Unitary Black hole radiation: Schwarzschild-global monopole background
aa r X i v : . [ g r- q c ] A ug Unitary Black hole radiation: Schwarzschild-global monopole background
Arpit Das ∗ and Narayan Banerjee † Department of Physical Sciences,Indian Institute of Science Education and Research Kolkata,Mohanpur, West Bengal 741246, India.
Black hole radiation from an infinitesimally thin massive collapsing shell, possessing a globalmonopole charge, which in turn leads to a Schwarzschild black hole with a global monopole chargehas been shown to be processed by a unitary evolution. The exterior metric of the collapsing shell isdescribed by the global monopole (GM) metric. The analysis is performed using the Wheeler-deWittformalism which gave rise to a Schr¨odinger-like wave equation. Existence of unitarity is confirmedfrom two independent lines of approach. Firstly, by showing that the trace of the square of thedensity matrix, of the outgoing radiation, from a quantized massless scalar field, is unity. Secondly,by proving that the conservation of probability holds for the wave function of the system.
Keywords: Global monopole, Black hole radiation, Unitarity, Density matrix, Conservation of probability,Semi-classical analysis
I. INTRODUCTION
Recently, in an attempt to shed some light on theresolution of the information loss paradox [1–6], it hasbeen shown by Das and Banerjee[7] that radiation froma collapsing charged shell is processed with a unitaryevolution. This was achieved in a Reissner-Nordstr¨ombackground using the Wheeler-deWitt formalism[8, 9]and unitarity checks were carried out using two indepen-dent lines of approach, density matrix and conservationof probability. We extend the result as given in [7]by performing the same kind of analysis for a notasymptotically flat spacetime. We adopt the formalismand method of analysis from [7] and apply it to aglobal monopole background metric [10]. It was shownin [11] that a Schwarzschild black hole with a globalmonopole charge Hawking radiation is Planckian innature. So, naturally it is a relevant theoretical questionto investigate unitarity issues in such backgrounds. Thisis the primary motivation of this work.The present work shows that the process of blackhole radiation, in a not asymptotically flat spacetime,is unitary. Saini and Stojkovic[12] worked with a notasymptotically flat spacetime before, specifically withan asymptotically AdS spacetime. However, the resultsobtained therein are based on numerical estimates. Fornot asymptotically flat spacetimes, our analysis andtherby the results obtained from them are more robustas they are done analytically.We work with a metric that includes a globalmonopole charge η . The Schwarzschild case as consid-ered in [13], is recovered trivially as a special case bysetting η = 0.In section 2 we describe the global monopole met-ric. Section 3 contains the description of the model. The ∗ [email protected] † [email protected] scalar field is discussed in section 4. The unitarity isascertained in section 5. The last section includes a dis-cussion of the results. II. THE GLOBAL MONOPOLE
