The Information Paradox and Bose-Einstein Condensate Black Holes

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University of Amsterdam

MSc Physics

Theoretical Physics

Master Thesis

The Information Paradox

and

Bose-Einstein Condensate Black Holes

by

Isa´ıas Rold´

an

10901388

Supervisor/Examiner:

Dr. Jan Pieter van der Schaar

Second Supervisor: Dr.Ben Freivogel

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Abstract

The semi-classical gravity approach for describing quantum field theory in fixed clas-sical space leads to the emission of Hawking radiation by black holes. The radiation energy can only come from the black hole mass and in consequence it will evaporate after a finite time. We will obtain this result from the Unruh effect and then show that it im-plies an information paradox that cannot be solved in the scope of semi-classical gravity. We will discuss a proposal by G.Dvali and C.Gomez[17][18] for describing black holes as Bose-Einstein condensates in order to avoid the information paradox.

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1 Introduction

It has already been four decades since Hawking derived that general relativity black holes radiate particles in the same form as a thermal body would do [9]. Even though this result was obtained long time ago, it is not completely understood yet. As a consequence of the emission of particles, the black hole will lose its mass and will reduce its size until it finally evaporates. This outcome is in contradiction with the principle of unitarity from quantum field theory [7], which will lead to a paradox where the equivalence principle is in contradiction with the unitarity principle, therefore we will have to drop one of them in the microscopic description of a black hole. As we will see, it is not easy to renounce any of this principles, since they play an important role in general relativity and quantum field theory respectively. In this thesis we will discuss the information paradox, which arises from black hole evapo-ration after emitting Hawking radiation. In order to do this, we will need some background in general relativity and quantum field theory, since the first is unavoidable when dealing with gravitation, and therefore with black holes, and the second one describes relativistic quantum phenomena, which when applied at the event horizon of the black hole, will lead to the radiation of quanta with a thermal spectrum. We will keep our discussion simple, introducing first the closely related Unruh effect in §3, in which an observer with a constant proper acceleration observes a thermal spectrum for a temperature proportional to its proper acceleration. We will find this discussion quite useful for arguing in §4 that, as a consequence, a black hole emits also radiation corresponding to the thermal spectrum of a black body, with temperature inversely proportional to the radius of the black hole. In this section, without diving into the details of how this radiation is described, we will arrive to the conclusion that a new physical description of the black hole will be needed at some point, in order to avoid the information paradox, after we discuss the main obstacles for solving it, following Mathur reference[7]. A proposal for this new description is a Bose-Einstein condensate description of the black hole, proposed by G. Dvali and C. Gomez [18][20][17]. We will introduce this proposal in §5 and discuss its effectiveness for coping with the information paradox. Finally we will end with a summary and some conclusions.

1.1 General Relativity

A fundamental ingredient which we will need for studying black holes is the theory of general relativity, as it is the main reference of any attempt to describe gravitational effects1. Its successful predictions over the last century, including the latest measurement of gravitational waves [10], support the geometric description of gravity that the theory poses. In this section, we will remark on the concepts of general relativity that we will need in our discussion about Hawking radiation. The reader who is unfamiliar with the subject can consult any general relativity introductory book, such as, for example, [1].

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We have not forgotten classical Newtonian gravity, but we will consider it a limiting case of general relativity.

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1.1.1 Formalism

General relativity describes gravity as a consequence of the relation between matter and space-time geometry. This relation can be briefly stated with the Einstein equation.

Rµν−

1

2Rgµν = 8πGTµν (1)

Of course in order to understand this statement it is needed to know what each one of its components means. The reader should have in mind that in the notation used in this project we will take the speed of light as c = 1. At the right hand side, G is the classical Newton gravitational constant and Tµν is the energy-momentum tensor, which gives us information

about the matter at a certain location, and it will be defined by the theory that describes the matter we are dealing with. At the left hand side we have the Ricci tensor Rµν and its trace,

the Ricci scalar R = Rµµ , which describe the “curvature” of space-time, their definition can

be found in any general relativity book. Finally, also at the left side, we have the space-time metric gµν , to which we will pay special attention, since it contains all the information about

the space-time geometry2,which is a central object in general relativity. We can get an idea about what the metric represents if we show how it is related to the distance between two points in any geometry. The distance between two points that are infinitesimally separated, is given by,

ds2 = gµνdxµdxν (2)

Where xµ are the coordinates that map the studied geometry. Then, the metric tells us how to measure distances in a given geometry and in particular we will be interested in space-time geometries. One of the most popular metrics is the Minkowski metric ηµν, which

corresponds to flat space.

ds2 = ηµνdxµdxν = −dt2+ dx2+ dy2+ dz2 (3)

In this expression, we are also fixing the signature of the metric that we will use.

In short, Einstein general relativity describes the geometry of the space-time studied, and tells us how it is intrinsically related to the matter that occupies it.

1.1.2 The Equivalence Principle

General relativity theory is constructed from the equivalence principle, which states : The outcome of any local non-gravitational experiment in a freely falling laboratory is in-dependent of the velocity of the laboratory and its location in space-time

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This principle tells us that any observer who is following a free-falling trajectory will observe the same physical laws as another free-falling observer, locally. Since an observer in flat space without any acceleration, i.e in an inertial frame, is indeed following a free-falling trajectory, any free-falling observer will observe locally the same physical laws as in an inertial frame. The term local is quite crucial in this statement,since the equivalence principle is really saying that there always will be a small enough region of space-time around the observer where the above will be true. How small this region is will depend on the specific geometry of the space-time studied. Intuitively, the less curved the space-space-time is, the more similar it will be to flat space in a small enough region. Therefore an important quantity to estimate how much we can trust the equivalence principle in certain region of space-time will be its curvature. We will apply these concepts in §4 in order to argue that black holes should emit particles with a thermal spectrum, and we will specify the conditions that should be satisfied for the validity of this result.

To sum up, we will have to use the general relativity formalism if we want to describe physics related to gravitational phenomena, such as a black hole. In this description, the equivalence principle will have to be accepted explicitly or implicitly.

1.2 Quantum Field Theory

The other fundamental ingredient which we will need for studying what happens at the mi-croscopic level near the event horizon of a black hole, is quantum field theory. This theory is usually used to describe many-particle quantum systems in terms of fields that can take different values at every point of space-time. Particles in this description are interpreted as possible excitations of these fields, and in order to ensure the quantum nature of the studied system, these fields are represented by quantum operators that satisfy the canonical commu-tation relations with their conjugated momenta, as defined in the Hamiltonian formalism. In this section, we will remark on some concepts of quantum field theory that are of interest for our discussion about Hawking radiation and the information paradox. Again, the reader who is not familiar with this formalism is recommended to read literature about the subject [2][3].

1.2.1 Formalism

As the reader can probably imagine, the fundamental object of quantum field theory are the fields. The field configurations will encode the information about the studied system. A simple case is a free real massless scalar field φ in d + 1 dimensions, which is described by the following action:

S = 1 2

Z

dxd+1∂µφ∂µφ (4)

As in classical field theory, we will apply the minimal action principle, leading to the following equation of motion for our field:

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which we immediately recognize as the wave equation. Therefore, the field φ can be written as a Fourier expansion of plane wave modes with different momenta k. The coefficients of this expansion will be quantum operators that satisfy the canonical commutation relations.

[ak, a †

k0] = δ(k − k0)

[ak, ak0] = 0

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Note that, in the notation used in this thesis, we set the Planck constant ~ = 1. Now we can expand our field.

