The Black Hole Information Paradox

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The Black Hole Information Paradox

15 EC Bachelor Thesis Physics- & Astronomy conducted between 01-04-2015 en 03-07-2015

Jonathan Derei (10168435)

Instutue: ITFA

Faculty: FNWI - UvA

Supervisor: Jan Pieter van der Schaar

2nd assessor: Jan de Boer

Abstract

The information paradox has arisen when Hawking wrote down his theory for radiating black holes. Because of the radiation process of the black hole, the previously pure state of the matter inside the black hole evolves into a collection of radiation quanta that are in a fundamentally mixed state. The problem here is that unitary evolution, as quantum mechanics demands, doesn’t allow pure states to evolve in that way. Later research revealed another problem, the three postulates of complementarity as suggested by Susskind and Thorlacius cannot simultaneously be valid for old black holes. Almheiri, Marolf, Polchinski and Sully suggested that after the Page time, when half the entropy of the black hole has been radiated, a firewall forms at the black hole horizon. In this thesis I will give an overview of all these concepts.

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Populair wetenschappelijke samenvatting

Zwarte gaten zijn al jaren het onderwerp van het wetenschappelijk debat. Ondanks dat men nog nooit dicht bij een zwart gat is geweest denken we vrij veel te weten over deze objecten. Maar hoe komen we aan al deze informatie? Gedachtenexperimenten en het analyseren van bekende natuurkundige en wiskundige theorien hebben er voor gezorgd dat we zwarte gaten beter kunnen begrijpen.

Wat wetenschappers hebben ontdekt over zwarte gaten is dat, wanneer de brandstof van een zware ster opraakt, de uitwaartse kracht door fusie de inwaartse zwaartekracht niet meer kan tegenwerken. De materie van de ster zal richting het centrum bewegen en dus steeds com-pacter worden. Omdat de kromming van ruimte-tijd dicht bij comcom-pacter wordende objecten steeds groter wordt zal uiteindelijk zelfs licht hier niet meer vandaan kunnen ontsnappen. Op dat moment spreken we van een zwart gat.

Het snelste deeltje, een foton, kan dus niet aan een zwart gat ontsnappen. Stephen Hawking ontdekte in 1974 echter, uit de formules van de quantum mechanica en relativiteitstheorie, dat er wel degelijk straling van zwarte gaten af komt. Deze straling, tegenwoordig Hawking straling genoemd, zorgt voor een paradox die tot nu toe nog niet is opgelost. Maar wat houdt deze paradox precies in? Laten we een ster nemen die 10 keer zo groot is als de zon. Wanneer deze ster nog geen zwart gat is zouden we, als we de technologie ervoor zouden hebben, alle aparte deeltjes kunnen bestuderen en hierover alles te weten kunnen komen. Als we alles te weten kunnen komen over een systeem noemen we het ook wel een pure toestand.

Het proces waarbij zwarte gaten stralen gaat als volgt; op de horizon van het zwarte gat (de plek waarboven een foton net zou kunnen ontsnappen) worden door het uitrekken van ruimtetijd deeltjesparen uit het vaccu¨um gecre¨eerd en uit elkaar getrokken. Een van de deelt-jes verlaat het zwarte gat en de ander valt naar binnen en verdwijnt in de singulariteit (het centrum van het zwarte gat). Deze deeltjes zijn met elkaar verstrengeld en bij elkaar vormen zij een pure toestand. Door dit proces verliest het zwarte gat massa en zal het uiteindelijk ver-dampen. Omdat de deeltjes samen een pure toestand vormden, maar de helft van de deeltjes nu verdampt zijn kunnen we niet meer alle informatie achterhalen. Nu bevinden de overge-bleven deeltjes zich in een zogenaamde gemixte toestand. Problematisch is dat de Quantum mechanica verbiedt dat pure toestanden evolueren tot fundamenteel gemixte toestanden. Dit probleem wordt de Informatie paradox genoemd.

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1 Introduction

Black holes, even though people had not yet coined this name, were first theorized in 1796 by Pierre-Simon Laplace. Playing with the idea of an object that had a gravitational pull so big not even light could escape, he laid the foundation for the theory of objects which we now call black holes.

Black holes are particularly interesting to study because they let us think about some of the most extreme environment in the universe. While thinking of black holes we need to consider gravitation and therefore Einsteins theory of relativity but also quantum mechan-ics plays a role here. These two theories, although separately being very successful, seem to contradict each other. This became, in the case of black holes, apparent when Stephen Hawking found theoretical evidence that black holes can radiate. Why this is a problem will be discussed in this paper, but this is in essence the phenomenon from which the information paradox originates.

Before getting too technical, the idea behind the information paradox is that it looks like information is getting destroyed through evaporation of black holes, even though this is for-bidden in quantum physical processes. In this thesis the details of the information paradox will be described. Many sources have written about different aspects of this subject but there are still few that have made a document where all these aspects are explained in a, for bach-elor students, complete and comprehensible manner. Through this bachbach-elor thesis I will try to create such a document.

To begin with, I will start this thesis with a section that contains the necessary informa-tion for understanding the rest of this document. This includes an overview of Hawking’s work on Hawking radiation and for instance quantum information. In the following chap-ters the information paradox will be explained and explored by looking at the evolution of a black hole from its formation till complete evaporation. Some thought experiments will also be explored to give a good overview of the emergence and consequences of the information paradox. The final chapter is about the firewall, an observation that has shook up the field of black hole research in 2012 and made the information paradox more precise.

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2 Theory

To understand the information paradox we first need to discuss some theory about black holes from the perspective of general relativity, but also quantum mechanics. These two subjects lay at the foundation of the information paradox, but we need also look at the quantum information formalism which describes quantum information in bits. The creation and evolution of a black hole will be used as a thread throughout this chapter.

