Eur. Phys. J. C (2018) 78:556https://doi.org/10.1140/epjc/s10052-018-6032-5

Regular Article - Theoretical Physics

Collapse of the wavefunction, the information paradox andbackreaction

Sujoy K. Modak1,2,a, Daniel Sudarsky3,4,b

1 Facultad de Ciencias, CUICBAS,Universidad de Colima, CP 28045 Colima, Mexico2 KEK Theory Center, High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan3 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, 04510 Distrito Federal, Mexico4 Department of Philosophy, New York University, New York, NY 10003, USA

Received: 9 November 2017 / Accepted: 27 June 2018 / Published online: 6 July 2018© The Author(s) 2018

Abstract We consider the black hole information prob-lem within the context of collapse theories in a scheme thatallows the incorporation of the backreaction to the Hawk-ing radiation. We explore the issue in a setting of the twodimensional version of black hole evaporation known as theRusso-Susskind-Thorlacius model. We summarize the gen-eral ideas based on the semiclassical version of Einstein’sequations and then discuss specific modifications that arerequired in the context of collapse theories when applied tothis model.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . 12 Semiclassical CGHS model with backreaction . . . 23 Review of the RST Model . . . . . . . . . . . . . . 3

3.1 Equations of motion . . . . . . . . . . . . . . . 33.2 Solving semiclassical equations . . . . . . . . 43.3 Dynamical case of black hole formation and

evaporation . . . . . . . . . . . . . . . . . . . 44 Quantization on RST . . . . . . . . . . . . . . . . . 75 Incorporating collapse mechanism in the RST model 8

5.1 Collapse of the quantum state and Einstein’ssemiclassical equations . . . . . . . . . . . . . 8

5.2 CSL theory . . . . . . . . . . . . . . . . . . . 115.3 Gravitationally induced collapse rate . . . . . . 115.4 Spacetime foliation . . . . . . . . . . . . . . . 125.5 CSL evolution and the modified back reaction . 12

6 Recovering the thermal Hawking radiation . . . . . 157 Discussion . . . . . . . . . . . . . . . . . . . . . . 168 Appendix A: The renormalized energy-momentum

tensor . . . . . . . . . . . . . . . . . . . . . . . . . 16

a e-mail: smodak@ucol.mxb e-mail: sudarsky@nucleares.unam.mx

9 Appendix B: The backreacted spacetime with GRWtype collapse . . . . . . . . . . . . . . . . . . . . . 18

References . . . . . . . . . . . . . . . . . . . . . . . . 19

1 Introduction

The black hole information question has been with us formore than four decades, ever since Hawking’s discovery thatblack holes emit thermal radiation and therefore evaporate,leading either to their complete disappearance or to a smallPlanck mass scale remnant [1]. The basic issue can be bestillustrated by considering an initial setting where an essen-tially flat space-time in which a single quantum field is in apure quantum state of relative high excitation correspondingto a spatial concentration of energy, that, when left on its ownwill, collapses gravitationally leading to the formation of ablack hole. As the black hole evaporates, the energy that wasinitially localized in a small spatial region, ends up in theform of Hawking radiation that, for much of this evolutionmust be almost exactly thermal [2]. The point, of course, isthat if this process ends with the complete evaporation of theblack hole (or even if a small remnant is left) the overwhelm-ing majority of the initial energy content would correspond toa state of the quantum field possessing almost no information(except that encoded in the radiation’s temperature) and it isvery difficult to reconcile this with the general expectationthat in any quantum process the initial and final states shouldbe related by a unitary transformation, and thus all informa-tion encoded in the initial state must be somehow present inthe final one. The issue, of course, is far more subtle and theabove should be taken as only a approximate account of theproblem.

There have been many attempts to deal with this conun-drum, with none of them resulting in a truly satisfactory res-

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olution of the problem [3,4]. In fact there is even a debateas to the extent to which this is indeed a problem or as somepeople like to call it a “paradox” [5,6].

In previous works [7–9] we helped to clarify the basis ofthe dispute, and proposed a scheme where the resolution ofthe issue is tied to a proposal to address another lingeringproblem of theoretical physics: the so called measurementproblem [10] in quantum theory.

The first task was dealt with [7–9] by noting that the trueproblem arises only when one takes the point of view that asatisfactory theory of quantum gravity must resolve the sin-gularity, and that, as a result of such resolution, there will beno need to introduce a new boundary of space-time in theregion where the classical black hole singularity stood. Oth-erwise the problem can be fully understood by noting that theregion in the black hole exterior, at late times correspondingto those where most of the energy takes the form of thermalHawking radiation, contains no Cauchy hypersurfaces andthus any attempt to provide a full description of the quan-tum state in terms of the quantum field modes in the blackhole exterior is simply wrongheaded. In order to provide acomplete description of such late quantum state one needsto include the modes that register on the part of the Cauchyhypersurface that goes deep into the black hole interior, inparticular one that treads close to the singularity, as describedin detail in [5].

The second task was carried out by considering the appli-cation of one particular dynamical collapse theory designedto address the measurement problem in quantum theory, toa simple two dimensional black hole model known as theCallan-Giddings-Harvey-Strominger (CGHS) model [11].The proposal was then to associate to the intrinsic breakdownof unitary evolution, which is typical of these dynamical col-lapse theories [12–18], (which were developed to deal withthe measurement problem in standard quantum mechanics)all the information loss that takes place during the formationand subsequent Hawking evaporation of the black hole. Thefirst concrete treatments along this line are [19,20].

In those works we noted that the treatment at that point leftvarious issues to be worked out, and that substantial progressin those would be required before the proposal could be con-sidered to be fully satisfactory. Among these issues that twomost pressing ones are the replacement of the treatment pre-sented, by one that is fully consistent with relativistic covari-ance, and to show how the important question of back reac-tion due to Hawking radiation on the spacetime and vice-versa can be incorporated in such a scheme (i.e., in presenceof the collapse of wavefunction type setting). A first step inthis direction was accomplished in [21] where the simpletwo dimensional problem is considered using a relativisticversion of collapse theories.

The objective of the present work is to continue theresearch path initiated in [19–22] and explore an example

where the remaining issue of backreaction in the settingof collapse theories. For this we will again consider a twodimensional black hole model known due to Russo-Susskind-Thorlacius (RST) [23,24] which presents a solution of thesemiclassical (Einstein) equation in 2D.

The paper is organized as follows: We start by reviewingthe semi-classical CGHS model in Sect. 2 and then moveto the RST model in Sect. 3 and discuss the quantization ofmatter fields on RST in Sect. 4. It is important to emphasizethat all those sections contain nothing novel and representjust a review, which is however needed in order to makesense of what follows. Section 5 contains necessary ingre-dients for the adaptation of collapse of the wave-function ina general setting as well as for the specific case of 2D RSTmodel. In Sect. 6 we discuss the Hawking radiation and theinformation paradox. There are two appendices (A and B)discussing important issues related with the renormalizationof the energy-momentum tensor and a specific example ofthe treatment of back reaction of the space-time metric (anddilaton field) to a discrete collapse of the wave-function.

2 Semiclassical CGHS model with backreaction

A natural way to incorporate backreaction effects of a quan-tum field on the background geometry is to modify the Ein-stein equations where the expectation value of the stress ten-sor is included on the right hand side of the equations ofmotion (E.O.M), so that,

Gab = TClassab + 〈�|Tab|�〉, (1)

where Gab is the Einstein tensor of the classical metric,TClassab represents the energy-momentum tensor of what-

ever matter is being described at the classical level, and〈�|Tab|�〉 is the renormalized expectation value of theenergy-momentum tensor of the matter fields that are treatedquantum mechanically, evaluated in the corresponding quan-tum state |�〉 of such fields.

In the two dimensional CGHS model with a single freelypropagating massless scalar field, characterized by the action[11]:

SCGHS = 1

2π

∫d2x

√−g[e−2φ(R + 4(∇φ)2 + 4�2) − (∇ f )2],(2)

where � is a constant. The dilaton field φ is usually treatedclassically, and the scalar field f is treated quantum mechan-ically.

Working in the conformal gauge with null coordinates themetric is described by:

ds2 = −e2ρdx+dx−. (3)

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The semiclassical E.O.M involve now the energy-momentumcontribution from the classical dilaton field and the cosmo-logical constant as well as the part coming from the expecta-tion value of the quantum field f . Those take now the follow-ing form (with respect to the appropriate variation mentionedon the left)

ρ : e−2φ(

2∂x+∂x−φ − 4∂x+φ∂x−φ − �2e2ρ)

−〈�|Tx+x−|�〉 = 0, (4)

g±± : e−2φ(−2∂2

x±φ + 4∂x±ρ∂x±φ)

+〈�|Tx±x±|�〉 = 0, (5)

φ : 2e−2φ∂x+∂x−(ρ − φ) + ∂x+∂x−e−2φ

+�2e2(ρ−φ) = 0. (6)

Note that even though the unperturbed metric has g±± =0 the general variations do not share this property in thesecoordinates, and their consideration results in Eq. (5).

In order to solve the above differential equations, it isnecessary to calculate the expectation value of various com-ponents of the renormalized energy-momentum tensor in aparticular state of the quantum field denoted by |�〉. Thestate is usually taken to be the “in vacuum state”. We reviewthis calculation, from a slightly different perspective that theusual one, in Appendix A.

