3.16)with the boundary radius R = r s , and the boundary Hamiltonian is˜ H bdy = 12 Z bdy ′ (cid:20) ( ∂ φ ) − r s r + (cid:16) − r s r (cid:17) ( ∂ r φ ) + 1 r (cid:18) ( ∂ θ φ ) + ( ∂ ϕ φ ) sin θ (cid:19) + M φ (cid:21) × r sin θ drdθdϕ (3.17)where the integration region ( bdy ′ ) is a thin sphere of depth 2 ǫ . If we take theboundary to be exactly at r = r s , the inside and the outside have different signs of(1 − r s r ). To avoid this ambiguity, we assume the boundary to be in r ∈ [ r s − ǫ, r s + ǫ ]and in this region ζ ( ~x ) = 1 is satisfied. Our definition of the Hawking radiation is R H := ∂H B ∂t = − i h Ω | [ H A , H B ] | Ω i = Z bdy ds r − g g rr r sin θ h Ω | Θ r | Ω i (4.1)where | Ω i is the ground state of H = ˜ H A + ˜ H B + ˜ H bdy . We sometimes call | Ω i the Hawking ground state. R H is clearly the amount of energy going through theboundary in the “vacuum state” which is nothing but the blackhole. This way ofinterpreting the Hawking radiation has two remarkable characters:– 8 –. The Hawking radiation process is the unitary process. Therefore, no violationof information conservation occurs. Explicitly, the above equation shows Z dt h Ω | R H | Ω i = h Ω | H B ( t ) | Ω i = h Ω | U † ( t ) H B (0) U ( t ) | Ω i (4.2)where U ( t ) = e iHt is the time evolution operator. Therefore, we obtain h Ω | i ∂U∂t | Ω i = −h Ω | U H A ( t ) | Ω i . (4.3)2. Since the time development of H B is completely given by the Hawking radia-tion, we can say that an observer in A and an observer in B are exchangingthe information using the Hawking radiation.Extending this situation to other entangled states such as the EPR system [7],we can say that Alice in A and Bob in B are exchanging the information usingthe Hawking radiation. In the case of EPR, there must be a “Hawking” radi-ation emitted to maintain the entanglement nature of A and B . An equationsimilar to (4.1) provides a way to calculate the Hawking radiation also in thiscase. In other words, we are claiming EPR = SH (Steven Hawking) rather thanEPR = ER [8] where the information is exchanged using a wormhole. Now we have to calculate R H in Eq. (4.1) explicitly. It is important to understandrather tricky nature of Eq. (4.1): To calculate the left hand side of Eq. (4.1), we mustbe careful about what happens at the boundary. We must use H = ˜ H A + ˜ H B + ˜ H bdy (4.4)where [ ˜ H A , ˜ H B ] = 0 and[ ˜ H B , ˜ H bdy ] = i Z bdy ds r − g g rr r sin θ Θ r (cid:12)(cid:12)(cid:12) r = r s + ǫ . (4.5)However, after taking the commutator and ending up with the expression on theright hand side, we may use the boundary value on the right hand side of Eq. (4.1)just by extending the following expression to the boundary r = r s ,˜ H A → H A , ˜ H B → H B . (4.6)– 9 –his means that the commutator singularity of [ ˜ H B , ˜ H bdy ] on the boundary is alreadytaken into account on the right hand side of (4.5). On the right hand side, we haveΘ r = − ∂ φ∂ r φ (4.7)where the scalar field φ is a free field and can be expanded in normal mode in theentire space, space A , or space B .We