UNIVERSITY OF AMSTERDAM

Master Thesis

Building traversable wormholes fromCasimir energy and non-local

couplings

by

Theodora Nikolakopoulou

11410310

Main Supervisor:

dr. Ben Freivogel

Second Supervisor:

dr. Diego Hofman

July 16, 2018

Abstract

The main purpose of this project is finding ways to construct traversable wormholes

(TW) and studying various aspects of them. It is now thirty years since Thorne and

Morris [1] understood that wormholes require the presence of exotic matter that violates

the null energy condition. In order to produce this necessary negative energy density

we mainly follow two different approaches.

First, we explore ways of making a wormhole traversable by using Casimir energy

[2]. We study the work of Butcher [3], in which he constructs a long throat TW by

using a non-minimally coupled quantum scalar field, and we also make an attempt to

construct an asymptotically AdS wormhole using a photon field.

The bigger part of this work is dedicated to another approach one can follow in

order to make a TW. Recently, Gao, Jafferis and Wall [4] showed that by coupling two

asymptotic boundaries of a maximally extended BTZ black hole, the Einstein-Rosen

bridge connecting the two asymptotic regions can be rendered traversable. In this

thesis, we study extensively these non-local couplings, first in flat space, and then in

the case of the BTZ black hole. We then perform explicit calculations in order to make

sure that any signal we send through this wormhole will indeed reach the other side

safely and that no violent events, such as the creation of another black hole, will take

place. Furthermore, we make some estimates about how much information we can send

through, from the bulk point of view, as well as from the quantum teleportation point

of view.

2

Contents

1 Introduction 7

1.1 Wormhole origins and 1st Renaissance . . . . . . . . . . . . . . . . . . . . . . 7

1.2 2nd Renaissance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Preliminaries 12

2.1 Energy conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Average Energy Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Proofs of ANEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Casimir effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 Electromagnetic Casimir effect . . . . . . . . . . . . . . . . . . . . . . 14

2.4.2 Topological Casimir effect . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Traversable wormholes from Casimir energy 17

3.1 An attempt to construct a wormhole using Casimir energy . . . . . . . . . . 17

3.2 Casimir energy of a long wormhole throat . . . . . . . . . . . . . . . . . . . 19

4 BTZ black hole 22

4.1 BTZ propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 The thermofield double formalism . . . . . . . . . . . . . . . . . . . . . . . . 24

5 BTZ shock-waves 25

6 Non-local coupling in 1+1 flat spacetime 29

6.1 First order calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.2 Smearing of the sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.3 Quantum Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7 Non-local couplings in black holes 36

7.1 BTZ with smeared sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7.2 AdS2 black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

7.2.1 Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

7.2.2 Gravity computation . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

7.2.3 Probe limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7.2.4 Bounds on information transfer . . . . . . . . . . . . . . . . . . . . . 47

7.3 BTZ with non-smeared sources . . . . . . . . . . . . . . . . . . . . . . . . . 48

7.3.1 Modified two-point function . . . . . . . . . . . . . . . . . . . . . . . 48

7.3.2 One-loop stress-energy tensor . . . . . . . . . . . . . . . . . . . . . . 50

7.3.3 Calculating the shift . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4

7.3.4 The center of mass energy of the collision . . . . . . . . . . . . . . . . 56

7.3.5 Bounds on the number of particles we can send through . . . . . . . . 59

8 Future directions 61

Acknowledgements 63

Appendix 65

A Electromagnetic Casimir effect in 3+1 . . . . . . . . . . . . . . . . . . . . . 65

B Second order stress tensor in 1+1 flat spacetime . . . . . . . . . . . . . . . . 70

C Refinement of the expression of the stress tensor TUU . . . . . . . . . . . . . 73

References 76

5

1 Introduction

1.1 Wormhole origins and 1st Renaissance

One of the most popular and exciting science-fiction concepts is that of a wormhole. Numer-

ous movies, series and books include such spacetime shortcuts that allow travellers to cover

distances that otherwise would take many lifetimes to travel. But the burning question is:

can we actually do this? Does physics allow for the existence of such objects?

Physicists have been puzzling over this question for almost a century. The simplest

theoretical wormholes are non-traversable, which means that even though they can exist we

cannot send anything through. The main problem is that they connect regions of space

that are spacelike separated. But, from a practical point of view there are more obstacles,

such as horizons and curvature singularities. If an astronaut decided to take a trip down

a black hole, all we would ever see is her moving slower and slower but never reaching the

horizon. Moreover, even if the astronaut could somehow escape a curvature singularity, the

tidal effects close to it would be extreme enough to tear her apart. All the above, as well as

other problems1 of the first theoretical wormholes, were discouraging physicists from taking

them seriously.

However, in 1988, Morris and Thorne found a way to construct a traversable wormhole

with pleasing characteristics. In order to avoid having both horizons and naked singularities,

which is the least one can ask in order to have a well-behaved object, they chose to con-

sider wormholes that have no curvature singularity. However, they did not follow the usual

approach (picking a Lagrangian with fields that hopefully support a wormhole, finding the

stress tensor and solving the Einstein equations), which was proven not to be a fruitful path.

They did the process in reverse: they chose a suitable metric describing a well-behaved

wormhole, they found the Einstein tensor and deduced what the stress tensor should be.

What they found was that the matter near the wormhole throat should not be the ordinary

matter that we constantly stumble upon in our universe, but an “exotic” kind of matter that

violates the null energy condition, along with all the other energy conditions (which we will

further explain in 2.1, 2.2). Now, if we were trying to make a purely classical wormhole (a

wormhole supported only by classical fields) the conclusions of Morris and Thorne should

make us sigh in despair. However, we know that in quantum field theories (QFTs) the energy

conditions we previously mentioned are not true any more, we can measure negative energy

locally, so not all is lost.

1One of the other problems we refer to is, for example, naked singularities, which were considered in order

to avoid having horizons in the way. However, such things violate the cosmic censorship hypothesis, which

leads to the failure of determinism.

6

1.2 2nd Renaissance

Many physicists throughout the years have tried to build traversable wormholes using dif-

ferent fields and techniques, but the construction that stood out and led to the comeback of

traversable wormholes was that of Gao, Jafferis and Wall (GJW) [4], in 2016. In this work,

they constructed a traversable wormhole in Anti-de-Sitter (AdS) spacetime. Their set-up

was the maximally extended AdS-Schwarzschild black hole2 in three dimensions (otherwise

called the BTZ black hole). This geometry has two asymptotically AdS regions, that are

connected by a non-traversable wormhole which collapses into a singularity. Starting from

this geometry, they coupled the right and left boundaries of the black hole at some time t0,

by adding in the action a term of this form:

δS =

∫dt dx hOR(t, x)OL(−t, x), (1.1)

where O is an operator dual to a scalar field ϕ in the bulk, and h is the coupling constant.

This resulted in the propagation of negative energy shock waves in the bulk. Thus, the

quantum matter stress tensor violates the averaged null energy condition. So, if we had sent

a light-ray really early (almost hugging the horizon) from the left/right boundary towards

the right/left boundary, it would not end up in the singularity. Instead, it would in principle

pass through the horizon, gain a time advance due to the encounter with the negative

energy density, and reappear at the right/left boundary. Thus, the wormhole is rendered

traversable. Of course, in real life it is not possible to connect two asymptotic regions since

they are spacelike separated. However, in a lab we could imagine building two copies of a

CFT on two plates and connect them with some “wire”. Thus, to consider a theory with

such a non-local coupling term is certainly something sensible.

This publication was followed by another, authored by Maldacena, Stanford and Yang

(MSY)[8], where they explored a similar set-up in AdS2. It is worth mentioning that in their

paper, they constructed a quantum teleportation protocol picture. Quantum teleportation

is the process during which we transmit a quantum state over long distances, while having

to transport only classical information. This process also requires previously shared entan-

glement between the sending and receiving region. So, we are actually moving one qubit

from one place to the other without having to physically transport the underlying particle to

which that qubit is normally attached. The actual protocol goes as follows. First, we split

an EPR pair and give one qubit to Alice and the other one to Bob. Alice also has a qubit

that she wants to teleport (we’ll call it the teleportee). She makes a Bell measurement of

the EPR pair qubit and the qubit to be teleported, and obtains a result. Then, she sends

this result to Bob through the classical channel. Finally, Bob modifies his qubit in order to

make it identical to the teleportee. MSY made a variation in the set-up of GJW, in order to

2Of course, one of the reasons to consider a black hole in AdS, instead of any other spacetime, is that

we can think of it in the context of AdS/CFT, in which the BTZ is dual to two copies of a CFT in the

thermofield double state

7

make it the same as the procedure we just described. Using this picture, they also calculated

some bounds on the information that can be sent through the wormhole, which is going

to be of interest for the purposes of this thesis. More publications inspired by the idea of

GJW followed [9],[10], [11], [12], [13] and thus, we might say that they sparked a second

Renaissance for traversable wormholes.

What is more, the set-up of GJW is not interesting just because of the obvious result of

making an Einstein-Rosen bridge traversable. It is important because it provides a model of

how a signal, and information in general, can escape a black hole.

1.3 Summary of Results

The main focus of this thesis is to study how to make traversable wormholes using non-

local couplings. In order to understand how they work we first apply them in the simplest

case we could think of: the free massless scalar in 1+1 dimensional flat spacetime. We

add to the action the interaction term δS = −gφLφR, where φL,R is the field operator

evaluated at uL,R, vL,R respectively, and calculate the stress tensor. What we find is that the

expectation value of the energy density, to first order in the coupling constant g, had the

form of two positive and two negative energy shock waves propagating in spacetime, as in

the left sub-figure below. We then smear out our sources in a diamond-shaped area around

(uL,R, vL,R) and re-calculate the expectation value of the energy density to first order in g.

This time, instead of localized shock waves, we obtain extended strips of positive/negative

energy density.

(uL, vL) (uR, vR)(0, 0)

a b c de

f g h i

(0, 0)(uL, vL) (uR, vR)

Figure 1: On the left we see the configuration without the smearing, whereas on the right we see

the configuration when we smear our sources. The blue/red lines or strips always represent the

negative/positive shock waves, respectively.

It’s easy to see that in both cases (non-smeared and smeared) the integral of the stress

tensor, along constant v (with vL < v < vR) or u (with uL < u < uR), is negative and thus

we violate the average null energy condition (ANEC). The motivation to smear our sources

is that we want to calculate the second order correction to the expectation value of the stress

tensor. In this calculation, we encounter the IR divergences of the scalar field, which are

cured if we smear our sources. For example, our result for the second order term in strip f

is:

〈Tuu〉f = − g

32A2πlog

(uL − u− AuL − u+ A

)− g2

32A2πlog (2Aµ) , (1.2)

8

where A is half the side of the diamond, and µ is the IR-cutoff. Furthermore, we check

whether the Quantum Inequalities (the other popular way of restricting negative energies)

are true for such a set-up. We find that (1.2) is violating these as well.

Of course, the most interesting part is when we apply this to the case of the BTZ black

hole. In our case, instead of smearing our operators OL and OR like GJW, we insert them

on a single instant of time. The reason behind this choice is that we want to have an analytic

expression for the stress tensor, which we calculate to be:

〈TUU 〉 = − ∆ sinπ∆

22∆+1/2π3/2

Γ(1−∆)

Γ(32 −∆)

h

`

U−(∆+1)√

2 (1/2−∆)

(1− U/U0)∆+1/2 (1 + U/U0)1/2(U2

0 + 1

U0

)−(∆+1)

F1

(−∆;

1

2,∆ + 1;

1

2−∆;

U − U0

U + U0,

U − U0

U(1 + U2

0

)) , (1.3)

where ∆ is the scaling dimension of O and U0 is the point of insertion at the boundary of

AdS. We also find that the integral of the stress tensor for ∆ < 1/2 is negative (if we choose

h > 0) and thus, 〈TUU〉 violates the ANEC. Hence, the Einstein-Rosen bridge is rendered

traversable. In addition, we calculate how much the wormhole opens-up, or otherwise the

shift that a signal would take upon collision with this negative energy, and find it to be of

order Planck scale.

In order for a signal to pass through such a small opening it has be highly boosted.

However, if this signal is very energetic we have to make sure that upon collision with the

negative energy shock wave there are no stringy effects that we should take into consideration,

and that no new black holes are created. For this reason, we assume that the signal and

the negative energy shock wave are particles, and we calculate the center of mass energy of

the collision. We find it to be of order 1√`P . Thus, we believe that we should not worry

about the aforementioned possible complications and moreover, there is even room to send

more than just one particle through. We calculate the number of different and same species

particles that we can send through until we reach Planck energy, and we find them to be

ndiffmax = h ``P

and nsamemax =√h ``P

respectively, which are both very big numbers. So, from the

bulk point of view it seems that we are allowed to send a lot of particles. In other words, if

we qubits to these particles we can send a big amount of information to the other side. This

seems to be clashing with the result of MSY. As we will see, if we naively use their result in

our case, we conclude that we are only allowed to send less than one particle.

1.4 Outline

The outline of this thesis is as follows. In chapter 2, we provide some background knowledge

needed for the better understanding of this thesis, such as the energy conditions, the Casimir

effect and a brief introduction in AdS/CFT. Next, in chapter 3 we review the work of Butcher

[3], in which he constructs a traversable wormhole that is supported by its own Casimir

9

energy, using a massive, non-minimally coupled, quantum, scalar field. Moreover, we make

an attempt to construct an asymptotically AdS traversable wormhole using a photon field.

In chapter 4 we switch gears and focus on the BTZ black hole, which is the main set-

up that we are going to work with in the rest of the thesis. We present some of its basic

characteristics, its propagators and its CFT dual. In addition, in chapter 5, we describe

the derivation of shock waves in the BTZ geometry, which is an essential concept for what

follows. Next, in chapter 6, we present an easy way of acquiring negative energy densities in

QFT, by the use of non-local couplings, which is essential in order to violate ANEC. When

this idea is applied in the case of the BTZ black hole even more interesting things happen.