One interesting feature of the Eqs. (4)–(6), is that one canwrite down a formal action, given by

S = SCGHS + SP , (7)

where SP is the Polyakov effective action [25]

SP = − h

96π

∫d2x

√−gR1

� R, (8)

and whose variation leads to the same set of Eqs. (4)–(6). Thisis because, in the effective action formalism, the expectationvalue of the renormalized energy-momentum tensor corre-sponding to the quantum field f , is given by the derivativeof the Polyakov term

− 2√−g

δSPδgab

= 〈ψ |Tab|ψ〉

= − h

48π

[∇aξ∇bξ − 2∇a∇bξ + gab

(2R − 1

2∇cξ∇cξ

)],

(9)

where ξ is an auxiliary scalar field constrained to obey theequation �ξ = R and |ψ〉 is the state of the quantum (scalar)field. We note that the freedom in the choice of the quantumstate, correspond, in the effective action treatment, to thefreedom of choice of boundary conditions for the solution ξ .We refer the interested reader to [26] for more discussionson the effective action formalism.

This is a very delicate issue that can generate serious con-fusion in our approach, and care must be taken to ensure one

goes back and forth from the two formalism in a consistentmanner. We will have to do so in particular if we want toconsider the changes in the quantum states of the f field (forwhich the treatment without the effective action is more con-venient) and at the same time consider explicitly solving forthe spacetime metric and dilaton field (for which the relianceon the effective action is most suitable). We will explore thisissue in detail in section V.A. and appendix B. In the mean-while we return to the review of the original RST model.

It has been found difficult to solve the set of differentialEqs. (4)–(6) without a numerical handle. The advantage ofusing “effective action formalism” is that it allows one toplay with the E.O.M without going into a rigorous quantumfield theory calculation, and indeed that approach was subse-quently exploited in Russo-Susskind-Thorlacius (RST) [23],where a local term was added in (7), allowing one to solvethe new semiclassical equations analytically. We review thismodel in the next section.

3 Review of the RST Model

In the RST model a local term is added to the CGHS andPolyakov actions such that the complete action, with a scalarfield f , which are however, treated via an effective term , isgiven by [23,26]

S = SCGHS + SP + SRST , (10)

where SCGHS is given by (2), SP is (8) and the local term is

SRST = − h

48π

∫d2x

√−g φR, (11)

which adds a direct coupling between the dilaton and theRicci scalar. Again, the above scheme should be seen aseffectively characterizing a model where the Polyakov termreplaces quantum effects of the massless scalar field.

3.1 Equations of motion

Next we present the equations of motion that result from themodel’s action (10).Varying (10) with respect to gab we obtain

e−2φ

[−2∇a∇bφ + 1

2gab(−4(∇φ)2 + 4∇2φ + 4�2)

]

(12)

= h

48π

(∇aξ∇bξ − 1

2gab(∇ξ)2

)

− Nh

24π(∇a∇bξ − gab�ξ) − h

24π(∇a∇bφ − gab�φ),

(13)

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On the other hand the E. O. M. for φ is:

e−2φ[−2R − 8�2 + 8(∇φ)2 − 8∇2φ

]− h

24πR = 0.

(14)

In 2D conformal gauge 1 the above equations take the follow-ing form (with respect to the appropriate variations indicatedbelow):

ρ : e−2φ(

2∂x+∂x−φ − 4∂x+φ∂x−φ − �2e2ρ)

+ h

12π∂x+∂x−ρ + h

24π∂x+∂x−φ = 0, (15)

g±± :(e−2φ − h

48π

)(−2∂2

x±φ + 4∂x±ρ∂x±φ)

+〈�|Tx±±|�〉 = 0, (16)

φ : 2e−2φ∂x+∂x−(ρ − φ) + ∂x+∂x−e−2φ

+�2e2(ρ−φ) + h

24π∂x+∂x−ρ = 0, (17)

where the expectation values of the energy-momentum tensorare those found in (97) and (98). The only choice yet toimplement is the selection of a particular state |�〉 to solvethe above set of equations. An important feature of theseequations is that if one uses (15) and (17) one still finds the“free field equation”:

∂x+∂x−(ρ − φ) = 0, (18)

which is typical of the CGHS model without backreaction.This feature is what in this model facilitates the finding of aspecific solution for the spacetime geometry in presence ofbackreaction.

3.2 Solving semiclassical equations

It is convenient to introduce the new variables [23]

� ≡√

κ

2φ + e−2φ

√κ

, (19)

χ ≡ √κρ −

√κ

2φ + e−2φ

√κ

, (20)

where κ = h12π

.In these variables (15)–(17) take the following form

∂x+∂x−� = − �2

√κe

2√κ(χ−�)

, (21)

∂x+∂x−χ = − �2

√κe

2√κ(χ−�)

, (22)

−∂x±χ∂x±χ + √κ∂2

x±χ + ∂x±�∂x±�

− κ

4x±2 + 〈�| : Tx±x± :in |�〉 = 0, (23)

1 One needs to replace ξ by solving �ξ = R, � = −4e−2ρ∂x+∂x− andR = 8e−2ρ∂x+∂x−ρ.

whereas the free field Eq. (18) becomes

∂x+∂x−(χ − �) = 0. (24)

The above equation allows us to write

χ − � =√

κ

2

(W+(x+) + W−(x−)

), (25)

where W+ and W− are arbitrary functions of x+ and x−respectively. Then (21) and (22) become

∂x+∂x−χ = − �2

√κeW++W− (26)

and

∂x+∂x−� = − �2

√κeW++W− . (27)

In the RST model one restricts oneself to the choice � = χ ,i.e., W+ = 0 = W− and then the solution is found to be

� = χ = D − �2x+x−√

κ− F(x+) + G(x−)√

κ, (28)

where D is an arbitrary constant and the functions F(x+),G(x−) can be found by substituting (28) in (23) and integrat-ing

F(x+) =∫ x+

dx ′+∫ x ′+

dx ′′+

×(

− 1

4x+2 + 〈�| : Tx+x+(x ′′+) :in |�〉)

, (29)

G(x−) =∫ x−

dx ′−∫ x ′−

dx ′′−

×(

− 1

4x−2 + 〈�| : Tx−x−(x ′′−) :in |�〉)

. (30)

Now using these expressions one can find particular solu-tions depending on the choice for the state of the quantumfield (|�〉). Specifically, we will focus on those ones whichcorrespond to solutions representing the formation and evo-lution of black holes.

For future convenience let us note that in new variablesthe Ricci scalar, R = 8e−2ρ∂+∂−ρ turns out to be

R = 8e−2ρ

�′

(∂+∂−χ − �′′

�′2 ∂+�∂−�

), (31)

where

�′ ≡ d�

dφ=

√κ

2− 2√

κe−2φ. (32)

3.3 Dynamical case of black hole formation andevaporation

Now we will consider the case where a sharp pulse of matterforms a black hole. This pulse can be well approximated by

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choosing |�〉 to be a coherent state build on top of the invacuum, corresponding to a wave packet peaked around aparticular classical value. In particular, we only need a leftmoving pulse to create a black hole, therefore, we can chose|�〉 = |Pulse〉L ⊗ |0in〉R where |Pulse〉L = O|0in〉L withO a suitable creation operator for the sharply peaked wavepacket. In this case the state dependent functions turns out tobe

〈�| : Tx+x+ :in |�〉 = m

�x+0

δ(x+ − x+

0

), (33)

〈�| : Tx−x− :in |�〉 = 0. (34)

This choice when used in (28) leads to the following solution

χ = � = −�2x+x−√

κ−

√κ

4ln

(−�2x+x−)

− m

�√

κx+0

(x+ − x+

0

)θ

(x+ − x+

0

). (35)

This solution contains a singularity. To see this we refer toEqs. (31) and (32). The singularity occurs when �′ = 0 and(32) gives e−2φs = κ

4 . As we have restricted ourselves tothe case ρ = φ, one can use the relation (20), to find the

value of �s =√

κ

4 (1 − ln κ4 ) associated with the singularity.

Therefore the location of the singularity turns out to be:

− �2x+√

κ

(x− + m

�3x+0

)−

√κ

4ln

(−�2x+x−)

+ m

�√

κ

=√

κ

4

(1 − ln

κ

4

). (36)

This singularity is hidden by the apparent horizon located at∂+φ = 0 which is given by

− �2x+(x− + m

�3x+0

)= κ

4. (37)

The apparent horizon and the singularity meet at

x+s = κ�x+

0

4m(e

4mκ� − 1), (38)

x−s = − m

�3x+0

1(1 − e− 4m

κ�

) . (39)

The physical meaning of this point is that it could be inter-preted as the end point of the black hole evaporation [23].This is confirmed by the fact that at x− = x−

s the solution(35) with x+ > x+

0 takes the form

χ = � = −�2x+√

κ

(x−s + m

�3x+0

)

−√

κ

4ln

(−�2x+

(x−s + m

�3x+0

)), (40)

which is nothing but the vacuum configuration, commonlyknown as the linear dilaton vacuum (L. D. V.).

Fig. 1 RST spacetime in Kruskal coordinates where a black hole iscreated due to the matter collapse and evaporated due to the Hawkingeffect

Thus the spacetime for x+ > x+s is given by

χ = � = −�2x+√

κ

(x− + m

�3x+0

)

+[−

√κ

4ln

(−�2x+x−)

+ m

�√

κ

]θ

(x−s − x−)

−√

κ

4ln

(−�2x+

(x− + m

�3x+0

))θ

(x− − x−

s

).

