Collapse of the wavefunction, the information paradox and backreaction
Eur. Phys. J. C
Collapse of the wavefunction, the information paradox and backreaction
Sujoy K. Modak 1 2
Daniel Sudarsky 0 3
0 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México , Apartado Postal 70543, 04510 Distrito Federal , Mexico
1 KEK Theory Center, High Energy Accelerator Research Organization (KEK) , Tsukuba, Ibaraki 3050801 , Japan
2 Facultad de Ciencias, CUICBAS,Universidad de Colima , CP 28045 Colima , Mexico
3 Department of Philosophy, New York University , New York, NY 10003 , USA
We consider the black hole information problem within the context of collapse theories in a scheme that allows the incorporation of the backreaction to the Hawking radiation. We explore the issue in a setting of the two dimensional version of black hole evaporation known as the RussoSusskindThorlacius model. We summarize the general ideas based on the semiclassical version of Einstein's equations and then discuss specific modifications that are required in the context of collapse theories when applied to this model. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2 Semiclassical CGHS model with backreaction . . . . 3 Review of the RST Model . . . . . . . . . . . . . . . 3.1 Equations of motion . . . . . . . . . . . . . . . . 3.2 Solving semiclassical equations . . . . . . . . . 3.3 Dynamical case of black hole formation and evaporation . . . . . . . . . . . . . . . . . . . . 4 Quantization on RST . . . . . . . . . . . . . . . . . . 5 Incorporating collapse mechanism in the RST model . 5.1 Collapse of the quantum state and Einstein's semiclassical equations . . . . . . . . . . . . . . 5.2 CSL theory . . . . . . . . . . . . . . . . . . . . 5.3 Gravitationally induced collapse rate . . . . . . . 5.4 Spacetime foliation . . . . . . . . . . . . . . . . 5.5 CSL evolution and the modified back reaction . . 6 Recovering the thermal Hawking radiation . . . . . . 7 Discussion . . . . . . . . . . . . . . . . . . . . . . . 8 Appendix A: The renormalized energymomentum tensor

Contents
The black hole information question has been with us for
more than four decades, ever since Hawking’s discovery that
black holes emit thermal radiation and therefore evaporate,
leading either to their complete disappearance or to a small
Planck mass scale remnant [
1
]. The basic issue can be best
illustrated by considering an initial setting where an
essentially flat spacetime in which a single quantum field is in a
pure quantum state of relative high excitation corresponding
to a spatial concentration of energy, that, when left on its own
will, collapses gravitationally leading to the formation of a
black hole. As the black hole evaporates, the energy that was
initially localized in a small spatial region, ends up in the
form of Hawking radiation that, for much of this evolution
must be almost exactly thermal [
2
]. The point, of course, is
that if this process ends with the complete evaporation of the
black hole (or even if a small remnant is left) the
overwhelming majority of the initial energy content would correspond to
a state of the quantum field possessing almost no information
(except that encoded in the radiation’s temperature) and it is
very difficult to reconcile this with the general expectation
that in any quantum process the initial and final states should
be related by a unitary transformation, and thus all
information encoded in the initial state must be somehow present in
the final one. The issue, of course, is far more subtle and the
above should be taken as only a approximate account of the
problem.
There have been many attempts to deal with this
conundrum, with none of them resulting in a truly satisfactory
resolution of the problem [
3,4
]. In fact there is even a debate
as to the extent to which this is indeed a problem or as some
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 scheme where the resolution of
the issue is tied to a proposal to address another lingering
problem of theoretical physics: the so called measurement
problem [
10
] in quantum theory.
The first task was dealt with [
7–9
] by noting that the true
problem arises only when one takes the point of view that a
satisfactory theory of quantum gravity must resolve the
singularity, and that, as a result of such resolution, there will be
no need to introduce a new boundary of spacetime in the
region where the classical black hole singularity stood.
Otherwise the problem can be fully understood by noting that the
region in the black hole exterior, at late times corresponding
to those where most of the energy takes the form of thermal
Hawking radiation, contains no Cauchy hypersurfaces and
thus any attempt to provide a full description of the
quantum state in terms of the quantum field modes in the black
hole exterior is simply wrongheaded. In order to provide a
complete description of such late quantum state one needs
to include the modes 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 dynamical collapse theory designed
to address the measurement problem in quantum theory, to
a simple two dimensional black hole model known as the
CallanGiddingsHarveyStrominger (CGHS) model [
11
].
The proposal was then to associate to the intrinsic breakdown
of unitary evolution, which is typical of these dynamical
collapse theories [
12–18
], (which were developed to deal with
the measurement problem in standard quantum mechanics)
all the information loss that takes place during the formation
and subsequent Hawking evaporation of the black hole. The
first concrete treatments along this line are [
19,20
].
In those works we noted that the treatment 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. Among these issues that two
most pressing ones are the replacement of the treatment
presented, by one that is fully consistent with relativistic
covariance, and to show how the important question of back
reaction due to Hawking radiation on the spacetime and
viceversa can be incorporated in such a scheme (i.e., in presence
of the collapse of wavefunction type setting). A first step in
this direction was accomplished in [
21
] where the simple
two dimensional problem 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 example
(1)
(2)
(3)
where the remaining issue of backreaction in the setting
of collapse theories. For this we will again consider a two
dimensional black hole model known due to
RussoSusskindThorlacius (RST) [
23,24
] which presents a solution of the
semiclassical (Einstein) equation in 2D.
The paper is organized as follows: We start by reviewing
the semiclassical CGHS model in Sect. 2 and then move
to the RST model in Sect. 3 and discuss the quantization of
matter fields on RST in Sect. 4. It is important to emphasize
that all those sections contain nothing novel and represent
just a review, which is however needed in order to make
sense of what follows. Section 5 contains necessary
ingredients for the adaptation of collapse of the wavefunction in
a general setting as well as for the specific case of 2D RST
model. In Sect. 6 we discuss the Hawking radiation and the
information paradox. There are two appendices (A and B)
discussing important issues related with the renormalization
of the energymomentum tensor and a specific example of
the treatment of back reaction of the spacetime metric (and
dilaton field) to a discrete collapse of the wavefunction.
2 Semiclassical CGHS model with backreaction
A natural way to incorporate backreaction effects of a
quantum field on the background geometry is to modify the
Einstein equations where the expectation value of the stress
tensor is included on the right hand side of the equations of
motion (E.O.M), so that,
Gab = TaCblass +
Tab
,
where Gab is the Einstein tensor of the classical metric,
TaCblass represents the energymomentum tensor of
whatever matter is being described at the classical level, and
Tab is the renormalized expectation value of the
energymomentum tensor of the matter fields that are treated
quantum mechanically, evaluated in the corresponding
quantum state  of such fields.
In the two dimensional CGHS model with a single freely
propagating massless scalar field, characterized by the action
[
11
]:
1
SCG H S = 2π
d2x√−g[e−2φ(R + 4(∇φ)2 + 4 2) − (∇ f )2],
where is a constant. The dilaton field φ is usually treated
classically, and the scalar field f is treated quantum
mechanically.
Working in the conformal gauge with null coordinates the
metric is described by:
ds2 = −e2ρ d x +d x −.
The semiclassical E.O.M involve now the energymomentum
contribution from the classical dilaton field and the
cosmological constant as well as the part coming from the
expectation value of the quantum field f . Those take now the
following form (with respect to the appropriate variation mentioned
on the left)
ρ : e−2φ 2∂x+ ∂x− φ − 4∂x+ φ∂x− φ −
2e2ρ
−
g±± : e−2φ
Note that even though the unperturbed metric has g±± =
0 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
components of the renormalized energymomentum tensor in a
particular state of the quantum field denoted by  . The
state is usually taken to be the “in vacuum state”. We review
this calculation, from 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 formal action, given by
S = SCG H S + SP ,
where SP is the Polyakov effective action [
25
]
h
¯
SP = − 96π
d2x √−g R
1
R,
and whose variation leads to the same set of Eqs. (4)–(6). This
is because, in the effective action formalism, the expectation
value of the renormalized energymomentum tensor
corresponding to the quantum field fˆ, is given by the derivative
of the Polyakov term
2 δSP
− √−g δgab = ψTabψ
h
¯
= − 48π
1
∇a ξ ∇bξ − 2∇a ∇bξ + gab 2R − 2 ∇cξ ∇cξ
where ξ is an auxiliary scalar field constrained to obey the
equation ξ = R and ψ is the state of the quantum (scalar)
field. We note that the freedom in the choice of the quantum
state, correspond, in the effective action treatment, to the
freedom of choice of boundary conditions for the solution ξ .