The metric for a Schwarzschild black hole with aglobal monopole charge η is given in natural units as,[10, 11, 14], ds GM = − (cid:18) − η − Mr (cid:19) dt + (cid:18) − η − Mr (cid:19) − dr + r d Ω , (1)where, η << M is the mass of the black hole.Note that the above metric is not asymptotically flat andeven with M = 0 the spacetime is not flat, as it has somenon-zero curvature [11], R = R = 0 = R , (2) and, R ∝ η r , (3)where the above terms are the components of the Riccitensor. The observational signature of a global monopoleis in the existence of a “solid angle deficit”.The event horizon is at, R GM = 2 M − η . (4)Let us also give below the stress-energy tensor corre-sponding to the Global monopole field [14], T = T = η πr , (5)where we see that the total energy is divergent andso solutions of such form as eq n (1) are unrealistic andperhaps appear in some instances of cosmic phasetransition [10].The surface gravity κ for the metric as given in eq n (1) is obtained by noting that the metric is of theform [15], ds = − f dt + f − dr + r d Ω , (6) implying, κ = f ′ ( r )2 , (7) implying, κ GM = (cid:0) − η (cid:1) M (8) (cid:18) as, f ( r ) = 1 − η − Mr (cid:19) . where κ GM is the surface gravity for the global monopolemetric.The semi-classical study of the metric as given in eq n (1) was done in [11] and it was show that theoutgoing Hawking radiation is thermal possessing aPlanck spectrum, N = 1 e πMω/ (1 − η ) − N is the number density of outgoing quanta ofparticles. The Hawking temperature is recovered to be, T GM = (cid:0) − η (cid:1) πM , (10)which can also be obtained from eq n (8) using the Hawk-ing relation T H = κ π (which holds here too). III. THE MODEL
In our model we have an infinitesimally thin mas-sive collapsing spherical shell with a global monopolecharge [11, 14], whose background metric is g µν . Thereis also a massless scalar field Φ whose dynamics we shallstudy. We assume that Φ couples to the gravitationalfield (which originates from the presence of a non-trivialbackground metric). However, Φ does not directly cou-ple to the shell. An asymptotic observer, at the futurenull infinity, is present to detect the outgoing flux witha detector and by assumption does not interact with the“shell-metric-scalar” system. Hence, the observer doesnot significantly affect the evolution of the system andsimilarly for the system vis-a-vis the observer. The ac-tion for the whole system is then given by [16], S tot = Z d x √− g (cid:20) − R π + 12 ( ∂ µ Φ) (cid:21) − σ Z d ξ √− γ + S obs , (11)where the first term denotes the usual Einstein-Hilbertterm for the background metric g µν , the second termrepresents the action for the massless scalar field, thethird term represents the shell’s action in terms of itsworld-volume coordinates ξ a ( a = 0 , , σ is the tension of the shell (or, the shell’s proper energy density per unitsurface area) and γ ab is the shell’s induced world-volumemetric, given by, γ ab = g µν ∂ a X µ ∂ b X ν , (12)where X µ ( ξ a ) determines the location of the shell.The Roman indices run over the internal world-volumecoordinates ξ a ( a = 0 , ,
2) while the Greek indices runover the usual spacetime coordinates.The last term S obs represents the action for the ob-server. A. Spacetime Foliation-GM coordinates
The mass and the global monopole charge is con-fined in an infinitesimally thin shell [14], as per our con-siderations. So that for an exterior observer the distri-bution would be spherical. However, the inside of theshell would be empty and would be described by theMinkowski metric. The exterior of the shell is describedby a Global Monopole metric. Thus, we have, ds out = − (cid:18) − η − Mr (cid:19) dt + (cid:18) − η − Mr (cid:19) − dr + r d Ω , (13) ds in = − dT + dr + r d Ω , (14) ds on − shell = − dτ + r d Ω , (15)for r > R ( t ), r < R ( t ) and r = R ( t ) respectively. Here r is the radial coordinate. So r = R ( t ) describes thecollapsing shell and R := R ( t ) is the radius of the shell. T , τ and t are the time coordinate inside the shell,proper time on the shell and time coordinate of the exte-rior observer respectively. d Ω is the standard S metric.An important consideration to observe