(41)

Now we can construct the complete spacetime metric sothat for x+ < x+

s one has (35) and for x+ ≥ x+s the appro-

priate expression of the metric is given by (41). We showthe overall spacetime in Kruskal coordinates in Fig. 1. Notethat there are two different linear dilaton vacuums—(1) forx+ < x+

0 and (2) for x+ ≥ x+s , x− ≥ x−

s . These L. D. V. sare glued together with the black hole regions (Reg. I and II)by the pulse of matter (for L. D. V.-I) and radiation (for L. D.V.-II). The pulse of the matter at x+ = x+

0 carries positiveenergy and forms the black hole, whereas, the pulse of theradiation at x− = x−

s , x+ ≥ x+s carries negative energy

associated with the singularity and usually called thunder-pop. The space-time metric, although, is continuous at thosegluing points but clearly it is not differentiable.

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Fig. 2 Penrose diagram for RST spacetime where a black hole is cre-ated due to the matter collapse and evaporated due to the Hawking effect

The Penrose diagram of the RST spacetime can be con-structed following Ref. [27]. The asymptotic past and futureregions are specified with respect to the Minkowskian coor-dinates. First, in the asymptotic past, where the metric cor-responds to a linear dilaton vacuum, so that, ds2 = 1

�2x+x− ,one can use the coordinates,

y+ = 1

�ln(�x+) − y+

0 , (42)

y− = − 1

�ln(−�x−) (43)

to write ds2 = −dy+dy−, where y+0 is introduced to set the

origin of the coordinates y+.There is also a subtle issue regarding the extensions of

the linear dilaton vacuum regions. The expression for thefor the Ricci scalar (31) implies that at �′ = 0 it diverges.Since even in the linear dilaton vacuum regions this valuecan be reached one has to put some boundary conditions sothat such an artifact does not show up in the solution [23].In the literature this issue is bypassed by putting reflectingboundary conditions there. The conformal/Penrose diagramfor the RST model is given by Fig. 2. For a discussion aboutthe boundary conditions to make finite curvature in �cri t see[24].

Let us now consider the asymptotic structure of the space-time. Particularly we want to check the asymptotic behaviorand viability of defining J +. Let us focus on the metric inReg. I and Reg. II (inside and outside the black hole apparenthorizon), given by

χ = � = −�2x+√

κ

(x− + m

�3x+0

)

−√

κ

4ln(−�2x+x−) + m

�√

κ. (44)

If we want to find out the physical metric coefficient in theconformal gauge (3.1) we need to use the relations (20) and(20). By using those we obtain the following equation√

κ

2ρ + e−2ρ

√κ

− � = 0, (45)

whose solution determines ρ. However, practically this equa-tion is not invertible and therefore we cannot find an exactsolution for ρ from the known expression of �. But we canperform certain analysis to unfold the asymptotic behavior.First, we check the staticity of the metric by expressing (44)in the standard Schwarzschild like coordinates (t, r ). Theseare related with Kruskal (x±) coordinates in the followingway

x+ = 1

�e�

(t+ 1

2�ln

(e2�r−m

�

)), (46)

x− = − m

�3x+0

− 1

�e−�

(t− 1

2�ln

(e2�r−m

�

)). (47)

Using these relations the metric function takes the followingform

χ = � = 1√κe2�σ

−√

κ

2�σ −

√k

4ln

(1 + m

�2x+0

e�(t−σ)

), (48)

σ = 1

2�ln

(e2�r−m

�

). (49)

Now note that for t = const. and r → ∞ (i.e., at spatialinfinity i0) the last term which is time dependent vanishesaltogether. This makes � time independent and thereforeany solution for ρ in (45) will be time independent. Thisguaranties the staticity of the metric at i0. Furthermore, asone moves up in J +

R where t → ∞, σ → ∞ keepingt − σ = finite, the time-dependent term becomes least dom-inant and one can approximately define an asymptotic time-like Killing vector field nearJ +

R . We shall use this asymptoticKilling time to be associated with the physical observers asusually done in the RST model. To talk about the asymptoticflatness let us focus on (48). Near J +

R as σ, t, x+ all tends toinfinity, from (44) and (45) and, by comparing the dominating

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coefficients of κ , we have limx+→∞ e2ρ = 1−�2x+(x−+ m

�3x+0),

which is the asymptotic form of the LDV-II. Thus in entireJ +R one has a flat metric. This is essential for discussions

related with Hawking radiation and information paradox.Up to this point we have presented the standard RST model

which incorporates the backreaction of the spacetime to theHawking evaporation, corresponding to the usual quantumevolution of the matter field, and which indicates that infor-mation is either lost at the singularity or somehow conveyedto the exterior in some unknown fashion encoded in the verylate outward flux of energy known as the thunderpop.

However, in the context of our proposal, the state of thequantum field will be affected by the modified quantum evo-lution prescribed by the modified dynamics involving spon-taneous collapse of the wavefunction. The specific model thatwe shall consider is given by the Continuous-Spontaneous-Collapse (CSL) theory, a brief introduction of which will beprovided in section V. In light of this modification the back-reaction of the quantum matter on the spacetime metric willbe modified as well. We will discuss this in the reminder ofthis work.

4 Quantization on RST

In order to discuss in some detail the modifications, broughtin by the CSL version of quantum theory, it is convenientto describe the two relevant constructions of the quantumtheory of the scalar field f on RST spacetime.

We note, however that the power of this model residesin the fact that one is able to obtain the whole space-time,including the back reaction of the space-time metric to thequantum energy momentum stress tensor, before one actuallydiscusses the construction of the quantum field theory for thematter field f . This, in turn, allows for that construction tobe carried out in the appropriate spacetime which alreadyincludes backreaction.

Thus one might think that one can safely ignore this part ofthe treatment and just go ahead with the usage of the effectiveaction and never actually carry out the explicit constructionof the quantum field. This would be correct except that in ourapproach we will need to further consider the changes in thestate of the quantum field brought about by the dynamicalcollapse theory. Doing that requires the quantum field theoryfor f and we proceed with this now.

We can express the scalar field both in the in region andthe out region. The out region consists of the modes havingsupport in the inside and outside of the event horizon. Sincein the discussion related with Hawking radiation one onlyconsiders the modes in the “right moving” sector we shallonly use them in various expressions here. In the in regionone can write

f =∑ω

(aωuω + a†

ωu∗ω

), (50)

where the in vacuum is defined by aω|0〉in = 0. In the outregion one has

f =∑ω′

(bω′vω′ + b†

ω′v∗ω′

)+

∑ω′

(cω′ vω′ + c†

ω′ v∗ω′

), (51)

where the modes with and without tildes respectively havesupports inside and outside the horizon. Vacuum within theFock spaces interior and exterior to black hole are respec-tively cω′ |0〉int = 0 and bω′ |0〉ext = 0. Using Bogolyubovcoefficients one can express the creation or annihilation oper-ators of the in region in terms of a linear combination ofcreation and annihilation operators (defined in either Fockspaces) of the out region. Specifically, there are two sets ofBogolyubov coefficients connecting the in region to blackhole interior and exterior regions (only for the right movingsector of the scalar field modes).

The field modes appearing in above expressions of f arerespectively given by

uω = 1√2ω

e−iωy+, (52)

vω′ = 1√2ω′ e

−iω′σ−, (53)

where y+ is defined in (42) and σ− = − 1�

ln(

�x−+πM/�κ

�x−s +πM/�κ

)[27]. Whereas the mode in the interior of the black hole canbe defined from the expression of vω, in the following way

vω′(y−) = v∗−ω′(−y−), (54)

with y− given by (43). One should also note that in thecontinuous basis (using ω,ω′) modes are in fact not ortho-normalized and to talk about particle creation in a particularquantum number one has to introduce a discrete basis to makesense of the particle definition. Moreover, the discrete basisallows for a relatively simple characterization of localizationof the modes which in the continuous basis would require theuse of wave packets. We will rely on the discrete basis in ourwork. It is easily obtained from the continuous counterpartby defining the modes,

v jn = 1√ε

∫ ( j+1)ε

jεdωe2πωn/εvoutω (55)

with n and j ≥ 0 are integer numbers. These wave pack-ets are peaked about uout = 2πn/ε with width 2π/ε. TheBogolyubov coefficients between the uω modes and vω′ (andits complex conjugate) modes turns out to satisfy the follow-ing relationship in the late time limit [27]

αωω′ ≈ e−πω/�βωω′ . (56)

It is also possible to express the Bogolyubov coefficients inthe discrete basis just by using the transformation (55).

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A standard calculation from the above expression imme-diately leads us to the Hawking radiation. Also, as it is wellknown, with this relation one can express the in vacuum(defined in the Fock space at J −

R ) as a superposition of theparticle states in the joint basis in the out region defined insideand outside the event horizon (i.e., interior to black hole andat J +

R ) [28,29]. Later, we need to consider the initial stateof the quantum field which will be evolved using the CSLevolution, with the CSL term taken as an interaction hamil-tonian. We will take this state, to be the in vacuum for rightmoving sector and a pulse for the left moving sector, as dis-cussed in subsection 3.3. Using Bogolyubov transformationwe can express

|�i 〉 = |0in〉R ⊗ |Pulse〉L ,

= N∑F

CF |F〉int ⊗ |F〉ext ⊗ |Pulse〉L , (57)

where the in vacuum for right moving sector is expressedas a linear combination of |F〉 states interior and exteriorto the black hole. Each of these states is characterized by aset of excitation numbers corresponding to various modes.Entanglement between the interior and exterior modes, in theabove state, implies that corresponding to a |F〉ext there isa single |F〉int with the same particle excitation number butwith negative energy/wave-vector. It is this initial quantumstate (57) that had led to the formation of the black hole. Weshould also state that using the late time characterization, theusual thermal nature of the radiation can be seen in the factthat CF = e−πEF/� , where EF = ∑

F ωnj Fnj is the totallate time energy of the |F〉 state.

