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University of Groningen

Bachelor Thesis

The Entanglement ”Thermodynamics”

and Gravity Duality

Author:

Max Weltevrede

Supervisor:

Dr. Kyriakos Papadodimas

A thesis submitted in fulfilment of the requirements for the degree of BSc Physics

in the

Theoretical High-Energy Physics Centre for Theoretical Physics

July 19, 2014

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“What I cannot create, I do not understand.”

Richard P. Feynman

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UNIVERSITY OF GRONINGEN

Abstract

Faculty Mathematics and Natural Sciences Centre for Theoretical Physics

BSc Physics

The Entanglement ”Thermodynamics” and Gravity Duality by Max Weltevrede

In this paper the linearized Einstein equations are obtained from the first law of entangle- ment ’thermodynamics’ in the AdS/CFT correspondence. This is done following the paper by Lashkari, McDermott and Van Raamsdonk. In order to follow the derivation by Lashkari et al., a few concepts are introduced first. Entanglement entropy and Rindler space-time are established in order to understand the first law of entanglement ’thermodynamics’. Further- more, the AdS/CFT correspondence is introduced along with the Ryu-Takayanagi conjec- ture. Finally, these concepts are combined in the derivation from Lashkari et al. to derive the linearized Einstein equations from the first law of ’entanglement’ thermodynamics.

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Acknowledgements

I would like to thank my supervisor dr. Kyriakos Papadodimas for his willingness to su- pervise my project, for his useful and enthusiastic guidance during the project and for his constructive comments on the manuscript. I would also like to thank dr. Diederik Roest for his preparedness to be the second evaluator for my thesis.

iii

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Contents

Abstract ii

Acknowledgements iii

Contents iv

Introduction 1

1 Entanglement Entropy 3

1.1 Mixed and Pure States . . . 3

1.1.1 The Difference . . . 3

1.1.2 Density Matrix . . . 4

1.2 Entanglement . . . 5

1.2.1 Entanglement . . . 5

1.2.2 EPR Paradox . . . 5

1.2.3 Reduced Density Matrix . . . 6

1.3 Entanglement Entropy . . . 7

1.3.1 Von Neumann Entropy . . . 7

1.3.2 Entanglement Entropy . . . 7

2 Rindler Space-time 10 2.1 Rindler Coordinates . . . 10

2.1.1 Hyperbolic motion . . . 10

2.1.2 Rindler Coordinates . . . 12

2.2 Rindler Space-Time . . . 14

2.2.1 Rindler Horizons . . . 14

2.2.2 Rindler Wedges . . . 15

2.3 Unruh Effect . . . 16

2.3.1 Bogoliubov Transformations . . . 17

2.3.2 Unruh Effect . . . 18

2.4 Entanglement Entropy . . . 22

3 AdS/CFT Correspondence 26 3.1 Anti-de Sitter Space . . . 26

3.1.1 Manifolds . . . 26

3.1.2 Anti-de Sitter Space . . . 29

3.2 Conformal Field Theories . . . 33

3.3 AdS/CFT Correspondence . . . 33 iv

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Contents v

3.3.1 Ryu-Takayanagi Conjecture . . . 34

4 Einstein’s Equations from δS = δE 39 4.1 Einstein’s Equations . . . 39

4.2 Proving that δS = δE . . . 41

4.2.1 Entanglement Entropy . . . 41

4.2.2 Hyperbolic Energy . . . 42

4.3 Perturbation in AdS/CFT . . . 43

4.3.1 Calculation of Entropy . . . 44

4.3.2 Calculation of Energy . . . 45

4.3.3 Recovering Einstein’s equations . . . 45

Conclusion 47

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Introduction

During the past 40 years, research on black holes in general relativity has established that black holes obey a set of laws that strongly resemble the classical laws of thermodynamics.

Jakob Bekenstein and others showed that a black hole can be assigned an entropy whose magnitude is proportional to the area of the event horizon[1]. Conceptually, a black hole’s entropy can be related to the internal information shielded by the event horizon. In classical theory, black holes only absorb particles but don’t emit any. However, Stephen Hawking showed that in quantum theory, black holes both create and emit radiation in a thermal spectrum[2]. Black holes can, therefore, also be assigned a temperature. The laws that these thermodynamic properties obey (black hole thermodynamics) were originally derived from the Einstein equations and greatly resemble the laws of classical thermodynamics.

According to Einstein’s equivalence principle, an observer cannot distinguish between be- ing at rest with respect to a black hole or accelerating with a uniform acceleration in flat, empty space. This implies that, at any point in empty space-time one can find black hole- like behaviour through a simple coordinate transformation to an accelerating frame. As stated, black hole thermodynamics was derived from the Einstein equations. This duality suggested that it might be possible to derive Einstein’s equations from thermodynamic laws in empty space. In 1995, Ted Jacobsen made an attempt at deriving Einstein’s equations from the first law of thermodynamics using the concept of entanglement entropy[3]. The paper showed promise, but was not yet completely clear. In 2013, Lashkari, McDermott and Van Raamsdonk[4] derived the linear form of the Einstein equations from the first law of

’entanglement’ thermodynamics in a more mathematically sound manner. Lashkari et al.

used the concept of holography and in particular the concept of the Anti-de Sitter/Confor- mal Field Theory (AdS/CFT) correspondence. They related the first law of ’entanglement’

thermodynamics in a Conformal Field Theory (CFT) with the linear Einstein equations in Anti-de Sitter (AdS) space. The goal of this paper is to understand the derivation that Lashkari, McDermott and Van Raamsdonk performed. But first, it is necessary to obtain a background knowledge on entanglement entropy, Rindler space-time and the AdS/CFT correspondence.