φ = Z _dkd √ 2π√2k0 h eikxak+ e−ikxa†_k i (7) The operators ak and a

†

k are known as the annihilation and creation operators, and they

are interpreted as eliminating and adding a particle respectively, when they act on a state. A relevant state in most of the theories will be the vacuum state |0i, defined as the state from which you cannot eliminate any more particles ak|0i = 0. The vacuum allows us to write a

basis of states for the system, in terms of the creation operators.

|N i = {N } Y k 1 √ nk! (a†_k)nk_|0i ₍₈₎

where nk holds for the number of particles of momentum k in the state, and N represents

a configuration of the system.

In summary, in quantum field theory particles are excitations of fields that fill the space. These fields have a quantum nature, and therefore they should be described in the quantum language of operators. To find these operators, the field operators are expanded in a set of operators whose coefficients are mode functions that solve the equations of motion of the field.3.

1.2.2 The Unitarity Principle

One of the postulates of quantum mechanics is the unitary principle[24]. The unitarity principle states that the evolution in time of any state of the system should be described by the transformation of a unitary operator |ψ(t)i = U (t0, t)|ψ(t0)i, such that U U†= 1. Then,

we can check that if the state |ψ(to)i was initially normalized hψ(t0)|ψ(t0)i = 1, then the

probability is conserved.

hψ(t)|ψ(t)i = hψ(t0)|U†(t0, t)U (t0, t)|ψ(t0)i = hψ(t0)|ψ(t0)i = 1 (9)

Conservation of probability is not the only property of unitary operators that we will be interested in. They also conserve the information of the state they act upon. We can see this property in the fact, that by definition, unitary operators are invertible, therefore after evolving an initial state with the unitary operator U as |ψfi = U |ψii, we can always

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find the original state from the final state, by acting over it with the inverse |ψii = U−1|ψi.

Finally, the property of unitary operators which we will recall further in our discussion of the information paradox in §4.4, is that they can only transform a pure state into another pure state.

A pure state is any quantum state that can be described by a wave-function |ψi. In contrast, a mixed state is a state with a probability pi of being in the pure state |ψii. In order to

describe mixed states, we have to use the density operator formalism, which we will introduce in appendix §A, but we will give an overview of some of its properties here . The density operator for a pure state |ψi is defined as,

ρp= |ψihψ| (10)

while for a mixed state it will be defined as a linear sum of pure density operators weighed by the probability pi of finding the corresponding pure state |ψii.

ρm=

X

i

pi|ψiihψi| (11)

where for consistency the probability factors have to satisfy P

ipi = 1. In general, the trace

of any density operator has to be one, i.e. Tr{ρ} = 1 , but only if a density operator is pure, the trace of the squared operator will also be one Tr{ρ2_p} = 1 . We will use this property to identify whether an operator represents a pure or a mixed state. We are going to be especially interested in this formalism because it allows us to describe what an observer, who can only perform measurements in a Hilbert subspace HA of a larger Hilbert space H = HA⊗ HB

observes for a state |ψi = |ψAi ⊗ |ψBi, in terms of a reduced density operator ρA, obtained

by performing the trace of the density operator ρ = |ψihψ| in the Hilbert subspace HB.

ρA= TrB{ρ} (12)

This will allow us to define the entanglement entropy between the two Hilbert spaces HA

and HB as,

SEnt = −ρAln ρA (13)

which will be different than zero when the obtained reduced density operator ρA is a mixed

state, and the bigger SEnt is, the “more mixed” we will consider the state described by ρAto

be.

In appendix §A, the density operator formalism for describing mixed states and its properties are explained in more detail. This formalism already allows us to verify that a mixed state cannot be obtained from applying a unitary transformation to a pure state. Because in that case the square of the operator which describes that mixed state ρm = U ρpU†cannot be one,

but,

1 6= Tr{ρ2_m} = Tr{U ρpU†U ρpU†} = Tr{ρ2p} = 1 (14)

So a contradiction is obtained, and we conclude that a unitary transformation cannot turn a pure state into a mixed state.

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Therefore, in quantum field theory, where relativistic many-particle systems are described in a quantum language, the unitarity principle has to be obeyed, and any evolution in time of a pure state should conserve probability and pureness. In §4.4, we will see that in the evaporation process of a black hole pureness is not conserved, so we will conclude that the process cannot be described by a unitary operator.

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2 QFT in Curved Space

It will not be enough to use our knowledge of general relativity and quantum field theories, but we also have to know what each one implies for the other one, if we want to use them both at once. General relativity tells us that we should write all our equations in a co-variant form, therefore the geometry of space- time will modify the solution of the equations of motion obtained in quantum field theory in flat space. On the other hand, quantum field theory tell us that we have to describe any dynamical field in terms of quantum operators. Since now gravity is included in our theory described by the metric gµν, which is closely related to

the matter field by the Einstein equation (1), the metric should be described in terms of quantum operators. However, this quantum description of gravity is problematic, as it leads to divergences that cannot be re-normalized[3]. Due to the lack of a quantum description of gravity, we will have to use an approximate description, which is called the semi-classical approximation. We will see that this approximation will be enough in order to obtain results that were not expected in quantum field theory when applied to flat space-time.

2.1 Semi-Classical Gravity

Although the Einstein equation (1) relates the matter fields with the space-time metric, the description of the last one as a quantum operator leads to problems. Therefore, if the matter fields are described by quantum operators, their relation with the space-time geometry should be in a “more classical” way, so that the metric is related with the expectation value of the energy-momentum tensor operator ˆTµν.

Rµν −

1

2Rgµν = 8πGh ˆTµνi (15)

Then, in this approximation, the metric will not suffer quantum fluctuations and will behave as a classical field. Quantum field theory will be applied over this fixed classical back-ground geometry. For example. if we have a real massless scalar field φ in 1 + 1 dimensions, which is described by the following action.

S = 1 2

Z

dx2√−g ∇_µφ∇µφ (16)

where g is the determinant of the metric gµν, which describes the space where this field

exists, and ∇µ stands for the co-variant derivative4. The equation of motion obtained from

this action will be given by ∇µ∇µφ = 0. It is straightforward to see that if we are dealing

with a Minkowski metric gµν = ηµν = diag(−1, 1), then the solution to the equation of motion

is given in form of plane waves in the quantum field theory formalism. φ = Z dk √ 2πp2|k| h e−i|k|t+ikxak+ ei|k|t−ikxa†_k i (17) We expect to obtain a more complicated solution in curved space for the mode func-tions fk, which determine the expansion of the field operator φ in creation and annihilation

operators.

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The definition of the co-variant derivative from the metric can be found in any introductory book of general relativity [1]

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φ = Z dkhfk(x, t)ak+ fk∗(x, t)a † k i (18) We would like to emphasize that the metric does not fluctuate not only because it is not described as a quantum field, but also because we have suppressed any effect that the matter quantum fluctuations would have in the metric. In other words, we have deactivated the gravitational coupling constant G → 0, which couples the space-time geometry and the matter of our theory. This may sound strange, as we have already stated that in this semi-classical approximation we will use the modified Einstein equation (15), where G appears. But, we are using this equation as a constraint to fix the classical background which we are going to work with, instead of a dynamical equation for the metric. If that was the case, our theory would be described by an action which is a sum of (16) and the Einstein-Hilbert action, SE-H = 1 16πG Z dx2√−gR (19)

From this action we would obtain the Einstein equation (1 as the equation of motion for the metric, therefore the appearance of the metric in the equation of motion for the matter field φ , would involve a back reaction dependence. Instead, in the semi-classical approximation, we turn off the gravitational coupling constant G → 0, and in the equation of motion for φ we use an already determined metric, whose value is fixed5_.