Now that we have characterized the initial quantum statewe move to the next section to incorporate the CSL evolutionon this state.

5 Incorporating collapse mechanism in the RST model

We will be addressing the question of the fate of informa-tion in the evaporation of the black hole by considering amodified version of quantum theory proposed to address the“measurement problem” of the standard quantum theory. Thespecific version that we will be using is the CSL theory pro-posed in [16]. This theory is a continuous version of the socalled Ghirardi-Rimini-Weber (GRW) theory [14,15] wherethe unitary evolution is accompanied by occasional discretecollapse of the wavefunction that happens for a very smallamount of time. As these theories were developed in the con-text of many particle non-relativistic quantum mechanics wewill need to adapt it to the present context involving quantumfield theory in curved spacetime2.

2 Ideally one would finally need to use a fully relativistic version ofcollapse theories such as [30–34] as was done in [21] for the non-backreacting case.

5.1 Collapse of the quantum state and Einstein’ssemiclassical equations

One of the main difficulties that must be dealt with whenconsidering a semi-classical treatment of gravitation in thecontext of modified quantum theories involving a collapse ofthe quantum state is the fact that Einstein’s equations simplywill not hold when the energy-momentum tensor is replacedby its quantum expectation value and the quantum state ofthe matter fields undergoes a stochastic collapse. In fact thisis connected to the intrinsic problem of treating gravitationin a classical language, and it is expected to be fully solvedonly in the context of a complete theory of quantum grav-ity. This is illustrated in [35] where it is argued that semi-classical gravity is either inconsistent when we assume thestate of quantum matter undergoes some sort of collapse, or,it is simply at odds with experiments when we do not makesuch an assumption. Unfortunately, we do not have at thispoint a fully workable quantum gravity theory to explorethese issues. Furthermore, even when, and if we eventuallyget our hands on such a theory, in which as expected theclassical spacetime metric is replaced by some more funda-mental set of quantum variables, the recovery the standardnotions of classical spacetime, a task that seems unavoidableif we want to be able to describe such things a formation andevaporation of black holes, can be expected to be a rathercomplex process, that, moreover, might only work in someapproximate sense.

These considerations lead us to adopt the followingapproach: We will consider semi-classical gravity as anapproximate and effective description, valid in limited cir-cumstances, of a more fundamental theory of quantum grav-ity including matter fields. This seems to be, in fact, the posi-tion that would be adopted in this regard by a good segment ofour community working on these questions, but we describeour posture explicitly in order to avoid misunderstandings.The analogy to keep in mind is the hydrodynamic descrip-tion of fluids, which as we know, works rather well in a largeclass of circumstances, but does not represent the behaviorof the truly fundamental degrees of freedom involved. Weknow, that at a deeper level fluids are made of moleculesthat interact in a complex manner, and that, there are onlycertain aspects of their collective behavior describable in thehydrodynamic language. The semi-classical Einstein equa-tions therefore cannot be trusted to hold precisely at the fun-damental microscopic level, just like the Navier-Stokes equa-tions, which cannot be thought to represent the true behaviorof the fluid molecules, but must be taken as holding only inan approximate sense. Moreover, just as the hydrodynamiccharacterization of a fluid is known to break down ratherdramatically in certain circumstances, such as when a oceanwave breaks at the beach, we can also expect that Einstein’ssemiclassical equations should become invalid under some

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situations. We thus must take the situations associated withthe collapse of the quantum state of matter fields to be oneof such circumstance.

Now we must consider a way out to formally implementsuch ideas in order to be able to further explore them andtheir consequences, and in particular to apply them to theproblem at hand.

Below, we discuss the issue, first in the realistic setting ofa 3+1 dimensional spacetime and second in the particularlysimple situation concerning the RST model in 1+1 dimen-sions.

Here we will describe an approach initially proposed in[36] in the context of inflationary cosmology and the problemof emergence of the primordial inhomogeneities [37]. Thestaring point is the notion of Semi classical Self- consistentConfigurations (SSC), defined for the case of a single matterfield (for simplicity), as follows:Definition The set {gab(x), ϕ(x), π(x),H, |ξ 〉 ∈ H} repre-sents a SSC if and only if ϕ(x), π(x) and H correspond ato quantum field theory constructed over a space-time withmetric gab(x) and the state |ξ 〉 in H is such that

Gab[g(x)] = 8πG〈ξ |Tab[g(x), ϕ(x)]|ξ 〉, (58)

where 〈ξ |Tμν[g(x), ϕ(x)]|ξ 〉 stands for the renormalizedenergy momentum tensor of the quantum matter field ϕ(x)(in the state |ξ 〉) constructed with the space-time metric gab.This corresponds, in a sense, to the general relativistic ver-sion of Schrödinger–Newton equation [38–43]. The point ofthis setting is to ensure a consistency between the descriptionof the quantum matter and that of gravitation by consideringtheir influences on each other3.

To this setting we want to add an extra element: the col-lapse of the wave function. That is, besides the unitary evo-lution describing the change in time of the state of a quan-tum field, we consider situations such as those envisaged indiscrete collapse theories such as GRW, where there will be,sometimes, spontaneous jumps in the quantum state. We willconsider the situation when we are given a dynamical col-lapse theory that, given an SSC (dubbed as SSC1 and consid-ered to describe the situation before the collapse), specifies,a space-like hypersurface �Collapse (perhaps through somestochastic recipe that we can overlook at this point) on whichthe collapse of the quantum state takes place, and also thefinal quantum state (generally, again in a similar stochasticmanner). The remaining task is now twofold – (i) to describethe construction of the new SSC, (to be called SSC2) thatwill be taken to describe the situation after the collapse and,

3 We consider this as a scheme to be used when all matter is treatedquantum mechanically, but one might add the contribution to the energy-momentum tensor from any fields which are treated classically, such asthe dilaton in the RST model.

(ii) to join the two SSC’s in a manner to generate somethingto call, in its closest sense, a “global space-time”.

In order to have a picture in our mind we can think ofthe above scheme as something akin to an effective descrip-tion of a fluid involving a situation where “instantaneously”,the Navier-Stokes equations do not hold. Let us think, forinstance, once more, about an ocean wave breaking at thebeach. The situation just before the wave breaks should bedescribable to a very good approximation by the Navier-Stokes equations, and should the situation, well after the wavebreaks and the water surface becomes rather smooth again.The particular regime where the breakdown of the wave istaking place, and its immediate aftermath, will, of course,not be the one where fluid description and the Navier Stokesequations can be expected to provide an accurate picture.This is because such regime involves large amounts of energyand information flowing between the macroscopic degrees offreedom that are well characterized in the fluid language, andthe underlying molecular degrees of freedom, (accompaniedby other complex process including such things as incorpora-tion of air molecules into the water, the mechanism by whichocean water is oxigenated). All these represent aspects wouldthat have been “averaged out” in passing from the molecularto the fluid description. If we now take the limit in which thiscomplex non-fluid characterization is essential to be instan-taneous, then we will be in possession of two regimes that aresusceptible to a fluid description using Navier-Stokes equa-tions, joined through an instantaneous collapse of the wave-function (to be identified with the space-like hypersurface�Collapse) where the said equations cannot hold.

The specific proposal for the effective characterization ofthese situations, that we will have in mind is based on the3 + 1 decomposition of the space-time associated with thehypersurface �Collapse and inspired by the application ofthese ideas in the specific case treated in [36].

The spacetime metric of the SSC1 defined on �Collapse

has the induced spatial metric h(1)ab , the unit normal na(1) and

the extrinsic curvature Kab(1). The fact that the SSC1 cor-

responds to a semi-classical solution of Einstein’s equationsthen ensures that the Hamiltonian and momentum constrainsare satisfied on �Collapse viewed as a hypersurface embeddedin the space-time of the SSC1.

The task at hand now, would be to specify the quantumstate and initial conditions for the construction of the SSCto the future of the collapse hypersurface, assuming that weare given the expectation of the energy-momentum tensorfor the SSC2, i.e., assuming that the collapse theory allowsus to determine 〈ξ |Tbc[g(x), ϕ(x)]|ξ〉(2). That is, as we areconsidering the treatment for the case where the collapse istaken to be instantaneous, we will assume that the collapsetheory, in our case CSL, determines (by some scheme involv-ing stochastic components) the quantum state at the hyper-

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surface lying “just to the future of the collapse hypesurface”(assuming that the initial, pre-collapse, state was given) on�Collpase, and given that, one would need is to constructthe complete, corresponding SSC2. The first thing would beto obtain suitable initial data for the space-time metric ofSSC2. That is we need to find h(2)

ab and the extrinsic cur-

vature Kab(2)satisfying the Hamiltonian and momentum

constrains, involving the expectation value of the energy-momentum tensor corresponding to the SSC2.

We have previously used the ansatz of taking h(2)ab = h(1)

ab

on �Collapse and finding a suitable expression K (2)ab , on

�Collapse determined so as to ensure the Hamiltonian andmomentum constraints for the SSC2 are satisfied.

This determination of the initial data for the SSC2 metricwill be referred as Step 2. Next one would have to carry out thecompletion of the SSC2, namely to specify the constructionof the quantum field theory, identify the quantum state andshow how the full space-time metric can be determined.

