Quantum fluctuating geometries and the information paradox

# Quantum fluctuating geometries and the information paradox

## Abstract

We study Hawking radiation on the quantum space-time of a collapsing null shell. We use the geometric optics approximation as in Hawking’s original papers to treat the radiation. The quantum space-time is constructed by superposing the classical geometries associated with collapsing shells with uncertainty in their position and mass. We show that there are departures from thermality in the radiation even though we are not considering back reaction. One recovers the usual profile for the Hawking radiation as a function of frequency in the limit where the space-time is classical. However, when quantum corrections are taken into account, the profile of the Hawking radiation as a function of time contains information about the initial state of the collapsing shell. More work will be needed to determine if all the information can be recovered. The calculations show that non-trivial quantum effects can occur in regions of low curvature when horizons are involved, as for instance advocated in the firewall scenario.

## I Introduction

Black hole evaporation is perhaps the salient problem of fundamental physics nowadays, since it tests gravity, quantum field theory and thermodynamics in their full regimes. Hawking’s calculation showing that black holes radiate a thermal spectrum initiated the study of this phenomenon. However, the calculation assumes a fixed given space-time, whereas it is expected that the black hole loses mass through the radiation and eventually evaporates completely. Associated with the evaporation process is the issue of loss of information, whatever memory of what formed the black hole is lost as it evaporates in a thermal state characterized by only one number, its temperature. Having a model calculation that follows the formation of a black hole and its evaporation including quantum effects would be very useful to gain insights into the process. Here we would like to present such a model. We will consider the collapse of a null shell. The associated space-time is very simple: it is Schwarzschild outside the shell and flat space-time inside. We will consider a quantum evolution of the shell with uncertainty in its position and momentum and we will superpose the corresponding space-times to construct a quantum space-time. On it we will study the emission of Hawking radiation in the geometric optics approximation. We will see that in the classical limit one recovers ordinary Hawking radiation. However, when quantum fluctuations of the collapsing shell are taken into account we will see that non vanishing off-diagonal terms appear in the density matrix representing the field. The correlations and the resulting profile of particle emission are modulated with information about the initial quantum state of the shell, showing that information can be retrieved. At the moment we do not know for sure if all information is retrieved.

The model we will consider is motivated in previous studies of the collapse of a shell (2); (3); (1). In all these, an important role is played by the fact that that there are two conjugate Dirac observables. One of them is the ADM mass of the shell. The other is related to the position along scri minus from which the shell was sent inwards. These studies are of importance because they show that the quantization of the correct Dirac observables for the problem lead to a different scenario than those considered in the past using other reduced models of the fluctuating horizon of the shell (see for instance (4)).

The organization of this paper is as follow. In the next section we review the calculation of the radiation with a background given by a classical collapsing shell for late times in the geometric optics approximation, mostly to fix notation to be used in the rest of the work. In section 3 we will remove the late time approximation providing an expression of the radiation of the shell for all times. We will also derive a closed expression for the distribution of radiation as a function of the position of the detector on scri plus. We will show that when the shell approaches the horizon the usual thermal radiation is recovered. We will see that the use of the complete expression for all times is useful when one considers the case of fluctuating horizons in the early (non-thermal) phases of the radiation prior to the formation of a horizon. This element had been missed in previous calculations that tried to incorporate such effects. In section 4 we will consider a quantum shell and the radiation it produces, we will proceed in two stages. First we will compute the expectation value of Bogoliubov coefficients. This will allow to explain in a simple case the technique that shall be used. However, the calculation of the number of particles produced requires the expectation value of a product of Bogoliubov coefficients. In section 5 we consider the calculation of the density matrix in terms of the product of Bogoliubov operators and show that the radiation profile reproduces the usual thermal spectrum for the diagonal elements of the density matrix, but with some departures due to the fluctuations in the mass of the shell. In section 6 we will show that it differs significantly from the product of the expectation values, particularly in the late stages of the process. In section 7 we will analyze coherences that vanish in the classical case and show they are non-vanishing and that allow information from the initial state of the shell to be retrieved. We end with a summary and outlook.

## Ii Radiation of a collapsing classical shell

Here we reproduce well known results (5) for the late time radiation of a collapsing classical shell in a certain amount of detail since we will use them later on. The metric of the space-time is given by

 ds2=−(1−2Mθ(v−vs)r)dv2+2dvdr+r2dΩ2, (1)

where represents the position of the shell (in ingoing Eddington–Finkelstein coordinates) and its mass 1. Throughout this paper we will be working in the geometric optics approximation (i.e. large frequencies). In this geometry, light rays that leave with coordinate less than can escape to and the rest are trapped in the black hole that forms. Therefore defines the position of the event horizon. We will use that a light ray departing from with reaches at an outgoing Eddington–Finkelstein coordinate given by

 u(v)=v−4Mln(v0−v4M0), (2)

where is an arbitrary parameter that is usually chosen as , stemming from the definition of the tortoise coordinate which involves a constant of integration.