2.1 The black hole

At the end of the life of a star the outward radiation pressure that is generated through fusion becomes insufficient to keep the material, from which the star is made, from falling inwards. When all of the material has traversed the Schwarzschild radius it will become a black hole, which means nothing can escape from it when past the black hole horizon, which resides at the Schwarzschild radius. In a vacuum we normally describe space with a Minkowski metric, this has the form

ds2= −c2dt2+ dx2+ dy2+ dz2. (1) This function is used to determine line element distances between points in flat space-time. But, in the case of a black hole a metric is needed that describes a space-time that is curved around a point like mass. Such a metric is called the Schwarzschild metric

ds2 = − 1 −2GM r dt2+ 1 −2GM r −1 dr2+ r dθ2+ sinθdφ (2) and can be used for a non rotating, uncharged black hole. Within this metric we find two values for r at which the metric becomes divergent. This happens at r = 0 at the singularity and at r = 2GM at the black hole horizon. The second singularity at the horizon is an apparent singularity that emerges in the Schwarzschild metric, but can be avoided by a change of coordinates, for example in Kruskal-coordinates. In Kruskal-coordinates nothing special happens at the horizon distance, this means that someone crossing the horizon would not notice anything at that point. However, the central singularity at r = 0 is real because there is no coordinate system where r = 0 is not singular. We can look at the curvature in a coordinate independent way at r = 0, here the curvature invariant blows up showing that this is truly a singularity, at the horizon this is not the case. From the metric it can also be seen that when r goes from r > 2GM to r < 2GM the signs of the first two terms of the metric change. Here r changes from space-like to being time-like and t becomes space-like. This means that when something outside the black hole crosses the horizon it cannot be kept at a constant r but will proceed moving in the direction of the singularity, as normally objects

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Figure 1: The tipping of lightcones becomes so extreme at the black hole horizon so that space and time start to switch roles inside the black hole.

move forward through time outside a black hole. So nothing, not even light, inside a black hole can ever escape the singularity. This can be visualized by figure 1 where we see that in Minkowski space, far from the black hole, the light cone stands in it’s usual position, but while approaching the horizon the cones start tipping sideways. After crossing the horizon we see that the light cone has rotated more than 45 degrees switching the role of space and time.

2.2 Black hole thermodynamics

We now know that a black hole can be described by a singularity and a horizon, the second at least in Schwarzschild coordinates, but is there more to a black hole than only these properties? In the 1970’s J.D. Bekenstein thought about the possibility that black holes could have entropy. Through this example we can understand what his conclusion was. So, at first, let us throw a container filled with gas into a black hole. Upon reaching the singularity the entropy of this gas would disappear and thus lower the amount of entropy in the universe. This contradicts the second law of Thermodynamics, which says that the entropy of the universe cannot be decreased

dStotal≥ 0. (3)

Something else must be going on. When a container with gas is thrown into a black hole the mass of the black hole increases. The radius of the black hole is proportional to its mass Rs∝ M and thus to its surface area. From Hawking’s findings that the area of the black hole

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found a relationship that relates the entropy of a black hole directly to its surface area SBek=

A

4G. (4)

Assigning an entropy to black holes would mean that the second law of thermodynamics is not violated. The black hole would now in the case of the container with gas increase its entropy so the total entropy in the black hole and in the universe combined will not change

dStotal = dSBek+ dSuniverse≥ 0. (5)

Entropy is a measure of the number of possible ways in which a system can be arranged, from statistical physics we know an expression for the entropy

S = lnΩ, (6)

where Ω is the number of microstates, in this case of the black hole. We can calculate the the amount of microstates for a black hole with for example a Schwarzschild radius Rs ∼ 3km.

For this example we get

Ω = e4GA ∼ 1010 77

. (7)

This huge number of microstates implies that there are an enormous amount of configurations of black holes with a certain radius. You might think that these configurations can be found as deformation modes of the horizon because a black hole is a lot of empty space, but that doesn’t seem to be the case. It was discovered that for black holes there is no information near the horizon, this is also called the no-hair theorem, so these deformation modes cannot exist there. If there were deformations, they would decay exponentially in time and are radiated away as Hawking quanta, which we will discuss later on. It seems that there is only one possible microstate that defines the black hole, leading to an entropy of 0, instead of ln(101077). It is more probable that the entropy is at the singularity and thus configurations of states are different through quantum effects at the singularity. A physical quantity that is related to entropy is temperature, a year after Bekenstein’s discovery Hawking found that black holes indeed have a temperature.

2.3 Hawking radiation

It was believed that black holes were the safest vaults in the universe, at least in a sense that nobody could extract its contents. In 1974 Stephen Hawking published his theory about black hole radiation which, since then, has weakened that believe. With the findings of W.G. Unruh that a uniformly accelerated observer, when measuring a vacuum that is created at

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rest and stays at rest, sees a thermal bath, Hawking was able to predict that at the black hole horizon particles are radiated.