1

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Introduction 2

This paper is organised as follows: in chapter 1 the concept of entanglement entropy is introduced after the introduction of mixed/pure states and density matrices. In chapter 2 the Rindler coordinate frame will be introduced. Using the Rindler coordinate frame, the Unruh effect will be derived, after which the entanglement entropy of the right Rindler wedge will be calculated. In chapter 3 the AdS/CFT correspondence will be introduced, after the introductions of Anti-de Sitter space and Conformal Field Theories separately.

After the explanation of the AdS/CFT correspondence, the Ryu-Takayanagi conjecture will be covered. In chapter 4 all these previously introduced concepts will be combined aiming to reproduce the derivation in the Lashkari, McDermott and Van Raamsdonk paper. First the thermodynamic relation between entanglement entropy and hyperbolic energy will be proven for perturbations to the vacuum state in a general conformal field theory. After that, the effects of an arbitrary perturbation in the Anti-de Sitter space on the conformal field theory boundary will be evaluated. Finally, these two will be combined to show that implying the thermodynamic relation for the arbitrary perturbation, is the same as implying the linear form of the Einstein equations.

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Chapter 1

Entanglement Entropy

In the paper from Lashkari et al. the concept of entanglement entropy is used to derive Einstein’s equations. In this chapter this concept will be explained, before we can do so however, we need to introduce some background knowledge. To start it off, the distinction between mixed and pure states in quantum mechanics needs to be emphasised. A very basic, but very important distinction. Following this, we need to explain what quantum entanglement is. We will discover that it can be defined in a convenient way using mixed and pure states. In the last section we will establish entropy as a possible measure of how entangled a system is, leading to the concept of entanglement entropy. As mentioned however, we will start with the distinction between mixed and pure states.

1.1 Mixed and Pure States

1.1.1 The Difference

In quantum mechanics, the state of a system is usually described by a state vector |Ψi living in some n dimensional Hilbert space H spanned by the orthonormal basis vectors |uki for k = 1, 2, 3, ..., n. This type of state is called a pure state. A pure state is, in general, a superposition of basis vectors[5]:

|Ψi =

n

X

k=1

ck|uki ,

n

X

k=1

|ck|2= 1 (1.1)

A more general type of state is the mixed state. A system is in a mixed state if, for example, there exists a ’probability’ p1 for it to be in the (pure) state |Ψ1i and a ’probability’ p2 for it to be in state |Ψ2i (with p1+ p2 = 1). There is a crucial difference between a pure state consisting of a superposition of states |Ψ1i and |Ψ2i and a mixed state consisting of

3

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Chapter 1. Entanglement Entropy 4

an ensemble of states |Ψ1i and |Ψ2i. On one hand, the superposition of the states |Ψ1i and |Ψ2i can be thought of as existing in both state |Ψ1i and |Ψ2i at the same time (with their respective amplitudes c1 and c2). Due to the wave-particle duality of the Copenhagen interpretation, these states can interfere with each other. On the other hand, a system in a statistical ensemble of the states |Ψ1i and |Ψ2i is really either in state |Ψ1i or in state |Ψ2i.

Since there is no longer a superposition there will no longer be interference between the two states. As claimed above, a mixed state is a more general state than a pure state: indeed a pure state is just a mixed state that consists of an ensemble of just one state (the pure state) with probability p = 1.

Because a mixed state is not a superposition of it’s components it is not possible to write it in the form of (1.1). Therefore, we use the concept of the density matrix.

1.1.2 Density Matrix

Operators in quantum mechanics can be defined in the following way[5]:

O = |φi hψ|ˆ

One useful operator is the projection operator:

ψk = |ψki hψk| (1.2)

It satisfies ˆPψ2

k = ˆPψk. Applying this operator to a pure state |Ψi =Pn

k=1ckki will return the state |ψki with the appropriate amplitude ck.

A mixed state cannot be written in the form of (1.1). It is possible, however, to use the projection operator to associate a density operator with a mixed state (a statistical ensemble of pure states)[5]:

ˆ ρ =X

i

piii hΨi| .

Where in this case all the states |Ψii are of the form in (1.1). By choosing a basis, for instance the basis of the Hilbert space |uki for k = 1, 2, 3, ..., n, the operator can be resolved into a matrix:

ρlm=X

i

pihul| |Ψii hΨi| |umi =X

i

pi(cl)i(cm)i

However, the choice of basis is completely arbitrary[5], so we need to be careful with calling the pi’s probabilities[6]. In the case above though, pi is indeed the probability of finding the system in the state |Ψii. The expectation value of an operator ˆA for a mixed state is:

h ˆAi =X

i

pii| ˆA |Ψii =X

i

X

k

ρi(ck)i(ck)ihuk| ˆA |uki =X

k

ρkkAkk= tr( ˆρ ˆA)

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Chapter 1. Entanglement Entropy 5

1.2 Entanglement

1.2.1 Entanglement

Entanglement is a very important and counter-intuitive concept in quantum mechanics. To illustrate the concept of entanglement, let’s consider a composite system composed out of the systems A and B, each with it’s own Hilbert spaces, respectivaly HA and HB. Suppose that the composite system is in a bipartite pure state |Ψi ∈ HAN HB. It is possible to distinguish two types of pure states for a composite system: seperable states and entangled states. If it is known that system A is in state |ΨiAand system B is in state |ΨiB, the state of the composite system is just the tensor product of the two seperate states[7]:

|Ψi = |ΨiAO

|ΨiB

This is an example of a seperable state. Seperable states are states of the composite system which can be written as a product of pure states of the underlying composing systems. In contrast to seperable states there exist entangled states. A state of a composite system is entangled if it cannot in any way be written as the product of pure states of the composing systems.