So, in chapter 7 we how to make a wormhole traversable using this idea. We follow the

chronological order and we first review the set-up of GJW and then that of MSY. Finally,

we choose a slightly different interaction term than the one of GJW, we calculate the matter

stress tensor at the horizon and we find that it violates the ANEC. We then do some checks

in order to make sure that the signal will travel through the wormhole safely and we calculate

the maximum number of particles that we can send through.

10

2 Preliminaries

2.1 Energy conditions

One concept that we have to get acquainted with in order to begin understanding wormholes

is the energy conditions. The essence of GR can be captured by the Einstein equations:

Gµν = 8πGNTµν , (2.1)

where Gµν is the Einstein tensor and Tµν is the stress-energy tensor. The l.h.s. represents

the curvature of spacetime and is determined by the metric and the r.h.s. represents the

matter/energy content of spacetime. So, the Einstein equations can be summarized as the

main relation between matter and the geometry of spacetime.

Despite their elegance, the Einstein equations have a great deal of arbitrariness when

it comes to deciding what Tµν is going to be. Since all metrics satisfy Einstein equations,

we can choose any metric we like, calculate the Einstein tensor and then demand that Tµν

is proportional to Gµν . However, this does not necessarily mean that the stress tensor we

found is going to describe a realistic source of energy.

In order to make sure that the stress tensors we deal with are physical we have to

impose some restrictions, namely, the energy conditions. As Carroll [5] explains: “The energy

conditions are coordinate-invariant restrictions on the energy-momentum tensor”. In order

to have coordinate-invariant quantities we construct scalars that contain the stress tensor by

contracting it to timelike or null vectors. There are many different energy conditions that

apply to different circumstances. In order to gain more intuition we will use the stress tensor

of the perfect fluid, which is:

Tµ = (ρ+ p)UµUν + pgµν , (2.2)

where Uµ is the four-velocity of the fluid, ρ is the energy density and p the pressure. So, let’s

now see the most frequently used energy conditions:

1. The Null Energy Condition (NEC): Tµν`µ`ν ≥ 0 for all null vectors `µ. This condition

is the hardest one to violate. For the perfect fluid it implies that ρ+ p ≥ 0.

2. The Weak Energy Condition (WEC): Tµνtµtν ≥ 0 for all timelike vectors tµ. WEC

includes NEC and it for the perfect fluid it implies that ρ ≥ 0 and ρ+ p ≥ 0.

3. The Dominant Energy Condition (DEC): Tµνtµtν ≥ 0 (WEC) and TµνT

νλtµtλ ≤ 0,

for all timelike vectors tµ. The second part of the condition, namely TµνTνλtµtλ ≤ 0,

means that Tµνtµ is a non-spacelike vector. As we see DEC includes WEC. For the

perfect fluid DEC means that ρ ≥ |p|, i.e. the energy density is bigger or equal than

the absolute value of the magnitude of pressure.

11

4. The Null Dominant Energy Condition (NDEC): Tµν`µ`ν ≥ 0 (WEC) and TµνT

νλ`µ`λ ≤

0, for all null vectors `µ. The second part of the condition, namely TµνTνλ`µ`λ ≤ 0,

means that Tµν`µ is a non-spacelike vector. The NDEC is the DEC for null vectors

only. The densities and pressures allowed are the same as for the DEC, except negative

energy densities are allowed as long as p = −ρ.

5. The Strong energy condition (SEC): Tµνtµtν ≥ 1

2T λλt

σtσ, for all timelike vectors tµ.

SEC does not imply WEC. However, it implies NEC but at the same time it does not

allow for very large negative pressures. For the case of the perfect fluid SEC means

ρ+ p ≥ 0 and ρ+ 3p ≥ 0.

2.2 Average Energy Conditions

The energy conditions are in general true for classical matter. There are some exceptions,

for example, it is possible to violate the SEC in the case of a classical free scalar field, but

especially the WEC and the NEC are indeed always obeyed.

However, upon entering the quantum realm these conditions, as well as the rest of the

energy conditions, cease to be true and observers can measure negative energy density.

Examples where this happens is the Casimir effect [2] and the squeezed photon states [14],

both of which have been experimentally observed. Moreover, the existence of negative energy

density is required for the Hawking evaporation of black holes [15]. However, having no

restrictions on how much negative energy density we are allowed to observe, can result to

the violation of cosmic censorship [16],[17] and the second law of thermodynamics [18], [19].

As a result, over the recent years there have been significant efforts in finding reasonable

constraints.

There have been two main approaches. The first one is the quantum inequalities, first

introduced by Ford [20] which are constraints on the magnitude and duration of the negative

energy fluxes and densities, measured by an inertial observer. The second one, which is the

subject of this chapter, is the average versions of the energy conditions, first discussed by

Tipler [21]. He thought of integrating the WEC over a whole worldline of some observer.

This can be done for other energy conditions as well. The success of this method is that

these non-local conditions do hold for quantum field theories, unlike their local counterparts.

Two of the most popular averaged energy conditions are the following:

1. The Averaged Null Energy Condition (ANEC):∫γT µν`µ`νdλ ≥ 0, for all null vectors

`µ. We integrate over a null curve γ. Moreover, λ is a generalized affine parameter for

the null curve.

2. The Averaged Weak Energy Condition (AWEC):∫γT µνtµtνdτ ≥ 0, for all timelike vec-

tors tµ. We integrate over a timelike curve γ and τ is the proper time parametrization

of the timelike curve.

12

2.3 Proofs of ANEC

All these energy conditions that we previously mentioned have been motivated by GR, and

in this context they are considered to hold in any spacetime, curved or flat. However, it is

highly non-trivial to prove them for quantum field theories, even in the case of flat spacetime,

let alone curved. Many physicists have worked on proving these conditions in free quantum

field theories. It has been established that ANEC holds in Minkowski space for free scalar

fields [[22], [23]], for Maxwell fields [23],and arbitrary two dimensional theories with positive

energy and a mass gap [24].

During the last few years it has been understood that ANEC is not just a true statement

for QFTs, but one of their fundamental properties. The latter has been understood from

three different angles for interacting QFTs, in flat spacetime. In 2014 Kelly and Wall [25]

proved ANEC for a class of strongly coupled conformal field theories using AdS/CFT. Later,

in 2016, Faulkner, Leigh, Parrikar and Wang [26] proved ANEC from the point of view of

quantum information and , in 2017, Hartman, Kundu and Tajdini [27] proved ANEC using

causality, i.e. using that commutators should vanish at spacelike separation. We encourage

the interested reader to look up these papers and also the lectures of Thomas Hartman for

the Spring School on Superstring Theory and Related Topics 2018, that can be found here

[28]. Finally, we must note that there are some proposals for curved spacetimes, but no

actual proof.

2.4 Casimir effect

2.4.1 Electromagnetic Casimir effect

In 1948 Hendrik Casimir showed that in the presence of two conducting plates distorts the

vacuum energy of the electromagnetic (EM) field [2]. In particular, it is found to be negative

relative to the normal zero point energy.

This can be explained as follows. The plates are acting as boundaries. Thus, they are

forcing the waves to be quantized due to the interactions between the atoms of the plates

and the EM field. The plates are separated by distance L. So, the modes that have longer

wavelength than L, will not be able to fit. This means that we are “missing” some modes

between the plates. Thus, the vacuum energy we calculate is essentially lower than the

vacuum energy of the Minkowski vacuum that contains all modes.

The explicit calculation of the electromagnetic Casimir stress tensor can be found in

Appendix A. The result is the following:

T µνCasimir =π2

720a4

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 −3

(2.3)

13

Let’s see which of the energy conditions are violated in the case of the EM Casimir effect.

Since the energy density (ρ = T tt) is negative the WEC is automatically violated. Moreover,

it’s easy to show that NEC is also violated since ρ+ pz < 0. The aforementioned violations

do not come as a surprise since Casimir is a quantum effect. Of course, in all the previous

analysis we have assumed that we have perfectly conducting plates. If the plates are realistic

their mass is always much larger than the Casimir energy density and the the averaged

energy conditions are not violated.

In Visser’s book [29] there is an explicit analysis about what happens with the averaged

energy conditions. In a nutshell, he defines another Casimir stress tensor, similar to (2.3),

that corresponds to having realistic metal plates:

T µνCasimir = σtµtν [δ(z) + δ(z − a)] + Θ(z)Θ(a− z)π2

720a4[ηµν − 4zµzν ] , (2.4)

where tµ is the unit vector in the time direction and the plates are not ideal and have surface

mass density σ. Then he writes down ANEC and immediately infers that the only way it can

be violated is when σ is physically unreasonable. So, it’s safe to say that ANEC is obeyed.

The only case that some averaged energy conditions, like the AWEC, are violated is when a

photon is travelling parallel to the plates (ANEC is still obeyed).

From the above discussion we see that the case of realistic plates does not seem very

promising for our ultimate goal to built a traversable wormhole. However, Casimir effect

may also arise from the topology of spacetime, for example if we impose periodic boundary

conditions

2.4.2 Topological Casimir effect

As we mentioned, a variation of the original electromagnetic Casimir effect, is the Casimir

effect that arises due to non-trivial topologies. For example, if we are at 1 + 1 flat spacetime

and we take our universe to be periodic in the spatial direction (with period L), the Casimir

stress tensor of some field will be non zero. Let’s consider the simplest example, the free

massless scalar. The momentum of this field will be quantized in the spatial direction, due

to the periodicity. The field modes are:

uk = (2Lω)−1/2ei(ωt−kx), (2.5)

where k = 2πnL, n = 0,±1,±2, · · · . We want to calculate 〈0L|T µν |0L〉, where |0L〉 is the

vacuum of the quantum field in the periodic universe and |0〉 is the vacuum of the same

quantum field in normal Minkowski spacetime. Of course, if we take L→∞ then we should

recover 〈0|T µν |0〉. The stress-energy tensor components of the scalar field in two dimensions

are:

Ttt = Txx =1

2

(∂φ

∂t

)2

+1

2

(∂φ

∂x

)2

Ttx = Txt =∂φ

∂t

∂φ

∂x.

(2.6)

14

The field can be expanded as:

φ =∑k

(akuk(t, x) + a†ku

∗k(t, x)

), (2.7)

where ak/a†k are the annihilators/creators and uk the field modes. Using (2.7) we find that:

∂µφ∂νφ =∑k

∑k′

(ak∂µuk + a†k∂µu

∗k

)(a′k∂µu

′k + a†k′∂µu

′∗k

)=∑

k

∑k′

(2Lω)−1/2(2Lω′)−1/2kk′(−akak′eikxeik

′x+

aka†k′e

ikxe−ik′x + a†kak′e

−ikxeik′x − a†ka

†k′e−ikxe−ik

′x),

(2.8)

where k is the momentum tensor. In order to go from the first to the second line we have

used equation (2.5). We can now find the expectation value of ∂tφ∂tφ:

〈0L|∂tφ∂tφ|0L〉 =∑k

∑k′

(2Lω)−1/2(2Lω′)−1/2ωω′〈0L|aka†k′eikx−ik′x|0L〉 =∑

k

∑k′

(2Lω)−1/2(2Lω′)−1/2ωω′〈0L|(δkk′ + a†k′ak

)eikx−ik

′x|0L〉 =∑k

ω

2L

(2.9)

By performing a similar calculation we also calculate ∂xφ∂xφ =∑

k|k|2L

. For a free massless

scalar in two dimensions we have ω = |k| and thus finally we can find the timelike component

of the stress tensor to be:

〈0L|Ttt|0L〉 =1

2〈0L|∂tφ∂tφ|0L〉+

1

2〈0L|∂xφ∂xφ|0L〉 =

∑k

|k|2L

=2π

L2

∑n

n (2.10)

Let’s regulate the sum by giving a penalty to the high frequency modes, using the Heat-

Kernel cutoff:

〈0L|Ttt|0L〉 =∑k

|k|2L

=2π

L2

∑n

ne−a|k| =2π

L2

∑n

ne−2πaL , (2.11)

where a is the regulator. In the end we are going to a → 0. For convenience, we define

ε = 2πaL

and proceed.

〈0L|Ttt|0L〉 =2π

L2

∑n

ne−εn =2π

L2

∑n

∂

∂ε

(e−εn

)=

2π

L2

∂

∂ε

∑n

e−εn =

2π

L2

∂

∂ε

(1

1− e−ε

)=

2π

L2

e−ε

(1− e−ε)2=

2π

L2

(1

ε2− 1

12+ · · ·

)=

1

2πa2− π

6L2,

(2.12)

15

In order to finally obtain the Casimir stress tensor we have to subtract the expectation value

of the stress tensor on the original Minkowski vacuum from the one on |0L〉. Thus:

TCasimirtt = 〈0L|Ttt|0L〉 − 〈0|Ttt|0〉 = 〈0L|Ttt|0L〉 − lim

L→∞〈0L|Ttt|0L〉 = − π

6L2(2.13)

Thus, the scalar field has a non zero Casimir stress tensor due to the periodicity of spacetime.

Of course similar effects exist in different dimensions.

3 Traversable wormholes from Casimir energy

3.1 An attempt to construct a wormhole using Casimir energy

The previous calculations have been in four dimensional Minkowski space. But one could

wonder, if we perform a Weyl transformation to our metric and assume that the parallel

planes we used before are the boundaries of AdS, could we get a metric that describes a

wormhole? Of course, the generalized second law (GSL) of causal horizons states that it’s

not possible to have traversable wormholes connecting two disconnected regions, but it is

worth to try. In this scenario, we have:

gµν = Ω(z)2ηµν , (3.1)

where Ω(z)2 is the conformal factor. The variable z is goes from 0 to a. As, we mentioned

before, a is the separation between the two ideal plates. Using the metric (3.1) we can

calculate the components of the Einstein tensor:

Gtt = −Gxx = −Gyy =(Ω(z)′)2 − 2Ω(z)Ω(z)′′

Ω(z)2

Gzz =3 (Ω(z)′)2

Ω(z)2.

