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Collapse of the wavefunction, the information paradox and backreaction

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Eur. Phys. J. C :556 Regular Article - Theoretical Physics Collapse of the wavefunction, the inforation paradox and backreaction Sujoy K. Modak 1,2,a, Daniel
Eur. Phys. J. C :556 Regular Article - Theoretical Physics Collapse of the wavefunction, the inforation paradox and backreaction Sujoy K. Modak 1,2,a, Daniel Sudarsky 3,4,b 1 Facultad de Ciencias, CUICBAS,Universidad de Colia, CP 2845 Colia, Mexico 2 KEK Theory Center, High Energy Accelerator Research Organization KEK, Tsukuba, Ibaraki 35-81, Japan 3 Instituto de Ciencias Nucleares, Universidad Nacional Autónoa de México, Apartado Postal 7-543, 451 Distrito Federal, Mexico 4 Departent of Philosophy, New York University, New York, NY 13, USA Received: 9 Noveber 217 / Accepted: 27 June 218 / Published online: 6 July 218 The Authors 218 Abstract We consider the black hole inforation proble within the context of collapse theories in a schee that allows the incorporation of the backreaction to the Hawking radiation. We explore the issue in a setting of the two diensional version of black hole evaporation known as the Russo-Susskind-Thorlacius odel. We suarize the general ideas based on the seiclassical version of Einstein s equations and then discuss specific odifications that are required in the context of collapse theories when applied to this odel. Contents 1 Introduction Seiclassical CGHS odel with backreaction Review of the RST Model Equations of otion Solving seiclassical equations Dynaical case of black hole foration and evaporation Quantization on RST Incorporating collapse echanis in the RST odel Collapse of the quantu state and Einstein s seiclassical equations CSL theory Gravitationally induced collapse rate Spacetie foliation CSL evolution and the odified back reaction Recovering the theral Hawking radiation Discussion Appendix A: The renoralized energy-oentu tensor a e-ail: b e-ail: 9 Appendix B: The backreacted spacetie with GRW type collapse References Introduction The black hole inforation question has been with us for ore than four decades, ever since Hawking s discovery that black holes eit theral radiation and therefore evaporate, leading either to their coplete disappearance or to a sall Planck ass scale renant [1]. The basic issue can be best illustrated by considering an initial setting where an essentially flat space-tie in which a single quantu field is in a pure quantu state of relative high excitation corresponding to a spatial concentration of energy, that, when left on its own will, collapses gravitationally leading to the foration of a black hole. As the black hole evaporates, the energy that was initially localized in a sall spatial region, ends up in the for of Hawking radiation that, for uch of this evolution ust be alost exactly theral [2]. The point, of course, is that if this process ends with the coplete evaporation of the black hole or even if a sall renant is left the overwheling ajority of the initial energy content would correspond to a state of the quantu field possessing alost no inforation except that encoded in the radiation s teperature and it is very difficult to reconcile this with the general expectation that in any quantu process the initial and final states should be related by a unitary transforation, and thus all inforation encoded in the initial state ust be soehow present in the final one. The issue, of course, is far ore subtle and the above should be taken as only a approxiate account of the proble. There have been any attepts to deal with this conundru, with none of the resulting in a truly satisfactory res- 556 Page 2 of 2 Eur. Phys. J. C :556 olution of the proble [3,4]. In fact there is even a debate as to the extent to which this is indeed a proble or as soe people like to call it a paradox [5,6]. In previous works [7 9] we helped to clarify the basis of the dispute, and proposed a schee where the resolution of the issue is tied to a proposal to address another lingering proble of theoretical physics: the so called easureent proble [1] in quantu theory. The first task was dealt with [7 9] by noting that the true proble arises only when one takes the point of view that a satisfactory theory of quantu gravity ust resolve the singularity, and that, as a result of such resolution, there will be no need to introduce a new boundary of space-tie in the region where the classical black hole singularity stood. Otherwise the proble can be fully understood by noting that the region in the black hole exterior, at late ties corresponding to those where ost of the energy takes the for of theral Hawking radiation, contains no Cauchy hypersurfaces and thus any attept to provide a full description of the quantu state in ters of the quantu field odes in the black hole exterior is siply wrongheaded. In order to provide a coplete description of such late quantu state one needs to include the odes that register on the part of the Cauchy