In Quantum Field Theory (QFT) one can describe a field which can be expanded into an infinite collection of quantum harmonic oscillators that are used to describe points in space and time. Exited states of these oscillators in this quantum field can be interpreted as parti-cles. The derivation of the form of the field and how this leads to Hawking radiation is quite tedious, so I will not explicitly show every step. A more detailed derivation of the following section can be found in the work of Mukhanov and Winitzki (Mukhanov and Winitzki 2004). The field can be written a form dependent on x and t through Fourier expansion

ˆ φ(x, t) = Z d3k (2π)3/2 1 √ 2ωk h ˆ a−_ke−iωkt+ik·x_{+ ˆ}_a+ ke iωkt−ik·xi_, ₍₈₎

where ˆa± are the creation and annihilation operators, this form is called the mode expansion of the field. To see what quantum effects appear in an accelerated frame, we can use Rindler coordinates τ and ˜ξ to rewrite the mode expansion in the following form

ˆ φ(τ, ˜ξ) = Z +∞ −∞ dk (2π)1/2 1 p2|k| h

e−i|k|τ +ik ˜ξˆb−_k + ei|k|τ −ik ˜ξˆb+_k i

, (9)

where ˆb±_k are the creation and annihilation operators defined for the frame of an accelerated observer. The goal of this exercise is to compare the vacuum created by an observer at rest with a vacuum created by an accelerated observer in their respective frames. We define the two vacua as

ˆ

a−_k|0_Mi = 0, and ˆb−_k|0_Ri = 0, (10) which says that the lowering operator ˆa−_k, acting on the Minkowski vacuum state results in 0, and lowering operator ˆb−_k acting on the Rindler vacuum state also gives 0. These two operators and vacua are inherently different and when acting on the other vacuum it would not give 0.

To see how the operators of one observer acts on the vacuum of the other these operators can be transformed through a Bogoliubov transformation that enables us to rewrite the ˆa±_k in terms of ˆb±_k and vice versa. If we now, for example, use this transformation on ˆb−_k and ˆb+_k we get ˆ_b− Ω = Z ∞ 0 dωαωΩˆa−ω + βωΩaˆ+ω (11)

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and ˆ_b+ Ω = Z ∞ 0 dωα∗ ωΩˆa+ω + βωΩ∗ aˆ−ω (12) where αωΩ= r Ω ωF (ω, Ω) (13) βωΩ= r Ω ωF (−ω, Ω) (14)

with F (ω, Ω) some auxiliary function. The measure for the amount of particles within a state is the occupation number operator which is defined as

ˆ

NΩ = ˆb+ˆb− (15)

and is needed to calculate the amount of particles an accelerated observer sees when looking at a Minkowski vacuum as created by a stationary observer.

By rewriting βωΩin terms of F (ω, Ω) it can be shown that

h ˆNΩi = nΩ= exp 2πΩ a − 1 −1 . (17)

From statistical physics this function can be identified as the Bose-Einstein distribution, with which the behavior of bosonic particles (integer spin) can be described. This function is normally written as nBE= exp E T − 1 −1 (18)

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here E = Ω and T =_2πa. What this means is that when an accelerated observer measures the Minkowski vacuum |0Mi it will be filled with particles and have a temperature expressed in

SI-units as

TU = a~

2πkBc

. (19)

This result is particularly useful because it has made it possible for us to calculate the temper-ature of a black hole. In the derivation of Mukhanov and Winitzki the coordinate systems of the stationary and the constantly accelerated observer are written in light cone coordinates, from which we get the following relations

¯

u ≡ t − x, ¯v ≡ t + x, u ≡ τ − ˜ξ, v ≡ τ + ˜ξ. (20) We can relate these coordinates by writing

¯ u = −e −au a , v =¯ eav a . (21)

In the case of a black hole we can write down a coordinate system that describes all space for r > 0 without a singularity at the horizon, this is called the Kruskal coordinate system and can be used, for example, for a freely falling observer. This coordinate system rewritten in light cone coordinates becomes

¯

u = ¯t − ¯r, v = ¯¯ t + ¯r (22) and is related to u and v by

¯ u = −4M exp − u 4M , ¯v = 4M exp v 4M . (23)

By comparing equation 21 to equation 23 we see that the acceleration of someone at the horizon as seen by a distant observer is a = _4M1 , this is called the surface gravity. With this acceleration Hawking calculated the temperature at the black hole horizon by using equation 19

TH = ~

8M πkBc

. (24)

So, according to a stationary observer far from the black hole, at the black hole horizon a thermal bath exists with a temperature TH. Particles that escape radially from the horizon

and travel into infinity are called Hawking quanta.

From Hawking’s conclusion that black holes radiate a paradox was created. Problems arise due to the entanglement between the constituents of the particle pairs. To see why this happens we first need to introduce some concepts of quantum information theory.

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2.4 Quantum information

We need to get some basic understanding about quantum information theory to comprehend the information paradox that will be discussed in the following chapters. In quantum infor-mation theory the quantum bit (qubit) is used as as a unit of quantum inforinfor-mation. The qubit is a two state system that can represent for instance the spin of a system like an electron which can be either up or down. A normal bit as used by computers can assume only one value ’0’ or ’1’ while a qubit can be in a superposition of the two states. The qubit

|ψi = α|0i + β|1i (25)

in this case represents a superposition of the state |0i and |1i with α and β the respective probability amplitudes. The probability, when measuring the qubit, of finding it in state |0i is |α|2 and |β|2 for it to be in state |1i. The sum of |α|2 and |β|2 must, as all probabilities do, add up to 1.

2.4.1 Density matrix formalism

In quantum mechanics a distinction is made between pure states and mixed states. A pure state is a state that can be written in the same way as equation 25 and can be written as a ket vector or as a sum of basis state vectors. For a pure state all information about that system is accessible. For mixed states this is different, the description of this state is a statistical one. To find out whether a state is pure or mixed the density matrix formalism can be used, where a density matrix is expressed as

ρ = |ψihψ|. (26)

These density matrices have some properties from which can be determined if a state is pure or mixed. For both mixed and pure states Trρ = 1. Trρ2 _{= 1 for pure states, and Trρ}2 _{< 1}

for mixed states. Now for example take the following composite state |φi = √1

2(|00i + |11i), (27)

where |00i is composed of two separate states |0Ai ⊗ |0Bi, where A and B represent the labels

for the Hilbert spaces HA and HB of the subsystems. The corresponding density matrix for

the state in equation 27 is ρ = 1

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It can be seen, when looking at the matrix form of the density matrix ρ2=     1 2 1 2 1 2 1 2     , (29)

that this is a pure state because Trρ2 = 1.