(3.2)

Moreover, the stress-energy tensor transforms as [30]:

Tµν = Ω(z)−2Tµν , (3.3)

where Tµν is the Casimir stress-energy tensor of the photon:

TCasimirµν =

π2

720a4

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 −3

. (3.4)

We can now use (3.2) and (3.3) in order to write down the Einstein equations. Since we

want our wormhole to be asymptotically AdS we need to also add a cosmological constant

to the Einstein equations. So, we have:

Gµν = 8πGNTµν − Λgµν , (3.5)

16

with Λ < 0. Equation (3.5) gives us two separate equations:

(Ω(z)′)2 − 2Ω(z)Ω(z)′′ = −8π3GN

720a4− ΛΩ(z)4 (3.6)

and

(Ω(z)′)2

= −8π3GN

720a4− ΛΩ(z)4. (3.7)

We can rewrite the second one as:

dΩ

dz= ±

√−ΛΩ4 − GNc0

a4, (3.8)

where c0 ≡ 8π3

720and we have dropped the argument of Ω for convenience. We want our space-

time to be asymptotically AdS. So, near the boundaries the metric should asymptotically

be:

ds2 =`2

z2

(−dt2 + dz2 + dx2 + dy2

), z > 0 (3.9)

which covers half of AdS and is conformally equivalent to half-space Minkowski spacetime.

As we see from (3.9), Ω should be infinite near the boundaries and take a minimum value

in the center. So, if we want to integrate dz from 0 to a/2 (the center), we have to pick the

negative sign in (3.8) since in this region Ω is decreasing:

−∫ Ωmin

∞

dΩ√−ΛΩ4 − GN c

a4

=

∫ a/2

0

dz ⇒ a

2=

∫ ∞Ωmin

dΩ√−ΛΩ4 − GN c0

a4

, (3.10)

where we integrate If we make a coordinate change and define u ≡ Ωa the above equation

takes the following form:1

2=

∫ ∞aΩmin

du√−Λu4 −GNc0

(3.11)

As we can see, the a’s dropped. So, we see that in principle we want the r.h.s to be of order

one. Then, we make a second change of coordinates and define x = −(

ΛGN c0

)1/4

u and thus

(3.11) becomes:

1

2=

(GNc0

Λ

)1/4 ∫ ∞xmin

dx√x4 − 1

(3.12)

where xmin = aΩmin

(Λ

GN c0

)1/4

. The integral at the r.h.s of (3.12) is an number of order one.

Thus, for (3.12) to be true we need GN ∼ Λ, which means that `P ∼ `. This is problematic

since we started by assuming that they were not of the same order. Moreover, the regime

that we know how to handle is when ` `P , since for distances of Planck scale order we

expect quantum gravity effects to appear, and we do not yet know how to treat them. Thus,

this does not seem to be the best approach in order to build a traversable wormhole from

Casimir energy.

17

3.2 Casimir energy of a long wormhole throat

As we saw, Casimir energy can arise without the use of actual plates but due to the topology

of spacetime. So, it seems like the topological Casimir effect might be a better candidate

for the construction of traversable wormholes. This is exactly what is discussed by Luke

Butcher in his paper “Casimir Energy of a Long Wormhole Throat” [3].

The idea is whether a wormhole, itself, can produce the Casimir energy it requires. In

order to achieve this the shape of the wormhole has to be optimized in a way as to produce

as much negative energy density as possible and require as little negative energy density

as possible. In order to achieve this we have to make the wormhole much longer that it is

wide. We start with a static spherically symmetric metric that can describe a traversable

wormhole:

ds2 = −dt2 + dz2 + A(dθ2 + sin2 θdφ2

), A =

√L2 + z2 − L+ a, (3.13)

where 2L is the length of the wormhole throat and a its radius. As we can see in figure 3.2

this metric represents a surgically constructed wormhole that connects two flat regions.

The stress tensor for this metric in the orthonormal basis is:

Tµν =Gµν

κ=

L2

(L2 + z2)A2κdiag

(1,−1,

A√L2 + z2

,A√

L2 + z2

)+

2L2

(L2 + z2)3/2Aκdiag (−1, 0, 0, 0)

(3.14)

We are going to assume that L ≥ a and consequently A√L2+z2 ≤ 1. If that’s the case then it’s

straightforward to see that the fist part of the stress tensor obeys all the energy conditions,

whereas the second does not. So, essentially the second part is the “exotic” matter that we

need in order to support the wormhole. Also, we can see that the second part of the stress

tensor takes its maximum value at z = 0 (in the center of the wormhole), and this value is :

ρmaxrequired ∼

2

Laκ, (3.15)

We can use this result as a measure of how much negative energy we need.

18

We would also like to know how much negative energy can be produced. As we saw, in

the center of the wormhole we need the most negative energy. We can find the radius of the

wormhole near the center if we Taylor expand A around z = 0. The result is:

A = a =z2

2L+O

(z4

L3

), (3.16)

and if L is very large comparing to z then A ∼ a. So, if there is a field in our spacetime it

will become quantized inside the wormhole throat. The field modes that have wavelength

shorter than L will fit in the wormhole throat, whereas the ones with longer wavelength will

not. Due to this, we expect Casimir energy to be produced, which will be of order:

ρmaxproduced ∼

~a4. (3.17)

It’s easy to see that if we keep a constant and make L very big we minimize the required

energy and make the produced energy approximately constant. Now, if ρmaxrequired ≈ ρmax

produced

we have that:2

Laκ∼ ~a4⇒ a ∼ (`p)

2 L

a, (3.18)

from which we can infer that if L a, both of L and a are bigger than the `p. So, by taking

L→∞ we have the following metric to work with:

ds2 = −dt2 + dz2 + a(dθ2 + sin2 θdφ2

). (3.19)

We are going to consider a non-minimally coupled massive scalar field in our spacetime, that

has the following action:

S =

∫dx4√−g((∇ϕ)2 + (m2 + ξR)ϕ2

), (3.20)

where ξ is the coupling constant for the interaction term between gravity and the scalar

field. For the case of the conformally coupled scalar field ξ = 16. The classical stress tensor

can be calculated by varying the action with respect to the metric:

Tµν =2√−g

δS

δgµν= ∇µϕ∇νδphi+ξ

(Rµνϕ

2 −∇µ∇ν(ϕ2))−gµν

1− 4ξ

2

((∇ϕ)2 + (m2 + ξR)ϕ2

)(3.21)

The field will become quantized inside the wormhole throat, so we will expand it as follows:

φ =∑n

(ϕ−n a

−n + ϕ+

n a+n

), (3.22)

where a+n = (a−n )

†are the creation/annihilation operators, satisfying the usual commutation

relations. As we saw in section 2.4.2, the goal is to calculate the difference between the

expectation value of the stress tensor of ϕ in the spacetime with the non-trivial topology

(here the wormhole throat) and the expectation value of the stress tensor of ϕ in normal

Minkowski spacetime, i.e.:

〈0a|Tµν |0a〉 − 〈0|Tµν |0〉, (3.23)

19

where |0〉 is the Minkowski vacuum and |0a〉 is the “throat” vacuum. The actual calculation

is a very extensive and involved one and goes beyond the scope of this analysis. We are

going to give the final result and discuss it. The Casimir stress tensor is:

TCasimirµν =

1

2880π2a4

[diag (−1, 1,−1,−1) 2 log

(a

a0

)+ diag (0, 0, 1, 1)

], (3.24)

where a0 is length scale introduced by the regularization scheme the author chose. However,

it has a simple meaning. It is clear from (3.24) that if our wormhole has radius a0 the

Casimir energy density becomes zero. Thus, we can interpret it as the radius that the

Casimir energy density vanishes. If a is sufficiently larger than a0 then the Casimir energy

density is negative. Consequently, the dominant and weak energy condition are immediately

violated because T00 < 0. Moreover, the required energy density ρmaxrequired, that we previously

defined, can be supplied by the stress tensor of the scalar field. So, we may write:

2

Laκ=

log (a/a0)

1440π2a4⇒ a2 = `2

p (L/a)log (a/a0)

360π. (3.25)

If L a, indeed the wormhole has a macroscopic throat-radius a `p. However, up to this

point we have completely ignored the non-“exotic” part of the stress tensor.

We saw before that for a > a0 the weak and dominant energy conditions are automatically

violated since the energy density is negative. Let’s now check what happens with the null

energy condition. We need that Tµνkµkν < 0, for some null vector kµ. So, we have:

Tµνkµkν = T00k

0k0 + T11k1k1 + T22k

2k2 + T33k3k3 < 0, (3.26)

where we have dropped the hats for convenience. We also know that in order for kµ to be a

null vector it needs to satisfy:

kµkµ = 0⇒ −k20 + k2

1 + k22 + k2

3 = 0⇒ k23 = k2

0 − k21 − k2

2. (3.27)

By using (3.27) and also that T11 = −T00 and T22 = T33, (3.26) becomes:(k2

0 − k21

)(T00 + T22) < 0. (3.28)

We substitute T22 and T00 and get:(k2

0 − k21

) 1

2880π2a4(−4 log (a/a0) + 1) < 0 (3.29)

From (3.27) we know that k20−k2

1 = k22 +k2

3 > 0 and thus −4 log (a/a0)+1 should be smaller

than zero. This happens when:

a > a0e1/4 (3.30)

Thus, the NEC is violated if a > a0e1/4 for all null vectors except for k0 = ±k1, i.e for

the null ray that is parallel to the throat. So, all possible observers travelling through this

20

wormhole will “see” negative energy except the null ray that is travelling directly parallel to

the throat

This particular null direction is the one causing problem with the stability of the worm-

hole. We would like to be able to solve Einstein’s equations, but with some additional

ordinary matter3. Then we have:

Gµν = 8πκ(TCasimirµν + T ordinary

µν

), (3.31)

and let’s contract this with kµ, a null or timelike vector as follows:

Gµνkµkν = 8πκ

(TCasimirµν kµkν + T ordinary

µν kµkν). (3.32)

For almost all null and timelike vectors it’s true that TCasimirµν kµkν < 0 , so we should be able

to accommodate negative values for Gµνkµkν . However, for kµ = (1,±1, 0, 0) we have that

TCasimirµν kµkν = 0 and thus ifGµνk

µkν < 0 then also T ordinaryµν kµkν < 0, which is a contradiction

since ordinary matter satisfies the null energy condition by definition. Consequently, it is

not possible to solve the Einstein equations for this kind of wormhole just by using the

Casimir stress energy tensor and some ordinary matter and the Casimir energy produced by

the wormhole itself is not enough to stabilize it permanently. The reason behind this it that

the wormhole throat is spherically symmetric which makes the Casimir stress tensor have

the form TCasimir = diag (ρ,−ρ, p, p). Thus, when contracted with kµ = (1,±1, 0, 0) it gives

zero. In order to avoid this effect, the author suggests inducing some symmetry breaking,

for example by “twisting” the throat.

However, even though the wormhole is not permanently stable it collapses slowly allowing

a null ray to cross it as is shown explicitely in [3]. So, this is an example where Casimir

energy allows for the creation of a traversable wormhole. Of course, it is slowly collapsing

and is not stable, but it is traversable nonetheless.

4 BTZ black hole

1+2-dimensional gravity, at first sight, looks trivial. In particular general relativity has no

Newtonian limit and the graviton has no propagating degrees of freedom. So, it came as a

surprise when Baados, Teitelboim and Zanelli discovered the BTZ black hole solution [31].

The BTZ black hole differs from the Kerr and Schwartzchild solutions in some important

aspects. Firstly it is asymptotically AdS, instead of asymptotically flat and secondly it

does not have a curvature singularity at the origin. However, it is indeed a black hole

with a horizon, it appears as the final state of collapsing matter, and it has thermodynamic

properties similar to these of the 1+3-dimensional black hole.

The uncharged, non-rotating BTZ black hole metric in “Schwartzchild” coordinates is:

ds2 = −r2 − r2

h

`2dt2 +

`2

r2 − r2h

dr2 + r2dφ2, (4.1)

3By ordinary matter, we mean matter that does not violate the energy conditions.

21

where rh is the horizon radius and ` is the radius of AdS. The φ coordinate has period 2π,

the mass of the black hole is M =r2h

8GN `2and its inverse temperature is β = 2π`2

rh. In some

cases it will be more convenient to use the metric in Kruskal coordinates, which smoothly

cover the maximally extended two-sided geometry. In Kruskal coordinates the metric has

the following form:

ds2 =4`2dudv + r2

h (1− uv)2 dφ2

(uv + 1)2 , (4.2)

where u > 0 and v < 0 in the right wedge (see figure below 2), uv = −1 at the boundaries

and uv = 1 at the singularities.

uvL R

Figure 2: On the right/left we see the Kruskal/Penrose diagram of the maximally extended BTZ

black hole.

As we see from figure 2, the maximally extended BTZ black hole has two asymptotically

AdS regions (L, R), that are connected by a non-traversable wormhole. That wormhole

collapses into a singularity in the future and in the past. The crossing lines represent the

horizons and separate the spacetime into four regions. The left and right regions are the

exterior of the black hole. The upper and lower regions are the future and past interior

respectively.

4.1 BTZ propagators

This discussion is based on [32]. In order to obtain the bulk-to-boundary propagator for the

BTZ black hole, we can exploit the fact that it is a quotient of AdS3. So, we only need to

add a sum over images on the bulk-to-boundary propagator of AdS3, in order to obtain the

BTZ propagators.

Let’s start from the bulk-to-boundary propagator. We need to specify a “source” point

on the boundary b′ and a “sink” point in the bulk x. For a scalar field of mass m the

22

bulk-to-boundary propagator in the right wedge, up to normalization, is:

K(x, b′)RR ∼∞∑

n=−∞

(−

√r2 − r2

h

r2h

cosh (rh(t− t′)) +r

rhcosh rh (φ− φ′ + 2πn)

)−∆

, (4.3)

where ∆ is the conformal dimension of the boundary operator dual to the massive scalar

field. Equation (4.3) is valid whenever the source and sink points are in the same region. If

we want the sink point to be in a the left wedge then we replace t→ t− iβ2

. Then we have:

K(x, b′)LR ∼∞∑

n=−∞

(−

√r2 − r2

h

r2h

cosh rh

(t− t′ − iβ

2

)+

r

rhcosh rh (φ− φ′ + 2πn)

)−∆

,

(4.4)

As we saw before β = 2π`2

rh. Assuming ` = 1, the propagator K(x, b′)LR takes the form:

K(x, b′)LR ∼∞∑

n=−∞

(√r2 − r2

h

r2h

cosh rh (t− t′) +r

rhcosh rh (φ− φ′ + 2πn)

)−∆

, (4.5)

We notice that (4.3) can be singular, whereas (4.5) is always finite. The reason for that

is that in the first case the points are timelike separated, whereas in the second they are

spacelike separated.