basically follow ’t Hooft’s formulation [3] in the following. To get the normalmode, we need to solve the equation of motion. This can be done by changingthe coordinate system to Rindler coordinate. Since the right hand side of Eq. (4.1)contains only the quantity at the boundary (i.e., horizon), the transformation from( t, r, θ, ϕ ) to the Rindler system ( τ, ρ, θ, ϕ ) becomes very simple: τ = 12 r s t , ρ = 2 r s r rr s − , (4.8)and g ττ = ∂t∂τ ∂t∂τ g = − r s r ρ → − ρ , g ρρ = ∂r∂ρ ∂r∂ρ g rr = rr s → , (4.9)in the limit of r → r s (i.e., ǫ → +0). The equation of motion for the Lagrangian(3.1) becomes "(cid:18) ρ ∂∂ρ (cid:19) − ∂ ∂τ + ρ (cid:18) ∂ ∂~z − M (cid:19) φ = 0 (4.10)where ~z := r s × ( θ, ϕ ) | r = r s ,θ ≈ π is defined. This means that we discuss only the regionwith sin θ ≈
1, satisfying1 r s (cid:18) θ ∂∂θ (sin θ ∂∂θ ) + 1sin θ ∂ ∂ϕ (cid:19) ≈ r s (cid:18) ∂ ∂θ + ∂ ∂ϕ (cid:19) = ∂ ∂~z , (4.11)instead of the whole region 0 ≤ θ ≤ π , − π ≤ ϕ ≤ π . In this way, we approximate thehorizon sphere by a flat plane. This approximation can be justified, since our systemhas the spherical symmetry for θ, ϕ directions and, of course, any small region nearthe horizon is approximately flat spacetime.The equation (4.10) gives a common time τ to both inside and outside of theblackhole. Since the first term does not change by going from outside of the blackhole(real positive ρ ) to inside the blackhole (imaginary negative ρ ), we must change thesign of µ := (cid:16) − ∂ ∂~z + M (cid:17) to keep the equation unchanged: this means µ must beimaginary. The Hamiltonian corresponding to the Rindler time τ can be commonlyused both inside and outside of the blackhole. This is the advantage of Rindler– 10 –oordinate compared to the ( r, t ) coordinates: in the latter case, we must exchange r and t when we go inside the blackhole, and also the Hamiltonian density changesfrom Θ to Θ rr .By putting φ ( τ, ρ, ~z ) =: e i ( ~k · ~z − ωτ ) ˜ φ ( ρ ), we obtain "(cid:18) ∂∂ρ (cid:19) + 1 ρ ∂∂ρ − (cid:18) µ − ω ρ (cid:19) ˜ φ = 0 . (4.12)The solution with energy ω and transverse momentum ~k is given by φ ( τ, ρ, ~z ) = 1 N J − iω ( iµρ ) e i ( ~k · ~z − ωτ ) =: K ( ω, µρ , µρ ) e i ( ~k · ~z − ωτ ) (4.13)where N is the normalization factor determined in Appendix B, and J is the Bessel(or Hankel) function with the integral formula1 iπ K ( ω, µρ , µρ ) = 1 iπN Z C dss s − iω e iµρ ( s − s ) (4.14)where the path C can be [0 , ∞ ] and µ = − ∂ ∂~z + M = ~k + M . (4.15)Going from outside to inside of the blackhole, we have µρ → − µρ , ~z → i~z . (4.16)By making ~k → − i~k , we have exactly the same equation as Eq. (4.12). Then we noteuseful identities satisfied both by Bessel and Hankel functions: K ( ω, α, β ) = K ∗ ( − ω, − α, − β ) . (4.17)For both Bessel and Hankel functions we also have K ( ω, α, β ) = e − πω K ∗ ( − ω, α, β ) (for positive α, β ) K ( ω, α, β ) = e πω K ∗ ( − ω, α, β ) (for negative α, β ) . (4.18)In the end, we have to choose Bessel function so that the boundary value is finite.