here is thatsince the above GM coordinates would lead to a coordi-nate singularity, at R = R GM (the event horizon), wemight face trouble using this for our analysis. However,observe that from the point of view of an asymptoticobserver, the event horizon is an infinitely red shiftedsurface. So, the observer can only observe the collapseof the shell approaching its event horizon in infinitetime as per his time t . Thus, the analysis would happenupto this limit which is relevant from an asymptoticviewpoint and the GM coordinates are well behavedupto this limit, that is just outside the event horizon.Similar to [7], we consider timelike unit vectors u α := dx αout dτ and v α := dx αin dτ , for ds out and ds in respec-tively. From their normalization, that is, u α u α = − v α v α = −
1, one obtains, at r = R ( t ), t τ = √ E + R τ E , T τ = p R τ and T t = q E − (1 − E ) R t E . In theabove expressions, a subscript indicates a differentiationw.r.t. that particular coordinate. x αout and x αin are thecoordinates pertaining to ds out and ds in respectively.Also, E := 1 − η − MR ( t ) . B. Mass of the shell
According to Israel’s formulation[14, 17, 18], themass M of the shell can be obtained as, M = 4 πσR hp R τ − πσR i − η R , (16)We shall show below that M would turn out to be aconstant of motion. So, there would be no conflict withthe fact that M is a constant of integration in the metricand can be identified as the mass of the shell. Similar tothe results given in [19], one can write, R ττ α = η πσR + 6 πσ − αR , (cid:16) where, α := p R τ (cid:17) . (17)Now, using eq n s (16) and (17), M τ = R τ (cid:20) πσR ( α − πσR ) − η (cid:21) + R τ (cid:20) πσR (cid:18) η πσR − αR + 6 πσ − πσ (cid:19)(cid:21) = 0 . Thus, we see that M is a constant of motion.Since, we have proven that M is a constant of mo-tion, we can have the following identification, H shell ≡ M, (18)where H shell is the Hamiltonian of the shell. H shell is tobe treated classically for our analysis. C. Action for the shell
The shell’s action is given as, S shell = − Z dT (cid:20) πσR (cid:20)q − R T − πσR (cid:21) − η R (cid:21) . (19)The Lagrangian corresponding to the shell’s actionyields the conjugate momentum as,Π shell = ∂ L shell ∂R T = 4 πσR R T p − R T ! . (20)Now the Hamiltonian is, H shell = Π shell R T − L shell = 4 πσR hp R τ − πσR i − η R . (21) H shell as obtained above matches with M as expressedin eq n (16). Hence, the action in eq n (19) is consistent(since, this action gives the correct H shell as expressedin eq n (18)). Now let us consider S shell in terms of time t , (using the expression for T t ), S shell = − Z dt " πσR "r E − R t E + Z dt " πσR " πσR r E − − EE R t + Z dt " η R r E − − EE R t . (22)Let us also consider the conjugate momentum and Hamil-tonian in terms of t ,Π shell = ∂ L shell ∂R t = 4 πσR R t √ E " p E − R t − πσR (1 − E ) p E − (1 − E ) R t − πσR R t √ E " η (1 − E )8 πσR p E − (1 − E ) R t , (23) H shell = Π shell R t − L shell = 4 πσE / R " p E − R t − πσR p E − (1 − E ) R t − πσE / R " η πσR p E − (1 − E ) R t . (24) D. Incipient Limit
We define the so-called incipient limit, R → R GM ,as the limit when the radius of the shell approaches theevent horizon. From eq n (23) and eq n (24) we note that,as R → R GM , Π shell = 4 πµR R t √ E p E − R t , (25) H shell = 4 πE / µR p E − R t , (26)where, µ := σ (cid:16) − πσR GM − η πσR GM (cid:17) . Then we have, H shell = [( E Π shell ) + E (4 πµR ) ] / ≡ [ q + m ] / , (27)where q := ( E Π shell ) and m := E (4 πµR ) . H shell as given in Eq n (27), is the Hamiltonian of arelativistic particle with a position dependent mass.This is how the shell behaves in