The boundary-to-boundary propagator can be acquired by sending the bulk point x to

the boundary. This is done in the following way:

P (b, b′) ∼ limr→∞

r∆K(x, b′). (4.6)

By doing the above, we obtain the boundary-to-boundary propagators:

P (b, b′)RR ∼∞∑

n=−∞

(− cosh rh(t− t′) + cosh rh(φ− φ′ + 2πn))−∆

(4.7)

P (b, b′)LR ∼∞∑

n=−∞

(cosh rh(t+ t′) + cosh rh(φ− φ′ + 2πn))−∆

(4.8)

In (4.8),we have assumed that in the second copy of the CFT the time increases towards the

future.

4.2 The thermofield double formalism

This discussion follows from [33] and [34].

The thermofield double formalism, developed by Takahashi and Umezawa [35], is a trick we

use to treat the thermal, mixed state ρ = e−βH as a pure state in a bigger system. If we

have a QFT with some Hamiltonian H, we first double the degrees of freedom by considering

two copies of this QFT. The states of this doubled QFT are |n〉1|m〉2. These two QFTs live

23

in different spacetimes and are non-interacting. Now, in this doubled system we consider a

particular pure state, i.e. the thermofield double state:

|TFD〉 =1√Z(β)

∑n

e−βEn/2|n〉1|n〉2, (4.9)

where the 1, 2 indicates the Hilbert space where the state is defined, Z(β) is the partition

function of one copy of the QFT with inverse temperature β. The density matrix of the

doubled QFT in this state is:

ρtot = |TFD〉〈TFD| (4.10)

The reduced density matrix of the first system is:

ρ1 =Tr2ρtot =∑

2

2〈m|

(1

Z(β)

∑n,n′

e−βEn/2|n〉1|n〉2 2〈n′|1〈n′|e−βE′/2

)|m〉2 =∑

n

e−βEn/2|n〉1 1〈n| = e−βH1

(4.11)

So, this pure state in the double system cannot be distinguished from a thermal state.

For the Hamiltonian of the doubled system we have two options. Either Htot = H1 +H2

or Htot = H1 −H2. We shall choose Htot, under which |TFD〉 is time independent since the

phases cancel.

We saw in the previous chapter that an eternal BTZ black hole has two asymptotic

boundaries. It has been proposed by Maldacena [34], that the BTZ is dual to two copies of

a CFT, in the thermofield double state. This was shown by performing the path integral on

the boundary CFT. We must note that even though the two CFTs are not interacting the

expectation value of two operators, each from one of the two independent CFTs is non zero.

This is due to the entanglement of the two theories, or equivalently, due to the presence

of the wormhole, as Maldacena and Susskind proposed [36]. Moreover, the Hamiltonian we

chose before, Htot, is dual to the Hamiltonian that generates the time evolution along the

isometry ∂t in the bulk.

5 BTZ shock-waves

In this chapter, we are going to mainly review the work of Shenker and Stanford on shock

waves in the BTZ black hole [37], [38]. Previously, we explored the case of an unperturbed

BTZ black hole and some of its properties. It is interesting to see what happens when we

mildly perturb it. So, starting form the thermofield double state we will add a perturbation

to it and see what happens. We can do this by adding some particle at the left boundary,

at some time t. Thus, we consider a CFT state of the form:

W (tw)|TFD〉, (5.1)

24

where W (tw) is a local operator that acts unitarily on the left CFT and raises the energy by

an amount E. We assume that the energy is much smaller than the mass of the black hole.

There are at least two ways of thinking of that state. The first one is to think of it as prepared

by changing the Hamiltonian at some time, which means that we start from the thermofield

double state and then we perturb it at time tw. This scenario is depicted in subfigure (a) of

figure 3. The second way is to think of it as a state with a time independent Hamiltonian,

in which case we have to follow this perturbation backwards through the past horizon. So,

in the second scenario, which is shown in subfigure (b) of figure 3, the perturbation comes

out of the white hole, approaches the left boundary at tw and falls in the black hole.

tw

L R

(a)

tw

L R

(b)

Figure 3: In this figure we see the insertion of particles at the left boundary at some early time.

The perturbation falls in through the future horizon. The double blue line

Naively, we would think that this perturbation will not have any effect on the geometry.

However, we may instead release the perturbation from the left boundary long in the past,

as in figure 4 . As we know, translation in Killing time acts as a boost at the near horizon

region. So, at the local frame of the timeslice t = 0, the energy we are going to measure is

going to be:

Ep ∼E`

Rerhtw/`

2

, (5.2)

where E is the initial energy of the particle. Thus, in this frame the particle is actually a

high energy shock wave that has a back reaction on the geometry. For simplicity, we consider

a spherically symmetric null shell of matter. The resulting geometry is obtained by gluing

two BTZ black holes of mass M and M +E accross the null surface vw = e−rhtw/`2. We will

use u, v coordinates for the past of the shell and u, v for its future. Since we are “tossing”

a positive energy object in our black hole we increase the mass and that makes the radius

grow. So, using that M =r2h

8GN `2, the new radius is:

rh =

√M + E

Mrh (5.3)

25

If we take a look at the metric (4.2), we see that the following has to hold:

rh1− uvw1 + uvw

= rh1− uvw1 + uvw

(5.4)

tw

L R

Figure 4: Here, we release the perturbation very early.

In order to solve (5.4) we will assume that vw = vw and we will define the new variables

x = uvw and x = uvw. Then (5.4) becomes:

rhrh

(1− x1− x

)=

1 + x

1 + x⇒(

1 +E

2M

)(1− x+ x− x

1− x

)=

(1 + x+ x− x

1 + x

)⇒(

1 +E

2M

)(1 +

x− x1− x

)=

(1 +

x− x1 + x

),

(5.5)

where we have Taylor expanded (5.3) around EM

= 0, and we have added and subtracted x

both in the left and right hand side terms, in order to go from the second line to the third.

Next, we add to both sides the term −(1 + x−x

1−x

)and get:

E

2M

(1− x1− x

)=

2(x− x)

(1− x)(1 + x)⇒

x− x1 + x

=E

4M(1− x).

(5.6)

Then, we substitute x, x:

vwu− vwu1 + vwu

=E

4M(1− vwu)⇒ u− u

v−1w + u

=E

4M(1− vwu)⇒

u− u =E

4M(1− vwu)(v−1

w + u)⇒ u =u+ E

4M(v−1w + u)

1 + E4M

(1 + vwu)⇒

u = u+E

4Mv−1w −

E

4Mvwu

2 +O(E2

M2

),

(5.7)

where we have expanded once again around EM

= 0 in order to from the second line to the

third. Since vw = e−rhtw/`2, for tw → ∞ we have that vw → 0. Thus, the term E

4Mvwu

2 is

26

going to be approximately zero and we can ignore it. Consequently, the solution is a simple

shift in the v coordinated, namely:

u = u+ a, a =E

4Merhtw/`

2

, (5.8)

where we have substituted vw in the last step of(5.7).

Figure 5: Here we see the (non-square) Penrose and Kruskal diagrams of the perturbed BTZ black

hole. The red parallel lines represent the shock wave. The horizons now are not touching any more.

They have separated by an amount of a.

We can write our new metric (after the backreaction) as:

ds2 =4`2dudv + r2

h [1− ((u+ aθ(v)) v]2 dφ2

[1 + (u+ aθ(v)) v]2, (5.9)

where θ(v) is the step function. What this means is that if we send a signal from the right

boundary towards the left boundary, it will suffer a time delay when it reaches the horizon

v = 0.

tw

L R

Figure 6: Here is the square Penrose diagram of the perturbed BTZ black hole. The double red

lines represent the shock wave. The orange line represents a signal that we send from the right

boundary.

27

As we see in figure 6, when the signal collides with the shock wave, it suffers a time delay

and essentially ends up in the singularity. So, for our purpose of constructing a traversable

wormhole, one could say that the shock wave make it even harder for a signal to cross to

the other side. In order for a signal to avoid falling in the singularity we would like to have

exactly the opposite effect, namely, our signal to gain a time advance instead of a time delay.

If we had an operator that could create a negative energy shock wave instead of a positive

one, the shift a would be negative. In this case, the signal would meet with the negative

energy shock wave, shift towards the opposite direction and reappear on the left boundary.

Below, we depict how such a hypothetical configuration would look.

tw

L R

6 Non-local coupling in 1+1 flat spacetime

In this chapter, we are now going to explore a simple way of getting negative energy density

in QFT. As we will see, if we add in the action of our system a term of the form δS = gφLφR

at a certain time, the resulting expectation value of the stress tensor consists of positive and

negative energy shock waves.

6.1 First order calculation

We consider a free massless scalar field in two dimensions, whose action is:

S = −∫

1

2∂µφ∂

µφ. (6.1)

We will deform the system by adding an interaction term of this form:

δS = gφLφR (6.2)

where φL,R is the field operator φ evaluated at xL and xR respectively, at a time-slice t = 0, in

two dimensional Minkowski space. We assume that the two points are spacelike separated.

For convenience, we will use light-cone coordinates. We are now going to calculate the

expectation value of the normal ordered stress-energy tensor on the state |Ψ〉 = eigφLφR |0〉,

28

to first order in g:

〈Ψ| : Tuu(u) : |Ψ〉 =⟨e−igφLφR : ∂uφ∂uφ : eigφLφR

⟩= 〈(1− igφLφR) : ∂uφ∂uφ : (1 + igφLφR)〉 =

ig 〈: ∂uφ∂uφ : φLφR〉 − ig 〈φLφR : ∂uφ∂uφ :〉 = ig 〈[: ∂uφ∂uφ :, φLφR]〉 .(6.3)

We define C ≡ 〈: ∂uφ∂uφ : φLφR〉. Then assuming φR, φL and ∂uφ are Hermitian, we can

recover the the commutator 〈[: ∂uφ∂uφ :, φLφR]〉 by taking the imaginary part of C:

〈Ψ| : Tuu(u) : |Ψ〉 = −2gIm (〈: ∂uφ∂uφ : φLφR〉) = −4gIm (〈∂uφφL〉 〈∂uφφR〉) , (6.4)

where in order to go from the second to the third equality we have performed the Wick

contraction. The correlators we need to calculate are non-time ordered. Therefore, we are

going to use the Wightman function rather than the Feynman propagator. The Wightman

function for the free massless scalar in two dimensions is [39]:

W (t, x; t′, x′) = 〈φ(t, x)φ(t′, x′)〉 = − 1

4π[log [iµ (∆t+ ∆x− iε)] + log [iµ (∆t−∆x− iε)]] ,

(6.5)

where µ is an infrared cutoff. In lightcone coordinates it takes the following form:

W (u, v;u′, v′) = 〈φ(u, v)φ(u′, v′)〉 = − 1

4π[log [iµ (∆u− iε)] + log [iµ (∆v − iε)]] . (6.6)

Then by using (6.6), we may compute (6.3) to be:

〈Ψ| : Tuu(u) : |Ψ〉 =

− 4g

16π2Im

(1

(u− uL)− iε· 1

(u− uR)− iε

)=

− g

4π2

(u− uL)

((u− uL)2 + ε2)· ε

((u− uR)2 + ε2)− g

4π2

(u− uR)

((u− uR)2 + ε2)· ε

((u− uL)2 + ε2),

(6.7)

and if we take the limit ε→ 0 we finally obtain:

〈Ψ| : Tuu(u) : |Ψ〉 = − g

4π

(δ(u− uR)

u− uL+δ(u− uL)

u− uR

), (6.8)

where we have used that:

δ(x) =1

πlimε→0

ε

x2 + ε2. (6.9)

Following exactly the same procedure we find that 〈Ψ| : Tvv(v) : |Ψ〉 = is:

〈Ψ| : Tvv(v) : |Ψ〉 = − g

4π

(δ(v − vR)

v − vL+δ(v − vL)

v − vR

). (6.10)

Hence, the resulting configuration of the energy density4 is:

4The energy density is defined as ρ ≡ T00(t, x), which in the case of the free massless scalar in 2d is equal

to Tuu(u) + Tvv(v)

29

v u

(uL, vL) (uR, vR)(0, 0)

Figure 7: The blue/red lines represent the regions of space where we have negative/positive energy

density.

A light ray travelling along u = 0 will only “pass through” negative energy density and

we will thus have∫∞−∞ 〈Tuu〉 du < 0.

Up to this point, everything we have calculated is to first order in g. We would rather

like to calculate the expectation value of the components of the stress tensor up to second

order in g,

〈Ψ| : Tuu(u) : |Ψ〉 =⟨e−igφLφR : ∂uφ∂uφ : φ eigφLφR

⟩=⟨(

1− igφLφR −g2φLφRgφLφR

2

): ∂uφ∂uφ :

(1 + igφLφR −

g2φLφRgφLφR2

)⟩=

− 4gIm (〈∂uφφL〉 〈∂uφφR〉) +

g2 〈φLφR : ∂uφ∂uφ : φLφR〉 −g2

2〈φLφRφLφR : ∂uφ∂uφ :〉 − g2

2〈: ∂uφ∂uφ : φLφRφLφR〉 ,

(6.11)

where we have omitted terms of higher order than g2. The final result is (for details of the

calculation, see Appendix B):

〈Tuu〉 = −4gIm (〈∂uφφL〉 〈∂uφφR〉) +

g2 (−2 〈φLφR〉 〈φL∂uφ〉 〈φR∂uφ〉 − 2 〈φLφR〉 〈∂uφφL〉 〈∂uφφR〉+ 2 〈φLφR〉 〈φL∂uφ〉 〈∂uφφR〉+2 〈φLφR〉 〈φR∂uφ〉 〈∂uφφL〉+ 2 〈φLφL〉 〈φR∂uφ〉 〈∂uφφR〉 − 〈φLφL〉 〈φR∂uφ〉 〈φR∂uφ〉− 〈φLφL〉 〈∂uφφR〉 〈∂uφφR〉+ 2 〈φRφR〉 〈φL∂uφ〉 〈∂uφφL〉 − 〈φRφR〉 〈φL∂uφ〉 〈φL∂uφ〉− 〈φRφR〉 〈∂uφφL〉 〈∂uφφL〉)

(6.12)

Similarly, we can compute the expectation value of the Tvv component of the stress tensor.