The completion of the process would then involve find-ing the state of the new Hilbert space such that its expec-tation value of the (renormalized) energy-momentum tensorcorresponds to the values given in Step 1 above. One mustthen ensure that with the integration of the space-time metricand the mode functions given those initial data can be doneeffectively. An explicit example showing the completion ofthis process in the inflationary cosmological context repre-senting a single mode perturbation with specific co-movingwave vector was presented in [36]. There one can see that ingeneral the tasks involved are rather non-trivial.

The point is that this scheme will allow the construction ofa space-time made of two four dimensional regions charac-terized by the SSC constructions and joined along a collapsehypersurface where Einstein’s equations do not hold.

In the case of the RST model, in 1 + 1 dimensions, thesituation is substantially simplified by the fact that the space-time is two dimensional and thus the Einstein tensor vanishesidentically. The violation of the equations of motion during acollapse of the quantum state are thus a bit more subtle and,at the same time easier to deal with.

Again, one can make use of the fact that the most generalspacetime (smooth) metric can be written as:

ds2 = −e2ρdx+dx−, (59)

where ρ is a smooth function. Thus both the space-time met-ric for SSC1 and SSC2 can be put in this form. The issue isnow joining these two space-times along �Collapse. Thus wecan regard, as a complete generalized space-time, the resultobtained by this gluing procedure where the price we havepaid by doing so is that now the function ρ will not nec-essarily be a smooth function. Note that something similarhappens when the linear dilaton vacuum region is glued withthe black hole space-time in CGHS or RST model.

Also the remaining SSC 2 construction, i.e., the specifica-tion of the corresponding Hilbert space and identification ofthe quantum state needs to be dealt with. The specificationof the mode functions that determine the Fock space, can betaken to be done at the level of initial data on �Collapse, andthis can be achieved by making use of the fact that the generalsolution of the Klein-Gordon equation for a massless scalarfield f on any space-time metric of the form (59) is of theform

f = f+(x+) + f−(x−). (60)

As we mentioned before, the idea is then to take take themodes used in the SSC1 Fock space as providing initial datafor the SSC2 Hilbert space construction. However it is easyto see that using such procedure on �Collapse with functionssatisfying (60) both in the regions to the past and future of�Collapse corresponds, simply, to functions satisfying (60)in all our generalized space-time ( i.e. after incorporating thediscontinuity in the derivative of ρ). All this will work fineas long as the spatial-metric is continuous along �Collapse

and the space-time metric is continuous as one crosses it.The point is that we can take the modes of the SSC2 and

the corresponding Fock space to be the same as those of theSSC1. This represents a very nice simplification provided bythe two dimensional nature of the situation of interest.

The last step would be the identification of the state in theHilbert space of SSC 2 with the appropriate expectation val-ues of the renormalized energy-momentum tensor, but againas the Hilbert space of two SSC’s are the same we can takethis as being provided by the collapse theory in Step 1.

This shows that the program described at the start of thissection, regarding a single instantaneous collapse of the stateof the quantum matter field, can be easily implemented inthe present situation. The details of the general applicationof that scheme for situations in higher dimensions is an openproblem.

The final issue that needs to be considered before apply-ing collapse theories to the problem at hand, has to do withgeneralizing the above procedure, from the case of a singleinstantaneous collapse to a continuous collapse theory. Thatis, if instead of considering a single collapse taking place onthe space-like hypersurface �Collapse we want to considera foliation of space-time by space-like “collapse” hypersur-faces �Collapse(τ ) parametrized by a real valued time func-tion τ on the space-time manifold, and a theory like CSL tobe described in the next section, describing the change in thequantum state, as one “passes form one hypersurface to theother”.

We can deal with this, first for the case of a finite interval[τstart , τend ] in the time function by considering a partition ofthe corresponding interval τ0 = τstart , τ1......τi ...τN = τend ,performing the procedure describing the individual discrete

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collapse at each step i in the partition, and eventually takingthe limit N → ∞.

With these considerations in hand we now turn to thedescription of the specific type of collapse theory will beusing in this work.

5.2 CSL theory

We first consider the theory in the non-relativistic quantummechanical setting in which it was first postulated. The CSLtheory is generically described in terms of two equations. Thefirst is a modified version of Schrödinger equation, whosegeneral solution, in the case of a single non-relativistic par-ticle is:

|ψ, t〉CSLw = T e− ∫ t

0 dt ′[i H+ 1

4λ[w(t ′)−2λ A]2

]|ψ, 0〉, (61)

where T is the time-ordering operator. A is a smeared posi-tion operator for the particle. w(t) is a random classical func-tion of time, of white noise type, whose probability is givenby the second equation, the Probability Rule:

PDw(t) ≡ 〈ψ, t |ψ, t〉t∏

ti=0

dw(ti )√2πλ/dt

. (62)

The state vector norm evolves dynamically (not equal 1), soEq. (62) says that the state vectors with largest norm are mostprobable. It is straightforward to see that the total probabilityis 1, that is∫

PDw(t) = 〈ψ, 0|ψ, 0〉 = 1. (63)

The way we will incorporate the CSL modifications inour situation is by relying on the formalism of interactionpicture version of quantum evolution, where the free part ofthe evolution corresponding to the standard quantum evolu-tion will be absorbed in the construction of the quantum fieldoperators, while the interaction corresponding to the CSLmodifications will be used to evolve the quantum states. Onemore thing that needs to be modified is related to the fact thatthe quantum field is a system with infinite number of degreesof freedom (DOF), and thus instead of a single operator Aand a single stochastic function w(t) we will have an infiniteset of those labeled by the index α. Thus in our case we willhave:

|�, t〉CSL{wα} = T e− ∫ t0 dt ′

[1

4λ

∑α[wα(t ′)−2λ Aα]2

]|�, 0〉, (64)

with a corresponding probability rule for the joint realizationof the functions {wα(t)}.

It is not our aim to review various physical and technicalfeatures of CSL theory here for which we refer the readerto our previous papers [19,20] as well as well establishedpapers and review articles in the literature [10]. However,we would like to add an important point that is worthy to

highlight here. Since CSL theory adds a non-linear, stochas-tic term to the otherwise deterministic Schrödinger equation,there is an inevitable loss of information associated with it.Given an initial quantum state we cannot predict the final stateafter CSL evolution with 100% accuracy even in Minkowskispace-time. Given the tiny numerical value of the collapseparameter λ ∼ 10−16sec−1 this departure is so small that noobservable effect can be found in practical situations whiledealing within laboratory systems and hence making the the-ory phenomenologically viable. Nevertheless, it is an impor-tant insight that stochasticity that was brought in by CSLtheory allows information destruction in quantum evolution.The major challenge that we overcame in our earlier propos-als [19,20] was making this tiny effect substantially largerinside a black hole in a sensible manner. We review thisimportant feature in the next subsection.

5.3 Gravitationally induced collapse rate

The basic hypothesis we want to consider in the specificmodels we are studying is that all the information that isencoded in the quantum state of the fields, that might be con-sidered as entering the black hole region, will eventually beerased by a CSL type mechanism before the singularity (ormore precisely the region that requires a full quantum grav-ity description) is reached. As in our previous works, we findthat this can be achieved by postulating that usually small rateof information loss of information controlled by a fixed λ isintensified as the singularity is approached. This is achievedthrough the hypothesis that λ is a function of local curvatureof the space-time. Mathematically we expressed [19,20]

λ(R) = λ0

(1 +

(R

μ

)γ ), (65)

where μ is an appropriate scale and γ ≥ 1. In flat spacetimethis reduces to standard CSL theory.

Further motivation for such idea comes from arguments ofPenrose [41] and Díosi [38] suggesting that spontaneous thecollapse itself has gravitational origins. Furthermore, thereare strong indications that, at the phenomenological level, atheory in which λ depends on the mass of the particle speciesis preferred over theories where it is taken to be constant [44],and such a mass dependence is very suggestive of a possi-ble connection to local curvature. In any event, the aboveassumption tunes the rate of collapse in a manner that natu-rally leads to the scenario we want to explore.

The large value of λ in highly curved regions of space-time in fact leads to the dominance of the stochastic termover that of the usual term in (61). The non-unitary termbreaks the linear superposition (in terms of the vector basisadapted to the collapse operators) and stochasticity bringsa high degree of indeterminism in the quantum evolution.These two effects together cause the destruction of informa-

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tion as the black hole singularity is approached. Of course atpresent, we cannot directly test the hypothesis behind (65),but future technological developments might one day provideevidence in favour or against our proposal. We consider thisas a clear advantage over other approaches existing in the lit-erature. It is however important to note here that our proposaldoes not depend on the particular function that we have cho-sen in (65) as long as it satisfies, the following conditions—(1) λ is a (sufficiently rapidly) increasing function of localcurvature (so that the collapse rate diverges as the singu-larity is approached), (2) it is not in contradiction with theflat space-time constraints and various constraints that comesfrom astrophysical and cosmological observations and, (3) isa manifestly Lorentz invariant quantity. In previous works wehave argued that λ could naturally be taken to a function of theWeyl curvature scalar (WabcdWabcd ) simply because in real-istic scenarios (in 4 dimensions) the singularity in the blackhole interiors can be approached through paths where R van-ished, but, all paths in the black hole interior that approachthe singularity involve a divergence of Weyl curvature scalar.However in two dimensions the Weyl tensor vanishes, and soin our model we substitute it by what seems as the simplestalternative, the scalar curvature R (65).

Now we need to mathematically implement the aboveideas in RST model and the first step towards this is to foliatethe spacetime with Cauchy slices.