– 11 –hen we have φ ( τ, ρ, ~z ) = Z + ∞−∞ dω Z d ~k p π ) K ( ω, µρ , µρ ) e i ( ~k · ~z − ωτ ) a ( ω, ~k ) + h.c. = Z + ∞ dω Z d ~k p π ) K ( ω, µρ , µρ ) e i ( ~k · ~z − ωτ ) h a ( ω, ~k ) + a ( ω, ~k ) i + h.c. (4.19)where the region µρ < a ( ω, ~k ) is the annihilation operator satisfying[ a ( ω, ~k ) , a ∗ ( ω ′ , ~k ′ )] = δ ( ω − ω ′ ) δ ( ~k − ~k ′ ) , otherwise = 0 , (4.20)and we define a ( ω, ~k ) := 1 √ − e − πω (cid:16) a ( ω, ~k ) + e − πω a ∗ ( − ω, − ~k ) (cid:17) ,a ( ω, ~k ) := 1 √ − e − πω (cid:16) e − πω a ∗ ( ω, ~k ) + a ( − ω, − ~k ) (cid:17) , (4.21)so that these operators satisfy[ a ( ω, ~k ) , a ∗ ( ω ′ , ~k ′ )] = [ a ( ω, ~k ) , a ∗ ( ω ′ , ~k ′ )] = δ ( ω − ω ′ ) δ ( ~k − ~k ′ ) , otherwise = 0 . (4.22)Note that a ( ω, ~k ) and a ( ω, ~k ) are defined only in the region ω ≥
0. Solving theseequations backwards, we obtain the relations a ( ω, ~k ) − e − πω a ∗ ( ω, ~k ) = √ − e − πω a ( ω, ~k ) ,a ( ω, ~k ) − e − πω a ∗ ( ω, ~k ) = √ − e − πω a ( − ω, − ~k ) . (4.23) The blackhole ground state (Hawking ground state) is defined as a ( ω, ~k ) | Ω i = 0 . (4.24)In terms of a ( ω, ~k ) and a ( ω, ~k ), using Eq. (4.23) we have h a ( ω, ~k ) − e − πω a ∗ ( ω, ~k ) i | Ω i = 0 . (4.25)– 12 –he solution is, as given by ’t Hooft [3], | Ω i = 1 N Ω e R + ∞ dω R r sd ~k √ π )4 e − πω a ∗ ( ω,~k ) a ∗ ( ω,~k ) | i | i (4.26)where N Ω is the normalization factor. Starting from this expression, we can easilycalculate the von Neumann (entanglement) entropy of the system: S EE = A π K Z + ∞ dω (cid:20) − log (cid:0) − e − πω (cid:1) + 2 πωe πω − (cid:21) (4.27)where A is the area of the blackhole horizon, and we define K := R d ~k √ π ) . Thedetail of this calculation is given in Appendix A. We now calculate Hawking radiation from Eq. (4.1): R H = − i h Ω | [ H A , H B ] | Ω i = r s (cid:16) − r s r (cid:17) Z bdy sin θdθdϕ h Ω | Θ r | Ω i (cid:12)(cid:12)(cid:12) r = r s . (4.28)We write down the result, leaving the details to Appendix B: R H = √ K Z + ∞ dω ωe πω − . (4.29)This is similar to the black body radiation formula with the temperature 1 / π .However, one notices a slight deviation of this expression from the Planck formulafor the black body radiation in which the integral must be ∼ Z + ∞ dω ω e πω − . (4.30)Our radiation is not a thermal process and therefore we should be surprised by thesimilarity of two formulae rather than their difference.To be more precise, we must take into account the back reaction of the radiationdue to the decrease of the energy; to say it simply, the time dependence of theblackhole mass due to the radiation. In this sense, Eq. (4.29) is just an approximation.Another way to clarify the difference is that our radiation is from a pure (althoughentangled) state | Ω i = 1 N Ω e R + ∞ dω R r sd ~k √ π )4 e − πω a ∗ ( ω,~k ) a ∗ ( ω,~k ) | i | i . (4.31)On the other hand, the Planck radiation equation (4.30) is from the mixed state | i i – 13 –hat can only be described by the density matrix ˆ ρ :ˆ ρ = X i e E i /T | i ih i | (4.32)where E i is energy of the state | i i and T is temperature. We can sum over states ofeither | i or | i in