It has exactly the same form with (6.12), except u→ v. Both of 〈Tuu〉 and 〈Tvv〉, to second

order in g, contain correlators of the form 〈φLφL〉 and 〈φRφR〉, which diverge. In order to

cure this and get some meaningful results we need to smear our sources. This means that we

will “spread” our source in time and space. Since we are working in lightcone coordinates,

a convenient choice is to make our sources diamond-shaped.

30

6.2 Smearing of the sources

From now on instead of φL and φR, we are going to use:

OL ≡∫ vL+A

vL−A

∫ uL+A

uL−Advdu φ(u, v), and OR ≡

∫ vR+A

vR−A

∫ uR+A

uR−Advdu φ(u, v), (6.13)

where 2A is the side of the diamond (see figure 8). Now, we have to compute all the smeared

two-point functions. For demonstration purposes we will perform one of the calculations :

〈∂uφ(u, v)OL〉 =1

4A2

∫ vL+A

vL−A

∫ uL+A

uL−Adv′du′ 〈∂uφ(u, v)φ(u′, v′)〉 =

− 1

16πA2

∫ vL+A

vL−A

∫ uL+A

uL−Adv′du′

1

u− u′ − iε=

− 1

8πA(log (uL − A− u+ iε)− log (uL + A− u+ iε)) ,

(6.14)

where we have used (6.6), and we have divided by the volume of the diamond, 4A2, in order

to have the correct dimensions.

Next, we will take the limit ε → 0. We need to be careful and divide the space into

different zones, since the real part of the logarithm arguments can be either negative or

positive, depending on where we are.

I II III IV V

(0, 0)uL uR

Figure 8: The gray diamond areas are the smeared sources.

Zone I (u < uL − A) :

〈∂uφ(u, v)OL〉I =− 1

8πAlimε→0+

(log (uL − A− u+ iε)− log (uL + A− u+ iε)) =

− 1

8πAlog

(uL − A− uuL + A− u

).

(6.15)

Zone II (uL − A < u < uL + A) :

〈∂uφ(u, v)OL〉II =− 1

8πAlimε→0+

(log (uL − A− u+ iε)− log (uL + A− u+ iε)) =

− 1

8πA

(iπ + log

(−uL − A− uuL + A− u

)),

(6.16)

Zone III− V (u > uL + A) :

31

〈∂uφ(u, v)OL〉III−V =− 1

8πAlimε→0+

(log (uL − A− u+ iε)− log (uL + A− u+ iε)) =

− 1

8πAlog

(uL − A− uuL + A− u

).

(6.17)

Finally, we can repackage everything as:

〈∂uφ(u, v)OL〉 = − 1

8πA

(iπ θ (u− uL + A) θ (uL + A− u) + log

(∣∣∣∣uL − A− uuL + A− u

∣∣∣∣)) .(6.18)

In a similar fashion, we can also calculate the rest of the correlators appearing in (6.12) :

〈OL∂uφ(u, v)〉 = − 1

8πA

(−iπ θ (u− uL + A) θ (uL + A− u) + log

(∣∣∣∣uL − A− uuL + A− u

∣∣∣∣)) ,(6.19)

〈∂uφ(u, v)OR〉 = − 1

8πA

(iπ θ (u− uR + A) θ (uR + A− u) + log

(∣∣∣∣uR − A− uuR + A− u

∣∣∣∣)) ,(6.20)

〈OR∂uφ(u, v)〉 = − 1

8πA

(−iπ θ (u− uR + A) θ (uR + A− u) + log

(∣∣∣∣uR − A− uuR + A− u

∣∣∣∣)) ,(6.21)

〈OLOR〉 =− 1

16πA

(−12A2 − 2(uL − uR)2 log

(uR − uL

A

)+

(2A+ uL − uR

A

)2

log

(−2A− uL + uR

A

)+

(2A− uL + uR) log

(2A− uL + uR

A

)+ 8A2 log (Aµ)

),

(6.22)

and

〈OLOL〉 = 〈OROR〉 = − 1

2πlog (2Aµ) . (6.23)

Similarly, we find the associated two-point functions for 〈Tvv〉. The energy density is the sum

of 〈Tuu〉 and 〈Tvv〉 and now that we have all the two-point functions at hand we can finally

calculate the energy density up to second order in g, with smeared sources. The resulting

configuration is in the figure below:

a b c d

e

f g h i

(0, 0)uL uR

Figure 9: The light blue/red strips represent the negative/positive energy density.

32

In the areas a, c, e, g, i the stress-energy tensor is zero. In the strip b the stress-energy

tensor is:

〈Tuu〉b = − g

32A2πlog

(uR − u− AuR − u+ A

)− g2

32A2πlog (2Aµ) , uL − A < u < uL + A (6.24)

In the strip f we have:

〈Tuu〉f = − g

32A2πlog

(uL − u− AuL − u+ A

)− g2

32A2πlog (2Aµ) , uR − A < u < uR + A (6.25)

In the strip h:

〈Tvv〉h = − g

32A2πlog

(vL − v − AvL − v + A

)− g2

32A2πlog (2Aµ) , vR − A < u < vR + A (6.26)

and finally in d:

〈Tvv〉d = − g

32A2πlog

(vR − v − AvR − v + A

)− g2

32A2πlog (2Aµ) , vL − A < u < vL + A (6.27)

From the above results we notice that the O(g) term of the stress-energy tensor depends on

the distance between the sources, as well as the side of the diamond. However, the O(g2)

term only depends on the side of the diamond and the IR cutoff.

In order to see the plot of the stress tensor against the u coordinate, we will pick some

values for uR, uL and A. So, for uR = −uL = 10 and A = 0.1, we plot 〈Tuu〉 in the strip b:

We notice that the stress tensor is almost linear in u inside the strip. Moreover, we see that

it is slightly increasing with u. Next, we plot 〈Tuu〉 in the strip f , where we know that it is

negative:

33

In strip f the stress tensor decreases as u increases. As we previously mentioned, this stress

tensor violates the ANEC.

However, there are other bounds to the amount of negative energy we are allowed to have,

that are commonly referred to as the quantum inequalities (QIs). In the next subsection

we will see what the QIs are and check whether or not the stress tensor that we calculated

obeys them.

6.3 Quantum Inequalities

In section 2.2,we saw, that there are two different approaches in order to find constraints

analogous to the pointwise energy conditions. One approach is the average versions of the

energy conditions. The second approach, which we are going to discuss here is the QIs, first

introduced by Ford.[20]. Since then, they have been proved and refined by Ford, as well

as others. The first versions of the QIs were constraints on the magnitude and duration

of the negative energy fluxes and densities, measured by an inertial observer [40]. They

resembled the uncertainty principle because they said that a pulse of negative energy cannot

be arbitrarily intense for an arbitrarily long time. More precisely, they say that the duration

of a negative energy pulse is inversely related to its magnitude. Later the QIs were also

proved for the expectation value of the energy density in arbitrary quantum states, in d-

dimensional Minkowski spacetime. Let’s assume we have:

ρ(t) = 〈Ttt(t)〉 , (6.28)

the expectation value of the timelike component of the stress tensor evaluated on some

arbitrary state, at time t. Then the QI for a massless field has the following form:∫dtρ(t)g(t, τ) ≥ −C

τ d, (6.29)

where d is the spacetime dimension. Moreover g(t, τ) is a sampling/test function and C is a

positive constant. The meaning of (6.29) is that if a negative energy pulse lasts for time of

34

order τ , then its magnitude is bounded by − Cτd

. The version that we are mostly interested

in this chapter is the QIs in two dimensional Minkwoski space [41]. In two dimensions for

any light-ray travelling along v = constant, the QI has the following form:∫du 〈Tuu(u)〉 ρ(u) ≥ − 1

48π

∫du

(ρ(u)′)2

ρ(u), (6.30)

where ρ(u) is a smooth, peaked smearing function that integrates to one (the sampling

function we mentioned before). We would like to know if the QIs still hold when we add

non-local couplings in our theory. Thus, we will use the stress tensor we previously derived

(6.25) and we will choose the Gaussian as our smearing function.

Now, let’s assume a light-ray is travelling along v = 0. It will only “pass through” the

negative strip of 〈Tuu(u)〉. Thus, when we integrate along the u direction we will only have

non-zero 〈Tuu(u)〉 in the strip f . Then we have:∫ uR+A

uR−A〈Tuu(u)〉 1

σ√

2πe

(− 1

2(u−µσ )2)≥ − 1

48π

1

σ2. (6.31)

These inequalities are supposed to hold for any smearing function with the aforementioned

attributes, and for any parameters. We immediately see that at the large σ limit the inequal-

ity is violated. If we bring everything to the left-hand side and forget about the constants,

we will have:

σ

∫ uR+A

uR−Adu 〈Tuu(u)〉 e

(− 1

2(u−µσ )2)≥ −1 (6.32)

At the large σ limit the exponential will be approximately equal to one. So, (6.32) takes the

form:

σ

∫ uR+A

uR−Adu 〈Tuu(u)〉 ≥ −1 (6.33)

Since∫ uR+A

uR−Adu 〈Tuu(u)〉 < 0 the left-hand side of (6.33) can be arbitrarily negative. Thus,

we may conclude that when we add non-local sources in flat space, the quantum inequalities

are no longer true.

As we saw, the addition of non-local coupling induces the violation of both of the available

ways to constrain the negative energy density, ie. averaged version of energy conditions, and

the QIs.

7 Non-local couplings in black holes

7.1 BTZ with smeared sources

This idea of the non-local couplings was first applied by Gao, Jafferis and Wall (GJW) in

the case of the BTZ black hole [4]. We are going to review their results briefly, without

demonstrating the details of the calculation. Starting with a maximally extended BTZ black

35

hole, at some time t0 they coupled the two asymptotic boundaries of the black hole by adding

to the action a term of the form:

δS = −∫dtdφ h(t, φ)OR(t, φ)OL(−t, φ), (7.1)

where O is a scalar primary operator with scaling dimension ∆, dual to a scalar field ϕ. As

we can see the operators have been smeared over time and angle. By doing some dimensional

analysis we can find some condition on the scaling dimension of O:

[dtdφ] + [h] + 2∆ = [S]⇒ −2 + [h] + 2∆ = 0, (7.2)

where have expressed everything in units of energy. From (7.2), we can infer that in order

for h to be bigger than zero, ∆ has to be smaller than one.

The ultimate goal of [4] is to calculate the expectation value of the stress tensor due to the

insertion of these operators OL,OR. In order to do so they first calculate the modified bulk-

to-bulk propagator in the right wedge and then use the point-splitting method to compute

the stress tensor, on the horizon V = 0.

The modified bulk-to-bulk two-point function in the right wedge in Kruskal coordinates

is:

Gh =C0

(2π

β

)2∆−2

rh

∫dU1

U1

dφ1 h(U1, φ1)

(1 + U ′V ′

(U ′U1 − V ′/U1) + (1− U ′V ′) cosh rh (φ′ − φ1)

)∆

×(

1 + UV

(U/U1 − V U1)− (1− UV ) cosh rh (φ− φ1)

)∆

+ (U, φ←→ U ′, φ′),

(7.3)

where C0 =r2−2∆h sinπ∆

2(2∆π)2

(2πβ

)2−2∆

and h(U1, φ1) = h(

2πβ

)2−2∆

. The authors have also set

the radius of AdS to one. For simplicity, we can take φ = φ′ and also set V = V ′ =

0, which means that they calculate the two-point function on the horizon. By doing the

aforementioned simplifications we end up with:

Gh = hC0

∫ U

U0

dU1

U1

∫ UU1

1

2dy√y2 − 1

(1

U ′U1 + y

)∆(U1

U − U1y

)∆

︸ ︷︷ ︸F (U,U ′)

+ (U ←→ U ′)︸ ︷︷ ︸F (U ′,U)

, (7.4)

where y = cosh rh(φ1 − φ). Then, the stress tensor is then acquired by point splitting:

〈TUU〉 = limU ′→U

∂U∂U ′ (F (U,U ′) + F (U ′, U)) = 2 limU ′→U

∂U∂U ′F (U,U ′). (7.5)

The authors obtain the stress tensor numerically. In figure 10 we can see the stress tensor

against the coordinate U .

36

Figure 10: On the left we see the stress tensor against U in the case where we turn on the coupling

at U0 = 1 and never turn it off. In the right sub-figure we see the stress tensor against U in the

case where we turn on the coupling at U0 = 1 and turn it off at Uf = 2. The coupling constant h

is assumed to be 1.

The stress tensor for ∆ < 1/2 is finite, but for ∆ > 1/2 it is divergent at the point

where we turn on/off the coupling. However, this divergence is not important because it is

integrable. Moreover, in sub-figure (b) of figure 10 we see that after the turning off of the

coupling the stress tensor becomes positive. However, we need not worry since the relevant

quantity in order to detect whether or not we have a traversable wormhole is the integral

of the stress tensor along a null path. However, even though the stress tensor eventually

becomes positive, its integral is always negative. The integrated stress tensor is:

∫ ∞U0

TUUdU = − hΓ(2∆ + 1)2

24∆(2∆ + 1)Γ(∆)2Γ(∆ + 1)2`

2F1

(12

+ ∆, 12−∆; 3

2+ ∆; 1

1+U20

)(1 + U0)∆+1/2

. (7.6)

As we can see from figure 11, the integral of the stress tensor is always negative. Even

when the stress tensor becomes positive after the turn off of the coupling, the integral is

still negative (green line). Thus, the ANEC is violated and if we send a signal from the

right boundary towards the left, it will pass through the horizon of the black hole, it will

encounter the negative energy, gain a time advance and finally reappear on the left boundary

(see figure 12).