5.4 Spacetime foliation

In order to describe the evolution of quantum states we willneed to foliate the space-time with Cauchy slices and intro-duce a suitable global time parameter labeling the foliation.We perform an analysis similar to that used in works [19,20]for CGHS model, with some needed modifications. The rele-vant patch of the black hole space-time is referred as RegionI and Region II in Fig. 1. Region I is further divided into tworegions Region I(a) and I(b) which are within the event hori-zon and apparent horizon respectively. A Cauchy slice hasthe following characteristics– R = const. curve in RegionsI(a) and I(b), joined with a t = const., curve in Region II.Note that this time t , defined with respect to an asymptoticobserver, is well defined in regions II and I(b), i.e., outside theevent horizon. The family of slices are determined once wespecify the intersecting points between the R = const. andt = const. curves. For that we have to find out a curve whichstays within Region I(b). This is needed to ensure that theCauchy slices are spacelike and forward driven with respectto the asymptotic Killing time t . There might be many suchcurves and any of them should be as good as others to do thejob. We choose the curve

−�2x+(x− + m

�3x+0

)

−κ

4

(1 + a0

(x− − x−

s

) (x+ − x+

0

)) = 0, (66)

where a0 is a constant and we shall use a fixed value for thisin our analysis.

Let us now comment on the quantitative aspect of makingthese slices. The targeted regions are I and II as shown inFig. 1. We start by calculating R in (31) by using (44) whichgives

R = − 16

�x−x+x+0

(κe2ρ − 4

)3

× (e2ρ

(�κ2x+

0 + 4κmx+ + 16�2x−x+2 (m + �3x−x+

0

))× +κ2�3e4ρx−x+x+

0 + 16�3x−x+x+0

). (67)

In principle it is straightforward to find the R = const. =R0 slices in the following way. First we can solve (67) fore2ρ as a function of coordinates x± and other parameters(κ,m,�, x+

0 ) since the equation R − R0 = 0 is a cubicequation in e2ρ . Among the three solutions for e2ρ one has totake the real solution and put it in (20) to find �c which nowcorresponds to R = R0. The next step is to equate (44) with�c and this in turn allows one to write x+ = fRc (x

−) wherefRc (x

−) is a function of x− on the collapse hypersurfacewith curvature R = Rc. The resulting analytic expressionfor the R = Rc curve is too cumbersome to put in a paperand therefore not included here.

However, it is possible to plot the R = const. curvenumerically along with other relevant curves, namely, thesingularity (36), the apparent horizon (37), the intersectioncurve (66) and t = const. lines to complete the foliation.This family of Cauchy slices is plotted in Fig. 3. It is clearthat slices with increasingly higher curvature are those thatare closer to the singularity. Making the collapse parametera function of the spacetime curvature intensify the collapseof the wave function near singularity. This, in effect, erasealmost all the information about the matter that had oncecreated the black hole.

5.5 CSL evolution and the modified back reaction

Now we are in a position to use the construction made sofar and evolve the initial quantum state as given in (57). Wehave assumed the curvature dependent collapse rate in (65)and prepared our Cauchy slices in the preceding subsection.The only thing that is missing is to define the set of collapseoperators appearing in (64). As we have indicated before thatit is convenient to use the discrete basis in order to have atour disposal simple notions of localized excitations or “par-ticles”. Therefore it is convenient to characterize the collapseoperators also in terms of this discrete basis. Following ourearlier works, and for simplicity we chose

Aα := Nnj = Nintnj ⊗ I

ext . (68)

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Fig. 3 Spacetime foliationplots for RST model. The familyof Cauchy slices are given byR = const. curves inside theapparent horizon joined witht = const. lines outside. Wehave setk = 1,m = 2,� = 1, x+

0 = 0.1for all plots. Values of curvatureR = 50, 30, 25 for blue curvesfrom top to bottom. As requiredfor this slicing to be Cauchy,hypersurfaces with larger R(more closer to the singularity)are joined together with largervalues of asymptotic Killingtime t

Here Nintn, j is the number operator in the discrete basis (which

was used to discretize the mode function in (55)) interior ofthe black hole and I

ext is the identity in the exterior of theblack hole basis. The number operator acts on the particlestates as follows

Nnj |F〉int ⊗ |F〉ext = Fnj |F〉int ⊗ |F〉ext , (69)

where Fnj is the particle excitation in a state correspondingto quantum numbers n, j .

At this point we must consider the fact that when we intro-duce the CSL modification of the evolution of the quantumstate of the field, we also have a modification of the expec-tation value of the renormalized energy-momentum tensor(REMT), and this will in turn modify the back reaction ofthe quantum field on the background spacetime.

One issue that needs to be mentioned here is that suchsimple collapse operator might not be physically appropriate(or acceptable) as it might fail to take Hadamard states intoHadamard states. The issue of what are the kind of operatorsthat when used as collapse generating operators would ensurethis essential property has been studied with some generalityin [34], however the results suggest that there are suitablechoices of acceptable operators that are very close to the onedescribed above.

In order to consider the modification of the back reactionwe should in principle study the expression for the REMTin the quantum state |ψCSL 〉 which according to (89) can bewritten in the following form:

〈ψCSL |Tμν(x)|ψCSL 〉R= 〈ψCSL | : Tμν(x) :in |ψCSL 〉 + Geometric terms, (70)

where the first term on the r.h.s is normal ordered with respectto the “in” quantization. Here we shall be considering theright moving sector of field modes and concentrate how theREMT evolves due to CSL evolution (for a brief account onthe issue see Appendix A).

In fact what we would need to do is to compute the quanti-ties 〈ψCSL | : T±±(x) :in |ψCSL 〉 that should be used in (97)or more specifically within the full RST model in (29) and(30).

The in normal ordered energy-momentum tensor can beeasily expressed in terms of the objects used to construct the“in” quantization region and takes the following form

: T±±(x) :in = limx→x ′ : 1

2(∂± f (x))(∂± f (x ′)) :in

= 1

2

∫dω1dω2

{limx→x ′

(aω1 aω2uω1,±(x)uω2,±(x ′)

+2a†ω1aω2uω1,±(x)∗uω2,±(x ′)

123

556 Page 14 of 20 Eur. Phys. J. C (2018) 78 :556

+ a†ω1a†ω2uω1,±(x)∗uω2,±(x ′)∗

}, (71)

where ,± represents ordinary derivative with respect to x±.Note that the normal ordering procedure has been explicitlyperformed in the above expression. Also it should be clearedthat although the modes appearing in the above expression arenaturally associated with the early flat dilaton vacuum regionthey are defined everywhere, so the expression is valid forevaluations of the expectation value anywhere in the space-time.

As we mentioned before we shall focus only on the rightmoving modes for CSL effect. The changes in the state due tothe CSL modified evolution are expected to be relevant onlyat late times, due to the fact that this is where the parameter λ

becomes large due to the large values of the curvature as thesingularity is approached. Thus in principle we only need toconsider the CSL modifications of the state after the pulse.

The state in fact can be written as

|ψCSL 〉 = T e− ∫ t0 dt ′

[1

4λ

∑nj [wnj (t ′)−2λNnj ]2

]N

×∑F

CF |F〉int ⊗ |F〉ext

= N∑F

CFe− ∫ t

0 dt ′[

14λ

∑nj [wnj (t ′)−2λFnj ]2

]

×|F〉int ⊗ |F〉ext (72)

withCF = e− πEF� in the late time limit. One of the complica-

tions in evaluating the quantity of interest is that while in theabove expression all the operators refer to the out quantiza-tion, the expression (71) uses the in quantization. In principleone could rewrite all the operators appearing in (71) usingthe Bogolyubov relations and end up with an expression for〈ψCSL | : T±±(x) :in |ψCSL 〉 where everything is expressedin terms of the out quantization. Furthermore once one con-siders a specific realization of the stochastic functions {wnj },one would have a well defined expression, which could, inturn, be used to compute the modifications of the spacetimemetric given by the functions F&G in (29) and (30). That isin (71) we can express the creation and annihilation operatorsdefined in the “in” region, by using Bogolyubov transforma-tions, by operators associated with the black hole interiorand the exterior region:

aω =∑j,n

[α jn,ωb jn + β∗

jn,ωb†jn

]+

∑j,n

[ζ j n,ω

c j n + θ∗j n,ω

c†j n

],

(73)

where we have only discretized the modes in the out regionwhich is sufficient for our purpose and expressions with andwithout tildes are again belong to the interior and exterior ofthe black hole event horizon. Using this and the correspond-ing expression for a†

ω we could write : T±±(x) :in as a sum

of quadratic terms in the operators {b jn, b†jn, c jn, c

†jn} which

act in a simple form on the states |F〉int ⊗ |F〉ext .Unfortunately such calculation turns out to be extremely

difficult, even when one takes some simple form for the func-tions {wnj } (we had considered the case where those func-tions are just constants).

As evident, so far, with the above analysis we can writedown a formal general expression of the backreacted metricin presence of wavefunction collapse. As we have mentioned,the normal ordering contributes to � in (28) via (29) and (30).Of course, if we consider (72), the CSL evolved state of thein vacuum for the right movers, we can no longer neglect thenormal ordered part and an appropriate expression for themshould be

〈ψCSL | : T±±(x) :in |ψCSL 〉 = t±±(x) (74)

and we shall end up with a backreacted spacetime, given by

χ = � = −�2x+x−√

κ−

√κ

4ln

(−�2x+x−)

− m

�√

κx+0

(x+ − x+

0

)θ

(x+ − x+

0

)

+ 1√κ

(∫ x+dx

′+∫ x

′+dx

′′+t++

+∫ x−

dx′−

∫ x′−dx

′′−t−−

)(75)

where new undetermined functions t±± appear due to theCSL excitation of the vacuum state. One shortcoming of nothaving the explicit expression of the normal order expecta-tion value in generalCSL state (72) is that we cannot computean exact backreacted spacetime which needs to be a contin-uous change over the standard RST space-time. However,for completeness, in Appendix B, we explicitly compute thebackreacted and modified RST space-time for a single, GRWtype of collapse event. Even in this situation the metric canbe glued continuously on the collapse hypersurface. The caseof continuous collapse would need to generalize this situa-tion for multiple collapse events and gluing the spacetimefor each of those events over the family of such collapsehypersurfaces. This task, however, is beyond the scope ofthis paper.