To illustrate consider a composite bipartite 2-level system consisting of systems A and B with basis vectors {|0iA, |1iA} and {|0iB, |1iB} respectively. An example of a seperable state would be:

|Ψi = 1

2(|0iA|0iB+|1iA|1iB+|0iA|1iB+|1iA|0iB) = (|0iA+ |1iA

2 )(|0iB+ |1iB

2 ) = |ΨiAO

|ΨiB

In contrast, an example of an entangled state would be:

|Ψi = 1

√2(|0iA|1iB+ |1iA|0iB) (1.3)

This state is entangled since a measurement on system A would completely determine the state of system B(for example, if A is in state |0iA, B has to be in state |1iB).

1.2.2 EPR Paradox

Entanglement gives rise to an apparent paradox called the EPR paradox, named after the creators Einstein, Podolsky and Rosen. It states that in the Copenhagen Interpretation the process of entanglement is in violation with a principle of general relativity[7]. This principle states that no information can travel with a speed faster than that of light. To illustrate this consider a composite system prepared in the state in (1.3) with the systems A and B seperated by a very large distance d. Suppose the state of system A is measured

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Chapter 1. Entanglement Entropy 6

and found to be |0iA. If now the state of system B is measured one will find it to be |1iB. However, suppose now that system A and subsequently B are measured in fast succesion, in particular with the time between measurements ∆t, ∆t < dc. According to the Copenhagen Interpretation, system A and B where both in a superposition of the states |0i and |1i. So, at the time of measurement, some information must have travelled with a speed faster than light in order for a measurement on B to always depend on the sleightly earlier measurement on A. This was in contradiction with general relativity and therefore was a paradox. It was later argued however, that since the state of A that will be measured (and therefore the state of B) cannot be predicted beforehand, entanglement cannot be used to transmit information with a speed faster than light, therefore solving the paradox.

1.2.3 Reduced Density Matrix

Closely related to entanglement is the concept of the reduced density matrix. Consider once more a composite system consisting of the systems A and B. As was stated in the previous section it is, in general, not possible to associate a pure state with the subsystem A or B. It is, however, possible to associate a more general mixed state with the subsystems using the concept of the density matrix. The reduced density matrix of subsystem A is defined as[8]:

ρA≡X

j

huj|B(|Ψi hΨ|) |ujiB = TrB( ˆPΨ)

where ˆPΨis the projection operator in (1.2) and TrBis the partial trace over the basis vectors of system B. The process of taking the partial trace over B is sometimes called ’tracing out’

system B.

The reduced density matrix of system A in the entangled bipartite 2-level system with state (1.3), is:

ρA= 1

2(|0iAh0|A+ |1iAh1|A)

This is the density matrix for a mixed state. So, for this example it is not possible to associate a pure state with the system A, however it is possible to associate a density matrix of a mixed state with A.

This result is true in general: if and only if the reduced density matrix of an arbitrary bipartite pure state is the density matrix of a mixed state, then the bipartite pure state of the composite system is entangled[9].

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Chapter 1. Entanglement Entropy 7

1.3 Entanglement Entropy

It is possible to associate entropy with entanglement. Entropy can be thought of as the amount of uncertainty present in a system. A mixed state in quantum mechanics is a sta- tistical ensemble of pure states, which therefore has an uncertainty with respect to which state it’s in. In the previous chapter it was stated that the composing states of an entangled composite system are mixed states. In this way, it is possible to associate entropy with the entanglement of the composite system.

1.3.1 Von Neumann Entropy

In classical mechanics the Shannon entropy is the entropy for a probability distribution, defined in the following way[7]:

SShannon(p1, p2, ..., pn) = −

n

X

i=1

piln(pi)

Something similar can be introduced for a mixed state, which is a probability distribution over an ensemble of pure states. This entropy is called the von Neumann entropy [7]:

SvN = −Tr(ρln(ρ)) (1.4)

where ρ is the density matrix for the mixed state. If ρ is the density matrix of a finite- dimensional Hilbert space with eigenvalues λ1, ..., λn, the Shannon entropy is recovered:

SvN = −Tr(ρln(ρ)) = −

n

X

i=1

λiln(λi)

1.3.2 Entanglement Entropy

The von Neumann entropy is a measure for the entanglement of a system. As stated before, the reduced density matrix of a subsystem is that of a mixed state if and only if the composite state is entangled. Therefore, for a non-entangled state, the reduced density matrix is that of a pure state. Since the von Neumann entropy is only dependent on the eigenvalues of ρ, it is possible to choose, without the loss of generality, a basis for a pure state that brings the matrix to a diagonal form with only one non-zero entry:

ρA=

1 0 · · · 0 0 0 · · · 0 ... ... . .. ...

0 0 · · · 0

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Chapter 1. Entanglement Entropy 8

The natural logarithm of this matrix is not zero for every entry, however, the ρln(ρ) term in the entropy (1.4) is. So the von Neumann entropy for a non-entangled state is zero.