37

Figure 11: Here, we see the integral of the stress tensor against the scaling dimension ∆ of O, for

U0 = 1 and U0 = 2, with blue and orange respectively. The green line is corresponds to the case

where we turn on the coupling at U0 = 1 and turn off at Uf = 2.

OL

φL

OR

φR

Figure 12: The red line represents the signal we send, from the right boundary. The blue regions

represent the negative energy density. Upon collision with the negative energy the signal shifts and

instead of ending up in the singularity it emerges on the left boundary. We must note that when

we have even number of dimensions the negative energy is always localized on light cones (as we

saw in 1+1 Minkowski spacetime), whereas in odd number of dimensions the negative energy is

inside the light cones as well (as we see here, in case of the BTZ).

So, using this non-local coupling the wormhole is rendered traversable. The authors find

find that the wormhole opens up by ∆V ∼ hGN`

, which is a very small number. So, the

wormhole stays open only for some small amount of proper time and then it closes again 5.

This is different than the usual static wormhole solutions. However, highly boosted signals

will be able to pass through such a small time window. We are going to see more on this

topic in 7.3.

5Remember U, V coordinates are related to the “Schwartzchild” coordinates t, r like this (7.58).

38

7.2 AdS2 black hole

7.2.1 Set-up

Another interesting discussion concerning traversable wormholes in AdS using non-local

coupling was published shortly after [4] by Maldacena, Stanford and Yang (MSY) [8]. Instead

of working in AdS3, they focused in the AdS-Schwartzchild black hole in AdS2. So, their

boundary theory is a quantum mechanical theory. Their set-up is very similar to [4]. The

authors start from the thermofield double state. Then at time tR = tL = 0 they add in the

path integral the following interaction term:

eigV = eigOL(0)OR(0) (7.7)

However, instead of inserting only one operator O on each boundary, they insert K such

operators, in order to amplify the effect and also simplify some computations. So, the

coupling constant can be written as g = gK

. By taking the large K limit and keeping g fixed,

g will be small. This, as in [4], results in negative energy in the bulk, as in figure 13.

OL

φL

OR

φR

Figure 13: The red line represents a signal we send, from the right boundary. The blue lines

represent the negative shock waves.

This set-up can be understood as a quantum teleportation protocol, by making a small

change to the set-up of GJW. Instead of applying the quantum operator eigV we can think

of a different story. We may replace this operator by one that requires only the transfer of

classical information. This is done by measuring the operator OR. From this measurement

we will get one of its possible values oj. Then we will act with eigOLoj on the left system. We

must note that the left density matrix will be exactly the same one as if we had applied eigV ,

whereas the right density matrix, as well as the final global state of the the two systems, will

be different. In this case, we have the following picture:

39

OL

φL

ΠOR

φR

7.2.2 Gravity computation

An observable that can give us some intuition about what is going on is the following

commutator:

〈[φR, φL]〉V ≡⟨[φR, e

−igV φLeigV ]⟩

=⟨φRe

−igV φLeigV⟩−⟨e−igV φLe

igV φR⟩, (7.8)

where the correlators are evaluated on the thermofield double state. The time arguments

have been omitted but it’s implied that φL = φL(tL) and φR = φR(tR). What (7.8) captures

is the response of φL to a perturbation on the right boundary, after we turn on the coupling of

the two boundaries. If this commutator is non-zero it means that there is indeed something

appearing on the left boundary. If we set −tR = tL = t, (7.8) can be written as:

〈[φR(−t), φL(t)]〉V = ig 〈[φL(t),OL][φR(−t),OR]〉+O(g2) (7.9)

As we know from [42], for large time t6 the wavefunctions created by φ and the ones

created by O have a large relative boost. However, t should not be very large because then

the relative boost is enormous and the scattering processes will be dominated by inelastic

effects. Back in our case, we can approximate the scattering of O and φ particles with a

shock wave amplitude which has the form:

Sgrav = eiGNe2πβ tp+q−, (7.10)

where q− is the momentum of the O particle and p+ is the momentum of the φ particle,

each in a different frame in which they are unboosted. This e2πβt in (7.10) is the boost factor

between these two frames and we may define a Mandelstam-like variable (the center of mass

energy) as s ∼ e2πβtp+q−. Equation (7.10) is true when we are still in the regime where

GN 1, t 1 and GNe2πβt ∼ 1. When GNe

2πβt ∼ 1 the inelastic effects are negligible.

These exponentially growing contributions come from specific orderings of the operators.

6The relative boost is related to the time difference in the insertion of φ and O. Since the O operators are

inserted at t′ = 0, this t is precisely the difference between the time we inserted φ and the time we inserted

O.

40

Ultimately, we would like to calculate (7.8). In order to do so we are going to start from

something simpler. We are going to compute just the second part of (7.8), i.e.:

C =⟨e−igV φLe

igV φR⟩

(7.11)

Assuming φL,R and V to be Hermitian we have that:

C† =⟨φRe

−igV φLeigV⟩, (7.12)

and thus, we can acquire the commutator of interest by computing the imaginary part of C:

〈[φR, φL]〉V = C − C† = 2Im (C) (7.13)

In order to compute C we are going to assume it is of the following form:

C =⟨e−igVB

⟩, (7.14)

where B = φLeigV φR. We will expand the exponential:

C =∞∑0

(−ig)n

n!

⟨(1

K

K∑j=1

OjLOjR

)n

B

⟩≈

∞∑0

(−ig)n

n!

(1

K

K∑j=1

⟨OjLO

jR

⟩)n

〈B〉 = eig〈V 〉 〈B〉 ,

(7.15)

where we have used the fact that we have a large number K of O fields, in order to factorize

the correlator. So, we can rewrite (7.11) as:

C = eig〈V 〉C, C =∞∑n=0

(ig)n

n!

⟨φL(OjLO

jR

)nφR

⟩(7.16)

where g is gK

. The ordering of the operators is such that the scattering between φ and

O particles is exponentially enhanced. In order to proceed, we want calculate the cor-

relator⟨φL(OjLO

jR

)nφR

⟩. First, we will assume n = 1, and calculate 〈φLOLORφR〉 =

〈φLORφROL〉. As in [42] we are going to define the “in” state:

|Ψ〉 = φROL|TFD〉, (7.17)

and the “out” state:

|Ψ′〉 = O†Rφ†L|TFD〉. (7.18)

Each of these states are two-particle states. Now, each φ particle can be expressed in

terms of a superposition of particles with momentum p+ and each O particle in terms of a

superposition of particles with momentum q−. So, we can write:

|Ψ〉 = φROL|TFD〉 =

∫dpR+dq

L−⟨pR+|φR

⟩ ⟨qL−|OL

⟩|pR+; qL−〉in

|Ψ′〉 = O†Rφ†L|TFD〉 = ORφL|TFD〉 =

∫dpL+dq

R−⟨pL+|φL

⟩ ⟨qR−|OR

⟩|pL+; qR−〉out,

(7.19)

41

where we have used that O and φ are Hermitian. By |pR,L+ ; qL,R− 〉 we mean the product

between the state |pR,L+ 〉 and |qL,R− 〉. Now, we take the overlap of these two states:

D = 〈Ψ′|Ψ〉 =

∫dpR+dq

L−dp

L+dq

R−⟨pL+|φL

⟩∗ ⟨qR−|OR

⟩∗ ⟨pR+|φR

⟩ ⟨qL−|OL

⟩out

⟨pL+; qR−|pR+; qL−

⟩in

=∫dpR+dq

L−dp

L+dq

R−⟨φL|pL+

⟩ ⟨OR|qR−

⟩ ⟨pR+|φR

⟩ ⟨qL−|OL

⟩out

⟨pL+; qR−|pR+; qL−

⟩in

(7.20)

Previously, we saw, that for large values of t, the wavefunctions created by φ and O have

a relatively large boost. Since that is the case, the biggest contribution will come from

the region of integration where the momenta pR+, qL−, p

L+, q

R− are large. Other null momenta

pR−, qL+, p

L−, q

R− will be approximately zero and thus we can infer that pR+ ≈ pL+ ≡ p+ and

qR− ≈ qL− ≡ q−. The amplitude is then essentially diagonal in the p+, q− momenta and we

can approximate:

|p+; q−〉out ≈ eiδ(s,b)|p+; q−〉in + |χ〉, (7.21)

where |χ〉 is the inelastic component of the scattering. As long as we are in the regime where

GNs ∼ 1 we can ignore inelastic effects. If we do so, we have that:

out 〈p+; q−|p+; q−〉in ≈ eiδ = eiGNe2πβtp+q− (7.22)

Using the above results, (7.20) can be written as:

D =

∫dp+dq− 〈φL|p+〉 〈OR|q−〉 〈p+|φR〉 〈q−|OL〉 eiGNe

2πβtp+q− (7.23)

In the case of n particles we have:

|Ψ〉 = φROL1 · · · OLn|TFD〉 =

∫dpR+dq

L1− · · · dqLn−

⟨pR+|φR

⟩ ⟨qL1− |OL

⟩· · ·⟨qLn− |OL

⟩|pR+; qL1

− , · · · qLn− 〉in

(7.24)

|Ψ′〉 = O†R1· · · O†Rnφ

†L|TFD〉 =OR1 · · · ORnφL|TFD〉 =

∫dpL+dq

R1− · · · dqRn−

⟨pL+|φL

⟩ ⟨qL1− |OR

⟩· · ·⟨

qLn− |OR⟩|pR+; qL1

− , · · · qLn− 〉out(7.25)

Again we are going to assume that pR+ ≈ pL+ ≡ p+ and also qRi− ≈ qLi− ≡ q− for i = 1, 2, · · ·n.

Hence, we can rewrite |Ψ〉 and |Ψ′〉 as:

|Ψ〉 =

∫dp+ 〈p+|φR〉

(∫dq− 〈q−|OL〉

)n|p+; q1

−, · · · qn−〉in,

|Ψ′〉 =

∫dp+ 〈p+|φL〉

(∫dq− 〈q−|OR〉

)n|p+; q1

−, · · · qn−〉out,(7.26)

42

and their overlap is:

D =

∫dp+ 〈p+|φR〉 〈p+|φL〉∗

∫dq− (〈q−|OR〉∗ 〈q−|OL〉)n out

⟨p+; q1

−, · · · qn−|p+; q1−, · · · qn−

⟩in

=∫dp+ 〈p+|φR〉 〈φL|p+〉

∫dq− (〈OR|q−〉 〈q−|OL〉)n out

⟨p+; q1

−, · · · qn−|p+; q1−, · · · qn−

⟩in.

(7.27)

As before, we have that:

|p+; q1− · · · qn−〉out ≈ einδ(s,b)|p+; q1

− · · · qn−〉in + |χ〉. (7.28)

The reason that n appears in the exponential is because we have n separate scattering events,

instead of just one as before. So, we finally the overlap is:

D =

∫dp+ 〈p+|φR〉 〈φL|p+〉

∫dq−

(〈OR|q−〉 〈q−|OL〉 eiGNe

2πβtp+q−

)n(7.29)

and

C =

∫dp+ 〈p+|φR〉 〈φL|p+〉

∫dq−

∞∑n=0

(ig)n

n!

(〈OR|q−〉 〈q−|OL〉 eiGNe

2πβtp+q−

)n=∫

dp+ 〈p+|φR〉 〈φL|p+〉∫dq−exp

(ig 〈OR|q−〉 〈q−|OL〉 eiGNe

2πβtp+q−

)=∫

dp+ 〈p+|φR〉 〈φL|p+〉 exp

(ig

⟨OReiGNe

2πβtp+P−OL

⟩),

(7.30)

where P− is the momentum operator acting on OL. The form of (7.30) in higher dimensions

is very similar. We would need to add extra labels for the transverse direction in the wave-

functions. Moreover, we need to make the replacement GN → GNf(x−x′), where f(x−x′) is

the profile of the shock wave in the transverse direction and x and x′ the transverse position

of the particle φ and O respectively. From [42] we know that f(x − x′) has the following

form at large µ|x|:

f(|x|) =µd−4

2

2(2π|x|) d−22

e−µ|x|. (7.31)

where |x| = |x− x′|. The quantity µ can be found from the following expression:

µ2 =2π(d− 1)rh

β, (7.32)

For d = 2 boundary dimensions (in AdS3) we have then:

µ =2πrhβ

, (7.33)

and

f(|x− x′|) =

√β

8πrhe−√

2πrhβ|x−x′|. (7.34)

43

Thus, equation (7.30) becomes:

CAdS3 =

∫dp+dx 〈p+, x|φR〉 〈φL|p+, x〉

∫dq−dx

′exp (ig 〈OR|q−, x′〉 〈q−, x′|OL〉

eiGN

√β

8πrhe−√

2πrhβ|x−x′|

e2πβtp+q−

),

(7.35)

and by using that β = 2π`2

rhwe can write (7.35) as:

CAdS3 =

∫dp+dx 〈p+, x|φR〉 〈φL|p+, x〉

∫dq−dx

′exp (ig 〈OR|q−, x′〉 〈q−, x′|OL〉

eiGN

`2rh

e−rh`|x−x′|e

rh`2tp+q−

),

(7.36)

Back to the case of AdS2, we have that:⟨OR(−tR)eiGNe

2πt βp+P−OL(tL)

⟩=

(2 cosh

(tL + tR

2

)+a−

2etR−tL

2

)−2∆

, (7.37)

with a− = −GNetp+. In order to derive this result, the symmetries of AdS2 have been used.