On the above metric we would like to study Hawking radiation corresponding to a scalar field. We consider the “in” vacuum associated with the mode expansion . The asymptotic form of the modes in is given by,

 ψlmω′(r,v,θ,ϕ)=e−iω′v4πr√ω′Ylm(θ,ϕ),

and the “out” vacuum corresponding to modes with asymptotic form in given by

 χlmω(r,u,θ,ϕ)=e−iωu4πr√ωYlm(θ,ϕ).

The geometric optics approximation consists of mapping the modes into as

 e−iωu(v)4πr√ωYlm(θ,ϕ),

where is determined by the path of the light rays that emanate from at time and arrive in at .

The Bogoliubov coefficients are given by the Klein-Gordon inner products,

 αωω′=⟨χlmω,ψlmω′⟩,
 βωω′=−⟨χlmω,ψ∗lmω′⟩.

They can be computed in the geometric optics approximation projecting the out modes in and substituting the expression for . Focusing on the beta coefficient we get,

 βωω′=−12π√ω′ω∫v0−∞dve−iω[v−4Mln(v0−v4M0)]−iω′v. (3)

Since we are considering modes that are not normalizable one in general will get divergences. This can be dealt with by considering wave-packets localized in both frequency and time. For example,

 χlmnωj=1√ϵ∫(j+1)ϵjϵdωeunωiχlmω, (4)

constitute an orthonormal countable complete basis of packets centered in time , and in frequency .

The original Hawking calculation assumes that the rays depart just before the formation of the horizon and arrive at at late times. In that case one can approximate,

 u(v)=v−4Mln(v0−v4M0)≈v0−4Mln(v0−v4M0).

Defining a new integration variable one gets

 βωω′=−4M02π√ω′ωlimϵ→0∫∞0dxe−iω[v0−4Mln(x)]−iω′(v0−4M0x)e−ϵx, (5)

where the last factor was added to make the integral convergent since we have used plane waves instead of localized packets as the basis of modes, following Hawking’s original derivation. Using the identity

 ∫∞0dxealn(x)e−bx=e−(1+a)ln(b)Γ(1+a),Re(b)>0, (6)

and the usual prescription for the logarithm of a complex variable we can take the limit and get

 βωω′=−i2πe−i(ω+ω′)v0√ωω′e−2πMωΓ(1+4Mωi)e−4Mωiln(4M0ω′). (7)

Now from the Bogoliubov coefficients we can calculate the expectation value of the number of particles per unit frequency detected at scri using

 ⟨NHω⟩=∫∞0dω′βωω′β∗ωω′=14π2ωe−4πMω|Γ(1+4Mωi)|2∫∞0dω′1ω′,

where we added the superscript “” to indicate this is the calculation originally carried out by Hawking.

The pre-factor is computed using the identity

 Γ(1+z)Γ(1−z)=zπsin(zπ),

 |Γ(1+4Mωi)|2=8Mπωe+4Mωπ−e−4Mωπ.

To handle the divergent integral we note that

 ∫∞0dω′1ω′=limα→0∫∞0dω′1ω′ei4Mαln(ω′)=[y=ln(ω′)dy=dω′ω′]=
 =limα→0∫∞0dyei4Mαy=14Mδ(0).

Therefore,

 ⟨NHω⟩=1e8Mωπ−14M2π∫∞0dω′1ω′=1e8Mωπ−1δ(0). (8)

Again, the results is infinite because we considered plane waves. The time of arrival has infinite uncertainty and we are therefore adding up all the particles generated for an infinite amount of time. To deal with this we can consider wave-packets centered in time and frequency for which the Bogoliubov coefficients are,

 βωjω′=1√ϵ∫(j+1)ϵjϵdωeunωiβωω′.