We have to be aware that we are working with an approximation, and as a consequence we should worry about whether it is correct or not. First of all, we already know that there is a limit of quantum gravity that reproduces the results of general relativity, since this classical theory describes a wide range of observed phenomena, which should be predicted by the quantum theory as well. In this semi-classical approximation, we are neglecting the gravity interaction, therefore it is required that the gravitational effects are small enough to be neglected. It is expected that quantum gravity effects appear at energy scales close to the Planck mass mP =

p

~c/(8πG). We can relate this energy with the Planck length LP =

p

~G/c3, and state the following conditions for the semi-classical gravity limit [7], in terms of geometric quantities from general relativity that should be smaller than the Planck length LP:

• We will work with quantum states that are defined on d-dimensional space-like slices. In the semi-classical limit, it will be required that the intrinsic curvature of the space-like slice chosen, should be smaller than the Planck scale at every point of the slice

(d)_{R 1/L}2 P.

• The chosen d-dimensional space-like slice has to be nicely embedded in a d + 1 dimen-sional space-time. This will be satisfied by demanding that the extrinsic curvature of the slice is small compared to the Planck length at every point of the slice K 1/L2_P. • In the neighborhood of the d-dimensional space-like slice, the d + 1 curvature of the full

space-time should also be small compared to the Planck length(d+1)R 1/L2_P

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We may determine this metric with the classical theory. Here the word fixed stand for “known”, but the metric is still a function of space and time.

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• Any quanta on the space-like slice (defined by our quantum state) should have a wave-length longer than the Planck wave-length λ LP , and the energy and momentum densities

should be small compared to the Planck density U, P L−4_P . We will also require that all matter in the slice satisfy the energy conditions for completeness.

• The quantum state in our initial slice will evolve to a final state, defined in another space-like slice. All the slices that contain our quantum state, through its evolution, should satisfy the above conditions. Furthermore, the evolution of these slices should be governed by a lapse and shift vector, small enough compared to the Planck length

dNi

ds , dN

ds LP.

In short, the description of gravity in terms of a quantum metric leads to problematic non-renormalizable divergences. In order to avoid them, we keep the metric classical and perform quantum field theory over it. This semi-classical approximation is only possible if the gravity effects are not too strong in the region of the space-time described, therefore the curvature of this region should be small compared to the Planck length LP in order to apply

this semi-classical description. Nevertheless, this approach already implies an unexpected behavior, as will be explained in the next section.

2.2 Particles in Curved QFT

One of the principal features of curved space that we will need in order to introduce Hawking radiation, is its ambiguity to define what a particle is. The main reason for this ambiguity comes from problems that already exist in general relativity, when we try to define the energy in a generic space-time.

We can have an intuition of why this is so, if we visualize a quantum mechanics system (e.g. a finite box, or harmonic oscillator) to which we will perform an abrupt change of its potential. So, imagine a system of constant potential inside a box of size L and infinite potential outside it. We know what the ground-state of this system is. If we wait enough time (due to unavoidable perturbations in a realistic system) the system will be in its ground-state. Then, if we change instantaneously the size of the box to L0 > L, the state of the system will still be the same, but it will not be a groundstate anymore. Instead it will be a superposition of different eigenstates of the new Hamiltonian. So, we started with a system in its ground state and we ended up with an excited one. Then, if we work in a QFT formalism where the vacuum is the ground-state, if the Hamiltonian of the system changes, we will find excitations of the system in a form of particles. Furthermore, if our system is such that it is not possible to properly define the energy, then we have that we will neither be able to have a good definition of what is a particle, nor of what is the vacuum.

In this section, we will see how this behavior of quantum field theory in curved space leads to vacuum states that contain particles for some observers.

2.2.1 Energy in Curved Space

Following the steps of the Lagrangian formalism, both in quantum field theory and in general relativity the energy is defined as the conserved quantity related to time translations. This is formally defined in terms of a time-like Killing vector which keeps the metric of the stud-ied space-time invariant under translations in the Killing vector direction. Since in general

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relativity different time-like coordinates can be chosen, an energy quantity can be defined in all the ones that under a translation keep the metric invariant. The later statement is true even in flat space, where different energies can be defined for inertial observers with different velocities, but in this case the energies are conserved for all the observers since they are related between them at any point of space-time by the Lorentz transformations [1].

In general, this is not true in curved space, where two observers following two different geodesic can have proper times that are related between them by a non-linear transformation, which would lead to a transformation tensor that depends on the point of space-time it is being evaluated. Therefore, the quantities that are conserved for one of the observers will not be conserved for the other one and will not be of physical importance for the former one. There is no reason to prefer any of the two conserved quantities as the one corresponding to the energy, but the useful quantity, for each one of the observers, is the one that they see conserved. Then we will conclude that in opposition to flat space energy cannot be uniquely defined.

Thus, in contrast to flat space, where any inertial observer is able to define an energy which is conserved in the whole space-time, in curved space it is not always possible to define an energy that is conserved in the whole space-time. This ambiguity will lead to a non-unique definition of particles in curved space as well.

2.2.2 Non-Definiteness of Particles

The lack of a well defined energy will not let us define particles in an unique way either. In curved space the field operator which describes a massless real field in 1 + 1 dimensions, corresponding to the action (16) is expanded in a mode expansion with the form:

φ = Z dk h fk(x, t)ak+ fk∗(x, t)a † k i (20) Where fk(x, t) are mode functions that solve the equations of motion for the field φ and

form an orthogonal set of solutions respect the following inner product (f, g) = −i Z Σ dx (f ∇µg∗− g∗∇µf ) nµ √ γ (21)

Where the integral is evaluated over a space-like curve Σ of our choice, nµis a vector nor-mal to this curve and γis the determinant of the induced metric g11. This inner product will

fulfill the properties expected for an inner product and the mode functions will be orthogonal under it.

(fk, fk0) = δ(k − k0)

(f_k∗, f_k∗0) = −δ(k − k0)

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the canonical commutation relations introduced in §1.2 with their corresponding Hermitian conjugates, the creation operators a†_k. We can define a vacuum, related to this set of operators, as the state that vanishes after applying over it any annihilation operator ak|0ai = 0.

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The set of mode functions fk(x, t) is not the only possible orthogonal set of solutions that

solve the equations of motion for the field φ, but we can also represent the field operator φ in terms of another set of mode functions gk(x, t).

φ = Z dkhgk(x, t)bk+ gk∗(x, t)b † k i (23) This set of mode functions are related to another set of annihilation and creation operators bk that satisfy the canonical commutation relations. Again we can define a vacuum state

respect this set of operators as bk|0bi = 0.

We may ask ourselves which one of the two vacuum states we shall consider as the real one, a state empty of particles. In order to do this, we shall see which set of mode functions allows us to define a frequency ωk, respect to the proper time of the observer, that satisfy

the following equation[1][5]6.

D

dτfk= −iωkfk (24)

Since, in curved space different observers can have different proper times, the appropriate mode functions for describing particles will be different for each observer. Then there is no reason for expecting that the vacuum states |0ai or |0bi are empty of particles for other

observers. We will confirm this result in the next section where we will see how we can relate two different sets of mode functions using the Bogoliubov transformations.