Eq. (4.26) and get a nontrivial density matrix, and then claimthat we have a mixed state. However, since the entire system is pure (coherent),the system utilizes the Hawing radiation (4.29) to communicate and recover thecoherence. The simplest example of EPR shows that the entanglement entropy hasnothing to do with thermodynamics. In this sense, we are claiming the EPR = SH(Steven Hawking) using the terminology similar to EPR = ER [8].Our approach, at least in the spirit, is shared by some of the recent publicationsin which they claim that Hawking radiation carries information [9–11] . We have already obtained Hawking radiation R H in Eq. (4.29) and entanglemantentropy S EE in Eq. (4.27).If we take the cutoff to be expressed in terms of the Planck scale, we have K = Z d ~k p π ) =: ξG (5.1)where G is the Newton’s constant. The Hawking radiation (4.29) becomes R H = √ ξG Z + ∞ dω ωe πω − √ ξG (5.2)and the entanglement entropy (4.27) becomes S EE = A π ξG Z + ∞ dω (cid:20) − log (cid:0) − e − πω (cid:1) + 2 πωe πω − (cid:21) = 124 ξ A G (5.3)where A is the area of the blackhole horizon.We note the following comments on our results:1. We have a cutoff independent relation between the Hawking radiation and the The authors of these references discuss information loss in the context of AdS/CFT. Theyargue that the information inside a blackhole can be transferred to the Hawking radiation, whilethe semi-classical analysis around the horizon is kept intact. The information is not lost after theevaporation of a blackhole. Their system is a combination of classical (with Gibbons-Hawkingentropy) and the quantum mechanical objects. – 14 –ntanglement entropy: S EE = 1 √ A R H . (5.4)Thus the entanglement entropy can be calculated in terms of observed totalHawking radiation R H and the surface area A of the blackhole horizon.2. If ξ ∼
1, Hawking radiation does not depend on any parameter of the blackhole;it depends only on the gravitational constant G : K ∼ G = Planck massPlanck time = 2 × − [kg]5 × − [sec] = 4 × [kg / sec] . (5.5)Since solar mass is ∼ × kg, a blackhole with solar mass can survivefor only 10 − sec. Even a blackhole of galactic mass with 10 solar mass cansurvive for only a few days. Therefore, the cutoff of ξ ∼ ξ ∼ ξ becomes ∼ − . In this case, theHawking radiation becomes ∼ kg/sec and most of the celestial blackholesbecome stable.4. The entanglement entropy has the expression ∼ ξ A G , which is consistent withthat of Bekenstein-Hawking entropy. As we have seen above, ξ must be a tinyconstant in the four-dimensional realistic celestial cases.5. Our calculation is free of conical singularity, which is encountered in someof the entanglement entropy calculations. The divergence that occurs in ourcalculation of Hawking radiation and the entanglement entropy is a regularhigh-energy divergence, and we can either cutoff or “renormalize” it to fit theexperimental data. This divergence is common to the Hawking radiation andthe entanglement entropy so that one of those experimental data determinesthe other. It seems that very high energy