One can think of a system to be maximally entangled when all components responsible for entanglement have equal amplitude. For example, |Ψi = 1

2|0iA|1iB+1

2|1iA|0iB can be thought of as more entangled then |Ψi =

q1

3|0iA|1iB+ q2

3|1iA|0iB. The former leads to a reduced density matrix of an ensemble with uniform probability distribution:

ρA=

1 2 · · · n 1 n1 0 · · · 0 2 0 1n · · · 0 ... ... ... . .. ...

n 0 0 · · · 1n

which has the most uncertainty and therefore the highest entropy[7].

To summarize these findings: the von Neumann entropy is zero if the state is not entangled and it has a maximum if the system is maximally entangled (equal amplitudes).

To illustrate that entangled systems are not as rare as one might think, consider a composite bipartite system with systems A and B entirely seperated. If the states of systems A and B are measured and found to be |ΨiA and |ΨiB , pure states in m + 1 and n + 1 dimensional Hilbert spaces HAand HBrespectively, there are m and n degrees of freedom correspondingly to the subsystems A and B. If the total system is considered in a classical way, the degrees of freedom would be found to simply be m + n. However, in quantum mechanics combining two systems leads to a Hilbert space that is the tensor product of the Hilbert spaces of the systems H = HAN HB. This means that the degrees of freedom for the composite system in quantum mechanics is not m + n but rather m × n. For a typical physical system the number of degrees of freedom (≈ number of microstates) is usually really large (a system of air molecules for example is of the order of Avogadro’s number ≈ 1023). This means that the number of states that are not existent classically (entangled states) is astronomically larger than the number of classical (seperable) states since m × n  m + n for m, n >> 1. So, if an arbitrary system in a pure state is divided in two arbitrary subsystems, the probability of finding the subsystems to be entangled tends to unity.

For a bipartite pure system Araki & Lieb (1970) [10] showed that the triange inequality holds for the entropy:

S(ρAB) ≥ |S(ρA) − S(ρB)| (1.5)

As was shown before the entropy of a pure system (S(ρAB)) is zero which together with (1.5) gives the result:

S(ρA) = S(ρB)

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Chapter 1. Entanglement Entropy 9

So system A is entangled to B for the same amount that B is entangled to A. Which is not such a surprising conclusion.

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Chapter 2

Rindler Space-time

2.1 Rindler Coordinates

So far we’ve learned how to calculate the entanglement entropy of a system divided into two subsystems. If we want to calculate the entanglement entropy of a system in a quantum field theory, we need to divide it in two subsystems as well. The big question is however, how do we create this division? In quantum field theory, the simplest field theory is the scalar field φ(~x, t) obeying the Klein-Gorden equation: (2 − m2)φ(~x, t) = 0. The vacuum state of this field is defined as the state that vanishes under action of the annihilation operator:

a~p|0i = 0. A n particle state is created by n succesive applications of the creation operator:

a~p

na~p

n−1· · · a~p

1|0i. The Hilbert space of this quantum field is the collection of all possible multi- and solo-particle states and is called the Fock space. So the questions is, how do we partition a quantum field theory in two parts where both parts have independent Fock spaces? One way to do this is to split up Minkowski space-time into right and left Rindler wedges. This is done by changing the Cartesian coordinates of an inertial frame to Rindler coordinates of an accelerating frame. An accelerating particle is said to follow hyperbolic motion in space-time.

2.1.1 Hyperbolic motion

Hyperbolic motion is the motion of a particle travelling in special relativity with a constant proper acceleration. To derive the hyperbolic motion[11], we need to define the velocity 4- vector as: uµ= (cγ, γ~v) and the 4-momentum vector as: pµ= muµ= (γmc, γm~v) = (Ec, ~p).

Here γ = q 1 1−~v2

c2

is the usual Lorentz factor. The proper time is given by the time dilation formula: dt = γdτ . It is possible to define the acceleration as:

αµ= duµ

dτ = γduµ

dt = γ(c ˙γ,d(γ~v)

dt ) = (α0, ~α) (2.1)

10

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Chapter 2. Rindler Space-time 11

In order to calculate the proper acceleration it is necesarry to evaluate equation (2.1) in the rest frame of the accelerated observer. In the rest frame the Lorentz factor equals γ = 1 and the 3-vector velocity is zero: ~v = 0. Let’s calculate α0 first:

α0RF = c ˙γ|~v=0 = c(1 −|v|2 c2 )32~v

c

~v|˙ ~v=0= 0

This is equal to zero. The proper 3-vector acceleration ~α is simply: ~α = γd(γ~dtv) = d~dtv = ddt2~x2

which results in a proper acceleration 4-vector:

αµRF = (0,d2~x dt2)

Let’s now restrict our calculations to 1 time dimension and 1 space dimension. This way it is possible to find a differential equation for the proper acceleration:

ηµνανRFαµRF = (α0)2− (α1)2= γ2((du0

dt )2− (dv

dt)2) = −γ2(dv

dt)2 = −α2 Without any loss of generality α can be taken to be positive:

α = d dt( v

q 1 −vc22

) = constant

⇒ αdt = d( v q

1 −vc22

)

Now if we assume v(t0 = 0) = 0, then:

αt = Z v(t)

0

d

dv0( v0 q

1 −vc022

)dv0= v(t) q

1 −v(t)c22

v(t) = αt q

1 + (αtc)2

From this it is possible to derive the space coordinate x:

v(t) = dx(t)

dt = αt

q

1 + (αtc)2

⇒ x(t) = Z t

0

αt0 q

1 + (αtc0)2

dt0 = c2 α

r

1 + (αt c )2

(2.2)

This results in a hyperbolic motion:

x2− c2t2= (c2

α)2 (2.3)

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Chapter 2. Rindler Space-time 12

Notice how in the limit of t >> 1, (2.2) becomes x(t >> 1) = ct. So the hyperbolic motion asymptotes towards the x = ±ct light curves.