2.2.3 Bogoliubov Transformation

In order to relate the different allowed mode expansions of a field we will have to use the Bogoliubov transformations. We will see that depending on the form of this transformation, some observers are able to measure particles when other observers are seeing an empty space. If we have a field φ described with two different expressions involving two different operator sets that satisfy the canonical commutation relations.

φ(x, t) = Z dkhfk(x, t)ak+ fk∗(x, t)a † k i = Z dk h gk(x, t)bk+ gk∗(x, t)b † k i (25)

And we want to write the mode functions gk as a linear combination of the fk and fk∗

mode functions.

gk=

Z

dk0[αkk0f_k0 + β_kk0f∗

k0] (26)

With the help of the inner product (21), we can find that the corresponding coefficients αkk0 and β_kk0 are determined by,

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Note that in this expression nor appears a conventional derivative, but a directional covariant derivative

D dλ =

dxµ

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αkk0 = (g_k, f_k0) β_kk0 = −(g_k, f_k∗0) (27)

We can check with an evaluation of the inner product (φ, fk) that this coefficients also

tell us the relation between the ak and bk operators

ak= Z dk0hαkk0b_k0+ β_kk0b† k0 i (28) And from the commutation relations [ak, a†_k0] = δ(k −k0) and [bk, b†_k0] = δ(k −k0) we obtain

the following conditions for the coefficients. Z

dqαkqα∗k0_q− β_kqβ_k∗0_q = δ(k − k0)

Z

dqαkqβk0_q− β_kqα_k0_q = 0

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Which allows us to find the inverse Bogoliubov transformation. bk = Z dk0hα∗_kk0a_k0− β∗ kk0a†_k0 i (30) Finally we can check that the vacuum state defined for one set of operators ak|0ai = 0 , is

not always the same for the other set bk|0bi = 0 . For example we can compute the number

of b particles in the |0ai vacuum.

So we arrive at the result that the |0ai vacuum will not be empty of b particles if βkq6= 0.

To conclude, the convenience of defining an energy with respect to a determined time-like Killing vector will depend on the proper time of the observer. Ambiguity in the definition of energy will lead to different definitions of particles between different observers. This will lead to situations where a space which is empty of particles for an observer will have particles for another observer.

In summary, in order to apply together general relativity and quantum field theory we will neglect the quantum gravity fluctuations when the curvature of space-time is small enough compared to the Planck Length. In this semi-classical approach in contrast with quantum field theory in flat space, there is not an unique definition of either energy or particles, then different observers will observe a different number of particles. In the next section we will introduce the Unruh effect and see that this already happens in flat space. The description of the Unruh effect will help us to introduce the Hawking radiation.

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3 The Unruh Effect

In order to order to understand Hawking radiation it will be useful to introduce first the Unruh effect, where an accelerated observer in flat space measures a thermal spectrum of particles when there is a field in the state that corresponds to the vacuum for an inertial observer.

For deriving the Unruh effect we will first need to know what is the more appropriate coordinate system to describe what an accelerated observer observes. We will find that he only has access to observe a restricted region of space-time that we will call Rindler space and in consequence he will be a “Rindler observer”. Then we will apply quantum field theory with the aim of obtaining the particle definition for the Rindler observer. The next step will be to find the corresponding Bogoliubov transformation that relates the Minkowski description of particles, corresponding to an inertial observer, with the Rindler one. This transformation will allow us to write the Minkowski vacuum in terms of accelerated observer description. Finally, with the help of the density operator formalism, we will obtain that the Rindler observer measures a thermal spectrum when the system is in the Minkowski vacuum.

3.1 The Rindler Space

We are interested in describing what an observer that is constantly accelerated will observe in flat space-time. A first step is to find a coordinate system where the time-like coordinate coincides with the proper time of the observer. In order to do this, first we will want to define properly what we mean with a “constantly accelerated” observer.

We will work in a D = 1 + 1 flat space-time with the Minkowski metric

ds2 = −dt2+ dx2 (32)

Of course an inertial observer cannot see a body with constant acceleration, since it cannot reach the speed of the light and as a consequence its acceleration will decrease. But an observer with the same acceleration is able to measure a constant acceleration in his co-moving frame. Then we will define the proper acceleration from the evolution of the trajectory of a particle xp(τ ) with its proper time τ .

aµaµ= α2 , where aµ= duµ dτ and u µ₌ dx µ p dτ (33)

Then we will say that if an object has an acceleration such that α is constant, then it has a constant acceleration. We can make sense of this statement noting that the normalization condition for a massive particle uµuµ= −1, if it is differentiated with respect to the proper

time it yields aµuµ= 0. Then if we evaluate this equality at some time τ0 when the particle

is in rest uµ _{= (1, 0) i.e in the comoving coordinates, we obtain that a}µ_{(τ ) = (0, α) . So}

in these comoving coordinates du_dτ1 = α , and α is the proper acceleration observed by the particle itself. We will be interested in finding what is the trajectory of such a constantly accelerated particle. From the definition of the proper acceleration, we have.

− a 0 α 2 + a 1 α 2 = 1 (34)

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Which is solved by

a0 = α sinh β(τ ) a1 = α cosh β(τ ) (35)

Where β(τ ) is a parameter function that depends on the proper time, since any β function will be valid (up to some conditions) for parameterizing the trajectory, we will choose the easiest one β = ατ , the linear one, where we have included an α factor for dimensional reasons. Now integrating the expressions for the acceleration and using the normalization condition uµuµ= −1, we obtain the velocity.

u0 = cosh(ατ ) u1 = sinh(ατ ) (36)

And finally we can integrate the velocity in order to obtain the trajectory. This trajectory up to a constant will be.

x0_p = 1 αsinh(ατ ) x 1 p = 1 αcosh(ατ ) (37)

Then we obtained a trajectory parameterized by τ , for a particle that has constant proper acceleration.

We are interested in how an observer following such trajectory would describe the space-time, in order to do it, we will introduce new coordinates (η, ξ).

t = 1 χe

χξ_sinh(χη) _{x =} 1

χe

χξ_cosh(χη) ₍₃₈₎

The range of this new coordinates is −∞ < η < ∞ and −∞ < ξ < ∞, and they only cover the region of Minkowski space where x > |t| corresponding to region R in fig.1. As we will see, the parameter χ is related with the acceleration of the observer who is using this coordinates. If we want to describe with these coordinates the trajectory (37) we have to choose. η = α χτ ξ = 1 χln χ α (39) Now we can see that in these coordinates the proper time of the accelerated particle τ is proportional to the time-like coordinate η. Furthermore, we can observe that if we choose χ = α then we are in the frame with time equal to the particle proper time.

η = τ ξ = 0 (40)

Finally we can obtain the metric corresponding to this coordinates.

ds2= e2χξ(−dη2+ dξ2) (41)

Which is conformal by an e2χξ factor to the Minkowski metric, what is useful to study its causal structure. This region of the Minkowski space with metric (41) is known as Rindler space. It is possible to define also some coordinates which cover the region L in fig.1 x < −|t| and share the metric (41) adding a minus sign to (38).