cutoff, such as the Planck scale,seems impossible as long as we accept the existence of celestial blackholes.Finally, we comment on the fate of a realistic blackhole.It is customary to consider the Schwarzschild solution corresponding to a black-hole. It has nothing inside the horizon except the singularity at the center. The– 15 –nformation loss is related to the existence of this singularity. If we regard a black-hole literally as the Schwarzschild solution, we cannot avoid the information lossbecause the singularity at the single point cannot contain all the initial information.However, in the previous sections, we showed that the blackhole is not a mixedstate and there must be no information loss. This strongly suggests that in ourapproach we cannot take the Schwarzschild solution as it is. In fact, we can showthat there exists a solution of the Einstein equation which has a horizon outside ofsome hard solid core, if we assume the density of matter has some finite upper bound,e.g., the Planck scale [12].Then, by assuming that there exists a solid core rather than a singularity insidethe blackhole horizon, we can show that such a realistic blackhole actually does notevaporate. We will explain it in detail in a future work [12]. In this paper, we calculate the Hawking radiation and the entanglement entropy ofa blackhole based on the local field theory. The Schwinger commutation relationwhich symbolized the locality of the quantum field theory is explicitly utilized. Boththe Hawking radiation and the entanglement entropy turn out to be proportional toa common divergent quantity thus making the ratio to be finite.The blackhole ground state is an entangled state of inside and outside, and itmakes sense to calculate the entanglement entropy. However, it is nevertheless apure state and the entropy is not that of a mixed state as is often claimed. We alsofound that the formula we obtain for the Hawking radiation is similar to the blackbody radiation from a fixed temperature but not exactly, so thus invalidating theinterpretation of a blackhole with an object of fixed temperature.As for the information loss, we do not have it since the local field theory isunitary. More concretely, our Hamiltonian of the quantum scalar field is Hermitian.The inside region and the outside region of a blackhole communicate each other usingthe Hawking radiation. This situation must be common to all the entangled statesincluding the simple EPR state.Therefore, a blackhole never evaporates in our approach. We will show therealistic model of such a blackhole in a future work [12]: it is the same as theSchwarzshild solution outside the horizon, but inside it has a solid core to avoid theinformation loss.
Acknowledgment
We thank Professor Jiro Arafune for his comments at early stage of this work. Wealso thank Professor Alex Kusenko in UCLA for his valuable comments.– 16 –
Calculation of entanglement entropy