In terms of the proper time of the accelerating observer:

τ = Z t

0

dt0 r

1 −v(t)2 c2 =

Z t 0

dt0 q

1 + (αtc0)2

= (c

α)sinh−1(αt c )

⇒ t = (c

α)sinh(ατ c )

Substituting this in (2.3) results in the space coordinate x in terms of the proper time τ :

x = (c2

α)cosh(ατ c )

2.1.2 Rindler Coordinates

We’ve learned that an observer with constant proper acceleration travels in a hyperbolic motion in space-time. Rindler coordinates (λ,ξ) are the coordinates that are usually used for this observer[11]. They are defined in the following way:

ct = (c2

a)ec2sinh(aλ c ) x = (c2

a)ec2cosh(aλ c )

(2.4)

Here ξ has the units of distance, λ the units of time and a the units of acceleration. Both ξ and λ extend from −∞ to +∞. The Rindler equations are equal to the equations for hyperbolic motion in the previous section with:

α = aec2 and ατ = aλ (2.5)

This definition of the Rindler coordinates (λ,ξ) is useful for deriving the Unruh tempera- ture(section 2.3). However, for a more intuitive understanding of the space-time the Rindler coordinates create, it’s usefull to change to dimensionless Rindler coordinates (η,ρ).

To make the change to dimensionless Rindler coordinates we set c = 1 and make the following substitutions in (2.4): η = aλ, ρ =a1e. Resulting in:

t = ρsinh(η) x = ρcosh(η)

(2.6)

The time coordinate η runs from −∞ to +∞. The space coordinate ρ however runs from 0 to +∞. From (2.5) we can also deduce that ρ is simply inversly proportional to the proper acceleration α: ρ = 1α. As can be deduced from (2.6), the coordinate lines of η (lines for

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Chapter 2. Rindler Space-time 13

which η = constant) are simply linear lines in Cartesian coordinates. The coordinate lines for ρ (ρ = constant) however, are hyperbols. The coordinate lines are illustrated in fig 2.1.

Figure 2.1: Rindler coordinate lines in a Cartesian frame.

One thing to note as well is that Rindler coordinates produce coordinate lines of ρ that are asymptotic to the light lines through the center of the Cartesian frame x = ±t. In this way the Rindler coordinates produce a coordinate frame for this triangularly shaped region of space-time called the right Rindler wedge. The space-time corresponding to this frame is a static space-time, meaning that the metric of the space-time is invariant under time translation transformations and time reversal. In order to see that it is static the induced metric needs to be derived.

The minkowski metric in Cartesian coordinates is given by ds2 = c2dt2− dx2. By calculating dt = (∂ηt)dη + (∂ρt)dρ and dx = (∂ηx)dη + (∂ρx)dρ from (2.6) we can derive the metric in Rindler coordinates (η,ρ):

ds2 = ρ22− dρ2 (2.7)

It is clear that the above metric is independent of the time coordinate η, this means the metric is invariant under time translation. The metric also has no terms of the form ∂ηρ, so it’s invariant under time reversal (change of sign of ∂η).

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Chapter 2. Rindler Space-time 14

2.2 Rindler Space-Time

Since the Rindler coordinates describe a static space-time, they can be thought of as being Cartesian coordinates for an observer in a bent space-time. Let’s now suppose there is a rod accelerated with it’s front end located at ρ = 1 and it’s back end located at ρ = 0.8 in fig 2.1. Throughout time the locations of the front and back end stay respectivally at ρ = 1 and ρ = 0.8. The metric in the accelerating frame is static which means that the proper distance between ρ = 1 and ρ = 0.8, and therefore the proper length of the rod, remains constant. So, in accordance with length contraction in special relativity and as can be seen in fig 2.1, a rod of constant length in the accelerating frame (Rindler coordinates) is not of constant length in the inertial frame (Cartesian coordinates). In fact, two points with different ρ in the accelerating frame have different proper accelerations, since ρ = α1. So, the back end of a rod at ρ = 0.8 is actually accelerating faster than the front end located at ρ = 1. This can be partially understood by realizing that for an observer holding the rod at ρ = 1, the back end looks like it’s lagging behind (due to the finite time it takes for information of the back end to reach the front end). In order for the (lagging) back end to keep up it needs to accelerate faster than the front end. Also, Rindler coordinates are analogues to polar coordinates for a family of concentric circles in Euclidian geometry. For an observer tracing out a circle in Euclidian geometry, the acceleration is dependent on the distance to the center, just like the acceleration of a Rindler observer is dependent on the spatial coordinate ρ. The dependence of proper acceleration on the spatial coordianate ρ is the Lorentzian analogue of a fact familiar to speed skaters: in a family of concentric circles, inner circles must bend faster (per unit arc length) than the outer circles.