the incipient limit as R → R GM . We shall show below that in this limit also, H shell would turn out to be a constant of motion. Since, d H shell dτ = ∂ H shell ∂τ , we have, ddτ πµ E / R p E − R t ! = 0 leading to, E / R p E − R t = H shell πµ =: h ( a constant ) , (28)( as τ doesn ′ t appear explicitly in H shell ) . These expressions can be arrived at independentlyusing an alternative approach (see appendix).Classically, we have from eq n (28) and from the ex-pression of T t , R t = ± E r − ER h ≈ ± E (cid:18) − ER h (cid:19) ≈ ± E (29)( as R → R GM ) ,T t = E r − E ) R h , (30)where solving eq n (29) in terms of t will give us theclassical behaviour of the shell as R ( t ) → R GM . E can be written as, E = (1 − η ) (cid:18) − R GM R (cid:19) = ǫ (cid:18) − R GM R (cid:19) , (31)where ǫ := (1 − η ).In the incipient limit, E → R ( t ) → R GM ).Then, in this limit, R t ≈ ± E . Now solving for R ( t ) weget (from eq n (29) and eq n (31)), ± ǫ RR − R GM dRdt ≈ ǫ R GM R − R GM dRdt ( upto leading order ) integrating, R GM ln (cid:18) R f − R GM R − R GM (cid:19) = ± ǫt f ( R := R (0) and R f := R ( t f )) thus, R f = R GM + ( R − R GM ) e ± ǫt f /R GM , (32)where the lower limit of integration w.r.t. t is t = 0 andthe upper limit is t = t f .Similar to as in [7], as R f → R GM and t f > ǫ > η << t f → ∞ . Thus, the negative sign for R ( t ) describes acollapsing model in the incipient limit. Eq n (32) alsoshows that from the viewpoint of an asymptotic ob-server, the formation of the event horizon takes infinitetime implying that the event horizon is an infinite redshifted surface, which matches with the classical result,as stated earlier while choosing the GM coordinates. IV. THE SCALAR FIELD Φ The action for the scalar field Φ can be written asa sum of the actions, S Φ = S Φ ) in + S Φ ) out = 2 π Z dt " − ( ∂ t Φ) Z R dr r T t ! + 2 π Z dt " ( ∂ r Φ) Z R dr r T t ! + 2 π Z dt " − ( ∂ t Φ) Z ∞ R dr r − η − Mr ! + 2 π Z dt (cid:20) ( ∂ r Φ) (cid:18)Z ∞ R dr r (cid:18) − η − Mr (cid:19)(cid:19)(cid:21) , (33)where the limits of the integration w.r.t. r for S Φ ) in arefrom 0 to R and for S Φ ) out are from R to ∞ . T t → E ( upto leading order ) in the incipient limit(from eq n (30)). Thus,lim R → R GM T t − η − Mr = R − η R − Mr − η r − M rR = 0 .T t vanishes faster than (cid:0) − η − Mr (cid:1) in the limit R → R GM . Thus, for the coefficients of − ( ∂ t Φ) , the T t termdominates. For the coefficients of ( ∂ r Φ) , the dominatingterm is (cid:0) − η − Mr (cid:1) . Therefore, in the incipient limit, S Φ → π Z dt " − E Z R GM dr r ( ∂ t Φ) + 2 π Z dt (cid:20)Z ∞ R GM dr r (cid:18) − η − Mr (cid:19) ( ∂ r Φ) (cid:21) . (34) A. Mode expansion for Φ For Φ, one can easily check from its equation ofmotion, that is ∂ Φ = 0, that for r < R ( t ) (from S Φ ) in ), ∂ Φ ∂r + 2 r ∂ Φ ∂r = 1 T t ∂ Φ ∂t − T tt T t ∂ Φ ∂t , (35)where T t , along with its powers and derivatives w.r.t. t ,are independent of r .Similarly, for r > R ( t ), we have (from S Φ ) out )), (cid:18) − η − Mr (cid:19) ∂ Φ ∂r + 2( r − M ) r (cid:18) − η − Mr (cid:19) ∂ Φ ∂r = ∂ Φ ∂t . (36)From eq n (35) and eq n (36), we notice the following modeexpansion (due to the separability property satisfied bythe above equations),Φ( r, t ) = X k a k ( t ) f k ( r ) , (37)where a k ( t ) are the modes and f k ( r ) are some real-valuedsmooth functions of r.Now S Φ in terms of modes a k is (as R → R GM ), S Φ = Z dt X k,k ′ (cid:20) − E da k dt A kk ′ da k ′ dt + 12 a k B kk ′ a k ′ (cid:21) , (38)where A kk ′ and B kk ′ are defined as, A kk ′ := 4 π Z R GM dr r f k ( r ) f k ′ ( r ) , (39) B kk ′ := 4 π Z ∞ R GM dr r (cid:18) − η − Mr (cid:19) f ′ k ( r ) f ′ k ′ ( r ) , (40)where, f ′ k ( r ) := ∂f k ( r ) ∂r . Observe that, both A kk ′ and B kk ′ are independent of r and t (as no R ( t ) appears inthem).Following [7], we define the conjugate momenta, π k s (to the modes a k ) as, π k := ∂ L Φ ∂ ˙ a k ≡ − i ∂∂a k , (41)where ˙ a k := da k dt , and from eq n (38), we have (with L Φ defined as the Langrangian for Φ), L Φ = X k,k ′ (cid:20) − E ˙ a k A kk ′ ˙ a k ′ dt + 12 a k B kk ′ a k ′ (cid:21) , (42) L Φ = − E ( ˙a T A ˙a ) + 12 ( a T Ba ) , (43)where A and B are non-singular linear operators, suchthat, A kk ′ ∈ A and B kk ′ ∈ B in the chosen bases, say { ˙ a k } and { a k } respectively. In the basis { a k } , a is acolumn vector, such that, a k ∈ a . One can similarlyexpress ˙a in the basis { ˙ a k } .For the Hamiltonian of Φ, H Φ , we obtain, H Φ = X k π k ˙ a k − L Φ = X k,k ′ (cid:20) E ˙ a k A kk ′ ˙ a k ′ dt + 12 a k B kk ′ a k ′ (cid:21) (44)= E Π T A − Π ) + 12 ( a T Ba ) , (45)where Π is a column vector, such that, π k ∈ Π , in achosen basis say { π k } and A − is the inverse of A . Following arguments similar to [7], note that, B and A are real and symmetric infinite dimensionalmatrices and hence are self-adjoint. Therefore, bythe Spectral Theorem , there exists orthonormal basesof position space and momentum space consisting ofrespective eigenvectors of B and A . Furthermore, allthe corresponding eigenvalues are real. Say, for instance,the bases for position space and momentum space are { b k } and { ˙ b k } respectively (where, each b k is a linearcombination of the original basis vectors a k and each ˙ b k is a linear combination of the original basis vectors ˙ a k ). B. The Schr¨odinger-like wave equation
If we study the equation for one eigenvector b ∈{ b k } , then our conclusion will be the same for all othereigenvectors (see [16]). So, we shall solve the Schr¨odinger-like wave equation for a wave functional Ψ( { b k } , t ), whichby the above assumption of equivalence is now a wavefunction ψ ( b, t ). Therefore, ψ ( b, t ) ≡ Ψ( { b k } , t ). Hence,using eq n (43), we write the Schr¨odinger-like wave equa-tion (for a single eigenvector b ) as, (cid:20) − (cid:18) − η − MR (cid:19) α ∂ ∂b + 12 βb (cid:21) ψ ( b, t ) = i ∂ψ ( b, t ) ∂t , (46)where, α and β are the eigenvalues of A and B respec-tively.We define a new time parameter, e η := Z t dt (cid:18) − η − MR (cid:19) (47) leading to, ∂ e η∂t = E, (48)and write eq n (46) as (cid:20) − α ∂ ∂b + β E b (cid:21) ψ ( b, e η ) = i ∂ψ ( b, e η ) ∂ e η . (49) Eq n (49) becomes, (cid:20) − α ∂ ∂b + 12 αω ( e η ) b (cid:21) ψ ( b, e η ) = i ∂ψ ( b, e η ) ∂ e η , (50)where, we have chosen to set e η ( t = 0) = 0 and ω isdefined as, ω ( e η ) := (cid:18) βα (cid:19) E =: ω E . (51)We observe that, eq n (50) is a time dependentSimple Harmonic Oscillator (SHO) equation with ω ( e η )as the frequency.In the incipient limit (using eq n (31) and eq n (29)), dEdt = 2 MR dRdt = ǫ MǫR dRdt ≈ − ǫE R GM R GM = − ǫER GM . (52)Integrating eq n (52) w.r.t. t one gets (as R → R GM ), E = 1 − η − MR ( t ) ∼ e − ǫt/R GM . (53)From eq n (53) we see that at late times, 1 − η − MR ( t ) ∼ e − ǫt/R GM . Since we are interested in the incipient limit,that is, in late times of the collapsing process, we canchoose the behaviour of R ( t ) at early times as per ourconvenience for simplifying the calculations. Therefore,we choose both past and future behaviour of R ( t ) to bestationary. Hence, we can take the metric to be flat forall t ∈ ( −∞ , t f for the collapse and then allowing t f → ∞ , thus going into the continual collapse case tillthe black hole is formed. Therefore, E = , f or t ∈ ( −∞ , e − ǫt/R