The point is that we know from construction of theCSL type of evolution, together with our assumption thatλ depends on curvature (65) and diverges as one approachesthe singularity and that as one considers a hypersurface veryclose to the singularity, the state there would have collapsedto a state with definite occupation number, giving

|ψCSL 〉 = NCF0 |F0〉int ⊗ |F0〉ext (76)

where F0 stands collectively for a complete set of the occu-pation numbers in each mode i.e. {Fnj

0 }. This state is associ-

123

Eur. Phys. J. C (2018) 78 :556 Page 15 of 20 556

ated with a flux of positive energy towards future null infinitygiven by E0 ≈ ∑

nj Fnj0 ω′

nj where ω′nj is the mean frequency

of the mode n, j . The state is also associated with a corre-sponding (equal) negative energy flux into the black hole.Therefore, associated with the black hole at late times wewould have a total energy corresponding the mass associatedwith the pulse that led to the formation of the black holeM − E0 while the Hawking radiation would have carriedto I+ the energy E0. If quantum gravity cures the singular-ity without leading to large violations of energy-momentum,and if we ignore the actual violation of energy-momentumassociated with CSL4 then the thunderpop which we knowin this model, associated with the final and complete evapo-ration of the black hole, would have to carry the extra energyin the amount Ethunderpop = M − E0.

Thus taking into account the assumption that quantumgravity resolves the singularity, and the above characteriza-tion of the thunderpop can describe the full evolution of theinitial state from asymptotic past to the null future infinity asstarting with the initial state

|0〉Rin ⊗ |Pulse〉L=

∑F

CF |F〉int,R ⊗ |F〉ext,R ⊗ |Pulse〉L , (77)

transforming as a result of the CSL collapse into

|F〉int,R ⊗ |F〉ext,R ⊗ |Pulse〉L , (78)

on an hypersurface extending to null infinity but stayingbehind and very close to the singularity ( or QG region) andeventually leading to the state at I+ given by

|F〉ext ⊗ |thunderpop, M − E0〉. (79)

The point however is that this state is undetermined becausewe can not predict which realization of the stochastic func-tions wnj (t) will occur in a specific situation.

6 Recovering the thermal Hawking radiation

In order to deal with the indeterminacy, brought in by thestochasticity of the CSL evolution, it is convenient to consideran ensemble of systems, all prepared in the same initial stateand, described by the pure density matrix

ρ0 = |�i 〉〈�i |, (80)

which can, by using (57), be written as

ρ0 = ρ(τ0) ⊗ |PulseL〉〈PulseL |, (81)

4 With the idea that eventually a realistic calculation will be done witha fully relativistic collapse theory [30–33] with no violation of energyconservation akin [21].

where

ρ(τ0) =∑F,G

CF,G |F〉int ⊗ |F〉ext 〈G|ext ⊗ 〈G|int , (82)

and CF,G are determined by the Bogolyubov coefficientsbetween various mode functions. Now time evolution accord-ing to the CSL dynamics suggests [19]

ρ(τ) = T e− ∫ τ

τ0dτ ′ λ(τ ′)

2

∑n, j

[NLn, j−N R

n, j

]2

ρ(τ0). (83)

In the late time limit we have

ρ(τ) =∑F,G

e− π�

(EF+EG )e−∑

n, j (Fn, j−Gn, j )2∫ ττ0

dτ ′ λ(τ ′)2

×|F〉int ⊗ |F〉out 〈G|out ⊗ 〈G|int . (84)

Taking into account the dependence of λ on R and the factthat inside the black hole and the foliation makes R a functionof τ , we conclude that for the evolution under consideration,λ becomes effectively that function of τ . Noting the mannerin which λ(τ) in (65) depends on R we conclude that theintegral diverges near the singularity. Therefore the only sur-viving terms are the diagonal ones and thus very close to thesingularity we will have,

limτ→τs

ρ(τ) =∑F

e− 2π�

EF |F〉int ⊗ |F〉out 〈F |out ⊗ 〈F |int .

(85)

Next we explicitly include the left moving pulse, so that thecomplete density matrix very close to the singularity is givenby

limτ→τs

ρ′(τ ) =∑F

e− 2π�

EF

×|F〉int ⊗ |F〉out 〈F |out ⊗ 〈F |int ⊗ |Pulse〉〈Pulse|.(86)

Note that EF represents the energy of state |F〉ext as mea-sured by late time observers. The operator given by eq.(86) represents the ensemble when the evolution has almostreached the singularity.

Next by taking into account the effects of the quantumgravity region,5 on each one of the components of the ensem-ble characterized by the above density matrix, we can writethe corresponding density matrix for the ensemble at I+namely:

ρ′(I+) =∑F

e− 2π�

EF |F〉out 〈F |out ⊗ |T, M − E0(F)〉

×〈T, M − E0(F)|, (87)

where T stands for the thunderpop and the state has energyM − E0(F).

5 That, would include for instance, the effects represented by the thun-derpop in RST model.

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556 Page 16 of 20 Eur. Phys. J. C (2018) 78 :556

As far as the information is concerned we do not believethat the correlations between the thunderpop energy and thatin the early parts of the Hawking radiation can be of any helpin restituting a unitary relation between the initial and finalstates. This is because the same considerations concerningthe possibility that a remnant might help in this regard, applyto the thunderpop. That is, the amount of energy available tothe thunderpop is expected to be rather small and to a largeextent independent of the initial mass of the matter that col-lapsed to form a black hole, and thus for large enough initialmasses, the overwhelming part of the initial energy wouldbe emitted in the form of Hawking radiation. The small, andessentially fixed, amount of energy available to the thunder-pop is not expected to be sufficient for the excitation of thearbitrarily large number of degrees of freedom necessary torestore unitarity to en entangle the Hawking radiation andthunderpop state. Thus the resulting picture emerging fromthe present work is consistent with a full loss of informationduring the evaporation of a black hole that was present in ourprevious treatments.

7 Discussion

The black hole information problem continuous to be a topicattracting wide spread interest. A proposal to deal with theissue in a scheme which unified it with the general measure-ment problem in quantum mechanics, has been advocated in[7,8] and initially studied in detail in [19,20]. Those initialproposals left out two important aspects: relativistic covari-ance of the proposal and the issue of back reaction. The for-mer was explored in [21] while the later is the object of thepresent work.

We have studied the incorporation of spontaneous collapsedynamics into the back reaction of an evaporating black holeusing the RST model. We have shown in detail how a singlecollapse leads to the modification of the space-time and dis-cussed in general how the full continuous collapse dynamicsmight be used in this case and might be expanded to dealwith more realistic Black hole models.

Acknowledgements Authors thank Leonardo Ortíz for useful help atthe initial stage of the work. Part of the research of SKM was carriedout when he was an International Research Fellow of Japan Society forPromotion of Science. He is currently supported by a start up researchgrant PRODEP-NPTC, from SEP, México. DS acknowledges partialfinancial support from the grants CONACYT No. 101712, and PAPIIT-UNAM No. IG100316 México, as well as sabbatical fellowships fromPASPA-DGAPA-UNAM-México, and from Fulbright-Garcia Robles-COMEXUS.

Open Access This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecomm

ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.Funded by SCOAP3.

8 Appendix A: The renormalized energy-momentumtensor

Here we review, and to a certain degree clarify, the methodto calculate 〈�|Tab|�〉 using the properties of CGHS modeland a result for the renormalized energy-momentum tensoras found by Wald [45].

Let us start with the expression for the renormalizedenergy-momentum tensor obtained in [45]. The result isdescribed using Penrose’s abstract index notation for clarity.It applies to a two dimensional spacetime where the metricis conformally related to the flat metric so that gab = �2ηaband where, in the past the metric (is or approaches asymp-totically) the flat Minkowski metric, so that there �2 = 1. Itoffers an expression of the renormalized expectation value ofthe energy-momentum tensor, in terms of the derivative oper-ator ∇(η)

a associated with the flat metric ηab. The expressionis:

〈�|Tab|�〉 = 〈�| : Tab :in |�〉 + 1

12π

(∇(η)a ∇(η)

b C

−ηabηcd∇(η)

c ∇(η)d C − ∇(η)

a C∇(η)b C

+ηab

2ηcd∇(η)

c C∇(η)d C

), (88)

where the first expression on the right hand side is the nor-mal ordering with respect to the construction of the quantumfield theory that leads to the in vacuum, ηab is the flat met-ric, and C = ln �. In [45], the derivation of this expressionis obtained using an axiomatic approach to renormalize thestress tensor. We emphasize that the above expression is fullycovariant when all the objects are properly understood, (inparticular ∇(η)

a is derivative operator associated with the flatmetric ηab), and it is valid in all regions of space-time andnot only in the flat region in the past (where the expressionwould simply be 〈�| : Tab :in |�〉 as there C = constant).