2.2.1 Rindler Horizons

The light lines to which all coordinate lines of ρ are asymptoting are called the Rindler horizons. To understand why they are called horizons we should take a closer look at the speed of light in the accelerating frame[12]. The inverse of the equations for the Rindler coordinates in (2.6) are:

η = arctanh(t x) ρ =p

x2− t2

A ray of light emmited on t = 0 at x = x0 travels a path given by x = x0+ ct, where c = ±1 depending on the direction of the light. In Rindler coordinates this path would be:

ρ = x2− t2= x20+ c2t2+ 2x0ct − t2

= x20+ 2x0ct

= x20+ 2x0cρsinh(η)

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Chapter 2. Rindler Space-time 15

Using the quadratic formula results in:

ρ = x0(c sinh(η) + cosh(η)) = x0e= x0e±η, for c = ±1

So, the path of the light ray is not a straight line in Rindler coordinates. The speed of light at t = 0 in Rindler coordinates is:

c(ρ) = dρ

dη|t=0= ±x0e±0= ±x0

Therefore points in the right Rindler wedge closer to the horizon have a decreased speed of light (according to the accelerating observer), at ρ = x0 = 0, c = 0. Now suppose an observer at ρ = 1 in the accelerating frame lets go of an object. The object would travel onwards with the velocity of that of the observer at the time he let’s go. This would be represented by a straight line in fig 2.1tangent to the ρ = 1 coordinate line at the moment of release. This straight line would travel closer and closer to the horizon. For the observer in the accelerating frame the objects ρ coordinate would keep decreasing, this results in the decrease of the speed of the light sent by the object towards the observer. The farther the object ’falls’ towards the horizon the longer it takes for the light to reach the observer. For the observer the object will never actually fall into the horizon as on the horizon itself the speed of light in the accelerating frame has decreased to zero. The observer will see the object falling into the horizon slower and slower and with an increasingly higher redshift and lower intensity. This is in complete analogy with an object falling into a Schwarzschild black hole as seen by a stationary observer somewhere outside the horizon. This analogy is not surprising when realising that according to Einsteins equivalence principle there’s no distinction between a uniform acceleration or the effects of gravity experienced locally while standing on a massive body. In this way an observer accelerating with a constant acceleration is no different to an observer sitting stationary with respect to a (Schwarzschild) black hole.

2.2.2 Rindler Wedges

In analogy to the derivation of Rindler coordinates for the right (R) Rindler wedge, it’s also possible to derive Rindler coordinates for the three other wedges in Minkowski space-time called the left (L), future (F) and past (P) Rindler wedges with coordinates ρ ∈ (0, +∞) and η ∈ (−∞, +∞)[11]. The transformations between Rindler and Cartesian coordinates and the metric can be found in table 2.1. In fig. 2.2 the four wedges are drawn with in every wedge a ρ coordinate line. The arrow on the coordinate lines indicates the direction of increasing η. The left wedge L is, just like R, a static space-time and it’s coordinate lines are time-like. Since the ρ coordinate line is time-like it can be a wordline for a particle, however, time (η) is moving in the opposite direction with respect to R. The particle can therefore be thought of as an antiparticle moving backwards in Minkowski time. In the F and P wedges

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Chapter 2. Rindler Space-time 16

Figure 2.2: All four Rindler wedges with a single ρ coordinate line.

x t x2− t2 xt ds2

R ρ cosh(η) ρ sinh(η) ρ2 tanh(η) ρ22− dρ2 L −ρ cosh(η) −ρ sinh(η) ρ2 tanh(η) ρ22− dρ2 F ρ sinh(η) ρ cosh(η) −ρ2 coth(η) dρ2− ρ22 P −ρ sinh(η) −ρ cosh(η) −ρ2 coth(η) dρ2− ρ22

Table 2.1: Relation of Rindler coordinates of the Rindler wedges to the Cartesian coordi- nates.

the ρ coordinate lines are space-like and can therefore no longer be wordlines of particles.

Interestingly enough, when crossing the future horizon from R to F all the time-like (Rindler) coordinates turn space-like and vice versa. This is once more in analogy to crossing the event horizon of a Schwarzschild black hole.

2.3 Unruh Effect

So far it’s clear that Rindler observers see space-time around them in a very different way than inertial observers do. It is therefore worthwile to solve the Klein-Gordon equation for a real scalar field in the Rindler space-time and compare it to the solutions in Minkowski space-time. In particular we will find that the Minkowski vacuum state is not a vacuum state in Rindler space-time. The Rindler observers will therefore observe particles even though in in the inertial frame there are none. This is called the Unruh effect[11]. To compare the two vacua we need to perform a so called Bogoliubov transformation.

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Chapter 2. Rindler Space-time 17 2.3.1 Bogoliubov Transformations

In general, a Bogoliubov transformation is a unitary transformation from a unitary repre- sentation of some canonical commutation relation algebra (or canonical anti-commutation relation algebra) into another unitary representation[13]. That is to say, a Bogoliubov trans- formation between operators ˆa, ˆa and ˆb, ˆb is a transformation that satisfies: [ˆa, ˆa] = [ˆb, ˆb] (or {ˆa, ˆa} = {ˆb, ˆb}).

Suppose now φ is a free, massless, real, 1+1 dimensional scalar field and U = {ui}i∈I and V = {vk}k∈Kare two complete orthonormal positive frequency solutions to the Klein-Gordon equation[11]. Here I and K are sets of indices. Due to the completeness of U , the solutions of V can be written as a linear combination of ui’s:

vk=X

i∈I

kiui+ βkiui) vk=X

i∈I

kiui + βkiui)

(2.8)

Where the ui’s and vk’s are negative frequency solutions corresponding to U and V respec- tively. Note that we’ve used discrete sets of indices for simplicity. The αki’s and βki’s are called the Bogoliubov coefficients. Note also that the positive frequency solution vkcan have negative frequency components from ui if βki 6= 0.