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Figure 1: The coordinates defined in (38) cover the region labeled by R, while the coordinates (42) cover the region labeled by L[6]

t = −1 χe

χξ_sinh(χη) _{x = −}1

χe

χξ_cosh(χη) ₍₄₂₎

Thus, it is possible to obtain a metric, with a time-like coordinate that coincides to the one of the accelerated observer. We can observe from (41) that this metric is invariant under time translation so a Killing vector can be defined in the direction of the proper time of the accelerated observer. We can also see in fig.1 that the accelerated observer will only have access to a region of space-time that we call the Rindler wedge. In the next section we will apply quantum field theory in this space-time to obtain a definition of particles for our

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accelerated observer.

3.2 Scalar Field in Rindler Space

Now that we know an appropriate coordinate system to describe flat space-time by an acceler-ated observer, we will apply quantum field theory. The metric (41) in this coordinate system is invariant under translations in the time-like coordinate η, therefore there is a Killing vector in its direction and it is possible to define an energy that will be conserved for the accelerated observer. We will be interested in finding what is the corresponding particle definition for an accelerated observer.

Since the action (16) is invariant under conformal transformations of the metric,

gµν → ˜gµν = Ω2(t, x)gµν (43)

Then, the action will have the same form in both coordinate systems (t, x) and (η, ξ) and as a consequence we will be able to expand the field φ for the region of the right wedge x > |t| in terms of planes waves in the (η, ξ) coordinates and set operators that satisfy [bk, b†_k0] = δ(k − k0). φ = Z dk √ 2π√2ω h e−iωη+ikξbk+ eiωη−ikξb†_k i (44) Where ω(k) = |k| and we can define a vacuum for this region as bk|0bi = 0. This

expression only describes the field at the right wedge, if we want to have an expression for φ in the left wedge, we can analogously expand it in terms of a set of operators that satisfy [ck, ck0] = δ(k − k0). φ = Z dk √ 2π√2ω h e−iωη+ikξck+ eiωη−ikξc†_k i (45) In this expansion we have modified a sign in the exponentials, because in this region the time-like Killing vector has the opposite sign as that in the R region described by (38). Again for this region we can define a new vacuum as ck|0ci = 0 . In the regions F and P of fig.1 we

cannot make an expansion of the field φ because we are missing a time-like Killing vector that let us define energies.

As the bk operators only describe particles in the R region and the ck operators only act

in the L one, we can describe the field φ in both regions with the expression. φ = Z dk √ 2π√2ω h

e−iωη+ikξbk+ eiωη−ikξb†_k+ eiωη+ikξck+ e−iωη−ikξc†_k

i

(46) Thus, we have obtained a mode expansion in terms of the bk that describe the particles

in the Rindler region R and the ck operator that describe particles in the left wedge L.

We would like to relate this expression through a Bogoliubov transformation with the one for Minkowski space (17). Instead of computing the Bogoliubov coefficients directly we will use a more elegant shortcut due to Unruh [8], in which we will describe the field φ in terms of two new sets of operators that share the Minkowski vacuum which will be easier to relate with our Rindler operators bk and ck.

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3.2.1 Light-Cone Coordinates

In order to find a set of mode functions that are easier to relate with the mode functions of the Rindler expansion (46) and whose related operators share the Minkowski vacuum with the operators of the expansion (17) in flat space-time, we will introduce the light-cone coordinates.

¯

u = t − x v = t + x¯ and u = η − ξ v = η + ξ (47)

In these coordinates the relation between the inertial frame and the accelerated frame, with proper acceleration α, is quite simple.

¯ u = ∓1 αe −αu ¯ v = ±1 αe αv ₍₄₈₎

Where the upper sign corresponds to the R region and the lower ones to the L region. And we can check that in these coordinates both frames are still related by a conformal transformation of the metric.

ds2= d¯ud¯v = eα(v−u)dudv (49)

These nice properties will make it easier to us to relate the expressions of the two different coordinate systems.

We can relate the mode functions of expression (46) with the light cone coordinates, for seeing this first we will write (46) as

φ = Z

dkhg(1)_k bk+ g(1)∗_k b_k†+ g_k(2)ck+ g_k(2)∗c†_k

i

(50) And we will note that

√

4πωg(1)_k = e−iωη+ikξ =

(

(−α¯u)iωα k > 0

(α¯v)−iωα = (−e−iπα¯v)−i ω α = e− πω α (−α¯v)−i ω α k < 0 (51)

And taking into account that the light-cone coordinates have a different relation in region R and L. √ 4πωg(2)∗_k = e−iωη+ikξ = ( (α¯u)iωα = (−e−iπα¯u)i ω α = e πω α (−α¯u)i ω α k > 0 (−α¯v)−iωα k < 0 (52)

We can write an extended mode function that is valid in both regions R and L .

Akh (1) k = g (1) k + e −πω_α _g(2)∗ −k = ( ₁ √ 4πω(−α¯u) iω_α _{k > 0} 1 √ 4πωe −πω α (−α¯v)−i ω α k < 0 (53)

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We have to be aware that g(1)_k = 0 in region L and g_k(2) = 0 in region R so they do not overlap. Ak is a normalization constant such that

δ(k − k0) = (h(1)_k , h(1)_k0 ) = 1 AkA∗_k0 h (g(1)_k , g_k(1)0 ) + e− π α(ωk+ωk0)_(g(2)∗ −k , g (2)∗ −k0) i = 1 − e −π α(ωk+ωk0) AkA∗_k0 δ(k − k0) (54) Then |A_k|2 = 1 − e−2πωα = 2 sinh( πω α ) eπωα (55) Therefore up to a phase constant

h(1)_k = 1 p2 sinh(πω α ) eπωα g(1) k + e −πω α g(2)∗ −k (56) Analogously we can define a second extended mode function.

h(2)_k = 1 p2 sinh(πω α ) eπωα g(2) k + e −πω α g(1)∗ −k = 1 p8πω sinh(πω α ) ( (−α¯v)iωα k > 0 e−πωα (−α¯u)−i ω α k < 0 (57)

This mode functions will have associated to them an expansion of the field φ again in two sets of operators that satisfy the canonical commutation relations.

φ = Z dk h h(1)_k dk+ h(1)∗_k d†_k+ h(2)_k lh+ h(2)∗_k l†_k i (58) Since the mode functions h(1)_k and h(2)_k share the analytic properties of the Minkowski mode functions of (17) that are analytic and bounded in lower-half complex plane for ¯u and ¯

v , the d and l operators will share the Minkowski vacuum[1][5]

dk|0Mi = lk|0Mi = 0 (59)

Thus, with the introduction of the light-cone coordinates we have obtained a set of op-erators that share the Minkowski vacuum corresponding to the mode expansion (17) and it will be easier to obtain a Bogoliubov transformation that relates this set of operators with the Rindler ones in (46). In the next section we obtain the transformation that relates them.

3.3 Relation between Rindler and Minkowski Operators

After the introduction of the light-cone coordinates we can relate the Minkowski operators of the mode expansion (58) with the Rindler operators of the mode expansion (46) with a Bogoliubov transformation. This relation will allow us to obtain an expression for the number of particles that the Rindler observer measures when the field φ is in its Minkowski vacuum. A further study of the Minkowski vacuum in terms of the Rindler operators will tell us that the accelerated observer actually measures a thermal spectrum of particles.