We start computing the density matrix

 ρHω1,ω2=∫∞0dω′βω1ω′β∗ω2ω′=14π2√ω1ω2e−i(ω1−ω2)v0e−2πM(ω1+ω2)Γ(1+4Mω1i)Γ(1−4Mω2i)×
 ×∫∞0dω′1ω′e−4M(ω1−ω2)ln(4M0ω′)=[y=ln(4M0ω′)dy=dω′ω′]=
 =14π2√ω1ω2e−i(ω1−ω2)v0e−2πM(ω1+ω2)Γ(1+4Mω1i)Γ(1−4Mω2i)∫∞−∞dye−4M(ω1−ω2)y=
 =14π2ω1e−4πMω1|Γ(1+4Mω1i)|22πδ(4M(ω1−ω2))=1e8Mω1π−1δ(ω1−ω2). (9)

Therefore,

 ⟨NHωj⟩=∫∞0dω′βωjω′β∗ωjω′=1ϵ∫∫(j+1)ϵjϵdω1dω2eun(ω1−ω2)iρHω1,ω2=1ϵ∫(j+1)ϵjϵ1e8Mω1π−1dω1∼1e8Mωjπ−1, (10)

which is the standard result for the Hawking radiation spectrum.

## Iii Calculation without approximating u(v)

We will carry out the computation of the Bogoliubov coefficients using the exact expression for . This will be of importance for the case with quantum fluctuations. This is because if one looks at the expression of the time of arrival,

 u(v)=v−4Mln(v0−v4M0), (11)

when one has quantum fluctuations, even close to the horizon, the second term is not necessarily very large. For instance, if one considers fluctuations of Planck length size and a Solar sized black hole, it is around . Therefore it is not warranted to neglect the first term as we did in the previous section. In this section we will not consider quantum fluctuations yet. However, using the exact expression allows to compute the radiation emitted by a shell far away from the horizon.

Starting with the expression:

 βωω′=−12π√ω′ω∫v0−∞dvei4Mωln(v0−v4M0)−iω′ve−iωv,

we change variables to and introduce a regulator . We get,

 βωω′=−4M02π√ω′ωe−i(ω+ω′)v0limϵ→0∫∞0dxei4Mωln(x)e−(ϵ−i[ω+ω′]4M0)x. (12)

For we recover Hawking’s original calculation. However, we can continue without approximating. Using again (6) we get,

 βωω′=−4M02π√ω′ωe−i(ω+ω′)v0Γ(1+4Mωi)limϵ→0e−(1+4Mωi)ln(ϵ−i[ω+ω′]4M0).

And taking the limit,

 βωω′=−i2π1ω′+ω√ω′ωe−i(ω+ω′)v0Γ(1+4Mωi)e−2πMωe4Mωiln(4M0[ω′+ω]). (13)

To compare with Hawking’s calculation we first compute

 ⟨NCSω⟩=∫∞0dω′βωω′β∗ωω′=14π21ω|Γ(1+4Mωi)|2e−4πMω∫∞0dω′ω′(ω′+ω)2,

where the superscript “” stands for classical shell. The difference with the calculation in the previous section is the argument of the last integral with no divergence in .

We can formally compute the divergent integral using the change of variable . We get,

 ∫∞0dω′ω′(ω′+ω)2=∫∞ln(ω)dye−y(ey−ω)=∫∞ln(ω)dy−1=
 =∫∞ln(ω)dyei4Mαy∣∣∣α=0−1=∫∞0dyei4Mαyei4Mαln(ω)∣∣∣α=0−1=14M(πδ(0)+p.v.(i0))−1,

with the principal value. Therefore,

 ⟨NCSω⟩=1e8Mωπ−14M2π∫∞0dω′ω′(ω′+ω)2=1e8Mωπ−1[(δ(0)2+p.v.(i2π0))−2Mπ]. (14)

This is an infinite result but it looks different from Hawking’s. To deal with the infinities it is necessary to compute for a wave-packet of frequency . We start by computing the density matrix:

 ρCSω1,ω2=∫∞0dω′βω1ω′β∗ω2ω′=14π2√ω1ω2e−i(ω1−ω2)v0Γ(1+4Mω1i)Γ(1−4Mω2i)e−2πM[ω1+ω2]×
 ×∫∞0dω′ω′(ω′+ω1)(ω′+ω2)e−4Mi[ω1ln(4M0[ω′+ω1])−ω2ln(4M0[ω′+ω2])]. (15)

Since the packet is centered in with width we introduce and . As a consequence, the last integral takes the form,

 ∫∞0dω′ω′e−4Mi[(¯ω−Δω2)ln(4M0[ω′+¯ω−Δω2])−(¯ω+Δω2)ln(4M0[ω′+¯ω+Δω2])](ω′+¯ω)2−(Δω2)2=
 =∫∞0dω′ω′e4MiΔωln(4M0[ω′+¯ω])(ω′+¯ω)2+O(Δω),

where we have not expanded the exponential since it controls the divergent part of the integral when . Changing variable to the integral becomes,

 ∫∞ln(4M0¯ω)dy(1−4M0¯ωe−y)e4MiΔωy+O(Δω)=
 Missing or unrecognized delimiter for \right
 =[πδ(4MΔω)+p.v.(i4MΔω)]e4MiΔωln(4M0¯ω)+O(Δω0).