We refer the interested reader to [
26
] for more discussions
on the effective action formalism.
This is a very delicate issue that can generate serious
confusion in our approach, and care must be taken to ensure one
(10)
(11)
(12)
(13)
,
(9)
goes back and forth from the two formalism in a consistent
manner. We will have to do so in particular if we want to
consider the changes in the quantum states of the fˆ field (for
which the treatment without the effective action is more
convenient) and at the same time consider explicitly solving for
the spacetime metric and dilaton field (for which the reliance
on the effective action is most suitable). We will explore this
issue in detail in section V.A. and appendix B. In the
meanwhile we return to the review of the original RST model.
It has been found difficult to solve the set of differential
Eqs. (4)–(6) without a numerical handle. The advantage of
using “effective action formalism” is that it allows one to
play with the E.O.M without going into a rigorous quantum
field theory calculation, and indeed that approach was
subsequently exploited in RussoSusskindThorlacius (RST) [
23
],
where a local term was added in (7), allowing one to solve
the new semiclassical equations analytically. We review this
model in the next section.
3 Review of the RST Model
In the RST model a local term is added to the CGHS and
Polyakov actions such that the complete action, with a scalar
field f , which are however, treated via an effective term , is
given by [
23,26
]
S = SCG H S + SP + SRST ,
where SCG H S is given by (2), SP is (8) and the local term is
h
¯
SRST = − 48π
d2x √−g φ R,
which adds a direct coupling between the dilaton and the
Ricci scalar. Again, the above scheme should be seen as
effectively characterizing a model where the Polyakov term
replaces quantum effects of the massless scalar field.
3.1 Equations of motion
Next we present the equations of motion that result from the
model’s action (10).
Varying (10) with respect to gab we obtain
e−2φ
1
−2∇a ∇bφ + 2 gab(−4(∇φ)2 + 4∇2φ + 4 2)
= 48h¯π ∇a ξ ∇bξ − 21 gab(∇ξ )2
N h h
− 24π¯ (∇a ∇bξ − gab ξ ) − 24¯π (∇a ∇bφ − gab φ),
whereas the free field Eq. (18) becomes
∂x+ ∂x− (χ −
) = 0.
The above equation allows us to write
√κ
χ − = 2 W+(x +) + W−(x −) ,
where W+ and W− are arbitrary functions of x + and x −
respectively. Then (21) and (22) become
2
∂x+ ∂x− χ = − √ eW++W−
κ
and
∂x+ ∂x−
2
= − √ eW++W− .
κ
In the RST model one restricts oneself to the choice
i.e., W+ = 0 = W− and then the solution is found to be
= χ = D −
,
where D is an arbitrary constant and the functions F (x +),
G(x −) can be found by substituting (28) in (23) and
integrating
×
×
x+
x−
x +
x −
1
− 4x +2 +
1
− 4x −2 +
G(x −) =
d x −
d x −
(18)
F (x +) =
d x +
d x +
 : Tx+x+ (x +) :in 
, (29)
 : Tx−x− (x −) :in 
. (30)
In 2D conformal gauge 1 the above equations take the
following form (with respect to the appropriate variations indicated
below):
ρ : e−2φ 2∂x+ ∂x− φ − 4∂x+ φ∂x− φ −
2e2ρ
h h
+ 12¯π ∂x+ ∂x− ρ + 24¯π ∂x+ ∂x− φ = 0,
g±± : e−2φ
h
¯
− 48π
where the expectation values of the energymomentum tensor
are those found in (97) and (98). The only choice yet to
implement is the selection of a particular state  to solve
the above set of equations. An important feature of these
equations is that if one uses (15) and (17) one still finds the
“free field equation”:
∂x+ ∂x− (ρ − φ) = 0,
which is typical of the CGHS model without backreaction.
This feature is what in this model facilitates the finding of a
specific solution for the spacetime geometry in presence of
backreaction.
3.2 Solving semiclassical equations
It is convenient to introduce the new variables [
23
]
√κ e−2φ
≡ 2 φ + √κ ,
χ ≡ √κρ − √2κ φ + e√−2κφ ,
where κ = 12h¯π .
In these variables (15)–(17) take the following form
On the other hand the E. O. M. for φ is:
e−2φ
−2 R − 8 2 + 8(∇φ)2 − 8∇2φ
h
¯ R = 0.
− 24π
(14)
(15)
(16)
(17)
(19)
(20)
(21)
(22)
(23)
2
∂x+ ∂x− = − √κ e √2κ (χ− ),
2
∂x+ ∂x− χ = − √ e √2κ (χ− ),
κ
√
−∂x± χ ∂x± χ +
κ
− 4x ±2 +
κ∂x2± χ + ∂x± ∂x±
Now using these expressions one can find particular
solutions depending on the choice for the state of the quantum
field ( ). Specifically, we will focus on those ones which
correspond to solutions representing the formation 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
∂+∂−χ −
2 ∂+
∂−
where
8e−2ρ
d √κ
≡ dφ = 2
2
− √ e−2φ .
κ
3.3 Dynamical case of black hole formation and
evaporation
Now we will consider the case where a sharp pulse of matter
forms a black hole. This pulse can be well approximated by
(24)
(25)
(26)
(27)
= χ ,
(28)
(31)
(32)
choosing  to be a coherent state build on top of the in
vacuum, corresponding to a wave packet peaked around a
particular classical value. In particular, we only need a left
moving pulse to create a black hole, therefore, we can chose
 =  Pulse L ⊗ 0in R where  Pulse L = Oˆ 0in L with
Oˆ a suitable creation operator for the sharply peaked wave
packet. In this case the state dependent functions turns out to
be
 : Tx+x+ :in 
 : Tx−x− :in 
=
m
This choice when used in (28) leads to the following solution
χ = 2√x+κx − − √4κ ln − 2x +x −
x + − x0+ θ x + − x0+ .
This solution contains a singularity. To see this we refer to
Eqs. (31) and (32). The singularity occurs when = 0 and
(32) gives e−2φs = κ4 . As we have restricted ourselves to
the case ρ = φ√, one can use the relation (20), to find the
value of s = 4κ (1 − ln κ4 ) associated with the singularity.
Therefore the location of the singularity turns out to be:
2x+
− √κ
√κ
= 4
x− +
κ
1 − ln 4
m
3x+
0
This singularity is hidden by the apparent horizon located at
∂+φ = 0 which is given by
χ =
−
2x + x − +
The physical meaning of this point is that it could be
interpreted as the end point of the black hole evaporation [
23
].
This is confirmed by the fact that at x − = xs− the solution
(35) with x + > x0+ takes the form
χ =
Thus the spacetime for x + > xs+ is given by
2x +
= − √κ
√κ
+ − 4 ln −
x − +
2x +x −
m
3x +
0
+
− √4κ ln − 2x + x − +
m
√κ
m
3x +
0
θ xs− − x −
θ x − − xs− .
(41)
Now we can construct the complete spacetime metric so
that for x + < xs+ one has (35) and for x + ≥ xs+ the
appropriate expression of the metric is given by (41). We show
the overall spacetime in Kruskal coordinates in Fig. 1. Note
that there are two different linear dilaton vacuums—(1) for
x + < x0+ and (2) for x + ≥ xs+, x − ≥ xs−. These L. D. V. s
are 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 + = x0+ carries positive
energy and forms the black hole, whereas, the pulse of the
radiation at x − = xs−, x + ≥ xs+ carries negative energy
associated with the singularity and usually called
thunderpop. The spacetime metric, although, is continuous at those
gluing points but clearly it is not differentiable.
The Penrose diagram of the RST spacetime can be
constructed following Ref. [
27
]. The asymptotic past and future
regions are specified with respect to the Minkowskian
coordinates. First, in the asymptotic past, where the metric
corresponds to a linear dilaton vacuum, so that, ds2 = 2x1+x− ,
one can use the coordinates,
1
1
y+ =
ln( x +) − y0+,
y− = −
ln(−
x −)
(42)
(43)
to write ds2 = −d y+d y−, where y0+ 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 the
for the Ricci scalar (31) implies that at = 0 it diverges.
Since even in the linear dilaton vacuum regions this value
can be reached one has to put some 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 conformal/Penrose diagram
for the RST model is given by Fig. 2. For a discussion about
the boundary conditions to make finite curvature in crit see
[
24
].