GM , f or t ∈ (0 , t f ) e − ǫt f /R GM , f or t ∈ ( t f , ∞ ) . (54)The above choice of R ( t ) may seem quite problematic as dRdt is discontinuous at 0 and t f , but references [16, 20]show that the particle production by the collapsing shellhappens in the range, 0 < t < t f and in the t f → ∞ regime, all the solutions obtained are smooth andwell-behaved. Therefore with the above considerations,the wavefunction ψ would capture the whole collapsescenario, and in the limit of t f → ∞ or R ( t ) → R GM ,black hole formation sets in.We note that, at early times, t ∈ ( −∞ , J − (past null infinity) are[ ? ] just the sim-ple harmonic oscillator ground states (this can be seenfrom the form of eq n (50), which with e η = 0, is the SHOequation). Thus, ψ ( b ) := ψ ( b, e η = 0) = (cid:16) αω π (cid:17) / e − mω b / , (55)where ψ ( b ) represents the SHO ground state and { ψ n ( b ) } will denote the SHO basis states at early times. Eq n (55) suggests that ω defined in eq n (51) canbe identified with the ground state frequency associatedwith the initial vacuum state.With the help of eq n (55), the exact solution to eq n (50) is, ψ ( b, e η ) = e iχ ( e η ) (cid:20) απζ (cid:21) / exp (cid:20) i (cid:18) ζ e η ζ + iζ (cid:19) αb (cid:21) , (56)where ζ is the solution of the equation, ζ e η e η + ω ( e η ) ζ = 1 ζ , (57)with the following initial conditions, ζ (0) = 1 √ ω , (58) ζ e η (0) = 0 , (59) and, χ ( e η ) is given by, χ ( e η ) := − Z e η d e η ′ ζ ( e η ′ ) . (60)Differential equations of the form eq n (50) have beenextensively studied in [21–25].From eq n s (51), (53) and (54), we have the following(for t > ω ( e η ( t )) = e ǫt/ R GM ω . (61)Using eq n (48) and eq n (61),Ω( t ) = (cid:18) ∂ e η∂t (cid:12)(cid:12)(cid:12)(cid:12) t> (cid:19) ω ( e η ) = e − ǫt/ R GM ω , (62)where Ω( t ) is defined to be the frequency w.r.t. time t .We note that at early times ( J − ), the states arethe initial vacuum states of SHO, as described by ψ ( b ).With time, the frequency of the states Ω( t ) evolve, asper eq n (62), and more and more states get excited. Fi-nally, when the observer measures them at J + (futurenull infinity), that is for some t ∈ ( t f , ∞ ), we have thefollowing mode expansion (following the evolution n theSchr¨odinger picture[26]), ψ ( b, t ) = X n c n ( t ) φ n ( b ) , (63)where c n ( t ) represent the probability amplitudes. Thefinal SHO states { φ n ( b ) } are with the frequency Ω f =Ω( t f ) (a constant), given by, φ n ( b ) = (cid:18) α Ω f π (cid:19) / e − α Ω f b / √ n n ! H n ( p α Ω f b ) , (64)where H n are the Hermite polynomials. Observe that,Ω( t f ) = e − ǫt f / R GM ω ; (65) c n can be computed from an overlap integral as (see ap-pendix), c n = ( ( − n/ e iχ (Ω f ζ ) / q P (cid:0) − P (cid:1) n/ n − √ n ! , f or even n , f or odd n, (66)where P := 1 − i Ω f (cid:16) ζ e η ζ + iζ (cid:17) . V. UNITARITYA. Density Matrix approach
We shall now calculate the density matrices, ˆ ρ i andˆ ρ f , for the initial ( J − ) and the final ( J + ) states respec-tively. ˆ ρ i and ˆ ρ f can be written as (see [12, 13]),ˆ ρ i = X m,n l m l ∗ n | ψ m ih ψ n | , (67)ˆ ρ f = X m,n c m c ∗ n | φ m ih φ n | , (68)where, l n and c n are the probability amplitudes appear-ing in the intial and final states respectively.Since initially the system was in the SHO eigen-states { ψ n } and the wavefunction was normalized, weobtain, T r (ˆ ρ i ) = 1 . (69)From eq n (66), with λ := (cid:12)(cid:12) − P (cid:12)(cid:12) , we have, T r (ˆ ρ f ) = X even n | c n | = 2 p Ω f ζ | P | X even n ( n − n ! λ n = 2 p Ω f ζ | P | √ − λ = 2 p Ω f ζ | P | q − (cid:12)(cid:12) − P (cid:12)(cid:12) . (70) P has been