Suppose now ˆai, ˆai are ladder operators corresponding to the solutions from U and ˆbk, ˆbk are ladder operators corresponding to V satisfying:

[ˆai, ˆaj] = δij, [ˆai, ˆaj] = [ˆai, ˆaj] = 0 [ˆbk, ˆbl] = δkl, [ˆbk, ˆbl] = [ˆbk, ˆbl] = 0

Note that the ladder operators satisfy these commutation relations by definition, a transfor- mation between the two sets of operators is thus a Bogoliubov transformations. The field φ can be expanded using these operators:

φ =X

i∈I

(ˆaiui+ ˆaiui) = X

k∈K

(ˆbkvk+ ˆbkvk) (2.9)

Using (2.8) and (2.9) we can write the ladder operators ˆai, ˆai in terms of ˆbk, ˆbk: ˆ

ai = X

k∈K

kiˆbk+ βkiˆbk) ˆ

ai = X

k∈K

kiˆbk+ βkiˆbk)

(2.10)

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Chapter 2. Rindler Space-time 18

If we define the expansion ui = P

k∈K(rikvk+ sikvk), we can calculate the inner products

< ui, vk >= αki = rik and < ui, vk >= βki = −sik. Using these we can invert (2.10) to receive:

ˆbk =X

i∈I

kii− βkii) ˆbk =X

i∈I

kii − βkiˆai)

As usual we can define the vacuum as the state that gets annihilated by the annihilation operator. In this way we can define normalized vacua for both U and V : |0ia and |0ib satisfying ah0|0ia= h0|0ib b= 1. The defining relations for the two are:

ˆ

ai|0ia= 0 and ˆbk|0ib = 0

Using the Bogoliubov coefficients from (2.10) we can calculate the amount of ’ˆb particles’ in the vacuum state |0ia using the number operator ˆnk= ˆbkˆbk:

h0|ˆbkˆbk|0i

a a= X

i,j∈I

h0|(αkiˆai − βkii)(αkij− βkiˆaj)|0i

a a

= X

i,j∈I

βkiβki h0|ˆaij|0i

a a= X

i,j∈I

βkiβkiδij =X

i∈I

ki|2

This is non-zero if there is atleast one βki6= 0, so if there is a negative frequency contribution from U in the positive frequency solution of V . In this case the observer using the ’V basis’

would see particles where the observer in the ’U basis’ sees none.

2.3.2 Unruh Effect

The next step is to calculate the Bogoliubov coefficients for the transformation from Minkowski to Rindler space-time. This can most easily be illustrated in the case of a massless, 1+1 di- mensional Klein-Gordon equation[11]. This takes the easiest form in the Rindler coordinates (λ, ξ) for the right Rindler wedge that we specified in (2.4). In these coordinates the metric takes the simple form:

ds2= e2aξ/c2(c22− dξ2) Which leads to the Klein-Gordon equation:

(c22

∂λ2 − ∂2

∂ξ2)φ(λ, ξ) = 0

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Chapter 2. Rindler Space-time 19

Which has the same form as the Klein-Gordon equation in Minkowski space-time. The solutions therefore are also of the same form:

φ(t, x) = Z +∞

−∞

dk

2πp2|k|(ˆake−i(ωkt−kx)+ ˆakei(ωkt−kx)) (2.11) φ(λ, ξ) =

Z +∞

−∞

dl

2πp2|l|(ˆble−i(ωlλ−lξ)+ ˆblei(ωlλ−lξ)) (2.12) where ωk= c|k|, ωl= c|l| and the ladder operators satisfy:

[ˆak, ˆak0] = 2πδ(3)(k − k0), [ˆak, ˆak0] = [ˆak, ˆak0] = 0 [ˆbl, ˆbl0] = 2πδ(3)(l − l0), [ˆbl, ˆbl0] = [ˆbl, ˆbl0] = 0

(2.13)

In both coordinate systems, t = 0 & λ = 0 is a Cauchy surface. A Cauchy surface is a subset of space-time intersected by all inextensible, causal curves exactly once[14]. In effect this means that if you know all values and first derivatives on a Cauchy surface, you can derive everything from the past and the future. So it’s sufficient to derive the Bogoliubov coefficients on the slice t = 0 & λ = 0. Also, since φ is a scalar field, it’s value doesn’t depend on which coordinate system is used: φ(t, x) = φ(λ, ξ). On the Cauchy surface this results in:

φ(0, x) = Z +∞

−∞

dk

2πp2|k|(ˆakeikx+ ˆake−ikx)

= Z +∞

−∞

dk

2πp2|k|(ˆakeikac2eaξ/c2 + ˆake−ikac2eaξ/c2) = φ(0, ξ)

(2.14)

Here we used (2.4). The Fourier transform of (2.12) is:

Z +∞

−∞

dξe−il0ξφ(λ, ξ) = Z +∞

−∞

dl

2πp2|l|(ˆble−i|l|λ Z +∞

−∞

dξei(l−l0+ ˆblei|l|λ Z +∞

−∞

dξe−i(l−l0)