Let us now consider the asymptotic structure of the
spacetime. Particularly we want to check the asymptotic behavior
and viability of defining J +. Let us focus on the metric in
Reg. I and Reg. II (inside and outside the black hole apparent
horizon), given by
χ =
2x +
= − √κ
√κ
− 4 ln(−
x − +
2x +x −) +
m
If we want to find out the physical metric coefficient in the
conformal gauge (3.1) we need to use the relations (20) and
(20). By using those we obtain the following equation
whose solution determines ρ. However, practically this
equation is not invertible and therefore we cannot find an exact
solution for ρ from the known expression of . But we can
perform 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 ). These
are related with Kruskal (x ±) coordinates in the following
way
= √1κ e2 σ
ln e2 r− m
.
t+ 21 ln e2 r − m
,
1 e− t− 21 ln e2 r − m
.
√k
σ − 4 ln 1 +
m
2x +
0
e (t−σ ) ,
Using these relations the metric function takes the following
form
Now note that for t = const. and r → ∞ (i.e., at spatial
infinity i0) the last term which is time dependent vanishes
altogether. This makes time independent and therefore
any solution for ρ in (45) will be time independent. This
guaranties the staticity of the metric at i0. Furthermore, as
one moves up in JR+ where t → ∞, σ → ∞ keeping
t − σ = finite, the timedependent term becomes least
dominant and one can approximately define an asymptotic
timelike Killing vector field near JR+. We shall use this asymptotic
Killing time to be associated with the physical observers as
usually done in the RST model. To talk about the asymptotic
flatness let us focus on (48). Near JR+ as σ, t, x + all tends to
infinity, from (44) and (45) and, by comparing the dominating
coefficients of κ, we have limx+→∞ e2ρ = − 2x+(x1−+ 3mx+ ) ,
0
which is the asymptotic form of the LDVII. Thus in entire
JR+ 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 the
Hawking evaporation, corresponding to the usual quantum
evolution of the matter field, and which indicates that
information is either lost at the singularity or somehow conveyed
to the exterior in some unknown fashion encoded in the very
late outward flux of energy known as the thunderpop.
However, in the context of our proposal, the state of the
quantum field will be affected by the modified quantum
evolution prescribed by the modified dynamics involving
spontaneous collapse of the wavefunction. The specific model that
we shall consider is given by the
ContinuousSpontaneousCollapse (CSL) theory, a brief introduction of which will be
provided in section V. In light of this modification the
backreaction of the quantum matter on the spacetime metric will
be modified as well. We will discuss this in the reminder of
this work.
4 Quantization on RST
In order to discuss in some detail the modifications, brought
in by the CSL version of quantum theory, it is convenient
to describe the two relevant constructions of the quantum
theory of the scalar field f on RST spacetime.
We note, however that the power of this model resides
in the fact that one is able to obtain the whole spacetime,
including the back reaction of the spacetime metric to the
quantum energy momentum stress tensor, before one actually
discusses the construction of the quantum field theory for the
matter field fˆ. This, in turn, allows for that construction to
be carried out in the appropriate spacetime which already
includes backreaction.
Thus one might think that one can safely ignore this part of
the treatment and just go ahead with the usage of the effective
action and never actually carry out the explicit construction
of the quantum field. This would be correct except that in our
approach we will need to further consider the changes in the
state of the quantum field brought about by the dynamical
collapse theory. Doing that requires the quantum field theory
for fˆ and we proceed with this now.
We can express the scalar field both in the in region and
the out region. The out region consists of the modes having
support in the inside and outside of the event horizon. Since
in the discussion related with Hawking radiation one only
considers the modes in the “right moving” sector we shall
only use them in various expressions here. In the in region
one can write
uω = √
vω = √
1
where the in vacuum is defined by aω0 in = 0. In the out
region one has
bω vω + bω† vω∗
+
cω˜ v˜ω˜ + cω†˜ v˜ω∗˜ , (51)
ω ω˜
where the modes with and without tildes respectively have
supports inside and outside the horizon. Vacuum within the
Fock spaces interior and exterior to black hole are
respectively cω˜ 0 int = 0 and bω 0 ext = 0. Using Bogolyubov
coefficients one can express the creation or annihilation
operators of the in region in terms of a linear combination of
creation and annihilation operators (defined in either Fock
spaces) of the out region. Specifically, there are two sets of
Bogolyubov coefficients connecting the in region to black
hole interior and exterior regions (only for the right moving
sector of the scalar field modes).
The field modes appearing in above expressions of f are
respectively given by
(50)
(52)
(53)
(54)
(55)
(56)
where y+ is defined in (42) and σ − = − 1 ln xxs−−++ππ MM// κκ
[
27
]. Whereas the mode in the interior of the black hole can
be defined from the expression of vω, in the following way
vω˜ (y−) = v−∗ω (−y−),
with y− given by (43). One should also note that in the
continuous basis (using ω, ω ) modes are in fact not
orthonormalized and to talk about particle creation in a particular
quantum number one has to introduce a discrete basis to make
sense of the particle definition. Moreover, the discrete basis
allows for a relatively simple characterization of localization
of the modes which in the continuous basis would require the
use of wave packets. We will rely on the discrete basis in our
work. It is easily obtained from the continuous counterpart
by defining the modes,
1
v jn = √
j
( j+1)
dωe2πωn/ vout
ω
with n and j ≥ 0 are integer numbers. These wave
packets are peaked about uout = 2π n/ with width 2π/ . The
Bogolyubov coefficients between the uω modes and vω (and
its complex conjugate) modes turns out to satisfy the
following relationship in the late time limit [
27
]
αωω ≈ e−πω/ βωω .
It is also possible to express the Bogolyubov coefficients in
the discrete basis just by using the transformation (55).
A standard calculation from the above expression
immediately leads us to the Hawking radiation. Also, as it is well
known, with this relation one can express the in vacuum
(defined in the Fock space at JR−) as a superposition of the
particle states in the joint basis in the out region defined inside
and outside the event horizon (i.e., interior to black hole and
at JR+) [
28, 29
]. Later, we need to consider the initial state
of the quantum field which will be evolved using the CSL
evolution, with the CSL term taken as an interaction
hamiltonian. We will take this state, to be the in vacuum for right
moving sector and a pulse for the left moving sector, as
discussed in subsection 3.3. Using Bogolyubov transformation
we can express
 i = 0in
R ⊗  P ulse L ,
= N
C F F int ⊗ F ext ⊗  P ulse L ,
(57)
F
where the in vacuum for right moving sector is expressed
as a linear combination of F states interior and exterior
to the black hole. Each of these states is characterized by a
set of excitation numbers corresponding to various modes.
Entanglement between the interior and exterior modes, in the
above state, implies that corresponding to a F ext there is
a single F int with the same particle excitation number but
with negative energy/wavevector. It is this initial quantum
state (57) that had led to the formation of the black hole. We
should also state that using the late time characterization, the
usual thermal nature of the radiation can be seen in the fact
that C F = e−π EF / , where E F = F ωn j Fn j is the total
late time energy of the F state.
Now that we have characterized the initial quantum state
we move to the next section to incorporate the CSL evolution
on this state.
5 Incorporating collapse mechanism in the RST model
We will be addressing the question of the fate of
information in the evaporation of the black hole by considering a
modified version of quantum theory proposed to address the
“measurement problem” of the standard quantum theory. The
specific version that we will be using is the CSL theory
proposed in [
16
]. This theory is a continuous version of the so
called GhirardiRiminiWeber (GRW) theory [
14, 15
] where
the unitary evolution is accompanied by occasional discrete
collapse of the wavefunction that happens for a very small
amount of time. As these theories were developed in the
context of many particle nonrelativistic quantum mechanics we
will need to adapt it to the present context involving quantum
field theory in curved spacetime2.
2 Ideally one would finally need to use a fully relativistic version of
collapse theories such as [
30–34
] as was done in [21] for the
nonbackreacting case.