Note that as (88) is expressed in terms of the derivativeoperator associated with the flat metric ηab, thus it can besimply written in terms of the ordinary derivative operator∂(y) associated with Minkowski coordinates yμ in which onecan write the flat metric components as ημν ( i.e. ηab =ημνdy

μa dyν

b = −dy0ady

0b + dy1

ady1b (because this operator

coincides with the covariant derivative operator associatedwith the flat metric). That is, we can write the expression interms of the explicit components in these coordinates as:

〈�|Tμν |�〉 = 〈�| : Tμν :in |�〉 + 1

12π

(∂(y)μ ∂(y)

ν C

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Eur. Phys. J. C (2018) 78 :556 Page 17 of 20 556

−ημνηαβ∂(y)

α ∂(y)β C − ∂(y)

μ C∂(y)ν C

+ημν

2ηαβ∂(y)

α C∂(y)β C

), (89)

where the derivative operators are just partial derivativeswith respect to the coordinates yμ above. In order to usethis expression at an arbitrary point, where the space-time isexpressed in other generic coordinates one needs to rely onin appropriate covariant form (88).

When evaluating the expectation value in any state wemust ensure that we use the normal ordering with respect tothe in quantization, so that the first term on the r.h.s will bezero if we chose |�〉 to be the in vacuum.

In our calculations, we will be using the relationshipbetween the derivative operators corresponding to the flatmetric and that corresponding to the general metric. Recallthat relationship between two derivative operators is repre-sented by is a tensor of type (1,2) denotedCc

ab which specifieshow it acts on a dual vector field Ab [46]

∇a Ab = ∇′a Ab − Cc

ab Ac. (90)

Such expression is of course valid in all coordinate systems.In order to use (88) for computing (89) we have to write the

ordinary derivative operator associated with the asymptoticpast “in” coordinates which as we noted is the same as thecovariant derivative operator ∇(η)

a associated with the flatmetric ηab in terms of the derivative operators associated withthe coordinates that cover the whole space-time (that is thecoordinates x±). We denote these latter derivative operatorsas ∂

(x)a .

Next we compute the Ccab which becomes the Christoffel

symbol relating the covariant derivative operator ∇(η)a with

the ordinary derivative operator ∂(x)a . In the x coordinate basis

it is

�ρμν = 1

2gρσ (∂μgσν + ∂νgμσ − ∂σ gμν). (91)

Note that here the Christoffel symbol is a tensor field asso-ciated with the derivative operator ∇(η)

a and the coordinatechart xμ associated with the ordinary derivative operators∂

(x)a .

The in flat metric can be expressed in the global coor-dinates x+, x− as ds2

ημν= dx+dx−

(−�2x+x−)and can also be

expressed in the in coordinates as ημνdyμdyν . The rela-tion between the coordinates x± and the coordinates y±is given by dy± = dx±/�x± while dy+ = dy0 + dy1

and dy− = dy0 − dy1. Thus we have the metric com-ponents gx+x+ = gx−x− = 0 and gx+x− = gx−x+ =−(2�2x+x−)−1.

Next we find the appropriate expression for the conformalfactor relating the flat and curved metrics. For that let us recallthe spacetime metric in the conformal gauge is

ds2 = −e2ρdx+dx−,

= e2ρ(−�2x+x−)

ds2ημν

, (92)

where is the flat linear-dilaton spacetime and the conformalfactor or subsequently C is found to be

�2 = e2ρ(−�2x+x−),

�⇒ C = ln � = ρ + 1

2ln(−�2x+x−). (93)

Using (90), (91), (92) and (93), a simple calculation yields

〈�|Tx±x±|�〉 = − h

12π

((∂x±ρ)2 − ∂2

x±ρ − 1

4x±2

)

+〈�| : Tx±x± :in |�〉, (94)

〈�|Tx+x−|�〉 = − h

12π∂x+∂x−ρ

+〈�| : Tx+x− :in |�〉. (95)

As it is well known [47,48], the covariant behavior of therenormalized stress tensor (which is a direct consequence ofsemi-classical Einstein equations) requires a specific nonzerotrace of the expectation value of the energy-momentum ten-sor that should be in fact state independent. That is :

gμν〈�|Tμν |�〉 = h R

24π. (96)

In the conformal gauge where the only non-vanishing met-ric components are gx

+x−one can easily find the off-

diagonal components of renormalized energy-momentumtensor 〈�|Tx±x∓|�〉 which matches with (95) given the factthat 〈�| : Tx+x− :in |�〉 must vanish (this vanishing is a con-sequence of the conservation law ∇μ〈�|Tμν |�〉 = 0). Thisis a direct consequence of the fact that trace of the energy-momentum tensor which appears in (96) is independent ofthe state of the quantum field.

Thus we now have the explicit expressions for variouscomponents of the renormalized energy-momentum tensorthat appear in semi-classical Einstein Eqs. (4) and (5), givenby

〈�|Tx±x±|�〉 = h

12π

(∂2x±ρ − (∂x±ρ)2 − 1

4x±2

)

+〈�| : Tx±x± :in |�〉, (97)

〈�|Tx+x−|�〉 = − h

12π∂x+∂x−ρ. (98)

As a check on the above expressions we consider themfor the in vacuum state in the linear dilaton vacuum region,where components of the renormalized energy-momentumtensor should vanish. The conformal factor in linear dilatonvacuum e2ρ(x±) = − 1

�2x+x− which implies that the first termin (97) vanishes and the normal ordered part is trivially zerosince we have chosen |�〉 to be the vacuum state. Similarlyit is easy to see that (98) vanishes as well. This provides oneconsistency check for the expressions of the renormalizedenergy-momentum tensor.

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556 Page 18 of 20 Eur. Phys. J. C (2018) 78 :556

Fig. 4 Modified RSTspacetime due to the collapse ofwavefunction in Kruskal frame.The energy-momentum tensordue to the collapse of the “in”vacuum state has support in theregion c ≤ x+ ≤ c + b whichmodifies the spacetime after thecollapse hypersurface. Themodification to RST spacetimeis the intersection of the futurelight cone of the point(x+ = c, x− = x−

1 ) and thecausal future of all the points onthe collapse hypersurface

9 Appendix B: The backreacted spacetime with GRWtype collapse

Here we want to explicitly calculate the backreacted space-time assuming a single collapse event of GRW type [15]on one of the collapse hypersurfaces chosen stochastically.Also, we would only consider a situation where the right-moving modes are subjected to collapse. In this situation thebackreacted metric due to collapse has the following form

χ = � = −�2x+x−√

κ

−√

κ

4ln(−�2x+x−)

− m

�√

κx+0

(x+ − x+

0

)θ

(x+ − x+

0

)

+ 1√κ

∫ x+

0dx

′+∫ x

′+

0dx

′′+t++ (99)

where t++ is a state dependent function, defined in (74), andit vanishes if and only if we chose the quantum state to bethe “in” vacuum. In that case we have the RST spacetime.However, if the state is different than vacuum, such as when

the wavefunction is modified by a the collapse dynamics, wemust find out the new spacetime.

We will consider here a single collapse event, associatedwith a certain space-like hypersurface, which we take here asgiven by one of the hypersurfaces of the folliation introducedin section V.D. That is the hypersurface that corresponds tosay to a specific value of R say R = Rc (which in the dis-tant exterior region is matched to something else as shownin Fig. 3). In order to further simplify matters we will chosechose t++ to be proportional to a (localized) function of com-pact support along x+. This is further motivated by the formof the collapse operators which are associated to the modesn, j which as we know are highly localized. However aswe noted, the meaningful use of the semi-classical setting,requires that the precise form of the collapse operators shouldbe such as to ensure the Hadamard nature of the states thatresult from the collapse dynamics.

We represent the fact that t++ is nonzero only to the futureof the collapse hypersurface characterized by the equationx+ = fRc (x

−), by including a theta function. Keeping theseconsiderations in mind we have

t++(x+, x−) = εh(x+)�[x+ − fRc (x

−)], (100)

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Eur. Phys. J. C (2018) 78 :556 Page 19 of 20 556

where ε is small number, x+− fRc (x−) = 0 specifies the col-

lapse hypersurface and the theta function makes sure that theenergy-momentum tensor vanishes to all the points past to thecollapse hypersurface6. The resulting backreacted spacetimeis then found by putting (100) in (99) and integrating.

As an example, where the calculation can be made explic-itly we considered the function

h(x+) =

⎧⎪⎪⎨⎪⎪⎩

0 x+ ≤ cα(x+ − c) c ≤ x+ ≤ c + b/2αb − α(x+ − c) c + b/2 ≤ x+ ≤ c + b0 b + c < x+

(101)

and checked that the resulting space-time metric is smooth,everywhere except on the collapse hypersurface where it isonly continuous. The spacetime is modified only to the futureof the support of t++.

The analysis proves that as we move along the x− axis(with fixed x+) the metric changes nontrivially as one crossesthe hypersurface, but, the change is always continuous.

As one moves along a line of fixed x− by changing x+, themodification appears only to the future of the collapse hyper-surface and change is continuous The pictorial description ofthese results is depicted in Fig. 4.

By observing Figs. 4 and 3 we note that the collapse canresult in modifications of the metric and quantum state atasymptotic infinity, in the black hole region and on the thun-derpop.

The actual changes will of course depend on the specificrealization of the stochastic parameters/functions that con-trol the collapse evolution, as these will determine the actualstate of the quantum field that results from the collapse. Thetreatment presented here is limited to a single instantaneouscollapse event and the treatment involving the continuousand multi-mode CSL dynamics, presented in section V. B.,is more complicated but can be generalized as a continuousversion of what we showed here.

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