computed explicitly and used in eq n (70) toobtain (see appendix), T r (ˆ ρ f ) = 1 . (71) eq n (71) shows that the necessary condition for the uni-tary evolution of states holds. For the sufficient condi-tion, we compute T r (ˆ ρ f ). From eq n (68),ˆ ρ f = X m,n c m c ∗ n | φ m ih φ n | leading to, ˆ ρ f = X m,n c m c ∗ n | φ m ih φ n | ! X i,j c i c ∗ j | φ i ih φ j | = X m,n,i,j c m c i c ∗ n c ∗ j | φ m ih φ n | φ i ih φ j | = X m,n,j c m c ∗ j | c n | | φ m ih φ j | = X m,j c m c ∗ j | φ m ih φ j | X n | c n | ! = X m,j c m c ∗ j | φ m ih φ j | as, X n | c n | ! = 1 by eq n (71) ! = ˆ ρ f . (72)Therefore, by eq n (72) we get, T r (ˆ ρ f ) = T r (ˆ ρ f ) = 1 . (73) Analytically, we have shown that the idempotency ofthe final density matrix holds indicating a pure quantumstate to pure quantum state transition. B. Conservation of Probability approach
The probability current 4-vector J µ can be definedas, J = | ψ | , (74) ~J = 12 αi [ ψ ∗ ~ ∇ ψ − ψ ~ ∇ ψ ∗ ] . (75)As b is an eigenfunction of B , it is independent of thespatial coordinates x i . Thus, we conclude that ~J = ~ ∇ µ J µ = ∂ | ψ | ∂t obs . (76)Writing t obs = t (for the observer’s time coordinate), wehave (from equation (48)), ∇ µ J µ = ∂ | ψ | ∂t = ∂ | ψ | ∂ e η ∂ e η∂t = E ∂ | ψ | ∂ e ηF or, R → R GM , ∇ µ J µ = 0 ( as, E →
0) (77)Again analytically, we have shown from ( eq n (77)), thatprobability is conserved in the system, in the incipientlimit of black hole formation. VI. CONCLUSION
We showed analytically and comprehensivelythat the black hole radiation, for a spacetime whichis not asymptotically flat, is processed with a unitaryevolution. This is confirmed from the density matrixconsideration as well as from the conservation of proba-bility consideration.The Schr¨odinger-like wave equations that weused bear resemblance to a minisuperspace versionof Wheeler-DeWitt equations[8]. Interestingly, suchequations have a present resurgence, in the context ofissues concerned with unitarity[27–29].Saini and Stojkovic[13] had showed that blackhole radiation is processed with a unitary evolution,for a Schwarzchild black hole, from the density matrixconsideration. However, they had achieved their conclu-sion through numerical estimates. We worked with amore general, metric, the global monopole metric, andresults for the Schwarzchild case is recovered from thisby putting η = 0.The computations on unitarity are all in theincipient limit, the limit of black hole formation. Hence,it does not really take care of the complete black holeevaporation process. However, if unitarity is preservedin this limit, it should be valid at every instant of time.In saying this, we further emphasize that, what wehave shown in this paper is that black hole radiationis unitary in a not asymptotically flat backgroundspacetime. The present result of unitarity in space-time that is not asymptotically flat, together with theresults obtained in [7] that the unitarity is preservedfor a Reissner-Nordstrom metric which is not globallyhyperbolic, settles the issue of conservation of unitarityin spherically symmetric, static (1+3) dimensionalspacetimes of the form as given in eq n (6). It alsodeserves mention that similar results for a Schwarzschildbackround obtained in [13] numerically, can be arrivedat as a special case from both of these more involvedexamples. So the results are quite consistent, and shouldhave significant implications towards the resolution ofthe information loss paradox. ACKNOWLEDGEMENTS
AD would like to thank the Department of Scienceand Technology, Government of India for providing theINSPIRE-SHE scholarship which helped immensely inthis research work.