hypersurface that goes deep into the black hole interior, in particular one that treads close to the singularity, as described in detail in [5]. The second task was carried out by considering the application of one particular dynaical collapse theory designed to address the easureent proble in quantu theory, to a siple two diensional black hole odel known as the Callan-Giddings-Harvey-Stroinger CGHS odel [11]. The proposal was then to associate to the intrinsic breakdown of unitary evolution, which is typical of these dynaical collapse theories [12 18], which were developed to deal with the easureent proble in standard quantu echanics all the inforation loss that takes place during the foration and subsequent Hawking evaporation of the black hole. The first concrete treatents along this line are [19,2]. In those works we noted that the treatent at that point left various issues to be worked out, and that substantial progress in those would be required before the proposal could be considered to be fully satisfactory. Aong these issues that two ost pressing ones are the replaceent of the treatent presented, by one that is fully consistent with relativistic covariance, and to show how the iportant question of back reaction due to Hawking radiation on the spacetie and viceversa can be incorporated in such a schee i.e., in presence of the collapse of wavefunction type setting. A first step in this direction was accoplished in [21] where the siple two diensional proble is considered using a relativistic version of collapse theories. The objective of the present work is to continue the research path initiated in [19 22] and explore an exaple where the reaining issue of backreaction in the setting of collapse theories. For this we will again consider a two diensional black hole odel known due to Russo-Susskind- Thorlacius RST [23,24] which presents a solution of the seiclassical Einstein equation in 2D. The paper is organized as follows: We start by reviewing the sei-classical CGHS odel in Sect. 2 and then ove to the RST odel in Sect. 3 and discuss the quantization of atter fields on RST in Sect. 4. It is iportant to ephasize that all those sections contain nothing novel and represent just a review, which is however needed in order to ake sense of what follows. Section 5 contains necessary ingredients for the adaptation of collapse of the wave-function in a general setting as well as for the specific case of 2D RST odel. In Sect. 6 we discuss the Hawking radiation and the inforation paradox. There are two appendices A and B discussing iportant issues related with the renoralization of the energy-oentu tensor and a specific exaple of the treatent of back reaction of the space-tie etric and dilaton field to a discrete collapse of the wave-function. 2 Seiclassical CGHS odel with backreaction A natural way to incorporate backreaction effects of a quantu field on the background geoetry is to odify the Einstein equations where the expectation value of the stress tensor is included on the right hand side of the equations of otion E.O.M, so that, G ab = T Class ab + T ab, 1 where G ab is the Einstein tensor of the classical etric, Tab Class represents the energy-oentu tensor of whatever atter is being described at the classical level, and T ab is the renoralized expectation value of the energy-oentu tensor of the atter fields that are treated quantu echanically, evaluated in the corresponding quantu state of such fields. In the two diensional CGHS odel with a single freely propagating assless scalar field, characterized by the action [11]: S CGHS = 1 2π d 2 x g[e 2φ R + 4 φ f 2 ], where is a constant. The dilaton field φ is usually treated classically, and the scalar field f is treated quantu echanically. Working in the conforal gauge with null coordinates the etric is described by: ds 2 = e 2ρ dx + dx. 3 2 Eur. Phys. J. C :556 Page 3 of The seiclassical E.O.M involve now the energy-oentu contribution fro the classical dilaton field and the cosological constant as well as the part coing fro the expectation value of the quantu field f. Those take now the following for with respect to the appropriate variation entioned on the left ρ : e 2φ 2 x + x φ 4 x +φ x φ 2 e 2ρ T x + x =, 4 g ±± : e 2φ 2 x 2 ± φ + 4 x ±ρ x ±φ + T x ± x ± =, 5 φ : 2e 2φ x + x ρ φ + x + x e 2φ + 2 e 2ρ φ =. 