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From the expressions (56) and (57) for Minkowski mode functions, and the Bogoliubov transformation expressions (26) and (28) we can infer an expression for the Rindler operators in terms of the Minkowki ones.

bk= 1 p2 sinh(πω α ) (eπω2αd_k+ e− πω 2αl† −k) ck= 1 p2 sinh(πω α ) (eπω2αl_k+ e− πω 2αd† −k) (60)

From this Bogoliubov transformation expression we can already see that the Minkowski vacuum is not empty of Rindler particles.

h0M|N_k(R)|0Mi = h0M|bkb†_k|0Mi = e−πωα 2 sinh(πω_α )δ(0) = 1 e2πωα − 1 δ(0) (61)

We will argue that we shall not worry about the δ(0) divergence as it appears due to the fact that we are measuring an infinite volume. Then we will define the number density of particles. hn(R)_ω i = hN R ki δ(0) = 1 e2πωα − 1 (62) This spectrum is reminiscent of the one corresponding to the thermal radiation emitted by a black body of temperature,

TU =

α

2π (63)

Where following the use of natural units we have set kB = c = ~ = 1. We will call

this temperature the Unruh temperature. We are interested in checking that an accelerated observer in the Rindler wedge measures the state |0Mi indeed as a thermal state. In order

to do it, we can arrive at an expression for the Minkowski vacuum |0Mi in terms of b†_k and

c†_k operators and the Rindler vacuum |0Ri = |0bi ⊗ |0ci if we use the expression (30) so we

obtain. dk= 1 p2 sinh(πω α ) (eπω2αb_k− e− πω 2αc† −k) lk= 1 p2 sinh(πω α ) (eπω2αc_k− e− πω 2αb† −k) (64)

That lead to two equations for the Minkowski vacuum as dk|0Mi = lk|0Mi = 0

dk|0Mi = 1 p2 sinh(πω α ) (eπω2αb_k− e− πω 2αc† −k)|0Mi = 0 lk|0Mi = 1 p2 sinh(πω α ) (eπω2αc_k− e− πω 2αb† −k)|0Mi = 0 (65)

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Thus, after obtaining the Bogoliubov transformation that relates the Minkowski operators with the Rindler one, we have arrived at these equations (65) that will allow us to describe the Minkowski vacuum in terms of the Rindler operators, so we can describe the Minkowski vacuum state corresponding to an inertial observer in an empty flat space, in terms of particle states of the Rindler observer and check if he really observes a thermal spectrum.

3.3.1 Minkoswki Vacuum in Terms of Rindler Operators

The description of the Minkowski vacuum in terms of Rindler operators have to satisfy equation (65), in this section an ansatz inspired in the resolution of a simpler problem in appendix §B will be proposed to find this description.

Following the spirit of the solution obtained for an easier equation in appendix §B we will propose the following solution for equation (65).

|0_Mi = C0e ˆ K_|0

Ri (66)

Where C0is a normalization constant, |0Ri is the Rindler vacuum defined as |0Ri = |0bi ⊗ |0ci,

with bk|0bi = 0 and ck|0ci = 0, and the operator ˆK is given by

ˆ K =

Z

dk dk0 b†_kMkk0c†

k0 (67)

So we have to determine the matrix Mkk0. Since,

bk|0Mi = C0 Z dk0Mkk0c† k0e ˆ K_|0 Ri = Z dk0Mkk0c† k0|0mi (68)

And a similar expression is obtained for ck|0Mi. Then the equations (65) become

dk|0Mi = 1 p2 sinh(πω α ) Z dk0eπω2αM_kk0c† k0 − e− πω 2αc† −k |0Mi = 0 lk|0Mi = 1 p2 sinh(πω α ) Z dk0eπω2αM_k0_kb† k0 − e− πω 2αb† −k |0Mi = 0 (69)

From where we obtain, for example, from the first expression.

Z dk0emω2αM_kk0c† k0 = e− πω 2αc† −k Z dk0Mkk0c† k0 = e− πω α c† −k (70) Which is solved by Mkk0 = e− πω α δ(k + k0) (71)

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ˆ K = Z dk e−πωα b† kc † −k (72)

And after plugging it in our proposed solution (66) we can write the Minkowski vacuum in a complete form as.

|0_Mi = C₀exp Z dk e−πωα b† kc † −k |0_bi ⊗ |0_ci (73)

Therefore, we have obtained how the Minkowski vacuum is described in terms of the Rindler operators. We can note that these operators appear in pairs with opposite momenta. From this expression we still cannot state that the Rindler observer is measuring a thermal spectrum. The Rindler observer will be only able to measure the particles in the R region of fig.(1), therefore we will need the density operator formalism to describe the state that he observes.

3.3.2 The Density Operator of the Minkowski Vacuum

Although the Minkowski vacuum (73) is defined in the whole flat space-time, the Rindler observer only have access to the right wedge R region of fig.1, therefore he will not be able to perform measurements in the Hilbert space corresponding to the ck operators. For the

purpose of describing what he will observe the density operator formalism will be required, An introduction to this formalism can be found in appendix §A.

In this formalism the Minkowski vacuum will be describe by the density operator,

ρMvac= |0Mih0M| (74)

We want to restrict ourselves to what is observed by a Rindler observer, this is only one wedge of fig.1 . We will describe what a constantly accelerated observer in region R measures in terms of the bk operators, for this we have to compute the reduced density operator.

ρb = Trc{ρMvac} =

X

{N }

hN_c|ρ_{M vac}|N_ci (75) Where there is a sum in all the states of an orthonormal basis {N }, for representing particle states (also is included the vacuum |0ci) in the left wedge L, represented by the

states |Nci which will have the form.

|N_ci = CN {N } Y k (c†_k)nk_|0 ci (76)

Where CN is a normalization constant that will be given by

CN = {N } Y k 1 √ nk! (77)

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It will be useful for computing (75) to compute first hN_c|0_Mi = C₀CNh0c| {N } Y k (ck)nke ˆ K_|0 bi ⊗ |0ci = C0CN {N } Y k e−πnkωkα (b† −k)nk|0bi = C0e− PN k πnkωkα C_N {N } Y k (b†_k)nk_|0 bi = C0e− π αEN|N_bi (78)

In the first step we have commuted the ck operators with the exponential of the operator

ˆ

K and kept the only term that survives after acting with creation or annihilation operators over the ckvacuum bra or ket respectively. Then after relabeling in the product k → −k and

nk → n−k, we write the product of exponentials as the exponential of a sum. Finally, we

identify a state similar to the one defined in (76) for the bk operators, so now nk represents

the number of particles in the Rindler wedge R with momentum k for the —Nbi state, and we

can recognize the total energy corresponding to such a state in the sum inside the exponential. Then it is straightforward to evaluate (75)

ρb= |C0|2

X

{N }

e−2παEN|N_bihN_b| (79)

As we have already obtained a diagonalized expression for the reduced density operator it is easy to compute the constant C0 from the property.

1 = Tr{ρb} = |C0|2

X

{N }

e−2παEN = |C₀|2Z (80)

Where we have defined in the last step the partition function Z =P

{N }e−

2π

αEN and now

our expression for the reduced density operator ρb is,

ρb = 1 Z X {N } e−2παEN|N_bihN_b| (81)

Which we can identify with a thermal state of temperature TU = _2πα. Then an observer

with a constant proper acceleration α in flat space will observe a thermal bath of particles corresponding to the massless real scalar field φ with temperature TU. From the density

operator appendix §A, we can check that, as expected from thermal states, the state observed by the accelerated observer is a mixed state.

ρbρb 6= ρb (82)

To conclude, the Minkowski vacuum state is formed by two entangled Hilbert spaces corresponding to the left wedge L and the Rindler region R respectively, as the Rindler observer only has access to one of them, he will observe a mixed state that will be described by a density operator (81. This density operator has the same form of a thermal spectrum of temperature TU.