5.1 Collapse of the quantum state and Einstein’s
semiclassical equations
One of the main difficulties that must be dealt with when
considering a semiclassical treatment of gravitation in the
context of modified quantum theories involving a collapse of
the quantum state is the fact that Einstein’s equations simply
will not hold when the energymomentum tensor is replaced
by its quantum expectation value and the quantum state of
the matter fields undergoes a stochastic collapse. In fact this
is connected to the intrinsic problem of treating gravitation
in a classical language, and it is expected to be fully solved
only in the context of a complete theory of quantum
gravity. This is illustrated in [
35
] where it is argued that
semiclassical gravity is either inconsistent when we assume the
state of quantum matter undergoes some sort of collapse, or,
it is simply at odds with experiments when we do not make
such an assumption. Unfortunately, we do not have at this
point a fully workable quantum gravity theory to explore
these issues. Furthermore, even when, and if we eventually
get our hands on such a theory, in which as expected the
classical spacetime metric is replaced by some more
fundamental set of quantum variables, the recovery the standard
notions of classical spacetime, a task that seems unavoidable
if we want to be able to describe such things a formation and
evaporation of black holes, can be expected to be a rather
complex process, that, moreover, might only work in some
approximate sense.
These considerations lead us to adopt the following
approach: We will consider semiclassical gravity as an
approximate and effective description, valid in limited
circumstances, of a more fundamental theory of quantum
gravity including matter fields. This seems to be, in fact, the
position that would be adopted in this regard by a good segment of
our community working on these questions, but we describe
our posture explicitly in order to avoid misunderstandings.
The analogy to keep in mind is the hydrodynamic
description of fluids, which as we know, works rather well in a large
class of circumstances, but does not represent the behavior
of the truly fundamental degrees of freedom involved. We
know, that at a deeper level fluids are made of molecules
that interact in a complex manner, and that, there are only
certain aspects of their collective behavior describable in the
hydrodynamic language. The semiclassical Einstein
equations therefore cannot be trusted to hold precisely at the
fundamental microscopic level, just like the NavierStokes
equations, which cannot be thought to represent the true behavior
of the fluid molecules, but must be taken as holding only in
an approximate sense. Moreover, just as the hydrodynamic
characterization of a fluid is known to break down rather
dramatically in certain circumstances, such as when a ocean
wave breaks at the beach, we can also expect that Einstein’s
semiclassical equations should become invalid under some
situations. We thus must take the situations associated with
the collapse of the quantum state of matter fields to be one
of such circumstance.
Now we must consider a way out to formally implement
such ideas in order to be able to further explore them and
their consequences, and in particular to apply them to the
problem at hand.
Below, we discuss the issue, first in the realistic setting of
a 3+1 dimensional spacetime and second in the particularly
simple situation concerning the RST model in 1+1
dimensions.
Here we will describe an approach initially proposed in
[
36
] in the context of inflationary cosmology and the problem
of emergence of the primordial inhomogeneities [
37
]. The
staring point is the notion of Semi classical Self consistent
Configurations (SSC), defined for the case of a single matter
field (for simplicity), as follows:
Definition The set {gab(x ), ϕˆ(x ), πˆ (x ), H, ξ ∈ H}
represents a SSC if and only if ϕˆ(x ), πˆ (x ) and H correspond a
to quantum field theory constructed over a spacetime with
metric gab(x ) and the state ξ in H is such that
Gab[g(x )] = 8π G ξ Tˆab[g(x ), ϕˆ(x )]ξ ,
(58)
where ξ Tˆμν [g(x ), ϕˆ(x )]ξ stands for the renormalized
energy momentum tensor of the quantum matter field ϕˆ(x )
(in the state ξ ) constructed with the spacetime metric gab.
This corresponds, in a sense, to the general relativistic
version of Schrödinger–Newton equation [
38–43
]. The point of
this setting is to ensure a consistency between the description
of the quantum matter and that of gravitation by considering
their influences on each other3.
To this setting we want to add an extra element: the
collapse of the wave function. That is, besides the unitary
evolution describing the change in time of the state of a
quantum field, we consider situations such as those envisaged in
discrete collapse theories such as GRW, where there will be,
sometimes, spontaneous jumps in the quantum state. We will
consider the situation when we are given a dynamical
collapse theory that, given an SSC (dubbed as SSC1 and
considered to describe the situation before the collapse), specifies,
a spacelike hypersurface Collapse (perhaps through some
stochastic recipe that we can overlook at this point) on which
the collapse of the quantum state takes place, and also the
final quantum state (generally, again in a similar stochastic
manner). The remaining task is now twofold – (i) to describe
the construction of the new SSC, (to be called SSC2) that
will be taken to describe the situation after the collapse and,
3 We consider this as a scheme to be used when all matter is treated
quantum mechanically, but one might add the contribution to the
energymomentum tensor from any fields which are treated classically, such as
the dilaton in the RST model.
(ii) to join the two SSC’s in a manner to generate something
to call, in its closest sense, a “global spacetime”.
In order to have a picture in our mind we can think of
the above scheme as something akin to an effective
description of a fluid involving a situation where “instantaneously”,
the NavierStokes equations do not hold. Let us think, for
instance, once more, about an ocean wave breaking at the
beach. The situation just before the wave breaks should be
describable to a very good approximation by the
NavierStokes equations, and should the situation, well after the wave
breaks and the water surface becomes rather smooth again.
The particular regime where the breakdown of the wave is
taking place, and its immediate aftermath, will, of course,
not be the one where fluid description and the Navier Stokes
equations can be expected to provide an accurate picture.
This is because such regime involves large amounts of energy
and information flowing between the macroscopic degrees of
freedom that are well characterized in the fluid language, and
the underlying molecular degrees of freedom, (accompanied
by other complex process including such things as
incorporation of air molecules into the water, the mechanism by which
ocean water is oxigenated). All these represent aspects would
that have been “averaged out” in passing from the molecular
to the fluid description. If we now take the limit in which this
complex nonfluid characterization is essential to be
instantaneous, then we will be in possession of two regimes that are
susceptible to a fluid description using NavierStokes
equations, joined through an instantaneous collapse of the
wavefunction (to be identified with the spacelike hypersurface
Collapse) where the said equations cannot hold.
The specific proposal for the effective characterization of
these situations, that we will have in mind is based on the
3 + 1 decomposition of the spacetime associated with the
hypersurface Collapse and inspired by the application of
these ideas in the specific case treated in [
36
].
The spacetime metric of the SSC1 defined on Collapse
has the induced spatial metric ha(1b), the unit normal na(1) and
the extrinsic curvature K ab(1). The fact that the SSC1
corresponds to a semiclassical solution of Einstein’s equations
then ensures that the Hamiltonian and momentum constrains
are satisfied on Collapse viewed as a hypersurface embedded
in the spacetime of the SSC1.
The task at hand now, would be to specify the quantum
state and initial conditions for the construction of the SSC
to the future of the collapse hypersurface, assuming that we
are given the expectation of the energymomentum tensor
for the SSC2, i.e., assuming that the collapse theory allows
us to determine ξ Tˆbc[g(x ), ϕˆ(x )]ξ (2). That is, as we are
considering the treatment for the case where the collapse is
taken to be instantaneous, we will assume that the collapse
theory, in our case CSL, determines (by some scheme
involving stochastic components) the quantum state at the
hypersurface lying “just to the future of the collapse hypesurface”
(assuming that the initial, precollapse, state was given) on
Collpase, and given that, one would need is to construct
the complete, corresponding SSC2. The first thing would be
to obtain suitable initial data for the spacetime metric of
SSC2. That is we need to find ha(2b) and the extrinsic
curvature K ab(2) satisfying the Hamiltonian and momentum
constrains, involving the expectation value of the
energymomentum tensor corresponding to the SSC2.
We have previously used the ansatz of taking ha(2b) = ha(1b)
on Collapse and finding a suitable expression Ka(2b), on
Collapse determined so as to ensure the Hamiltonian and
momentum constraints for the SSC2 are satisfied.
This determination of the initial data for the SSC2 metric
will be referred as Step 2. Next one would have to carry out the
completion of the SSC2, namely to specify the construction
of the quantum field theory, identify the quantum state and
show how the full spacetime metric can be determined.
The completion of the process would then involve
finding the state of the new Hilbert space such that its
expectation value of the (renormalized) energymomentum tensor
corresponds to the values given in Step 1 above. One must
then ensure that with the integration of the spacetime metric
and the mode functions given those initial data can be done
effectively. An explicit example showing the completion of
this process in the inflationary cosmological context
representing a single mode perturbation with specific comoving
wave vector was presented in [
36
]. There one can see that in
general the tasks involved are rather nontrivial.
The point is that this scheme will allow the construction of
a spacetime made of two four dimensional regions
characterized by the SSC constructions and joined along a collapse
hypersurface where Einstein’s equations do not hold.