The integrals in Eq. (4.26) are discretized as Z dω ω → X i ω i , Z r s d ~k p π ) → X ~k . (A.1)Then we obtain the discrete version of Eq. (4.26): | Ω i = 1 N Ω Y ω i ,~k e e − πωi a ∗ ( ω i ,~k ) a ∗ ( ω i ,~k ) | i | i = Y ω i ,~k √ − e − πω i ∞ X n =0 e − nπω i | n i | n i . (A.2)The total density matrix isˆ ρ = Y ω i ,~k √ − e − πω i ∞ X n =0 e − nπω i Y ω ′ i ,~k ′ p − e − πω ′ i ∞ X n ′ =0 e − n ′ πω ′ i | n i | n i h n ′ | h n ′ | . (A.3)Taking the trace for the states | n i , we get the density matrix for the system 1. Inaddition, we apply the replica method here:ˆ ρ N = X ω i ,~k (cid:0) − e − πω i (cid:1) N ∞ X n =0 e − Nnπω i | n ih n | . (A.4)Then the entanglement entropy is calculated as S EE = − Tr ρ log ρ = − Tr ∂ρ N ∂N (cid:12)(cid:12)(cid:12)(cid:12) N =1 = − Tr ∂∂N Y ω i ,~k ∞ X n =0 (cid:2)(cid:0) − e − πω i (cid:1) e − nπω i (cid:3) N (cid:12)(cid:12)(cid:12)(cid:12) N =1 | n ih n | = − ∂∂N Y ω i ,~k (1 − e − πω i ) N − e − Nnπω i (cid:12)(cid:12)(cid:12)(cid:12) N =1 . (A.5)– 17 –n the continuous limit, and we obtain S EE = − ∂∂N e R dω R r sd ~k √ π )4 log ( − e − πω ) N − e − Nnπω (cid:12)(cid:12)(cid:12)(cid:12) N =1 = − ∂∂N Z dω Z r s d ~k p π ) log (1 − e − πω ) N − e − Nnπω (cid:12)(cid:12)(cid:12)(cid:12) N =1 = Z dω Z r s d ~k p π ) (cid:20) − log (cid:0) − e − πω (cid:1) + 2 πωe πω − (cid:21) . (A.6)Finally, we define K := R d ~k √ π ) and obtain Eq. (4.27): S EE = A π K Z dω (cid:20) − log (cid:0) − e − πω (cid:1) + 2 πωe πω − (cid:21) (A.7)with the surface area A = 4 πr s for the sphere with radius r = r s . B Calculation of Hawking radiation
In our definition, the Hawking radiation is given as Eq. (4.28): R H = r s (cid:16) − r s r (cid:17) Z bdy sin θdθdϕ h Ω | Θ r | Ω i (cid:12)(cid:12)(cid:12) r = r s = 4 πr s (cid:16) − r s r (cid:17) h Ω | Θ r | Ω i (cid:12)(cid:12)(cid:12) r = r s ,θ ≈ π . (B.1)As discussed in Sec. 4.2, due to the rotational symmetry, we only consider the regionof sin θ ≈ π comes from R sin θdθdϕ = 4 π . Then we have, using Eq. (4.19),Θ r (cid:12)(cid:12)(cid:12) r ≈ r s ,θ ≈ π = ∂τ∂t ∂ρ∂r ( − ∂ τ φ∂ ρ φ ) = − ∂ τ φ∂ ρ φ r s p rr s −
1= 12 r s p rr s − Z + ∞−∞ dω Z d ~k p π ) iωK ( ω, µρ , µρ ) e i ( ~k · ~z − ωτ ) a ( ω, ~k ) + h.c. ! × Z + ∞−∞ dω ′ Z d ~k ′ p π ) ∂ ρ K ( ω ′ , µρ , µρ ) e i ( ~k ′ · ~z − ω ′ τ ) a ( ω ′ , ~k ′ ) + h.c. ! . (B.2)Then we find that Θ r (cid:12)(cid:12) r ≈ r s ,θ ≈ π is described as a combination of the four terms pro-portional to a ( ω, ~k ) a ( ω ′ , ~k ′ ) , a ∗ ( ω, ~k ) a ( ω ′ , ~k ′ ) , a ( ω, ~k ) a ∗ ( ω ′ , ~k ′ ) , a ∗ ( ω, ~k ) a ∗ ( ω ′ , ~k ′ ) , (B.3)– 18 –here the normal product must be taken in a and a . This is because of thefollowing reason: the normal product is a way to renormalize the vacuum energy,which is defined in the entire system described by the variable a . The normal productin the subsystem 1 and 2 vanishes, since the entire system is described by a variablein which the zero point energy is taken into account. Then we should not repeat thevacuum energy defined in entire system in the subsystems.The first term in Eq. (B.3) satisfies, for ω, ω ′ ≥ N ( a ( ω, ~k ) a ( ω ′ , ~k ′ ))= 1 √ − e − πω √ − e − πω ′ : (cid:16) a ( ω, ~k ) − e − πω a ∗ ( ω, ~k ) (cid:17) (cid:16) a ( ω ′ , ~k ′ ) − e − πω ′ a ∗ ( ω ′ , ~k ′ ) (cid:17) := a ( ω, ~k ) a ( ω ′ , ~k ′ ) . (B.4)For ω, ω ′ <