For a detailed analysis see Appendix A of [8]. By substituting tL = −tR = 0 in (7.37) we

get: ⟨OR(0)e−ia

−P−OL(0)⟩

=1(

2 + a−

2

)2∆, (7.38)

We see that if our φ particle has negative momentum p+, then a− > 0 and the correlator

(7.38) is suppressed. We can now obtain 〈OR|q−〉 〈q−|OL〉 (the wavefunctions in momentum

space) by Fourier transforming. In a frame where the O particle is unboosted we have:

〈OR|q−〉 〈q−|OL〉 =

∫ ∞−∞

da−

2πeia−q−⟨OR(0)e−ia

−P−OL(0)⟩

=∫ ∞−∞

da−

2πeia−q−

1(2 + a−

2+ iε

)2∆=

1

Γ(2∆)

(2iq−)2∆

(−q−)e−i4q−Θ(−q−)

(7.39)

In order to find 〈p+|φR〉 〈φL|p+〉 we Fourier transform a correlator of the form:⟨φR(t)e−a

+P+φL(t)⟩

=1(

2 + a+

2+ iε

)2∆, (7.40)

where a+ = −gGNe2πβt∆

22∆+1 . In the frame where the φ particle is unboosted we get:

〈p+|φR〉 〈φL|p+〉 =1

Γ(2∆)

(2ip+)2∆

(−p+)e−i4p+Θ(−p+) (7.41)

44

Thus, C becomes:

C =1

Γ(2∆)

∫ ∞−∞

dp+

−p+

(2ip+)2∆e−i4p+Θ(−p+)exp

[ig

1(2 + a−

2

)2∆

]=

1

Γ(2∆)

∫ 0

−∞

dp+

−p+

(2ip+)2∆e−i4p+exp

ig(2− p+GNe

2πβt/2)2∆

, (7.42)

where we have assumed that the scaling dimensions of φ and O are equal. Also, from the

theta function we infer that p+ is negative. We can also generalize (7.42) for the case of

many φ particles. Schematically, it looks as follows:

C =

∫ ∏l

dpl+ψL(pl+)ψR(pl+)exp

ig(2− ptotal

+ GNe2πβt/2)2∆

(7.43)

where we have the wavefunctions of the multiparticle state.

7.2.3 Probe limit

It is interesting to investigate the limit where we can ignore the backreaction of the φ particle.

We are going to assume that GNet 1 and g 1. In this limit (7.42) becomes:

Cprobe = e−ig

22∆ C =1

Γ(2∆)

∫ 0

−∞

dp+

−p+

(2ip+)2∆e−i4p+exp

(igp+GNe

2πβt∆

22∆+1

), (7.44)

where we have expanded the argument of the exponential to first order in GNe2πβt and we

have used that eig〈V 〉 is equal to e−ig

22∆ . The latter is of zeroeth order in GNe2πβt and it will

not matter in this particular calculation, but let’s see how we obtain it. We start from:

eig〈V 〉 = e−igK

∑Kj=1〈OjLOjR〉, (7.45)

and we know that 〈OL(tL)OR(−tR)〉 =(2 cosh

(tL+tR

2

))−2∆. For tL = −tR = 0 this correlator

becomes:

〈OL(tL)OR(−tR)〉 =1

22∆. (7.46)

Consequently, eig〈V 〉 = e−igK

∑Kj=1

1

22∆ = e−ig

22∆ . By looking at equation (7.41) we under-

stand that Cprobe, to first order in GNe2πβt, is the Fourier transform of the momentum space

wavefunctions:∫ 0

−∞dp+ 〈p+|φR〉 〈φL|p+〉 e−ia

+p+ =1

Γ(2∆)

∫ 0

−∞

dp+

−p+

(2ip+)2∆e−i4p+e−ia+p+ =⟨

φLe−a+P+φR

⟩=

1(2 + a+

2

)2∆,

(7.47)

45

and for general tL,R we have:

Cprobe =

(2 cosh

(tL + tR

2

)+a+

2etL−tR

2

)−2∆

. (7.48)

What the above result means is that in this limit we can consider φ particle as a probe. The

OO insertions are creating a backreaction on our initial geometry (the AdS2-Schwartzchild

black hole) and φ is moving in this background. However, it is not backreacting itself on the

geometry. As we see from (7.47), the insertion of OO in our spacetime is making φ shift in

the x+ direction by an amount of a+. For g > 0 we have a+ < 0 and thus φ is gaining a

time advance.

As a+ becomes more negative we see that the correlator (7.48) is becoming bigger, which

we can also interpret as making the boundaries of AdS come closer to each other. We see

that at a+ = −4 there is a pole, but this can be taken care of if we smear φL,R over a

small timeband. As a+ becomes smaller than −4 the correlator (7.48) becomes imaginary,

since 0 < ∆ < 1, and thus, the separation between the points of φR and φL becomes

timelike. Since (7.47) is imaginary, the commutator 〈[φR, φL]〉V = 2Im (C) becomes non-

zero and consequently φL appears on the left boundary. This means that we have rendered

the wormhole traversable.

7.2.4 Bounds on information transfer

Finally, it is very interesting to investigate if there are some bounds on the informations that

we can send through the traversable wormhole. We will mention the conclusion and then

explain it. We cannot send more information through the wormhole than we transferred in

order to make the wormhole traversable in the first place. We are going to imagine we follow

the protocol described in the beginning of 7.2. So, we are going to measure OR and we are

going to assume that it has two eigenvalues, ±1 and hence, each measurement corresponds

to one bit. If we have K of these OR operators we need K number of bits. So, for the

coupling constant g = gK we can write:

g . K = Nbits needed (7.49)

Now, we can send φ particles through. As we said before, due to the OO insertions the φ

particles take a shift in the x+ direction equal to ∆x+ ≡ a+ ∼ gGNet. Then the spread of

the wavefunction of the φ particles has to be smaller or equal to a+ in order for it to pass to

the other side. So, using the uncertainty principle we have:

− peach+ ≤ 1

∆x+=

1

gGNet, (7.50)

where we have labelled the momentum with the word each meaning each φ particle we send

through. If we send several φ particles, their momenta add up and in order not to suppress

(7.43) we have to assume that:

− ptotal+ GNe

t . 1. (7.51)

46

By diving (7.51) with (7.50), we get:

Nbits send ∼−ptotal

+

−peach+

≤ g. (7.52)

It is clear that the number of bits we can send is less than the number of bits we used in

order to make the wormhole traversable.

7.3 BTZ with non-smeared sources

7.3.1 Modified two-point function

Instead of smearing OL and OR along a time-band as GJW did, we insert them at an instant

of time. The reason behind this choice is that we want to acquire an analytic expression for

the expectation value of the stress tensor. Thus, the term we add to the Hamiltonian has

the following form:

δH(t) =−∫dφ δ

(rh(t− t0)

`2

)h

(2π

β

)2−2∆

OR(t, φ)OL(−t, φ) =

− `2

rh

(2π

β

)2−2∆

h

∫dφ δ (t− t0)OR(t, φ)OL(−t, φ).

(7.53)

Notice that the argument of the delta function isrh(t− t0)

`2. In order for our coupling

term to have the correct units our Dirac delta function7 has to be dimensionless and the

aforementioned term achieves exactly that. We can now write down the time evolution

operator in the interaction picture:

U(t, t0) =e−i∫ tt0dt δH(t)

= ei `

2

rh( 2πβ )

2−2∆h∫ tt0dt∫dφ δ(t−t0)OR(t,φ)OL(−t,φ)

=

ei `

2

rh( 2πβ )

2−2∆h∫dφ OR(t0,φ)OL(−t0,φ)

(7.54)

Using (7.54) we can compute the bulk-to-bulk two-point function:

Gh =⟨φHR (t, r, φ)φHR (t′, r′, φ′)

⟩=⟨U−1(t, t0)φR(t, r, φ)U(t, t0)U−1(t′, t′0)φR(t′, r, φ)U(t′, t′0)

⟩=

2 sinπ∆`2

rh

(2π

β

)2−2∆

h

∫dφ0 K∆ (t′ + t0 − iβ/2)Kr

∆ (t− t0) + (t←→ t′)

(7.55)

where β = 2πl2

rh, K∆ is the bulk-to-boundary correlator and Kr

∆ is the retarded correlator.

The form of the bulk-to-boundary correlation function in the BTZ black hole is:

K∆ (t, r, φ) =r∆h

2∆+1π`2∆

(−(r2 − r2

h)1/2

rhcosh

rh`2t+

r

rhcosh

rh`φ

)−∆

(7.56)

7Remember that the Dirac delta function has the inverse units of its argument.

47

So, here we have:

K∆ (t′ + t0 − iβ/2) =r∆h

2∆+1π`2∆

(−(r2 − r2

h)1/2

rhcosh

rh`2

(t′ + t0 − iβ/2) +r

rhcosh

rh`

(φ′ − φ0)

)−∆

(7.57)

Then, it is more convenient to switch to Kruskal coordinates:

e2rh`2t = −U

V,

r

rh=

1− UV1 + UV

, (7.58)

and compute the form of the bulk-to-boundary correlator. From (7.58) it follows that:

(r2 − r2h)

1/2

rh= 2

√−U ′V ′

1 + U ′V ′. (7.59)

Therefore, the cosh rh`2

(t′ + t0 − iβ/2) can be written as:

coshrh`2

(t′ + t0 − iβ/2) =1

2

(erh`2

(t′+t0−iβ/2) + e−rh`2

(t′+t0−iβ/2))

=

1

2

(√U ′U0

V ′V0

e−irh`2β/2 +

√V ′V0

U ′U0

eirh`2β/2

).

(7.60)

For ` = 1 we have that β = 2π/rh, and thus we find that:

e−irhβ/2 = e−iπ = −1, eirhβ/2 = eiπ = −1 (7.61)

By using (7.59),(7.60) and (7.61), K∆ becomes:

K∆ =r∆h

2∆+1π`2∆

(1

1 + U ′V ′

(√U ′2U0

V0

+

√V ′2V0

U0

)+

(1− U ′V ′

1 + U ′V ′

)cosh

rh`

(φ′ − φ0)

)−∆

(7.62)

Since we are at the right wedge, we know that U ′ > 0 and V ′ < 0. So, we get:

K∆ =r∆h

2∆+1π`2∆

(1

1 + U ′V ′

(U ′√U0

V0

− V ′√V0

U0

)+

(1− U ′V ′

1 + U ′V ′

)cosh

rh`

(φ′ − φ0)

)−∆

(7.63)

We know that U0V0 = −1 because U0 is at the boundary. Hence, we multiply and divide

inside the square roots with U0 and acquire:

K∆ =r∆h

2∆+1π`4∆

(1 + U ′V ′

(U ′U0 − V ′/U0) + (1− U ′V ′) cosh rh`

(φ′ − φ0)

)∆

(7.64)

Similarly, we find that Kr∆ is:

Kr∆ =

r∆h

2∆+1π `4∆

(1 + UV

(U/U0 − V ′U0)− (1− UV ) cosh rh`

(φ− φ0)

)∆

(7.65)

48

We may finally write down the bulk-to-bulk two-point function:

Gh =C

∫dφ0

(1 + U ′V ′

(U ′U0 − V ′/U0) + (1− U ′V ′) cosh rh`

(φ′ − φ0)

)∆

×(

1 + UV

(U/U0 − V U0)− (1− UV ) cosh rh`

(φ− φ0)

)∆

+ (U, φ←→ U ′, φ′)

(7.66)

where C ≡ h sinπ∆

22∆+1π2

r2∆h

`4∆

`2

rh

(2π

β

)2−2∆

=h sin π∆

22∆+1π2

rh`2

. We want to focus on the TUU com-

ponent of the stress tensor on the horizon V = 0. So, we set V = V ′ = 0 and obtain:

Gh = C

∫dφ0

(1

U ′U0 + cosh rh`

(φ′ − φ0)

)∆(1

U/U0 − cosh rh`

(φ− φ0)

)∆

+(U, φ←→ U ′, φ′)

(7.67)

For simplicity, we can also take φ′ = φ. Then, we define a new variable y = cosh rh (φ0 − φ)

and Gh becomes:

Gh =C`

rh

∫ UU0

1

dy√y2 − 1

(1

U ′U0 + y

)∆(1

U/U0 − y

)∆

︸ ︷︷ ︸F (U,U ′)

+ (U ←→ U ′)︸ ︷︷ ︸F (U ′,U)

(7.68)

7.3.2 One-loop stress-energy tensor

Now, we are ready to compute the stress tensor. Using the point-splitting method we have

that 〈TUU〉 = limU ′→U ∂U∂U ′ (F (U,U ′) + F (U ′, U)) = 2 limU ′→U ∂U∂U ′F (U,U ′). Since the

limits of integration do not contain U ′, we can immediately take the derivative with respect

to U ′ and get:

〈TUU〉 = −2∆C`

rhlimU ′→U

∂U

∫ U/U0

1

dy√y2 − 1

U∆+10

(U ′U0 + y)∆+1

1

(U − U0y)∆(7.69)

49

Then, we define a new variable z = y−1U/U0−1

, in order to have limits of integration from 0 to

1 and get:

〈TUU〉 =

− 2∆C`

rhlimU ′→U

∂U

[U∆+1

0 (U/U0 − 1)

∫ 1

0

dz1(

(U/U0 − 1)2 z2 + 2 (U/U0 − 1) z)1/2

× 1

(U ′U0 + (U/U0 − 1) z + 1)∆+1

1

(U − U0 (U/U0 − 1) z − U0)∆

]=

−√

2∆C`

rhlimU ′→U

∂U

[U0

(U ′U0 + 1)∆+1 (U/U0 − 1)∆−1/2

∫ 1

0

dz1

z1/2(1− 1

2(1− U/U0) z

)1/2

× 1(1− (1−U/U0)

U ′U0+1z)∆+1

1

(1− z)∆

,(7.70)

and by using the integral representation of Appell hypergeometric function, we may rewrite

(7.70) as:

〈TUU〉 =−√

2∆C`

rhlimU ′→U

∂U

[U0

(U ′U0 + 1)∆+1 (U/U0 − 1)∆−1/2

Γ(12)Γ(1−∆)

Γ(32−∆)

×F1

(1

2;1

2,∆ + 1;

3

2−∆;

U0 − U2U0

,U0 − U

U0 (1 + U ′U0)

)].