In the case of the RST model, in 1 + 1 dimensions, the
situation is substantially simplified by the fact that the
spacetime is two dimensional and thus the Einstein tensor vanishes
identically. The violation of the equations of motion during a
collapse 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 general
spacetime (smooth) metric can be written as:
where ρ is a smooth function. Thus both the spacetime
metric for SSC1 and SSC2 can be put in this form. The issue is
now joining these two spacetimes along Collapse. Thus we
can regard, as a complete generalized spacetime, the result
obtained by this gluing procedure where the price we have
paid by doing so is that now the function ρ will not
necessarily be a smooth function. Note that something similar
happens when the linear dilaton vacuum region is glued with
the black hole spacetime in CGHS or RST model.
(59)
Also the remaining SSC 2 construction, i.e., the
specification of the corresponding Hilbert space and identification of
the quantum state needs to be dealt with. The specification
of the mode functions that determine the Fock space, can be
taken to be done at the level of initial data on Collapse, and
this can be achieved by making use of the fact that the general
solution of the KleinGordon equation for a massless scalar
field f on any spacetime metric of the form (59) is of the
form
f = f+(x+) + f−(x−).
(60)
As we mentioned before, the idea is then to take take the
modes used in the SSC1 Fock space as providing initial data
for the SSC2 Hilbert space construction. However it is easy
to see that using such procedure on Collapse with functions
satisfying (60) both in the regions to the past and future of
Collapse corresponds, simply, to functions satisfying (60)
in all our generalized spacetime ( i.e. after incorporating the
discontinuity in the derivative of ρ). All this will work fine
as long as the spatialmetric is continuous along Collapse
and the spacetime 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 the
SSC1. This represents a very nice simplification provided by
the two dimensional nature of the situation of interest.
The last step would be the identification of the state in the
Hilbert space of SSC 2 with the appropriate expectation
values of the renormalized energymomentum tensor, but again
as the Hilbert space of two SSC’s are the same we can take
this as being provided by the collapse theory in Step 1.
This shows that the program described at the start of this
section, regarding a single instantaneous collapse of the state
of the quantum matter field, can be easily implemented in
the present situation. The details of the general application
of that scheme for situations in higher dimensions is an open
problem.
The final issue that needs to be considered before
applying collapse theories to the problem at hand, has to do with
generalizing the above procedure, from the case of a single
instantaneous collapse to a continuous collapse theory. That
is, if instead of considering a single collapse taking place on
the spacelike hypersurface Collapse we want to consider
a foliation of spacetime by spacelike “collapse”
hypersurfaces Collapse(τ ) parametrized by a real valued time
function τ on the spacetime manifold, and a theory like CSL to
be described in the next section, describing the change in the
quantum state, as one “passes form one hypersurface to the
other”.
We can deal with this, first for the case of a finite interval
[τstart , τend ] in the time function by considering a partition of
the corresponding interval τ0 = τstart , τ1......τi ...τN = τend ,
performing the procedure describing the individual discrete
collapse at each step i in the partition, and eventually taking
the limit N → ∞.
With these considerations in hand we now turn to the
description of the specific type of collapse theory will be
using in this work.
5.2 CSL theory
We first consider the theory in the nonrelativistic quantum
mechanical setting in which it was first postulated. The CSL
theory is generically described in terms of two equations. The
first is a modified version of Schrödinger equation, whose
general solution, in the case of a single nonrelativistic
particle is:
ψ, t CwSL = T e− 0t dt i Hˆ + 41λ [w(t )−2λAˆ]2 ψ, 0 ,
where T is the timeordering operator. Aˆ is a smeared
position operator for the particle. w(t ) is a random classical
function of time, of white noise type, whose probability is given
by the second equation, the Probability Rule:
P Dw(t ) ≡ ψ, t ψ, t
t
ti =0
dw(ti )
√2π λ/dt
.
The state vector norm evolves dynamically (not equal 1), so
Eq. (62) says that the state vectors with largest norm are most
probable. It is straightforward to see that the total probability
is 1, that is
P Dw(t ) = ψ, 0ψ, 0 = 1.
The way we will incorporate the CSL modifications in
our situation is by relying on the formalism of interaction
picture version of quantum evolution, where the free part of
the evolution corresponding to the standard quantum
evolution will be absorbed in the construction of the quantum field
operators, while the interaction corresponding to the CSL
modifications will be used to evolve the quantum states. One
more thing that needs to be modified is related to the fact that
the quantum field is a system with infinite number of degrees
of freedom (DOF), and thus instead of a single operator Aˆ
and a single stochastic function w(t ) we will have an infinite
set of those labeled by the index α. Thus in our case we will
have:
 , t C{wSαL} = T e− 0t dt 41λ α[wα(t )−2λAˆα]2  , 0 ,
(64)
with a corresponding probability rule for the joint realization
of the functions {wα(t )}.
It is not our aim to review various physical and technical
features of CSL theory here for which we refer the reader
to our previous papers [
19,20
] as well as well established
papers and review articles in the literature [10]. However,
we would like to add an important point that is worthy to
(61)
(62)
(63)
highlight here. Since CSL theory adds a nonlinear,
stochastic 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 state
after CSL evolution with 100% accuracy even in Minkowski
spacetime. Given the tiny numerical value of the collapse
parameter λ ∼ 10−16sec−1 this departure is so small that no
observable effect can be found in practical situations while
dealing within laboratory systems and hence making the
theory phenomenologically viable. Nevertheless, it is an
important insight that stochasticity that was brought in by CSL
theory allows information destruction in quantum evolution.
The major challenge that we overcame in our earlier
proposals [
19,20
] was making this tiny effect substantially larger
inside a black hole in a sensible manner. We review this
important feature in the next subsection.
5.3 Gravitationally induced collapse rate
The basic hypothesis we want to consider in the specific
models we are studying is that all the information that is
encoded in the quantum state of the fields, that might be
considered as entering the black hole region, will eventually be
erased by a CSL type mechanism before the singularity (or
more precisely the region that requires a full quantum
gravity description) is reached. As in our previous works, we find
that this can be achieved by postulating that usually small rate
of information loss of information controlled by a fixed λ is
intensified as the singularity is approached. This is achieved
through the hypothesis that λ is a function of local curvature
of the spacetime. Mathematically we expressed [
19,20
]
λ(R) = λ0 1 +
R γ
μ
,
(65)
where μ is an appropriate scale and γ ≥ 1. In flat spacetime
this reduces to standard CSL theory.
Further motivation for such idea comes from arguments of
Penrose [
41
] and Díosi [
38
] suggesting that spontaneous the
collapse itself has gravitational origins. Furthermore, there
are strong indications that, at the phenomenological level, a
theory in which λ depends on the mass of the particle species
is preferred over theories where it is taken to be constant [
44
],
and such a mass dependence is very suggestive of a
possible connection to local curvature. In any event, the above
assumption tunes the rate of collapse in a manner that
naturally leads to the scenario we want to explore.
The large value of λ in highly curved regions of
spacetime in fact leads to the dominance of the stochastic term
over that of the usual term in (61). The nonunitary term
breaks the linear superposition (in terms of the vector basis
adapted to the collapse operators) and stochasticity brings
a high degree of indeterminism in the quantum evolution.
These two effects together cause the destruction of
information as the black hole singularity is approached. Of course at
present, we cannot directly test the hypothesis behind (65),
but future technological developments might one day provide
evidence in favour or against our proposal. We consider this
as a clear advantage over other approaches existing in the
literature. It is however important to note here that our proposal
does not depend on the particular function that we have
chosen in (65) as long as it satisfies, the following conditions—
(1) λ is a (sufficiently rapidly) increasing function of local
curvature (so that the collapse rate diverges as the
singularity is approached), (2) it is not in contradiction with the
flat spacetime constraints and various constraints that comes
from astrophysical and cosmological observations and, (3) is
a manifestly Lorentz invariant quantity. In previous works we
have argued that λ could naturally be taken to a function of the
Weyl curvature scalar (Wabcd W abcd ) simply because in
realistic scenarios (in 4 dimensions) the singularity in the black
hole interiors can be approached through paths where R
vanished, but, all paths in the black hole interior that approach
the singularity involve a divergence of Weyl curvature scalar.
However in two dimensions the Weyl tensor vanishes, and so
in our model we substitute it by what seems as the simplest
alternative, the scalar curvature R (65).