For such a composite state all information is known when evaluating the whole system. This is not the case when we only look at one of the two substates. By taking a partial trace, in which you let the degrees of freedom of the other system act on the full density matrix, you get a reduced density matrix

ρA= TrBρAB. (30)

If we apply this on the full state in equation 28 to get the reduced density matrix for subsystem A we get

ρA=

1

2(|0ih0| + |1ih1|). (31)

By taking the reduced density matrix ρA we look only at subsystem A, information about

subsystem B has been lost through this operation of tracing over the states of subsystem B. This can again be written in matrix form

ρA=     1 2 0 0 1₂     . (32)

From this it can be seen that Trρ2 < 1 and is thus mixed, or an ensemble of pure states. Subsystem A in this way is not pure, if we want to examine this state we can no longer make claims without any uncertainty. So, knowing everything of the overall system, which is in a pure state, does not necessarily mean that we know anything about its constituents.

2.4.2 Von Neumann entanglement entropy

A measure of quantum mechanical entropy that is contained by some system is the Von Neumann entropy

S(ρ) = −Tr(ρlnρ). (33)

It can give a measure of how mixed a system is and, as we will see in later chapters, how much information is available.

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For the following chapters about the information paradox we want to evaluate the Von Neu-mann entanglement entropy, here we look at separate subsystems and the entanglement en-tropy between these systems. Again we have a bipartite system split into two Hilbert spaces HA and HB. The entanglement entropy that for instance system A contains is

S(ρA) = −Tr(ρAlnρA). (34)

From this entanglement entropy we can learn some features of the state in question • For S(ρ) = 0 the state is pure

• For S(ρ) > 0 the state is mixed

• For S(ρ) = ln(N) the state maximally mixed (N is the dimension of the system, for an electron with spin up and spin down N=2)

In the case of the density matrix in equation 32 we can see that the entanglement entropy S(ρA) = −1₂ln(1₂) − 1₂ln(1₂) = ln(2), because it is a 2 state system, it is maximally mixed.

We have discussed some theory that will help understand the following chapters. What we have seen is how we describe space time around black holes and that they have are two distinct features, namely, the singularity and the horizon. Bekenstein assigned an entropy to these objects in order not to violate the second law of thermodynamics. The information paradox that arises is a consequence of what Hawking found out in 1974; black holes radi-ate. To see how this exactly happens we will use our understanding about the von Neumann entanglement entropy and apply this to the hawking radiation.

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3 The information paradox

Hawking’s realisation that black holes could radiate gave rise to a paradox that will be ex-plored throughout the following sections. Though in the last forty years many insights have been gained about the problem, till this day it hasn’t been fully solved, or there is at least not yet a clear answer or widespread consensus about what solves this paradox.

3.1 Black hole evaporation

In section 2.3 we saw that around the black hole horizon particles exist. An explanation of the underlying mechanism is that virtual particle pairs are created from vacuum fluctuations and because of stretching space-time around the horizon the particle pairs are able to separate and become real. So, a number of particles are able to escape the black hole, while their counterparts will fall back into the black hole. Before going further, let us demarcate two regions that play an essential role in the information paradox. The region of space inside of the black hole is HA, with A the reservoir of particles behind the black hole horizon. The

collection of particles in the region just outside the horizon HB we call B.

The Hawking radiation particles can be described in the qubit formalism. As we have seen, observers in spaces of different curvature can define a vacuum, but the vacuum for an ob-server in the accelerated frame, which we call D, would contain a number of particles when observed by the stationary observer ’B’, depending on his acceleration. The vacuum of D can be described in terms of the states that B can be in

|0i_D = C|0iB+ C2ˆb+ˆb+|0iB+ C4ˆb+ˆb+ˆb+ˆb+|0iB+ .... (35)

Where each ˆb+ working on the vacuum defines a particle, each term contains an even amount of particles because they are created in pairs. For Hawking radiation we can differentiate between the region inside the horizon and outside the horizon so that instead of ˆb+ˆ_b+ _we

write ˆa+ˆb+ and |0iB as |0ia⊗ |0ib.

We can now, for the first pair(s), write the state of the Hawking radiation |Ψi = C(|0i_b₁ ⊗ |0i_a₁ + γˆb+₁|0i_b₁⊗ γˆa+₁|0i_a₁ +γ

2 2 ˆb + 1ˆb + 1|0ib1 ⊗ γ2 2 ˆa + 1ˆa + 1|0ia1 + ...) (36)

When the creation operators act on the vacuum state it will become an excited state, we write this as

ˆ_b+

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We only use the leading order terms of this process because the factor γ will suppress the higher order terms

|Ψi = √1

2(|0ia1|0ib1 + |1ia1|1ib1). (38) The subscripts a1 and b1 denote the first emitted particles respectively inside and outside

the horizon. This process continues until the black hole has completely evaporated into N particles

|Ψi = √1

2(|0ia1|0ib1 + |1ia1|1ib1) ⊗ 1 √

2(|0ia2|0ib2+ |1ia2|1ib2)⊗ (39) ... ⊗ √1

2(|0iaN|0ibN + |1iaN|1ibN). (40) The entanglement between the first two particles a1 and b1 can be calculated by plugging

equation 38 into 26 and then using equation 34 to obtain

S(ρa1) = ln2. (41)

As we can see, the entropy is equal to ln(N), so there exists a maximal entanglement between a1 and b1. After N particles have been radiated the entanglement between a1, ..., aN and

b1, ..., bN is

S(ρa1,...aN) = N ln2. (42)