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In summary, in the theory of a massless real scalar field φ in flat space-time an observer with a constant proper acceleration will observe a thermal spectrum of particles where an observer in an inertial frame observes an empty space. This thermal spectrum is a mixed state and therefore we are able to measure how mixed is this state if we compute its entanglement entropy, as it is done in appendix §C. The entanglement entropy computation in that appendix is done as a warm up exercise to get familiar with computing the entanglement, and it is not required to continue to the introduction of the Hawking radiation in §4.

In short, the Unruh effect stands for the observation of a thermal spectrum, with temper-ature TU = _2πα, by any observer who has a constant proper acceleration α, when there exist

a field that is measured in its vacuum state by an inertial observer.

The Unruh effect will help us to introduce in the next section the Hawking radiation and to discuss how it leads to the information paradox.

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4 Black Holes and Hawking Radiation

The equivalence principle tells us that an observer falling in a black hole will (locally) observe that she is in an inertial frame. On the other hand, if our observer is at a fixed distance not too far from the black hole, she will be in a non-inertial frame which (locally) can be related with a constant acceleration. Therefore, as we have seen in §3 the observer will see a thermal bath due to the Unruh effect. In this section we will develop this argument that leads to the conclusion that black holes emit thermal radiation known as Hawking radiation [9][1]. This emission of particles will decrease the radiating black hole mass until it disappears. This result will be problematic, as the entanglement of the radiated quanta with the interior of the black hole will end up with an information loss paradox when the black hole evaporates.

4.1 Schwarzschild Black Hole

In order to keep our discussion simple, we will work with the simple spherically symmetric Schwarzschild black hole. We will introduce the metric that describes this black hole and its causal structure. From its metric we will be able to find an appropriate Killing vector that allows us to define an energy that is conserved in this space-time.

The Schwarzschild black hole of mass M is described in spherical coordinates by the metric [1] : ds2 = − 1 −2GM r dt2+ 1 −2GM r −1 dr2+ r2 dθ2+ sin2θ dφ2 (83) Since the metric (83) has no dependence on the time coordinate t we can define the time-like Killing vector Kµ= (∂t)µ which allows us to define a conserved energy.

E = −pµKµ= 1 −2GM r p0 (84)

This conserved quantity will allow us to compute shifts in the wave-length of radiation due to gravity. The norm of this Killing vector KµKµwill vanish at the Schwarzschild radius

RS = 2GM indicating that the surface r = RS is an event horizon, as can be seen in fig.2

where the region I and II, corresponding to the exterior and the interior of the black hole respectively, are separated by an event horizon at RS. Therefore an observer who is outside

of the black hole will not be able to observe anything that is happening inside the black hole.

The existence of this event horizon will tell us that we are dealing with a black hole. The Schwarzschild metric (83) is the only possible solution for a spherical symmetric black hole, therefore any black hole of this type will be characterized uniquely by its mass. This result in known as the No-hair theorem and it can be generalized to the statement that any black hole will be described by its mass, electromagnetic charge and angular momentum[1]. So although black holes are macroscopic objects they are uniquely characterized by a small number of parameters. When we define the black hole entropy and compute it we will find that it suggests a greater degeneracy than the No-hair theorem suggests, but before doing

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Figure 2: A diagram where the causal structure of a Schwarzschild black hole, formed of collapsing matter, can be observed. The interior of the black hole and its exterior are repre-sented by the regions I and II , an event horizon separates this two regions being possible to go to the interior of the black hole from the exterior but not the other way.Image from the Pittsburgh University website of John D. Norton

that we need to know how frequency shifts in the Schwarzschild space-time in order to obtain Hawking radiation.

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We can use the energy defined for the Schwarzschild metric for computing shifts in the frequency of particles that move through this space-time. The frequency of a particle with momentum pµ measured by an observer with velocity uµ can be obtained as ω = −pµuµ ,

so when an observer at rest at r → ∞ detects a photon, its energy will be E = −p0 = ω . If we define a static observer as the one whose velocity uµ is proportional to the Killing vector Kµ= V (x)uµ , then we can calculate the blue-shift, measured by a static observer, of a photon emitted at r → ∞ with energy E = ω1, at another point of space where energy will

be conserved E = −pµKµ , since ω2= −pµuµ= − 1 V (x2) pµKµ= 1 V (x2) E = ω1 V (x2) (85) Then the shift factor V (x) is telling us how the wavelength of the photon will change as it moves through the space-time. We can obtain its value from its definition as Kµ= V (x)uµ if we use the normalization condition uµuµ= −1 .

V =p−KµKµ (86)

And we obtain E =√−g00ω is the energy of the photon at any fixed radius outside the event

horizon.

Black holes are characterized by a small number of parameters. In our description we will only work with black holes described by their mass and are described by the Schwarzschild metric. We have obtained how energy shall be defined around one. This will be useful for describing the Hawking radiation as a consequence of shifting the wave-length of an Unruh spectrum. We have also seen that an observer who is outside the black hole cannot observe what is inside it, so in order to describe a quantum state in this space we will need to use the density operator formalism.

4.2 Hawking Radiation

In this section we will obtain the Hawking radiation emitted by a black hole with the help of the equivalence principle and the definition of energy obtained in the previous section for a Schwarzschild black hole. First we will use the equivalence principle to argue that an observer that keep still close to the event horizon will observe an Unruh spectrum. Then we will compute the frequency shift of this spectrum so we obtain how it is observer by an observer at an infinite distance from the black hole. The temperature obtained for the thermal spectrum radiated by the black hole will suggest us that we can apply thermodynamics laws to black holes, what will lead to the definition of the Bekenstein entropy.

We can derive the Hawking radiation if we invoke the principle of equivalence close to the event horizon of the black hole. A more formal derivation can be found at [9][13][5]. We will apply the equivalence principle close to the event horizon because the time scale related to the proper acceleration t ∼ a−1of an observer at a fixed distance is smaller the closer she is to the event horizon. Then compared to the curvature radius ∼ 2GM , the time scale is so small that we can consider that the radius of curvature is infinite, i.e. flat space. Therefore the equivalence principle tells us that our fixed observer will observe the same that an accelerated

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observer in flat space-time. In the black hole case if we choose to describe a point distant from the black hole horizon we would have to be careful with the gravity effects in this region between the event horizon and our point where curvature varies. Therefore we will choose a point close to the event horizon that minimize this region, so we can consider in enough small region a constant curvature.

At this point a static observer, since she is not following the expected radial geodesic toward the black hole, will feel a proper acceleration given by [1].

a = κ(R

∗ S)

V (R_S∗) (87)

Where κ is the surface gravity, and R∗_S = Rs+ δr is a point close to the Schwarzschild

radius, but separated by a small distance 0 < δr << Rs .We apply it close to the horizon

because there the acceleration scale is bigger than the curvature, therefore space-time in this region seems flat. As we derived in §3 an observer that is suffering a constant acceleration in flat space, will observe a thermal spectrum of massless particles, given by (62)

< n(R)_ω >= 1

eB_{− 1} (88)

Where, for our observer,

B =2πωRS

a = 2πωRS

V (R∗_S)

κ(R∗_S) (89)

We can relate the frequency of a particle at the event horizon to the frequency of the same particle in the asymptotic flat space at r → ∞ as ω∞V (∞) = ωRSV (R

∗

S), therefore,

B = 2π

κ(R∗_S)ω∞ (90)

Then an observer at r → ∞ will measure a thermal spectrum of massless particles coming from the black hole corresponding to the temperature.