So, the divergent part of the density matrix when is

 ρCSω1,ω2∼14π2¯ωeiΔωv0|Γ(1+4M¯ωi)|2e−4πM¯ω[πδ(4MΔω)+p.v.(i4MΔω)]e4MiΔωln(4M0¯ω).
 Missing or unrecognized delimiter for \left (16)

We proceed to compute by integrating both Bogoliubov coefficients in an interval around using the approximation that factors depending on are constant since the interval of integration is very small as it ranges between ,

 ⟨NCSωj⟩=1ϵ∫(j+1)ϵjϵ∫(j+1)ϵjϵdω1dω2eunΔωiρCSω1,ω2∼
 Missing or unrecognized delimiter for \right
 ×14M[πδ(Δω)+p.v.(ie4MΔωln(4M0¯ω)iΔω)]=
 ∼12πϵ1e8Mωjπ−1∫(j+1)ϵjϵ∫(j+1)ϵjϵdω1dω2e−[un−v0−4Mln(4M0¯ω)]Δωi[πδ(Δω)+p.v.(iΔω)].

Changing variables to and we get,

 Missing or unrecognized delimiter for \left
 ∼1e8Mωjπ−1[12+i2π∫ϵ−ϵd(Δω)p.v.(ϵ−|Δω|ϵΔω)e−αΔωi],

where we defined

 α≡un−v0−4Mln(4M0ωj). (17)

Notice that there appears the indeterminate parameter . This corresponds to the choice of origin of the affine parameter at scri plus.

A further change of variable leads us to

 ⟨NCSωj⟩=1e8Mωjπ−1[12+1πSi(ϵα)+1πcos(αϵ)−1αϵ] (18)

where is the sine integral. When we have that and the expression goes to

 ⟨NCSωj⟩→1e8Mωjπ−1.

This happens when either or . That is, at late times or in the deep infra-red regime. On the other hand, when (a detector close to spatial infinity or very early times) we have that and therefore

 ⟨NCSωj⟩→0.

We have obtained a closed form for the spectrum of the radiation of the classical shell along its complete trajectory. It only becomes thermal at late times. This agrees with previous numerical results (6). Previous efforts had differing predictions on the thermality or not of the radiation (7).

## Iv Radiation from the collapse of a quantum shell

### iv.1 The basic quantum operators

A reduced phase-space analysis of the shell shows that the Dirac observables and are canonically conjugate variables (2). We thus promote them to quantum operators satisfying,

 [ˆM,ˆvs]=iℏˆI, (19)

with the identity operator. It will be more convenient to use the operator which is also conjugate to . We call the expectation values of these quantities and .

In terms of them we define the operator

 ^u(v,ˆv0,ˆM)=vˆI−2[ˆMln(ˆv0−vˆI4M0)+ln(ˆv0−vˆI4M0)ˆM], (20)

where is a real parameter and an arbitrary scale. This operator represents the variable . Given a value of the parameter the operator is well defined in the basis of eigenstates of only for eigenvalues . This is the relevant region for the computation of Bogoliubov coefficients. It is however convenient to provide an extension of the operator to the full range of so that one can work in the full Hilbert space of the shell. The (quantum) Bogoliubov coefficients are independent of such extension. For instance, defining the function one can construct the operator

 ^uϵ(v,ˆv0,ˆM)=vˆI−2[ˆMfϵ(ˆv0−vˆI4M0)+fϵ(ˆv0−vˆI4M0)ˆM], (21)

which extends to the full Hilbert space. To understand the physical meaning, we recall that for values of less than the packets escape to scri, whereas for larger than they fall into the black hole. The extension corresponds to considering particle detectors that either live at scri or live on a time-like trajectory a small distance outside the horizon. As we shall see, the Bogoliubov coefficients will have a well-defined limit.

Next we seek for the eigenstates of . We work with wave-functions . The operator (conjugate to ) is,

 ⟨v0|^Mψ⟩=iℏ∂ψ∂v0. (22)

The eigenstates of of are given by the equation

 ⟨v0|^uϵψu⟩=uψu(v0),

that is,

 vψu(v0)−2iℏ∂∂v0[fϵ(v0−v4M0)ψu(v0)]−2iℏfϵ(v0−v4M0)∂ψ∂v0=uψu(v0), (23)
 vψu(v0)−4iℏfϵ(v0−v4M0)∂ψ∂v0−2iℏ4M0f′ϵ(v0−v4M0)ψu(v0)=uψu(v0).