= Z +∞

−∞

dl

2πp2|l|(ˆble−i|l|λ2πδ(3)(l − l0) + ˆblei|l|λ2πδ(3)(l + l0))

= 1

p2|l0|(ˆbl0e−i|l0+ ˆb−l0ei|l0) For λ = 0:

ˆbl+ ˆb−l=p 2|l|

Z

−∞

dξφ(0, ξ)e−ilξ Now using (2.14):

ˆbl+ ˆb−l= Z

−∞

dk 2π

s|l|

|k|(ˆak Z

−∞

dξei(c2 kaeaξ/c2−lξ)+ ˆak Z

−∞

dξe−i(c2 kaeaξ/c2+lξ)) (2.15)

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Chapter 2. Rindler Space-time 20

To solve this we need one more equation in ˆbl, ˆb−land ˆak, ˆak. So we take the first derivative from φ(λ, ξ) with respect to λ:

∂λφ(λ, ξ)|λ=0= −i Z

−∞

dl 2π

r|l|

2(ˆbleilξ− ˆble−ilξ) The Fourier transform is:

Z

−∞

dξe−il0ξ

∂λφ(λ, ξ)|λ=0= −i r|l0|

2 (ˆb0l− ˆb−l0) This leads to:

ˆb0l− ˆb−l0 = Z

−∞

dk 2π

s

|k|

|l|(ˆak Z

−∞

dξeaξ/c2ei(c2 kaeaξ/c2−lξ)− ˆak Z

−∞

dξeaξ/c2e−i(c2 kaeaξ/c2+lξ)) (2.16) To make the equations more readable we define some notations:

< l, k > ≡ Z

−∞

dξei(c2 kaeaξ/c2−lξ) (l, k) ≡

Z

−∞

dξeaξ/c2ei(c2 kaeaξ/c2−lξ)

They have the following properties:

< l, k >=< −l, −k >, < l, −k >=< −l, k > (l, k) = −ia

c2

∂k < l, k >, (l, −k) = ia c2

∂k < l, −k > (2.17) Using this notation and solving (2.15) and (2.16) we get:

ˆbl= 1 2

Z

−∞

dk 2π(ˆak(

s

|l|

|k|− i a c2

q|k|

|l|

∂λ

) < l, k > +ˆak( s

|l|

|k|− i a c2

q|k|

|l|

∂λ

) < l, −k >)

ˆb−l= 1 2

Z

−∞

dk 2π(ˆak(

s

|l|

|k|+ i a c2

q|k|

|l|

∂λ

) < l, k > +ˆak( s

|l|

|k|+ i a c2

q|k|

|l|

∂λ

) < l, −k >) (2.18) Now we need to solve < l, k >. To do so we split the integral in two parts:

< l, k >= I1+ I2 Where I1 and I2 are:

I1 = Z

0

dξei(c2 kaeaξ/c2−lξ) = c2 a

Z 0

dxeic2(kaexalx) = c2 a(−ik

ac2)ic2 laΓ(−ic2l

a, −ic2k a) I2 =

Z 0

dξei(c2 kaeaξ/c2+lξ) = c2 a

Z 0

dxeic2(kae−x+alx)= c2 a(−ik

ac2)ic2 laγ(−ic2l

a, −ic2k a)

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Chapter 2. Rindler Space-time 21

Where Γ(α, x) =R

x dte−ttα−1and γ(α, x) =Rx

0 dte−ttα−1are the incomplete Γ-functions[15].

We can use the property of these functions Γ(α, x) + γ(α, x) = Γ(α) for l, k > 0 to calculate

< l, k >:

< l, k >= kic2 la 1

c2a−ic2 la−1eπlc22a Γ(−ic2l a) With this we can now calculate the derivative in (2.17):

(l, k) = l

k < l, k >

Using this in (2.18):

ˆbl= Z

0

dk 2π

rl

k(ˆak< l, k > +ˆak< l, −k >), l > 0 ˆbl=

Z 0

dk 2π

r −l

k (ˆa−k < l, k > +ˆa−k < l, −k >), l < 0 ˆbl =

Z 0

dk 2π

rl

k(ˆak< −l, −k > +ˆak < −l, k >), l > 0 ˆbl =

Z 0

dk 2π

r −l

k (ˆa−k < −l, −k > +ˆa−k < −l, k >), l < 0

Here < l, k >, < l, −k > etc. are the Bogoliubov coefficients for the transformation between the ladder operators for Minkowski and Rindler space-time.

Using the relation |Γ(iy)|2 = y sinh(πy)π and the commutation relations in (2.13), it is now possible to calculate the average occupation number of Rindler particles in the Minkowski vacuum:

h0|ˆblˆbl|0i

M M = zc2

a × 1

e2πkbcωla − 1 Where z is a constant infinite factor. So:

h0|ˆblˆbl|0i

M M ∝ 1

e2πkbcωla − 1

(2.19)

(2.19) is the occupation number in Bose-Einstein statistics for a system with temperature:

T = ~a

2πkbc (2.20)

This temperature is called the Unruh temperature. Note that while everything we have done so far was for the right Rindler wedge, the same calculation can be done for solutions in the left Rindler wedge resulting in the same Unruh effect.

So, in summary, a Rindler observer does not experience the same vacuum that an inertial observer does. For a system in a Minkowski vacuum, a Rindler observer will experience the system in a thermal equilibrium with temperature (2.20). This is called the Unruh effect.

We already showed the similarities between a Rindler observer and an observer standing still

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