We see that after each radiated particle the entanglement of subsystem A with subsystem B grows with ln2. For A or B to be in a pure state the entanglement entropy has to be 0, but we see that after complete evaporation of the black hole B is in a mixed state. Here lies the paradox. The radiated quanta from the black hole are entangled with something non-existent, because the black hole and thus also subsystem A have disappeared. What this means is that a pure state, for example the system of all known particles that make up the black hole, after some time evolved to a fundamentally mixed state that can only be described by a density matrix. This is something that is not allowed by the rulebook of quantum mechanics. Quantum mechanics demands unitary evolution for states, but a pure state evolving into an inherently mixed state cannot be described by any unitary evolution operator. To avoid this problem we could avoid unitarity and thus allow pure states to evolve into fundamentally mixed states. Let us explore two other propositions that have been made to avoid this problem, remnants and corrections to the Hawking state.

3.1.1 Remnants

Hawking, after discovering this peculiarity, proposed that quantum mechanics had to be altered and would become a theory that allows non-unitary evolution operations to exist and

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mixed states to be a basic feature when evaluating quantum mechanics in gravity. The general view nowadays is that quantum mechanics is correct and thus an other principle has to give. It has also been proposed that remnants could exist. After a great deal of evaporating, the black hole could become a plank scale remnant. This remnant would contain the information that would purify the radiation system, which we argued to be in a mixed stated. The radiation would still be in a mixed state, but it is mixed with something that is not accessible to us. This is not a fundamental problem in physics, just a technical one. The problem with this is the fact that this plank scale remnant would have an enormous amount of possible states in such a small region. This possibility is not expected behaviour in normal physics and leads to loop divergences (Mathur 2011). A black hole is an maximum entropy object, if we take a part of space the maximum possible entropy in that part is the same as a black hole would have if it covered that same space. When we would allow black holes to evolve into remnants this maximum entropy bound is exceeded.

3.1.2 Corrections of the leading order state

The Hawking state given in equation 38 is the first order state. The Hawking state can be made much more accurate by writing down corrections to the leading order. Some believe that these corrections could solve the information paradox, but this is incorrect. To show that the paradox is not solved by these corrections we follow a crude version of the proof of Mathur (Mathur 2011). Mathur argues that we can define the following; corrections are small when the second order term of the Hawking state is much smaller than one

|Ψcorri = |ΨHawkingi + |Ψ2nd−orderi, 1. (43)

He also argues that the corrections must be small, or else the solar system limit breaks down (Mathur 2008). He gives a definition of the solar system limit being:

There must exist a set of niceness conditions N containing a small parameter such that when is made arbitrarily small then physics can be described to arbitrarily high accuracy by a known, local, evolution equation. That is, under conditions N we can specify the quantum state on an initial spacelike slice, and then a Hamiltonian evolution operator gives the state on later slices. Furthermore, the influence of the state in one region on the evolution in another region must go to zero as the distance between these regions goes to infinity (locality)

The ’niceness conditions’ restrict curvature to become too large, so that quantum gravity does not need to be considered. This is also what we expect when we do experiments in the solar system, because normal physics works and the effect of quantum gravity are thought

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to be too small in this limit, so we normally do not worry about this. In short, what this comes down to is that in the solar system limit we can expect normal physics and therefore corrections are small.

Let us first split the black hole into different subsystems. We can differentiate between three subsystems A, B and C.

• Subsystem A: All the previously emitted Hawking quanta up until the time we evaluate the black hole.

• Subsystem B: The newly created Hawking quantum that is to be radiated from outside the black hole horizon.

• Subsystem C: The newly created Hawking quantum behind the black hole horizon that shall fall further into the black hole and eventually into the singularity.

We want to see if corrections, applied to this process, would eliminate the entanglement between all the radiated particles (subsystem A and B) with the interior of the black hole (subsystem C) so that SA+B = 0, or that these corrections would not decrease the

entangle-ment entropy substantially. We define the entropy of subsystem A, which contains all the radiated particles up to this moment in a first order approximation, as

SA= S0. (44)

In his derivation he obtains the entropy bounds on the different systems and the combination of systems. When combining all his results he shows that the entanglement entropy of all the quanta outside the black hole, subsystems A and B, with the rest of the black hole meets the following relation

SA+B ≥ S0+ ln2 − 2. (45)

What this inequality tells us is that the entropy of the combined system of all the previously radiated particles plus the newly radiated particle will increase with at least ln2 − 2. Be-cause 1 the entanglement entropy of the radiation cannot decrease substantially, due to these small corrections to the leading order Hawking state. This means that the state of the radiation does not become pure.

Small corrections, as we have seen in this section, will not resolve the information para-dox, the entanglement seems to only increase and the state of the radiation stays mixed or will eventually become a remnant. At least, this is the case, unless we assume that we can discard the restriction on the corrections to be 1.

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3.2 Black hole complementarity

Having a pure state evolving into a fundamentally mixed state is not a desirable event, it is even forbidden by the quantum mechanical principles that demand unitary evolution of a state. We have seen that small corrections will not change anything about this and remnants also come with some problems. Susskind, Thorlacius and ’t Hooft proposed a solution to this paradox. They propose that an observer that stays outside a black hole sees the horizon as a hot membrane that stores in-falling quanta and eventually re-emits them as Hawking radiation. A free falling observer will not observe the same thing, complementarity assumes the equivalence principle not to be violated and for sufficiently large black holes the curvature of space is negligible, thus the region of the horizon is not a special place. An in-falling observer wouldn’t see any particles or even notice crossing the horizon. The following postulates have been set up to describe complementarity (Susskind 2012):

• Postulate1 : The process of formation and evaporation of a black hole, as viewed by a distant observer, can be described entirely within the context of standard quantum theory. In particular, there exists a unitary S-matrix which describes the evolution from infalling matter to outgoing Hawking-like radiation.