It is useful to make a change of variable which leads to

 −4iℏ4M0fϵ(x)∂ψ∂x−2iℏ4M0f′ϵ(x)ψu(x)=(u−v)ψu(x).

Defining by we get,

 ∂ϕu∂x=iM0ℏu−vfϵϕu,

with general solution

 ϕu(x)=ϕ0exp(iM0ℏ(u−v)∫dsfϵ(s)).

Substituting and going back to the original variables

 ψu(x)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩ψI0√|ln(x)|exp(iM0ℏ(u−v)li(x)),x≥ϵ,ψII0√|ln(ϵ)|exp(iM0ℏ(u−v)xln(ϵ)),x<ϵ,

where and are independent, complex, constants and

 li(x)=∫x0dtln(t), (24)

is the logarithmic integral, which is plotted in figure (2).

The discontinuity of in introduces a degeneracy in the eigenstates of . For each eigenvalue we can choose two independent eigenstates,

 ψ1u(x)=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩1√8πℏ|ln(ϵ)|exp(iM0ℏ(u−v)x−ϵln(ϵ)),x<ϵ1√8πℏ|ln(x)|exp(iM0ℏ(u−v)[li(x)−li(ϵ)]),ϵ≤x<10,x≥1 (25)
 ψ2u(x)=⎧⎨⎩0,x≤11√8πℏ|ln(x)|exp(iM0ℏ(u−v)[li(x)−li(ϵ)]),x>1 (26)

which we have chosen as orthonormal. We will adopt the notation with for these states.

### iv.2 Operators associated with the Bogoliubov coefficients and their expectation values

On the previously described quantum space-time we will study Hawking radiation associated with a scalar field. We will assume that the scalar field sees a superposition of geometries corresponding to different masses of the black hole. Therefore, to measure observables associated with the field one needs to take their expectation value with respect to the wave-function of the black hole. In this subsection we will apply these ideas to the computation of the Bogoliubov coefficients and in the next we will extend it to compute the density matrix. We will go from the usual Bogoliubov coefficient to the operator . We will then compute its expectation value on a wave-function packet associated to the black hole and centered on the classical values and . We start with the expression (3) and promote it to a well defined operator

 ^βωω′=−12π√ω′ωlimϵ→0∫+∞−∞dvθ(ˆv0−vˆI)e−iω^uϵ(v)−iω′vθ(ˆv0−vˆI). (27)

We then consider a state associated with the black hole and compute the expectation value,

 Missing or unrecognized delimiter for \left
 ×∑J=1,2∫+∞−∞du|u,J⟩ϵϵ⟨u,J|∫+∞−∞dv′0∣∣v′0⟩⟨v′0∣∣θ(ˆv0−vˆI)|Ψ⟩,

where we have introduced bases of eigenstates of and .

Given,

 ⟨^β⟩ωω′=−12π√ω′ωlimϵ→0∫∞−∞∫∞−∞∫∞−∞∫∞−∞dvdv0dv′0duΨ∗(v0)Ψ(v′0)θ(v0−v)θ(v′0−v)×
 ×e−iωu−iω′v∑J=1,2ψu,J(v0)ψ∗u,J(v′0),

and changing variables and we get

 ⟨^β⟩ωω′=−(4M0)22π√ω′ωlimϵ→0∫∞−∞dve−iω′v∫∞0∫∞0dx1dx2Ψ∗(4M0x1+v)×
 ×Ψ(4M0x2+v)∫+∞−∞due−iωu∑J=1,2ψJu(x1)ψJ∗u(x2). (28)

The definition of the eigenstates reduces the integral in to

 ∫ϵ0∫ϵ0dx1dx2+∫ϵ0∫1ϵdx1dx2+∫1ϵ∫ϵ0dx1dx2+∫1ϵ∫1ϵdx1dx2+∫∞1∫∞1dx1dx2.

In the appendix we show that the first 3 integrals do not contribute in the limit . Therefore the calculation reduces to,

 ⟨^β⟩ωω′=−(4M0)22π8πℏ√ω′ωlimϵ→0∫∞−∞dve−iω′v(∫1ϵ∫1ϵdx1dx2+∫∞1∫∞1dx1dx2)Ψ∗(4M0x1+v)×
 ×Ψ(4M0x2+v)∫∞−∞due−iωu1√