TH =

κ(R∗_S)

2π =

1

8πGM (91)

Where we have used κ =p∇µV ∇µV = 1/(4πGM ) .

We have used the results obtained in §3 to argue that a black hole shall radiate with the Hawking temperature TH.

This temperature allows to define an entropy for the black hole from the equation that describes the Penrose process. In this process an object falls toward a rotating Kerr black hole and when it is close enough to the black hole it splits in two parts, one falls inside the black hole and the other one escape from the black hole, following a geodesic, with more energy that the initial object. Therefore in the Penrose process one extracts energy from the black hole[1]. It is described by,

δM = κ

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Where the variation of the black hole mass M is related to its area A and its angular momentum J through its surface gravity κ and the angular momentum of its event horizon ΩH. This equation is reminiscent to the first law of thermodynamics,

dE = T dS + dW (93)

Where the energy E , temperature T , entropy S and work W of a thermodynamic system are related. This similarity together with the area theorem, which states that the area of a black hole in general relativity can not decrease[1] suggests that equation (92) is really a thermodynamic law for black holes. In this law the black hole energy M is related with the black hole entropy, which has a linear dependence with the black hole area. The Hawking temperature TH allows to obtain a definition of the entropy from this thermodynamic law.

SBH=

A

4G (94)

This entropy is known as the Bekenstein entropy and associates to a Schwarzschild black hole, with an event horizon area A = 4πR_S2, an entropy proportional to its area [11]. This result is surprising when we recall the statistical physics connection of entropy with all the possible configurations in which a system can be with a certain probability pi.

S =X

i

piln pi (95)

This equation tell us how many degenerate states a system can have. The surprise comes from the high degeneracy obtained for a black hole from the Bekenstein entropy (94) in opposition to the small number of parameters that characterize a black hole following the No-hair theorem introduced in §4.1.

In short, we have used the equivalence principle to obtain that a black hole will emit thermal radiation from our previous derivation of Unruh effect. This radiation is known as Hawking radiation and is a thermal spectrum corresponding to the Hawking temperature TH = _8πGM1 . This temperature suggest that the black hole is a thermodynamic object that

will follow the thermodynamics laws. Therefore the black hole will have an entropy called the Bekenstein entropy that is proportional to its area. In the next section we will see that Hawking radiation leads to the evaporation of the black hole, which, as we will see in §4.4, implies the information paradox.

4.3 Black Hole Evaporation

In previous section we obtained, with an intuitive approach, that an observer at an infinite distance from a black hole will observe a thermal spectrum of particles. Due to conservation of energy, the particles energy only can come from the distant black hole, and in consequence its mass shall decrease with time as the black hole is emitting particles. In this section, we will show how the emission rate of a Schwarzschild black hole leads to its evaporation and the number of quanta emitted by this process will be obtained, following the derivation of [12]. This result, as we will see in the next section, will be problematic.

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If the black hole is radiating a thermal spectrum of particles, then it will obey the Ste-fan–Boltzmann power law [12] which gives us the following luminosity for the black hole.

L = gπ

2

120AT

4

H (96)

Where g is the number of degrees of freedom of the radiated particles, A the area of the event horizon surface and TH is the Hawking temperature. The energy of the black hole is

given by its mass, then,

dM

dt = −L = −

g

15 · 211_πG2_M2 (97)

We can note from this expression that the black hole will radiate more energy when its mass is smaller, therefore it will lose mass continuously, until there is no more mass left (or new physics appears) and the black hole will disappear in a finite time. If we integrate (97), we obtain the evaporation time of the black hole.

tev =

5 · 211

g G

2_M3 ₍₉₈₎

We can obtain more properties of the Hawking radiation from its emission rate, which can be derived from the Planck formula.

Γ = gζ(3) 4π2 AT 3 H = gζ(3) 128π4 1 GM (99)

Where ζ is the Riemann zeta function. Since the emission rate is defined as the number of particles emitted by time unit Γ = dN/dt ,

dN dM = dN dt dM dt = 240π 3 ζ(3) GM (100)

Then, if we integrate the above expression, N = 240π

3

ζ(3) GM

2 ₍₁₀₁₎

So it has been obtained that the total number of radiated quanta due to Hawking radiation is proportional to the radiating black hole event horizon area.

Thus, we have obtained that the black hole will evaporate with a rate given by (97) and the number of quanta emitted after the black hole is evaporated. We will recall this expression in §5 as it will be obtained also by the model proposed by G. Dvali and C. Gomez (without fixing the value of the constants)[17]. As we will see in the next section, the evaporation of the black hole will lead to the information paradox

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4.4 The Information Paradox

In this section we will argue that the process of evaporation of a black hole leads to a loss of information which enters in conflict with the unitarity principle. This inconsistency between the equivalence and unitarity principles is known as the black hole information paradox.

We have to be aware that our results have been obtained in the semi-classical approxima-tion, so it is not expected that they represent the complete description of what is happening at the black hole horizon. However, there is no reason for believing that we cannot perform this approximation, so we cannot discard the results obtained just because we do not like them. We will try to emphasizes what are the limitations of this description, so we can be careful posing the problem.

The black hole, in its process of radiation, similarly to the Unruh effect, creates pairs of particles, one inside the event horizon and another one outside. These two particles will be entangled between them in such a way that the outgoing ones will be in a mixed state similar to a thermal one. The complete information of the system can be obtained if we know the state of the radiated particles and the one of the black hole that now is modified due to the in-going particles. We keep obtaining radiated quanta from the black hole, but the states that are inside it keep stacking until the black hole evaporates completely. It is not described where the information remaining inside the black hole goes, which lead to a problematic situation where the physics are described in terms of mixed states. This description is not compatible with the unitarity principle, since a process that transforms a pure state in a mixed state cannot be unitary.

4.4.1 Semi-Classical Gravity

First of all, we have to be aware of the limitations of the tools we have used for obtaining the Hawking radiation result. Although we have given here an intuitive derivation, a stricter one can be done [9][13][5], so we shall not worry about minor limitations of our derivation.

Instead we will remind the reader of the requirements needed to apply semi-classical approximation from §2.1. This discussion will emphasize what is the limit where these results are valid and what type of effects do not appear in this description.

Once that we are aware of these limitations a simplified state for representing the particle pair created in Hawking radiation will be proposed to obtain the problematic consequences of Hawking radiation in a clearer way.

The first restriction we have to be aware of is that although we are working with the quan-tum field theory, we treat the gravitational field gµν classically, that is, we do not associate

any quantum operator to the metric field, but only with the matter fields in 1. Rµν−

1

2Rgµν = 8πG

c4 h ˆTµνi (102)

This semi-classical description, as explained in §2.1, is valid because the gravitational field we are dealing with is not too “strong”, this is posed in the language of general relativity comparing the curvature of space-time with the Planck length LP =

p

~G/c3. This condition will require in §4.2 that the studied black hole has a large enough mass M compared to the Planck mass MP, so the conditions presented in §2.1 are satisfied..

The Information Paradox and Bose-Einstein Condensate Black Holes

University of Amsterdam

MSc Physics

Theoretical Physics

Master Thesis

The Information Paradox

and

Bose-Einstein Condensate Black Holes

by

Isa´ıas Rold´

an

10901388

Contents

1

Introduction

2

QFT in Curved Space

3

The Unruh Effect

4

Black Holes and Hawking Radiation