• Postulate2 : Outside the stretched horizon of a massive black hole, physics can be described to good approximation by a set of semi-classical field equations.

• Postulate3 : The global event horizon of a very massive black hole does not have large curvature or energy density. Therefore, an infalling observer will see nothing extraordinary when crossing the horizon, in accordance with the equivalence principle of general relativity.

When we follow these postulates it looks like information exists simultaneously inside and outside the black hole. The infalling observer carries information into the black hole, while the external observer sees it accumulate on the horizon. Finally the external observer will see the information coming out of the black hole for him to retrieve. At first we could say that this violates the quantum mechanical no-cloning theorem, which says there cannot exist two copies of a quantum state, This violation is not the case because the Hilbert space inside the black hole does not overlap with that outside the black hole. In other words an observer cannot see both copies simultaneously.

We can try to understand complementarity and give some arguments why the no-cloning theory is not violated by setting up a thought experiment involving Bob and Alice, who both start out outside the black hole. If we let Alice carry a qubit into a black hole she will observe

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it as being in her possession. Bob waits just outside the black hole in order to capture and extract the Hawking radiation to decode Alice her bit. Now Bob jumps into the black hole to check if he has a precise copy of Alice her bit. The problem is that after Alice has jumped in she has only limited time to send information about her bit before reaching the singularity. Hayden and Presskil (Hayden and Preskill 2007) say that because we do not know enough about quantum gravity, we cannot with certainty say whether or not it is possible for Alice to send her message in order for Bob to receive. They do show that it is probably unlikely that she can successfully send it without using super-planckian frequencies, which to our current understanding we cannot analyze, transmit or receive. If this is true then the no-cloning theorem would be abided, because Bob and Alice are not able to compare the qubits and find out that they are the same, and complementarity could be a solution.

3.3 Where is the information?

We are interested in finding out how to retrieve information from the black hole. To see from what system and at what time we can retrieve information we use the findings of D. Page (Page 1993a) about entropy in subsystems and his work about information in black hole radiation (Page 1993b). We can treat the complete system of the black hole and its Hawking radiation as a bipartite system that is in a pure state.

H_r,bh= Hr⊗ Hbh. (46)

We now define n as the dimension of the black hole and m as the dimension of the radiation system. The combined system ’mn’ is in an overall pure state (Tr(ρ2

r,bh) = 1), but the two

subsystems separately are in a mixed state that can be described by the reduced density matrices ρr and ρbh. These systems respectively have an entanglement entropy of

Sr= −Tr(ρrlnρr) = Sbh= −Tr(ρbhlnρbh). (47)

That Sbh= Sr follows from the triangle inequality

|Sr− Sbh| ≤ Sr,bh≤ Sr+ Sbh, (48)

where Sr,bh= 0 because the overall system is in a pure state. The dimension of the radiation

subsystem m∼ esr _{with s}

r the thermodynamic radiation entropy. The same is valid for

the black hole subsystem but with the black hole entropy sbh. The way that Page defines

information is the difference between the maximum entropy and the entanglement entropy Ir = lnm − Sr' sr− Sr Ibh= lnn − Sbh' sbh− Sbh. (49)

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The question that is important for a better understanding of the information paradox is: how much information does the radiation subsystem contain at certain times during evaporation? According to Page, for the configuration when m ≤ n, the average information in the radiation subsystem is Ir = lnm + m − 1 2n − mn X k=n+1 1 k. (50)

In the case of 1 m ≤ n this becomes Ir'

n 2m∼ e

sr−sbh_, ₍₅₁₎

while for m ≥ n we find

Ir = lnm − lnn + Ibh∼ lnm − lnn +

n

2m. (52)

In fig 2 we see the the average entanglement entropy and average information of the radiation subsystem plotted against the thermodynamic entropy lnm. This plot has been made for an overall random pure system with a dimension mn = 291600 where the subsystems m and n are coupled.

When the dimension of the Hilbert space associated with the radiation subsystem is small compared to that of the black hole, it can be seen from equation 51 that there is little infor-mation available in the radiation. If after some evaporating the radiation subsystem becomes much larger than the black hole, it contains most of the information. This however only becomes apparent when all the radiation is being observed at the same time. If we look only at a small collection of the radiation quanta it of course carries no information, it is namely concealed in the correlations between the other constituents of the radiation subsystem. From figure 2 it can be seen that information starts leaving the black hole after half of the entropy has been radiated away. The time at which this happens is called the Page time.

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Figure 2: The dotted lines show the average entanglement entropy and the information con-tained by the radiation subsystem, this is plotted against the thermodynamic entropy of the radiation on the horizontal axis. The thermodynamic entropy of the radiation system is equal to the dissipated black hole thermodynamic entropy sbh= _4GA during radiation, this graph is

thus read from left to right.

We can now make a distinction between what we call young and old black holes which are respectively younger or older than the Page time. It seems, through this discovery, that the Hawking radiation subsystem will eventually be purified, allowing information to leave the black hole. Purity, which was the first of the three postulates of complementarity, would in this sense be satisfied after the black hole has radiated half of its thermodynamic entropy sbh= _4GA away and reaches its Page time.