Now we need to mathematically implement the above
ideas in RST model and the first step towards this is to foliate
the spacetime with Cauchy slices.
5.4 Spacetime foliation
In order to describe the evolution of quantum states we will
need to foliate the spacetime with Cauchy slices and
introduce 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
relevant patch of the black hole spacetime is referred as Region
I and Region II in Fig. 1. Region I is further divided into two
regions Region I(a) and I(b) which are within the event
horizon and apparent horizon respectively. A Cauchy slice has
the following characteristics– R = const. curve in Regions
I(a) and I(b), joined with a t = const., curve in Region II.
Note that this time t , defined with respect to an asymptotic
observer, is well defined in regions II and I(b), i.e., outside the
event horizon. The family of slices are determined once we
specify the intersecting points between the R = const. and
t = const. curves. For that we have to find out a curve which
stays within Region I(b). This is needed to ensure that the
Cauchy slices are spacelike and forward driven with respect
to the asymptotic Killing time t . There might be many such
curves and any of them should be as good as others to do the
job. We choose the curve
− 2x + x − +
m
3x +
0
κ
− 4
1 + a0 x − − xs−
x + − x0+
= 0,
(66)
where a0 is a constant and we shall use a fixed value for this
in our analysis.
Let us now comment on the quantitative aspect of making
these slices. The targeted regions are I and II as shown in
Fig. 1. We start by calculating R in (31) by using (44) which
gives
R = −
16
x−x+x0+ κe2ρ − 4 3
× e2ρ
κ2x0+ + 4κmx+ + 16 2x−x+2 m +
3x−x0+
× +κ2 3e4ρ x−x+x0+ + 16 3x−x+x0+ .
(67)
In principle it is straightforward to find the R = const. =
R0 slices in the following way. First we can solve (67) for
e2ρ as a function of coordinates x ± and other parameters
(κ, m, , x0+) since the equation R − R0 = 0 is a cubic
equation in e2ρ . Among the three solutions for e2ρ one has to
take the real solution and put it in (20) to find c which now
corresponds to R = R0. The next step is to equate (44) with
c and this in turn allows one to write x + = f Rc (x −) where
f Rc (x −) is a function of x − on the collapse hypersurface
with curvature R = Rc. The resulting analytic expression
for the R = Rc curve is too cumbersome to put in a paper
and therefore not included here.
However, it is possible to plot the R = const. curve
numerically along with other relevant curves, namely, the
singularity (36), the apparent horizon (37), the intersection
curve (66) and t = const. lines to complete the foliation.
This family of Cauchy slices is plotted in Fig. 3. It is clear
that slices with increasingly higher curvature are those that
are closer to the singularity. Making the collapse parameter
a function of the spacetime curvature intensify the collapse
of the wave function near singularity. This, in effect, erase
almost all the information about the matter that had once
created the black hole.
5.5 CSL evolution and the modified back reaction
Now we are in a position to use the construction made so
far and evolve the initial quantum state as given in (57). We
have 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 collapse
operators appearing in (64). As we have indicated before that
it is convenient to use the discrete basis in order to have at
our disposal simple notions of localized excitations or
“particles”. Therefore it is convenient to characterize the collapse
operators also in terms of this discrete basis. Following our
earlier works, and for simplicity we chose
Aα := Nnj = Nni njt ⊗ Iext .
(68)
Fig. 3 Spacetime foliation
plots for RST model. The family
of Cauchy slices are given by
R = const. curves inside the
apparent horizon joined with
t = const. lines outside. We
have set
k = 1, m = 2, = 1, x0+ = 0.1
for all plots. Values of curvature
R = 50, 30, 25 for blue curves
from top to bottom. As required
for this slicing to be Cauchy,
hypersurfaces with larger R
(more closer to the singularity)
are joined together with larger
values of asymptotic Killing
time t
Here Nni n,jt is the number operator in the discrete basis (which
was used to discretize the mode function in (55)) interior of
the black hole and Iext is the identity in the exterior of the
black hole basis. The number operator acts on the particle
states as follows
In order to consider the modification of the back reaction
we should in principle study the expression for the REMT
in the quantum state ψC SL which according to (89) can be
written in the following form:
ψC SL Tμν (x )ψC SL R
Nnj F int ⊗ F ext = Fnj F int ⊗ F ext ,
(69)
= ψC SL  : Tμν (x ) :in ψC SL + Geometric terms, (70)
where Fnj is the particle excitation in a state corresponding
to quantum numbers n, j .
At this point we must consider the fact that when we
introduce the CSL modification of the evolution of the quantum
state of the field, we also have a modification of the
expectation value of the renormalized energymomentum tensor
(REMT), and this will in turn modify the back reaction of
the quantum field on the background spacetime.
One issue that needs to be mentioned here is that such
simple collapse operator might not be physically appropriate
(or acceptable) as it might fail to take Hadamard states into
Hadamard states. The issue of what are the kind of operators
that when used as collapse generating operators would ensure
this essential property has been studied with some generality
in [
34
], however the results suggest that there are suitable
choices of acceptable operators that are very close to the one
described above.
where the first term on the r.h.s is normal ordered with respect
to the “in” quantization. Here we shall be considering the
right moving sector of field modes and concentrate how the
REMT evolves due to CSL evolution (for a brief account on
the issue see Appendix A).
In fact what we would need to do is to compute the
quantities ψC SL  : T±±(x ) :in ψC SL that should be used in (97)
or more specifically within the full RST model in (29) and
(30).
The in normal ordered energymomentum tensor can be
easily expressed in terms of the objects used to construct the
“in” quantization region and takes the following form
1
: T±±(x) :in = xl→imx : 2 (∂± fˆ(x))(∂± fˆ(x )) :in
dω1dω2 xl→imx aˆω1 aˆω2 uω1,±(x)uω2,±(x )
+2aˆω†1 aˆω2 uω1,±(x)∗uω2,±(x )
+ aˆω†1 aˆω†2 uω1,±(x)∗uω2,±(x )∗ ,
(71)
where ,± represents ordinary derivative with respect to x ±.
Note that the normal ordering procedure has been explicitly
performed in the above expression. Also it should be cleared
that although the modes appearing in the above expression are
naturally associated with the early flat dilaton vacuum region
they are defined everywhere, so the expression is valid for
evaluations of the expectation value anywhere in the
spacetime.
As we mentioned before we shall focus only on the right
moving modes for CSL effect. The changes in the state due to
the CSL modified evolution are expected to be relevant only
at late times, due to the fact that this is where the parameter λ
becomes large due to the large values of the curvature as the
singularity is approached. Thus in principle we only need to
consider the CSL modifications of the state after the pulse.
The state in fact can be written as
ψC SL = T e− 0t dt 41λ nj [wnj (t )−2λNˆnj ]2 N
×
= N
F
CF F int ⊗ F ext
CF e− 0t dt 41λ nj [wnj (t )−2λFnj ]2
with CF = e− π EF in the late time limit. One of the
complications in evaluating the quantity of interest is that while in the
above expression all the operators refer to the out
quantization, the expression (71) uses the in quantization. In principle
one could rewrite all the operators appearing in (71) using
the Bogolyubov relations and end up with an expression for
ψC SL  : T±±(x ) :in ψC SL where everything is expressed
in terms of the out quantization. Furthermore once one
considers a specific realization of the stochastic functions {wnj },
one would have a well defined expression, which could, in
turn, be used to compute the modifications of the spacetime
metric given by the functions F &G in (29) and (30). That is
in (71) we can express the creation and annihilation operators
defined in the “in” region, by using Bogolyubov
transformations, by operators associated with the black hole interior
and the exterior region:
aˆω =
j,n
α jn,ωbˆ jn + β ∗jn,ωbˆ†jn +
j˜,n˜
j˜n˜,ω ˆ†j˜n˜ ,
ζ j˜n˜,ωcˆ j˜n˜ + θ ∗ c
(73)
where we have only discretized the modes in the out region
which is sufficient for our purpose and expressions with and
without tildes are again belong to the interior and exterior of
the black hole event horizon. Using this and the
corresponding expression for aˆ ω† we could write : T±±(x ) :in as a sum
F
×F int ⊗ F ext
(72)
of quadratic terms in the operators {bˆ jn, bˆ†jn, cˆ jn, cˆ†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
functions {wnj } (we had considered the case where those
functions are just constants).