0, according to Eq. (4.23), we have N ( a ( ω, ~k ) a ( ω ′ , ~k ′ ))= 1 √ − e πω √ − e πω ′ : (cid:16) a ( | ω | , − ~k ) − e πω a ∗ ( | ω | , − ~k ) (cid:17) (cid:16) a ( | ω ′ | , − ~k ′ ) − e πω ′ a ∗ ( | ω ′ | , − ~k ′ ) (cid:17) := a ( ω, ~k ) a ( ω ′ , ~k ′ ) . (B.5)In all the cases including ωω ′ <
0, we obtain the same result: This term remains thesame if the normal product for a , a is taken. Therefore, it does not contribute tothe Hawking radiation.Using the same method, the second term in Eq. (B.3) is computed as N ( a ∗ ( ω, ~k ) a ( ω ′ , ~k ′ )) = a ∗ ( ω, ~k ) a ( ω ′ , ~k ′ ) − e π | ω | − δ ( ω − ω ′ ) δ ( ~k − ~k ′ ) (B.6)for all the cases of ω, ω ′ . The second term in the right hand side is the contributionto the radiation.Similarly, the third and last terms in Eq. (B.3) are N ( a ( ω, ~k ) a ∗ ( ω ′ , ~k ′ )) = a ( ω, ~k ) a ∗ ( ω ′ , ~k ′ ) − e π | ω | − δ ( ω − ω ′ ) δ ( ~k − ~k ′ ) ,N ( a ∗ ( ω, ~k ) a ∗ ( ω ′ , ~k ′ )) = a ∗ ( ω, ~k ) a ∗ ( ω ′ , ~k ′ ) . (B.7)– 19 –ollecting these contributions and substituting them into Eq. (B.2), we obtain R H = πρ h Ω | Θ r | Ω i (cid:12)(cid:12)(cid:12) r = r s ,θ ≈ π = − πiρ Z + ∞ dω Z d ~k π ) ωK ( ω, µρ , µρ ) ∂ ρ K ∗ ( ω, µρ , µρ ) 1 e πω − (cid:12)(cid:12)(cid:12)(cid:12) r = r s ,θ ≈ π + c.c. (B.8)Here we note that r → r s is equivalent with ρ →
0. Using the expression of Besselfunction (4.13),lim ρ → ρK ( ω, µρ , µρ ) ∂ ρ K ∗ ( ω, µρ , µρ ) = 1 N lim ρ → ρJ iω ( iµρ ) ∂ ρ J ∗ iω ( iµρ )= − iωe ± πω N , (B.9)where we use the formula for the Bessel function on the boundary:lim ρ → J iω ( iµρ ) = (cid:18) iµρ (cid:19) iω , lim ρ → ∂ ρ J ∗ iω ( iµρ ) = − iωρ (cid:18) − iµρ (cid:19) − iω . (B.10)Therefore, we obtain R H = − K√ π Z dω e ± πω N ω e πω − , (B.11)where K := R d ~k √ π ) as before.To determine the normalization factor N , we use the commutation relation (3.7):[ ∂ φ, φ ] = − ig δ ( ~x − ~x ′ ) = ir s r (cid:18) rr s − (cid:19) δ ( r − r ′ ) δ ( ~z − ~z ′ ) (B.12)where ~z = r s × ( θ, ϕ ) | r = r s ,θ ≈ π , and at the boundary we obtain[ ∂ τ φ, φ ] = ∂t∂τ [ ∂ φ, φ ] = iρδ ( ρ − ρ ′ ) δ ( ~z − ~z ′ ) . (B.13)On the other hand, using Eq. (4.19) and the Bessel function formula (B.10), we obtain[ ∂ τ φ, φ ] = − i π Z + ∞−∞ dω π ωK ( ω, µρ , µρ ) K ∗ ( ω, µρ ′ , µρ ′ ) δ ( ~z − ~z ′ ) + h.c. ρ → = − i π Z + ∞−∞ dω π ωe ± πω N e iω (log ρ − log ρ ′ ) δ ( ~z − ~z ′ ) + h.c. (B.14)– 20 –hen we find that Eqs. (B.13) and (B.14) are coincident when N = − ωe ± πω π (B.15)is satisfied. Finally, we obtain R H = √ K Z + ∞ dω ωe πω − . (B.16) References [1] S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. (1975) 199; Erratum ibid. (1976) 206.[2] J. B. Hartle and S. W. Hawking, “Path Integral derivation of black hole radiance,”Phys. Rev. D13 (1976) 2188.[3] Excellent review article is by G. ’t Hooft, “The scattering matrix approach for thequantum black hole: an overview,” J. Mod. Phys.
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