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4 Bob and Alice

Before we continue we ought to get a good overview of the information and insights from the previous chapters. We have seen that, because of Hawking’s discovery that black holes radiate, a paradox has arisen. Throughout the process of evaporation the Hawking quanta that are radiated become entangled with the black hole interior, but when the black hole has evaporated completely, the radiation will still be entangled. Except, there is nothing to be entangled with, leaving the radiation in a mixed state. This is not allowed in quantum mechanics because a state is bound to unitary evolution. Later on we saw that through the analysis of Page it becomes apparent that in the case of the radiation and black hole sub-systems that the entanglement entropy, after the Page time, would decrease and thus purify the system of the radiation quanta. After this Page time, because the radiation subsystem becomes a pure state, information is released from the black hole. Instead of summarizing all of the main points, a story or thought experiment, containing all the concepts that have been discussed, might be more didactical:

Bob Marley, having shot the sheriff, denies having also pulled the trigger on the deputy. The FBI doesn’t believe him and instructs detective Alice to keep an eye on Bob. One night Bob drives to a hiding place where he hid his spaceship. Bob embarks the ship and the detec-tive follows him. After a short faster than light (ftl)-trip Bob makes a stop at a black hole, where he quickly dumps the deputy’s body as well as the murder weapon. Bob, having to fuel, stops at the nearest inhabited planet and decides to spend the night. Alice sneaks out of the spaceship and not knowing a lot about black holes decides to visit the local police station to explain the situation. They arrange a meeting with the best black hole expert of the planet.

Back at the bureau Alice is known for her impatience, so she asks the expert if she can’t just jump into the black hole and retrieve the information herself. The expert advises her to hold her horses and just think this through first. He points out that if you cross the black hole horizon there is no going back. The expert tells the detective not to worry, because black holes have no secrets for him. The BH-experts before him have observed the black hole for a long time, even before formation. In all those years they have collected all the Hawking radi-ation ever emitted by the black hole. The expert promises that if Alice waits for two weeks he can have all information about the evidence of the murder reconstructed. The detective asks him why it takes two weeks. The expert tells her she is very lucky because the black hole has almost lived past its Page time, where half its entropy has been radiated away.

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At the moment that the black hole formed it was in a pure state, while the Hawking pairs created at the horizon after creation where in a maximally entangled state. The Hawking radiation that the BH-experts have been meticulously collecting was useless, because it didn’t carry any information about the interior of the black hole. But luckily for Alice the Page time was near and the new radiation quanta would gradually purify the quanta in their collected reservoir and thus information could be extracted. In order for the expert to extract and decode the information from the radiation he must examine all the radiation quanta at the same time, because the information can be found in correlations between all the radiation quanta. After two weeks the expert found out that Bob indeed killed the deputy, with this information Alice returned home and put Bob behind bars!

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5 The Firewall

Almheiri, Marolf, Polchinski and Sully (AMPS) in their paper (et al. 2013) argue that the pos-tulates of complementarity cannot always be simultaneously correct. Their findings started a big discussion in the scientific world. The argument of an emerging firewall at the hori-zon, which we will talk about in the next paragraphs, sharpened our understanding of the information paradox.

5.1 What is the firewall

To understand the argument of AMPS we must first construct a setup. I will extend the ex-ample of the previous paragraphs by adding another system R, containing the early radiation quanta. As we can see in figure 3, which represents the black hole and the space around it, A denotes the collection of quanta inside the horizon, B those just outside and R now contains all quanta that have been radiated earlier. We assume unitary evolution, this means that the system of matter that formed the black hole was in a pure state and that the radiation from that black hole must ultimately be in a pure state. We have seen in the work of Page that the radiation subsystem evolves to a pure state after the Page time.

Figure 3: The three subsystems around the black hole; A (particle modes inside the horizon), B (particle modes outside the horizon) and R (the early radiation quanta).

At first, after formation of the black hole, the subsystems A and B are maximally entangled, with S= N ln(2), as they are assumed to be in normal Minkowski vacuum. We know from Page’s argument that in an overall random pure state a small subsystem is always nearly maximally entangled. This means that subsystems B, after the Page time, when more than half of the thermodynamic entropy of the black hole has been radiated away, must be max-imally entangled with subsystem R. But is this really possible? Monogamy of entanglement

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says that when a system is maximally entangled with another system there cannot exist any entanglement between any of those systems with a third system. What this suggests is that if B and R, so the whole system outside the black hole, are in a pure state and thus maximally entangled, there cannot be any entanglement between subsystems A and B anymore. The absence of entanglement between subsystem B and A means that this region of space-time is not the Minkowski vacuum state but it is rather in a state of excitation. To show this I will use the lectures on black holes of D. Harlow (Harlow 2014). When we evaluate points in a vacuum by correlation functions we see that one point functions contain no correlation

hΩ|φ(x, t)|Ωi = 0. (53)

This is expected, we don’t see a correlation when evaluating one point, but this is different for a two point function evaluated at the same time

hΩ|φ(x, 0)φ(y, 0)|Ωi 6= 0. (54) These correlations are strongest between closely neighbouring parts in space, meaning that the amount of entanglement entropy between points like A and B must be high. When two of such points in space become uncorrelated there arises a discontinuity at x = 0 (the horizon in the case of black holes). Because the Hamiltonian

ˆ H = 1

2 Z

d3x π(x)2+ ¯∇φ(x) · ¯∇φ(x) + m2φ(x)2 (55) contains a gradient of φ(x) which becomes divergent at x = 0 we obtain an infinite energy solution. We now see that after the Page time, when the entanglement between A and B decreases, there must be an enormous amount of energy located at the horizon. This is what AMPS calls the firewall.

A traveler who would want a (arguably) save passage into a black hole must, with this knowledge, do measurements on the subsystems to decide to go in or escape. Subsystem A lies behind the horizon and would not be a good choice to measure first, the traveler should rather measure subsystem R first. When all the quanta in the early radiation have been collected one can search for correlations between these quanta and those of the late radiation in subsystem B. If he finds no correlations whatsoever it means that subsystem A and B must still be entangled and at the horizon he will find the Minkowski vacuum, but when there are correlations it means that B is entangled with R. In case the radiation systems are entangled the traveler should turn around and escape from the black hole, because he would find the space around the horizon in a highly excited state where he, upon crossing the horizon, would be terminated by a firewall.