APPENDIXAlternate motivation for S shell In this section, we present a different action than S shell . We shall call it S new . We shall further show thatin the incipient limit it will give rise to H shell and Π shell .Since we know that the shell behaves like a relativisticparticle, we define the new action to be, S new = − Z dτ M = − Z dT MT τ , = − πσ Z dT R (cid:20) − πσR q − R T (cid:21) + Z dT η R q − R T , = − πσ Z dt R "r E − − EE R t − πσR r E − R t E + Z dt η R r E − R t E . (78) Then, L new = − πσR "r E − − EE R t − πσR r E − R t E + η R r E − R t E , (79)Π new = ∂ L new ∂R t = 4 πσR R t √ E " − E p E − (1 − E ) R t − πσR p E − R t − η R R t √ E p E − R t , (80) H new = Π new R t − L new = 4 πσE / R " p E − (1 − E ) R t − πσR p E − R t − η R E / p E − R t . (81)In the incipient limit we have, H new = 4 πE / µR p E − R t , (82)Π new = 4 πµR R t √ E p E − R t , (83)where, µ := σ (cid:16) − πσR GM − η πσR GM (cid:17) . Observe thatthese are the exact same equations we had obtained be-fore in this incipient limit. Computation of c n Now let us compute the c n ’s explicitly. We knowthat, ψ ( b, t ) = X n c n ( t ) φ n ( b ) , (84)From the overlap integral we have, c n = Z db φ ∗ n ψ = (cid:18) α Ω f π ζ (cid:19) / e iχ ( e η ) √ n n ! Z db exp (cid:20) − α Ω f b i (cid:18) ζ e η ζ + iζ (cid:19) αb (cid:21) H n (cid:16)p α Ω f b (cid:17) , (85)= (cid:18) f π ζ (cid:19) / e iχ ( e η ) √ n n ! Z dx exp (cid:20) − x x i Ω f (cid:18) ζ e η ζ + iζ (cid:19)(cid:21) H n ( x )( with, x := p α Ω f b ) , = (cid:18) f π ζ (cid:19) / e iχ ( e η ) √ n n ! Z dx e − P x / H n ( x ) (cid:18) with, P := 1 − i Ω f (cid:18) ζ e η ζ + iζ (cid:19)(cid:19) (86)= (cid:18) f π ζ (cid:19) / e iχ ( e η ) √ n n ! I n (cid:18) with, I n := Z dx e − P x / H n ( x ) (cid:19) . (87)To compute I n , let us consider the following generatingfunction for the H n ( x ), J ( z ) = Z dx e − P x / e − z +2 zx = r πP e − z (1 − /P ) ,since, e − z +2 zx = ∞ X n =0 z n n ! H n ( x ) , Z dx e − P x / H n ( x ) = d n dz n J ( z ) (cid:12)(cid:12)(cid:12)(cid:12) z =0 ,thus, I n = r πP (cid:18) − P (cid:19) n/ H n (0) ,as, H n (0) = ( ( − n/ √ n n ! ( n − √ n ! , f or even n , f or odd n. Thus we have, c n = ( ( − n/ e iχ (Ω f ζ ) / q P (cid:0) − P (cid:1) n/ n − √ n ! , f or even n , f or odd n. (88) Explicit computation of
T r (ˆ ρ f ) We know that,
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