6 Note that even though the unperturbed etric has g ±± = the general variations do not share this property in these coordinates, and their consideration results in Eq. 5. In order to solve the above differential equations, it is necessary to calculate the expectation value of various coponents of the renoralized energy-oentu tensor in a particular state of the quantu field denoted by. The state is usually taken to be the in vacuu state. We review this calculation, fro a slightly different perspective that the usual one, in Appendix A. One interesting feature of the Eqs. 4 6, is that one can write down a foral action, given by S = S CGHS + S P, 7 where S P is the Polyakov effective action [25] S P = h d 2 x gr 1 R, 8 96π and whose variation leads to the sae set of Eqs This is because, in the effective action foralis, the expectation value of the renoralized energy-oentu tensor corresponding to the quantu field f ˆ, is given by the derivative of the Polyakov ter 2 δs P g δg ab = ψ T ab ψ = h [ a ξ b ξ 2 a b ξ + g ab 2R 1 ] 48π 2 cξ c ξ, 9 where ξ is an auxiliary scalar field constrained to obey the equation ξ = R and ψ is the state of the quantu scalar field. We note that the freedo in the choice of the quantu state, correspond, in the effective action treatent, to the freedo of choice of boundary conditions for the solution ξ. We refer the interested reader to [26] for ore discussions on the effective action foralis. This is a very delicate issue that can generate serious confusion in our approach, and care ust be taken to ensure one goes back and forth fro the two foralis in a consistent anner. We will have to do so in particular if we want to consider the changes in the quantu states of the fˆ field for which the treatent without the effective action is ore convenient and at the sae tie consider explicitly solving for the spacetie etric and dilaton field for which the reliance on the effective action is ost suitable. We will explore this issue in detail in section V.A. and appendix B. In the eanwhile we return to the review of the original RST odel. It has been found difficult to solve the set of differential Eqs. 4 6 without a nuerical handle. The advantage of using effective action foralis is that it allows one to play with the E.O.M without going into a rigorous quantu field theory calculation, and indeed that approach was subsequently exploited in Russo-Susskind-Thorlacius RST [23], where a local ter was added in 7, allowing one to solve the new seiclassical equations analytically. We review this odel in the next section. 3 Review of the RST Model In the RST odel a local ter is added to the CGHS and Polyakov actions such that the coplete action, with a scalar field f, which are however, treated via an effective ter, is given by [23,26] S = S CGHS + S P + S RST, 1 where S CGHS is given by 2, S P is 8 and the local ter is S RST = h d 2 x g φ R, 11 48π which adds a direct coupling between the dilaton and the Ricci scalar. Again, the above schee should be seen as effectively characterizing a odel where the Polyakov ter replaces quantu effects of the assless scalar field. 3.1 Equations of otion Next we present the equations of otion that result fro the odel s action 1. Varying 1 with respect to g ab we obtain e [ 2 2φ a b φ + 1 ] 2 g ab 4 φ φ = h a ξ b ξ 12 48π g ab ξ 2 N h 24π a b ξ g ab ξ h 24π a b φ g ab φ, 13 556 Page 4 of 2 Eur. Phys. J. C :556 On the other hand the E. O. M. for φ is: [ ] e 2φ 2R φ φ h 24π R =. 14 In 2D conforal gauge 1 the above equations take the following for with respect to the appropriate variations indicated below: ρ : e 2φ 2 x + x φ 4 x +φ x φ 2 e 2ρ + h 12π x + x ρ + h 24π x + x φ =, 15 g ±± : e 2φ h π x ± φ + 4 x ±ρ x ±φ + T x ±± =, 16 φ : 2e 2φ x + x ρ φ + x + x e 2φ + 2 e 2ρ φ + h 24π x + x ρ =, 17 where the expectation values of the energy-oentu tensor are those found in 97 and 98. The only choice yet to ipleent is the selection of a particular state to solve the above set of equations. An iportant feature of these equations is that if one uses 15 and 17 one still finds the free field equation : x + x ρ φ =, 18 which is typical of the CGHS odel without backreaction. This feature is what in this odel facilitates the finding of a specific solution for the spacetie geoetry in presence of backreaction. 3.2 Solving seiclassical equations It is convenient to introduce the new variables [23] 2 φ + e 2φ, 19 χ ρ 2 φ + e 2φ, 2 h 12π. where = In these variables take the following for x + x = 2 e 2 χ, 21 x + x χ = 2 e 2 χ, 22 x ±χ x ±χ + x 2 ± χ + x ± x ± 4x + :T ±2 x ± x ± : in =, 23 1 One needs to replace ξ by solving ξ = R, = 4e 2ρ x + x and R = 8e 2ρ x + x ρ. whereas the free field Eq. 18 becoes x + x χ =. 24 The above equation allows us to write χ = W+ x + + W x, 25 2 where W + and W are arbitrary functions of x + and x respectively. Then 21 and 22 becoe x + x χ = 2 e W ++W 26 and x + x = 2 e W ++W. 27 In the RST odel one restricts oneself to the choice = χ, i.e., W + = = W and then the solution is found to be = χ = D 2 x + x Fx+ + Gx, 28 where D is an arbitrary constant and the functions Fx +, Gx can be found by substituting 28in23 and integrating x + x + Fx + = dx + dx + 1 4x +2 + :T x + x +x + : in, 29 x x Gx = dx dx 1 4x 2 + :T x x x : in. 3 Now using these expressions one can find particular solutions depending on the choice for the state of the quantu field. Specifically, we will focus on those ones which correspond to solutions representing the foration and evolution of black holes. For future convenience let us note that in new variables the Ricci scalar, R = 8e 2ρ + ρ turns out to be R = 8e 2ρ + χ 2 +, 31 where d dφ = 2 2 e 2φ Dynaical case of black hole foration and evaporation Now we will consider the case where a sharp pulse of atter fors a black hole. This pulse can be well approxiated by Eur. Phys. J. C :556 Page 5 of choosing to be a coherent state build on top of the in vacuu, corresponding to a wave packet peaked around a particular classical value. In particular, we only need a left oving pulse to create a black hole, therefore, we can chose = Pulse L in R where Pulse L = Ô in L with Ô a suitable creation operator for the sharply peaked wave packet. In this case the state dependent functions turns out to be :T x + x + : in = x + δ x + x +, 33 :T x x : in =. 