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It seems that the postulates of complementarity cannot all be true at the same time for old black holes (after its Page time). When we insist on unitarity (postulate 1) and thus eventually on purification of the complete radiation subsystem, it is not possible to main-tain the general relativistic idea that a freely falling observer would notice anything upon crossing the black hole horizon (postulate 3). The arguments of AMPS have sharpened our understanding of the information paradox. That after the Page time, some say at even earlier times, a firewall would exist at the horizon distance is not believed by everyone to be physical, but might be some phenomenon that arises because of our lack of understanding of quantum gravity. In any case, it shows us much knowledge still is to be gained through studying black holes.

5.2 Entanglement; a bigamist’s game

We can again set up some thought experiment to recap the concepts of the previous chapters. Let us think about a machine that can extract and separate particle pairs from the vacuum, let us call it the ’Vacuum Energy Extraction Device’ (VEED). The VEED separates entan-gled particles and subsequently emits them in opposite directions. Somewhere extremely far from the device on both sides these particles accumulate and form a black hole. Because the constituents inside the black hole on the left are maximally entangled with the constituents inside the black hole on the right we say that the two black holes are maximally entangled. When we have sufficiently large black holes the VEED is turned off. After the black holes have formed, ingoing and outgoing Hawking quanta will be produced at the horizon. These quanta are in their turn assumed to also be maximally entangled, in order for the space around the horizon to be in the vacuum state. This is a problem, namely, systems may not indulge in bigamy but must always stay monogamous to each other. We see that the black holes are maximally entangled, this means that the Hawking quanta inside and outside the black hole cannot be maximally entangled. In the Minkowski vacuum the Hawking quanta, or virtual particle pairs are entangled, this means that the state of the space between the quanta must be excited. Someone falling into one of these black holes would encounter this excited states and would be incinerated by the firewall.

As we have seen here, there is a conflict between the theory of quantum mechanics and general relativity. As AMPS has shown, two postulates from these theories cannot be simultaneously true. The phenomenon that follows from this contradiction is called the firewall.

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6 Discussion & conclusion

In this concluding summary I give a short recap about all the concepts described in this the-sis. We have found out, from the work of Bekenstein, that black holes have an entropy. This had to be the case, because else black holes would violate the 2nd law of thermodynamics by destroying entropy. Stephen Hawking confirmed this when he wrote down his theory about radiation from black holes. What he found was that at the horizon of black holes particle pairs could be created, which as we have seen leads to Hawking radiation. But this, in its turn, lead to the information paradox, which has still not been solved. With every creation of a particle pair the entanglement entropy between the radiation and the black hole seems to grow. The purity of the radiation system, after the black hole has evaporated is needed, in order to comply to unitary evolution, but this seemed not to be the case. The radiation seemed to be in a fundamentally mixed state, because the entanglement entropy of the ra-diation remains, but there is nothing to be entangled with. The propositions for solutions that have been given have still been unsuccessful in solving the paradox. We have seen rem-nants and corrections to the first order hawking state as propositions, corrections can’t solve the problem as Mathur shows, but remnants could be possible, although there are still some problems. Complementarity, for a long time, was thought to solve the information paradox, but in 2012 AMPS argued that the postulates of complementarity couldn’t be simultaneously correct. Their arguments lead to the birth of the firewall, which incinerates all that crosses the black hole horizon. This restarted the debate on the information paradox. We see that every time the paradox changes a little and becomes better defined, but with each change we hopefully get closer to the solution.

Some proposed solutions of the paradox are mentioned in this document, but there is a vast amount of other possible solutions that have been proposed. In the field of string theory, for instance, there are propositions I have not mentioned, like the fuzzball. In the two texts of S. Mathur, which you can find in the references, these propositions are described. This could, for a bachelor student who had no prior knowledge of string theory, be too complex, therefore I have not included this in this bachelor thesis. For further research or an extension on this text I would recommend a study on these different propositions.

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Acknowledgements

I would like to dedicate some words to Jan Pieter van der Schaar who has guided me through and provided me with relevant papers about the information paradox. I thank him for his time and help. Another word of thanks goes out to my friends, family and of course my lovely girlfriend for their support.

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References

A. Almheiri et al. Black holes: Complementarity or firewalls? April 2013. arXiv:1207.3123v4 [hep-th].

D. Harlow. Jerusalem lectures on black holes and quantum information. November 2014. arXiv:1409.1231v2 [hep-th].

P. Hayden and J. Preskill. Black holes as mirrors: quantum information in random subsys-tems. JHEP 0709:120,2007, September 2007. arXiv:0708.4025v2 [hep-th].

S.D. Mathur. What exactly is the information paradox? Lect.Notes Phys.769:3-48,2009, March 2008. arXiv:0803.2030 [hep-th].

S.D. Mathur. The information paradox: A pedagogical introduction. Class.Quant.Grav.26:224001,2009, January 2011. arXiv:0909.1038v2 [hep-th].

V.F. Mukhanov and S. Winitzki. Introduction to quantum fields in classical backgrounds. 2004.

D. Page. Average entropy of a subsystem. Phys.Rev.Lett.71:1291-1294,1993, August 1993a. arXiv:gr-qc/9305007v2.

D. Page. Information in black hole radiation. Phys.Rev.Lett. 71 (1993) 3743-3746, August 1993b. arXiv:hep-th/9306083v2.

L. Susskind. Singularities, firewalls, and complementarity. August 2012. arXiv:1208.3445 [hep-th].

The Black Hole Information Paradox