As evident, so far, with the above analysis we can write
down a formal general expression of the backreacted metric
in 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 the
in vacuum for the right movers, we can no longer neglect the
normal ordered part and an appropriate expression for them
should be
ψC SL  : T±±(x ) :in ψC SL = t±±(x )
and we shall end up with a backreacted spacetime, given by
χ = 2√x+κx − − √4κ ln − 2x +x −
= −
m
where new undetermined functions t±± appear due to the
CSL excitation of the vacuum state. One shortcoming of not
having the explicit expression of the normal order
expectation value in general CSL state (72) is that we cannot compute
an exact backreacted spacetime which needs to be a
continuous change over the standard RST spacetime. However,
for completeness, in Appendix B, we explicitly compute the
backreacted and modified RST spacetime for a single, GRW
type of collapse event. Even in this situation the metric can
be glued continuously on the collapse hypersurface. The case
of continuous collapse would need to generalize this
situation for multiple collapse events and gluing the spacetime
for each of those events over the family of such collapse
hypersurfaces. This task, however, is beyond the scope of
this paper.
The point is that we know from construction of the
CSL type of evolution, together with our assumption that
λ depends on curvature (65) and diverges as one approaches
the singularity and that as one considers a hypersurface very
close to the singularity, the state there would have collapsed
to a state with definite occupation number, giving
ψC SL = N CF0 F0 int ⊗ F0 ext
where F0 stands collectively for a complete set of the
occupation numbers in each mode i.e. {F0nj }. This state is
associ(76)
ated with a flux of positive energy towards future null infinity
given by E0 ≈ nj F0nj ωnj where ωnj is the mean frequency
of the mode n, j . The state is also associated with a
corresponding (equal) negative energy flux into the black hole.
Therefore, associated with the black hole at late times we
would have a total energy corresponding the mass associated
with the pulse that led to the formation of the black hole
M − E0 while the Hawking radiation would have carried
to I+ the energy E0. If quantum gravity cures the
singularity without leading to large violations of energymomentum,
and if we ignore the actual violation of energymomentum
associated with CSL4 then the thunderpop which we know
in this model, associated with the final and complete
evaporation of the black hole, would have to carry the extra energy
in the amount E thunder pop = M − E0.
Thus taking into account the assumption that quantum
gravity resolves the singularity, and the above
characterization of the thunderpop can describe the full evolution of the
initial state from asymptotic past to the null future infinity as
starting with the initial state
0 iRn ⊗  Pulse L
F int,R ⊗ F ext,R ⊗  Pulse L ,
on an hypersurface extending to null infinity but staying
behind and very close to the singularity ( or QG region) and
eventually leading to the state at I+ given by
F ext ⊗ t hunder pop, M − E0 .
The point however is that this state is undetermined because
we can not predict which realization of the stochastic
functions wnj (t ) will occur in a specific situation.
6 Recovering the thermal Hawking radiation
In order to deal with the indeterminacy, brought in by the
stochasticity of the CSL evolution, it is convenient to consider
an ensemble of systems, all prepared in the same initial state
and, described by the pure density matrix
ρ0 =  i
i ,
which can, by using (57), be written as
ρ0 = ρ(τ0) ⊗  PulseL
PulseL ,
4 With the idea that eventually a realistic calculation will be done with
a fully relativistic collapse theory [
30–33
] with no violation of energy
conservation akin [21].
CF,G F int ⊗ F ext Gext ⊗ Gint ,
(82)
where
ρ(τ0) =
F,G
ρ(τ ) =
F,G
and CF,G are determined by the Bogolyubov coefficients
between various mode functions. Now time evolution
according to the CSL dynamics suggests [
19
]
ρ(τ ) = T e− ττ0 dτ λ(2τ )
In the late time limit we have
n, j NnL, j −NnR, j 2 ρ(τ0).
e− π (EF +EG )e− n, j (Fn, j −Gn, j )2 ττ0 dτ λ(2τ )
(83)
(84)
(85)
(86)
(87)
(80)
(81)
×F int ⊗ F out Gout ⊗ Gint .
Taking into account the dependence of λ on R and the fact
that inside the black hole and the foliation makes R a function
of τ , we conclude that for the evolution under consideration,
λ becomes effectively that function of τ . Noting the manner
in which λ(τ ) in (65) depends on R we conclude that the
integral diverges near the singularity. Therefore the only
surviving terms are the diagonal ones and thus very close to the
singularity we will have,
lim ρ(τ ) =
τ →τs
F
e− 2π EF F int ⊗ F out F out ⊗ F int .
Next we explicitly include the left moving pulse, so that the
complete density matrix very close to the singularity is given
by
lim ρ (τ ) =
τ→τs
e− 2π EF
F
×F int ⊗ F out Fout ⊗ Fint ⊗ Pulse Pulse.
F
× T , M − E0(F ),
Note that E F represents the energy of state F ext as
measured by late time observers. The operator given by eq.
(86) represents the ensemble when the evolution has almost
reached the singularity.
Next by taking into account the effects of the quantum
gravity region,5 on each one of the components of the
ensemble characterized by the above density matrix, we can write
the corresponding density matrix for the ensemble at I+
namely:
ρ (I+) =
e− 2π EF F out F out ⊗ T , M − E0(F )
where T stands for the thunderpop and the state has energy
M − E0(F ).
5 That, would include for instance, the effects represented by the
thunderpop in RST model.
As far as the information is concerned we do not believe
that the correlations between the thunderpop energy and that
in the early parts of the Hawking radiation can be of any help
in restituting a unitary relation between the initial and final
states. This is because the same considerations concerning
the possibility that a remnant might help in this regard, apply
to the thunderpop. That is, the amount of energy available to
the thunderpop is expected to be rather small and to a large
extent independent of the initial mass of the matter that
collapsed to form a black hole, and thus for large enough initial
masses, the overwhelming part of the initial energy would
be emitted in the form of Hawking radiation. The small, and
essentially fixed, amount of energy available to the
thunderpop is not expected to be sufficient for the excitation of the
arbitrarily large number of degrees of freedom necessary to
restore unitarity to en entangle the Hawking radiation and
thunderpop state. Thus the resulting picture emerging from
the present work is consistent with a full loss of information
during the evaporation of a black hole that was present in our
previous treatments.
7 Discussion
The black hole information problem continuous to be a topic
attracting wide spread interest. A proposal to deal with the
issue in a scheme which unified it with the general
measurement problem in quantum mechanics, has been advocated in
[
7, 8
] and initially studied in detail in [
19, 20
]. Those initial
proposals left out two important aspects: relativistic
covariance of the proposal and the issue of back reaction. The
former was explored in [
21
] while the later is the object of the
present work.
We have studied the incorporation of spontaneous collapse
dynamics into the back reaction of an evaporating black hole
using the RST model. We have shown in detail how a single
collapse leads to the modification of the spacetime and
discussed in general how the full continuous collapse dynamics
might be used in this case and might be expanded to deal
with more realistic Black hole models.
Acknowledgements Authors thank Leonardo Ortíz for useful help at
the initial stage of the work. Part of the research of SKM was carried
out when he was an International Research Fellow of Japan Society for
Promotion of Science. He is currently supported by a start up research
grant PRODEPNPTC, from SEP, México. DS acknowledges partial
financial support from the grants CONACYT No. 101712, and
PAPIITUNAM No. IG100316 México, as well as sabbatical fellowships from
PASPADGAPAUNAMMéxico, and from FulbrightGarcia
RoblesCOMEXUS.
Open Access This article is distributed under the terms of the Creative
Commons 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 credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
8 Appendix A: The renormalized energymomentum tensor
Here we review, and to a certain degree clarify, the method
to calculate Tab using the properties of CGHS model
and a result for the renormalized energymomentum tensor
as found by Wald [
45
].
Let us start with the expression for the renormalized
energymomentum tensor obtained in [
45
]. The result is
described using Penrose’s abstract index notation for clarity.