34 This choice when used in 28 leads to the following solution χ = = 2 x + x 4 ln 2 x + x x + x + x + θ x + x This solution contains a singularity. To see this we refer to Eqs. 31 and 32. The singularity occurs when = and 32 givese 2φ s = 4. As we have restricted ourselves to the case ρ = φ, one can use the relation 2, to find the value of s = 4 1 ln 4 associated with the singularity. Therefore the location of the singularity turns out to be: 2 x + x + 3 x + 4 ln 2 x + x + = 1 ln This singularity is hidden by the apparent horizon located at + φ = which is given by 2 x + x + 3 x + = The apparent horizon and the singularity eet at x + s = x+ 4 x s = 3 x + 4 e 1, e 4 The physical eaning of this point is that it could be interpreted as the end point of the black hole evaporation [23]. This is confired by the fact that at x = xs the solution 35 with x + x + takes the for χ = = 2 x + x s + 3 x + 4 ln 2 x + xs + 3 x +, 4 which is nothing but the vacuu configuration, coonly known as the linear dilaton vacuu L. D. V.. Fig. 1 RST spacetie in Kruskal coordinates where a black hole is created due to the atter collapse and evaporated due to the Hawking effect Thus the spacetie for x + x + s χ = = 2 x + x + [ + 4 ln 3 x + 4 ln 2 x + x + 2 x + is given by ] θ xs x x + 3 x + θ x xs. 41 Now we can construct the coplete spacetie etric so that for x + x s + one has 35 and for x + x s + the appropriate expression of the etric is given by 41. We show the overall spacetie in Kruskal coordinates in Fig. 1. Note that there are two different linear dilaton vacuus 1 for x + x + and 2 for x+ x s +, x xs.thesel.d.v.s are glued together with the black hole regions Reg. I and II by the pulse of atter for L. D. V.-I and radiation for L. D. V.-II. The pulse of the atter at x + = x + carries positive energy and fors the black hole, whereas, the pulse of the radiation at x = xs, x+ x s + carries negative energy associated with the singularity and usually called thunderpop. The space-tie etric, although, is continuous at those gluing points but clearly it is not differentiable. 556 Page 6 of 2 Eur. Phys. J. C :556 Fig. 2 Penrose diagra for RST spacetie where a black hole is created due to the atter collapse and evaporated due to the Hawking effect The Penrose diagra of the RST spacetie can be constructed following Ref. [27]. The asyptotic past and future regions are specified with respect to the Minkowskian coordinates. First, in the asyptotic past, where the etric corresponds to a linear dilaton vacuu, so that, ds 2 = one can use the coordinates, 1 2 x + x, y + = 1 ln x+ y +, 42 y = 1 ln x 43 to write ds 2 = dy + dy, where y + is introduced to set the origin of the coordinates y +. There is also a subtle issue regarding the extensions of the linear dilaton vacuu regions. The expression for the for the Ricci scalar 31 iplies that at = it diverges. Since even in the linear dilaton vacuu regions this value can be reached one has to put soe boundary conditions so that such an artifact does not show up in the solution [23]. In the literature this issue is bypassed by putting reflecting boundary conditions there. The conforal/penrose diagra for the RST odel is given by Fig. 2. For a discussion about the boundary conditions to ake finite curvature in crit see [24]. Let us now consider the asyptotic structure of the spacetie. Particularly we want to check the asyptotic behavior and viability of defining J +. Let us focus on the etric in Reg. I and Reg. II inside and outside the black hole apparent horizon, given by χ = = 2 x + x + 3 x + 4 ln 2 x + x If we want to find out the physical etric coefficient in the conforal gauge 3.1 we need to use the relations 2 and 2. By using those we obtain the following equation 2 ρ + e 2ρ =, 45 whose solution deterines ρ. However, practically this equation is not invertible and therefore we cannot find an exact solution for ρ fro the known expression of. But we can perfor certain analysis to unfold the asyptotic behavior. First, we check the staticity of the etric by expressing 44 in the standard Schwarzschild like coordinates t, r. These are related with Kruskal x ± coordinates in the following way x + = 1 e t+ 2 1 ln e 2 r, 46 x = 3 x + 1 e t 2 1 ln e 2 r. 47 Using these relations the etric function takes the following for χ = = 1 e 2 σ k 2 σ 4 ln x + e t σ, 48 σ = 1 2 ln e 2 r. 49 Now note that for t = const. and r i.e., at spatial infinity i the last ter which is tie dependent vanishes altogether. This akes tie independent and therefore any solution for ρ in 45 will be tie independent.
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