It applies to a two dimensional spacetime where the metric
is conformally related to the flat metric so that gab = 2ηab
and where, in the past the metric (is or approaches
asymptotically) the flat Minkowski metric, so that there 2 = 1. It
offers an expression of the renormalized expectation value of
the energymomentum tensor, in terms of the derivative
operator ∇a(η) associated with the flat metric ηab. The expression
is:
Tab
=
 : Tab :in 
+ η2ab ηcd ∇c(η)C ∇d(η)C ,
−ηabηcd ∇c(η)∇d(η)C − ∇a(η)C ∇b(η)C
(88)
where the first expression on the right hand side is the
normal ordering with respect to the construction of the quantum
field theory that leads to the in vacuum, ηab is the flat
metric, and C = ln . In [
45
], the derivation of this expression
is obtained using an axiomatic approach to renormalize the
stress tensor. We emphasize that the above expression is fully
covariant when all the objects are properly understood, (in
particular ∇a(η) is derivative operator associated with the flat
metric ηab), and it is valid in all regions of spacetime and
not only in the flat region in the past (where the expression
would simply be  : Tab :in  as there C = constant).
Note that as (88) is expressed in terms of the derivative
operator associated with the flat metric ηab, thus it can be
simply written in terms of the ordinary derivative operator
∂ (y) associated with Minkowski coordinates yμ in which one
can write the flat metric components as ημν ( i.e. ηab =
ημν d yaμd yν b = −d ya0d yb0 + d ya1d y1b (because this operator
coincides with the covariant derivative operator associated
with the flat metric). That is, we can write the expression in
terms of the explicit components in these coordinates as:
=
 : Tμν :in 
1
+ 12π
∂ μ(y)∂ (y)C
ν
where the derivative operators are just partial derivatives
with respect to the coordinates yμ above. In order to use
this expression at an arbitrary point, where the spacetime is
expressed in other generic coordinates one needs to rely on
in appropriate covariant form (88).
When evaluating the expectation value in any state we
must ensure that we use the normal ordering with respect to
the in quantization, so that the first term on the r.h.s will be
zero if we chose  to be the in vacuum.
In our calculations, we will be using the relationship
between the derivative operators corresponding to the flat
metric and that corresponding to the general metric. Recall
that relationship between two derivative operators is
represented by is a tensor of type (1,2) denoted Cacb which specifies
how it acts on a dual vector field Ab [
46
]
∇a Ab = ∇a Ab − Cacb Ac.
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 asymptotic
past “in” coordinates which as we noted is the same as the
covariant derivative operator ∇a(η) associated with the flat
metric ηab in terms of the derivative operators associated with
the coordinates that cover the whole spacetime (that is the
coordinates x ±). We denote these latter derivative operators
as ∂a(x).
Next we compute the Cacb which becomes the Christoffel
symbol relating the covariant derivative operator ∇a(η) with
the ordinary derivative operator ∂a(x). In the x coordinate basis
it is
μρν = 21 gρσ (∂μgσ ν + ∂ν gμσ − ∂σ gμν ).
Note that here the Christoffel symbol is a tensor field
associated with the derivative operator ∇a(η) and the coordinate
chart x μ associated with the ordinary derivative operators
∂a(x).
The in flat metric can be expressed in the global
coordinates x +, x − as dsη2μν = (−dx2+xd+xx−−) and can also be
expressed in the in coordinates as ημν d yμd yν . The
relation between the coordinates x ± and the coordinates y±
is given by d y± = d x ±/ x ± while d y+ = d y0 + d y1
and d y− = d y0 − d y1. Thus we have the metric
components gx+x+ = gx−x− = 0 and gx+x− = gx−x+ =
−(2 2x +x −)−1.
Next we find the appropriate expression for the conformal
factor relating the flat and curved metrics. For that let us recall
the spacetime metric in the conformal gauge is
(89)
(90)
(91)
ds2 = −e2ρ d x +d x −,
Tx±x± 
,
.
As it is well known [
47,48
], the covariant behavior of the
renormalized stress tensor (which is a direct consequence of
semiclassical Einstein equations) requires a specific nonzero
trace of the expectation value of the energymomentum
tensor that should be in fact state independent. That is :
gμν
In the conformal gauge where the only nonvanishing
metric components are gx+x− one can easily find the
offdiagonal components of renormalized energymomentum
tensor Tx±x∓  which matches with (95) given the fact
that  : Tx+x− :in  must vanish (this vanishing is a
consequence of the conservation law ∇μ Tμν  = 0). This
is a direct consequence of the fact that trace of the
energymomentum tensor which appears in (96) is independent of
the state of the quantum field.
Thus we now have the explicit expressions for various
components of the renormalized energymomentum tensor
that appear in semiclassical Einstein Eqs. (4) and (5), given
by
Tx±x± 
As a check on the above expressions we consider them
for the in vacuum state in the linear dilaton vacuum region,
where components of the renormalized energymomentum
tensor should vanish. The conformal factor in linear dilaton
vacuum e2ρ(x±) = − 2x1+x− which implies that the first term
in (97) vanishes and the normal ordered part is trivially zero
since we have chosen  to be the vacuum state. Similarly
it is easy to see that (98) vanishes as well. This provides one
consistency check for the expressions of the renormalized
energymomentum tensor.
= e2ρ − 2x +x − dsη2μν ,
where is the flat lineardilaton spacetime and the conformal
factor or subsequently C is found to be
Using (90), (91), (92) and (93), a simple calculation yields
(92)
(93)
(94)
(95)
(96)
(97)
(98)
Fig. 4 Modified RST
spacetime due to the collapse of
wavefunction in Kruskal frame.
The energymomentum tensor
due to the collapse of the “in”
vacuum state has support in the
region c ≤ x+ ≤ c + b which
modifies the spacetime after the
collapse hypersurface. The
modification to RST spacetime
is the intersection of the future
light cone of the point
(x+ = c, x− = x1−) and the
causal future of all the points on
the collapse hypersurface
9 Appendix B: The backreacted spacetime with GRW type collapse
Here we want to explicitly calculate the backreacted
spacetime 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
rightmoving modes are subjected to collapse. In this situation the
backreacted metric due to collapse has the following form
χ =
= −
√κ
− 4 ln(−
m
√κ x0+
1
+ √κ 0
x+
where t++ is a state dependent function, defined in (74), and
it vanishes if and only if we chose the quantum state to be
the “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, we
must find out the new spacetime.
We will consider here a single collapse event, associated
with a certain spacelike hypersurface, which we take here as
given by one of the hypersurfaces of the folliation introduced
in section V.D. That is the hypersurface that corresponds to
say to a specific value of R say R = Rc (which in the
distant exterior region is matched to something else as shown
in Fig. 3). In order to further simplify matters we will chose
chose t++ to be proportional to a (localized) function of
compact support along x +. This is further motivated by the form
of the collapse operators which are associated to the modes
n, j which as we know are highly localized. However as
we noted, the meaningful use of the semiclassical setting,
requires that the precise form of the collapse operators should
be such as to ensure the Hadamard nature of the states that
result from the collapse dynamics.
We represent the fact that t++ is nonzero only to the future
of the collapse hypersurface characterized by the equation
x + = f Rc (x −), by including a theta function. Keeping these
considerations in mind we have
t++(x +, x −) =
h(x +)
x + − f Rc (x −) ,
(100)
where is small number, x + − f Rc (x −) = 0 specifies the
collapse hypersurface and the theta function makes sure that the
energymomentum tensor vanishes to all the points past to the
collapse hypersurface6. The resulting backreacted spacetime
is then found by putting (100) in (99) and integrating.
As an example, where the calculation can be made
explicitly we considered the function
h(x +) =
⎧ 0
⎪⎪⎨ α(x + − c)
αb − α(x + − c)
⎪⎪⎩ 0
x + ≤ c
c ≤ x + ≤ c + b/2
c + b/2 ≤ x + ≤ c + b
b + c < x +
and checked that the resulting spacetime metric is smooth,
everywhere except on the collapse hypersurface where it is
only continuous. The spacetime is modified only to the future
of the support of t++.
The analysis proves that as we move along the x − axis
(with fixed x +) the metric changes nontrivially as one crosses
the hypersurface, but, the change is always continuous.
As one moves along a line of fixed x − by changing x +, the
modification appears only to the future of the collapse
hypersurface and change is continuous The pictorial description of
these results is depicted in Fig. 4.
By observing Figs. 4 and 3 we note that the collapse can
result in modifications of the metric and quantum state at
asymptotic infinity, in the black hole region and on the
thunderpop.
The actual changes will of course depend on the specific
realization of the stochastic parameters/functions that
control the collapse evolution, as these will determine the actual
state of the quantum field that results from the collapse. The
treatment presented here is limited to a single instantaneous
collapse event and the treatment involving the continuous
and multimode CSL dynamics, presented in section V. B.,
is more complicated but can be generalized as a continuous
version of what we showed here.
(101)
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