(7.71)

For convenience, we define A ≡√

2∆C`

rh

Γ(

12

)Γ(1−∆)

Γ(32−∆)

=∆ sinπ∆

22∆+1/2π3/2

Γ(1−∆)

Γ(32−∆)

h

`. Now, if

we take the derivative and the limit we finally obtain:

〈TUU〉 =

− A(1− U/U0)∆+1/2 (1 + UU0)∆+1

[(∆− 1

2

)F1

(1

2;1

2,∆ + 1;

3

2−∆;

U0 − U2U0

,U0 − U

U0 (1 + UU0)

)+

U − U0

4(2∆− 3)U0 (1 + UU0)

(−4(∆ + 1)F1

(3

2;1

2,∆ + 2;

5

2−∆;

U0 − U2U0

,U0 − U

U0 (1 + UU0)

)−(1 + UU0)F1

(3

2;3

2,∆ + 1;

5

2−∆;

U0 − U2U0

,U0 − U

U0 (1 + UU0)

))](7.72)

The expression for the expectation value of the stress tensor can be refined by using some

properties of the Appell hypergeometric functions (for full derivation see Appendix C). The

refined expression for the stress tensor is:

〈TUU〉 =

− AU−(∆+1)√

2 (1/2−∆)

(1− U/U0)∆+1/2 (1 + U/U0)1/2

(U2

0 + 1

U0

)−(∆+1)

F1

(−∆;

1

2,∆ + 1;

1

2−∆;

U − U0

U + U0

,U − U0

U (1 + U20 )

).

(7.73)

50

At first sight, (7.73) looks like a complicated function. In order to get a better idea about

the form of (7.73) we study it in two limits. First, we find 〈TUU〉 in the limit where U → U0

(remember we are on the V = 0 horizon). We do this by Taylor expanding around this point.

The result is the following:

〈TUU〉U→U0= −A

√2 (1/2−∆)

1

(U/U0 − 1)∆+1/2(7.74)

From (7.74) we immediately see that at U = U0 the expectation value of the stress tensor is

divergent. However, this is not a problem because for ∆ < 0.5 this is an integrable divergence,

meaning that when we integrate (7.73) for ∆ < 0.5 we get a finite result. We would also like

to know he behaviour of 〈TUU〉 as U →∞. This time things are more complicated.

In order to get an idea about the form of 〈TUU〉U→∞ we first plotted the hypergeometric

function F1

(−∆; 1

2,∆ + 1; 1

2−∆; U−U0

U+U0, U−U0

U(1+U20 )

)against U and we saw that for every 0 <

∆ < 0.5 it was behaving like a logarithmic function, for U 1. As a result, we expected

our final result to include a log (U/U0).

We start from equation (7.70) and instead of using the Appell integral representation we

take the derivative and the limit U ′ → U :

〈TUU〉 = −√

2∆C`

rh

∫ U/U0−1

0

dx(−1− UU0 + x+ x2 + ∆ (x+ 2) (1 + UU0 + 2x))

(U/U0 − 1)∆+1 (1 + UU0 + x)∆+2(

1− U0xU−U0

)∆

(x(x+ 2))3/2,

(7.75)

where we have also changed to a variable x = zU/U0−1

. We then define a = U/U0 − 1 and

b = UU0 + 1 and rewrite (7.75) as:

〈TUU〉 = −√

2∆C`

rh

∫ a

0

dx(−b+ x+ x2 + ∆ (x+ 2) (b+ 2x))

a (b+ x)∆+2 (a− x)∆ (x(x+ 2))3/2. (7.76)

In the limit U →∞ b is approximately equal to a. We make yet another change of variables

by defining y ≡ xa

and (7.76) takes the following form:

〈TUU〉U→∞ = −√

2∆C`

rh

∫ 1

0

dya−2∆(1− y)−∆(1 + y)−2−∆y (ay2 + ∆(2 + ay))

(ay (ay + 2))3/2. (7.77)

The terms (1−y)−∆ and (1+y)−2−∆, after integration, only contribute a number of order 1 to

our stress tensor, since ∆ < 1/2. Thus, we can ignore them. Furthermore, we Taylor expand

the integrand around a→∞. However, we do not Taylor expand the whole integrand. We

leave the term 1

(ay(ay+2))1/2 as it is, and expand the rest. The reason we do this is because

we know that this term is going to give the logarithm we found when we plotted the stress

tensor. After the above steps, we get:

〈TUU〉U→∞ = −√

2∆C`

rh

∫ 1

0

dy∆a−1−2∆

(ay (ay + 2))1/2= −√

2∆2C`

rh

log(2a)

a2+2∆. (7.78)

51

Thus, finally the stress tensor at U →∞ is:

〈TUU〉U→∞ ∼ −√

2C`

rh∆2U2−2∆

0

log(

2 UU0

)U2+2∆

(7.79)

In the plot below, we see 〈TUU〉U→∞ and 〈TUU〉 for some values of the scaling dimension and

for U0 = 1.

As we see, the bigger the scaling dimension, the better our approximation is. We can now

plot the expectation value of the stress tensor against the coordinate U . For convenience,

we will use the coordinate x = UU0

, which indicates how far we are from the insertion point

U0.

Figure 14: ` 〈TUU (x)〉 /h versus x, for U0 = 1, on the horizon V = 0. TUU (x) is negative, after we

turn on the coupling, for scaling dimensions ∆ ≤ 0.3, and flips sign for ∆ > 0.3.

52

If we take a look at figure (14), we see that for 0.3 < ∆ < 0.5 ,(7.73) is not always

negative. This might seem alarming at first, since our goal is to create negative energy

density in order to make the wormhole traversable. However, the relevant quantity, in order

to determine whether or not a signal that starts from the right boundary will emerge at the

left one, is the integral of the stress tensor along the signal’s path. If the aforementioned

quantity is negative, then the signal will indeed suffer a time advance and reappear at the

left boundary. So, we want: ∫ ∞U0

dU 〈TUU(U)〉 < 0, (7.80)

where we start form U0, and not −∞, because before we turn on the coupling 〈TUU(U)〉 is

zero. We performed this integral by numerical methods and we plot the result below:

Figure 15: `/h∫∞U0dU 〈TUU (U)〉 versus ∆, for different insertion points U0.

In this figure, we see that the integral of 〈TUU(U)〉 is invariant under U0 going to 1/U0.

Furthermore, it is clear that the optimum insertion point in order to violate the ANEC as

much as possible is U0 = 1. The reason for this is that the 〈OLOR〉 correlator takes its

maximum value for U0 = 1(t = 0).

53

Figure 16: `/h∫∞U0dU 〈TUU (U)〉 versus U0, for some values of the scaling dimension. In order to

violate ANEC as much as possible we should choose ∆ close to 0.5.

Both figures 15 and 16 show that even though for 0.3 < ∆ < 0.5 the expectation value

of the stress tensor eventually becomes positive, its integral is always negative and thus, we

can, in principle, traverse the wormhole.

7.3.3 Calculating the shift

Now, as we did with the positive energy shock waves, we would like to calculate how much is

the signal going to shift upon encountering the negative energy. Assuming we send a signal

from the left boundary along a constant V line, we want to compute ∆V . Using equations

(1.4) and (1.5) from [4] we find that:

∆V = 4πGN

∫ ∞U0

dU 〈TUU(U)〉 . (7.81)

We may further define:

a ≡ `

h〈TUU(U)〉 , (7.82)

which is the dimensionless quantity that we plot in figure 15 and 16, and write (7.81) as:

∆V =4πGNha

`. (7.83)

In three dimensions GN is equal to the Planck length. Moreover, since our calculation is

pertubative h 1. So, ∆V is a very small number, below Planc scale. In order to get a

clearer picture, we are going to borrow a figure from [4] since the cases we study are very

similar.

54

Figure 17: Here, we see the Kruskal diagram of the BTZ black hole, including the backreaction of

the non-local coupling of the two boundaries. The grey curve is the past horizon. The orange curve

represents the future horizon. E1 is the original bifurcation point and E2 is the point where the

future horizon is moved to, after the turning on of the coupling. The two horizons are not “touching”

any more and thus, the wormhole is rendered traversable. The signal is depicted with pink. It shifts

upon encountering the negative energy shock wave and reappears at the right boundary.

From the figure we see that the wormhole opens up by ∆V , which we found that is very

small. Consequently, the amount of time that the wormhole remains open is also of the

same order. One might worry that the signal we send will not be able to make it through.

However, if we send this signal early enough, by the time it is close to the point E1 it will

be highly boosted and it will have no problem passing through such a small time window.

7.3.4 The center of mass energy of the collision

It is true that we can boost something to fit through a very small time window. However,

if the signal is highly boosted we have to check that upon collision with the negative energy

shock wave, there won’t be any violent effects, such as the creation of another black hole. In

order to exclude this possibility we assume that the signal we send from the left boundary is a

positive energy particle and the negative energy shock wave is a negative energy particle, and

we calculate their center of mass energy (the square root of the usual Mandelstam variable

s). If we calculate this energy to be less than 1/`P we need not worry. In order to do that

we first zoom into the central region of figure 17, and go to some intertial coordinates (t, x):

55

∆V`∆t

∆x

E2

E1

Figure 18: The diamond is the region between the point E1 and E2, in figure 17. The side of the

diamond is equal to ∆V `. ∆t is the amount of time that the wormhole remains open. The yellow

region represents a signal that passes through the wormhole throat.

From the figure we immediately see that:

∆x = ∆t =√

2∆V `. (7.84)

In order to find a constraint on the energy and momentum of the signal we shall use the

uncertainty principle:

∆E ∆t ≥ 1

2, ∆p ∆x ≥ 1

2. (7.85)

In our case, this means:

E ≥ 1

2√

2∆V `, p ≥ 1

2√

2∆V `. (7.86)

By using (7.83) we may rewrite the above result as follows:

E ≥ 1

8√

2π`Pha, p ≥ 1

8√

2π`Pha. (7.87)

Naively, we would be worried for this result, since the minimum energy/momentum that a

signal should have in order to pass through the diamond region is of order Planck energy.

However, this is the energy that we measure in the frame of the diamond and not some

invariant quantity. Thus, it can be arbitrarily large. The crucial diagnostic for violent

events is the center of mass energy, which is indeed invariant. We are now ready to compute

it.

56

Figure 19: The particle on the left represents the highly energetic positive energy particle and the

particle on the right represents the negative energy particle.

Since distances are smaller towards the central region of AdS, we can assume that the

positive energy particle has the same energy at the frame of the collision, as in the frame of

the diamond. So, the momentum tensor of the positive energy particle is:

P µr =

(1

8√

2π`P ha

~pr

). (7.88)

When we calculated the expectation value of the stress tensor, we found it to be of order `

(see definition of A in (7.73)). Thus, we may assume that the negative energy particle will

have the following momentum tensor :

P µb =

(− c0

`

~pb

), (7.89)

where c0 is an order one constant. The center of mass energy is defined as:

√s = (P µ

r + P µb ) (7.90)

In the relativistic limit we can ignore the masses of the particles and assume that |~pr| =1

8√

2π`Phaand |~pb| =

c0

`and thus, (7.90) takes the following form:

√s =√

2P µr Pbµ =

√−(− c0

4√

2πa

1

h`P `

)+ 2~pr · ~pb√

c0

4√

2πa

√1

h`P `(1 + cos θ),

(7.91)

where θ is the angle between the two particles. The only value of the angle that would play

a significant role in the expression of√s is θ = π, which is not the case here. In the scenario

we consider θ ∼ π/2, and thus the center of mass energy is:

√s ∼

√1

h`P `(7.92)

which is less than 1`P

. That is an indication that we do not need to worry about black hole

creation or stringy effects.

57

7.3.5 Bounds on the number of particles we can send through

As was previously mentioned, the energy that we usually start feeling uneasy is the Planck

energy. We found that the center of mass energy of the collision is approximately√

1/`P ,

so it seems that we can even send a lot more positive energy particles through the wormhole

until we reach Planck scale energy.

We may think of these particles as a message that we want to send from one boundary

to the other. So, we would like to be able to detect each of them individually on the target

boundary. We consider two cases. We can either send n number of different species particles

or n number of the same species particles. In the first case, we need n different detectors

to be able to detect the particles. So, we can allow them to coincide. Thus, if we send n

positive energy particles the momentum tensor is:

P µr =

(n

8√

2π`P ha

~pr

), (7.93)

and the√s becomes:

√s ∼

√n

h`P `. (7.94)

Since we want that to be less than 1`P

, we finally find that the maximum number of particles

we can send is:

nmaxdiff = h

`

`P. (7.95)

In the second case, however, since we want to detect each of them individually we have

to make sure they are not travelling on top of each other. We have to send them with some

time difference. Hence, we have to divide the side of the diamond in n “slots” so that no

particle “falls” on another.

∆v`

E2

E1· · ·

︸ ︷︷ ︸n

particles

Figure 20: Here, we see how the same species particles pass through the diamond region. Each of

them should pass through its own “slot”.

In that case we find the energy from the uncertainty principle to be:

E ≥ n

2√

2∆V `, (7.96)

58

and thus, the center of mass energy is:

√s ∼

√n2

h`P `, (7.97)

and the maximum number of same species particles:

nmaxsame =

√h`

`P(7.98)

In both cases, the number of particles that we an send through the wormhole until we reach

Planck energy is a very big one. So, it seems that we are allowed to send a large number of

signals before we encounter any problems, from the bulk point of view.

Finally, we can do a cross-check of the previous result with the teleportation picture we

discussed in 7.2.4. Of course, their analysis is done in AdS2 but the general idea should be

similar. In our case we only have one O field, instead of K. Thus, following the logic of [8]

in order to make the coupling we need:

Nbits needed = 1 > h, (7.99)

and using the uncertainty principle, as before, we find that:

peach &1

`Ph. (7.100)

Moreover, since we are in the regime where GNs 1, we can also assume that:

ptotal .1

`P, (7.101)

were by ptotal we mean the momentum of all the particles combined. Hence, the number of

particles that we can send through the wormhole is:

Nsend ∼ptotal

peach

≤ h. (7.102)

Taking the above into consideration, it seems that from the teleportation point of view we

can send less than one particle through. This seems to be in conflict with our previous result.

It would be useful to understand why there is such a difference in these two descriptions and

how it can be reconciled. We leave this for future work.

59

8 Future directions

The straightforward continuation of this study is to understand why there is a clash between

the amount of particles we can send through the wormhole from the bulk point of view and

the teleportation picture. In order to make a rough comparison we used the AdS2 result of

MYS for the bound from the teleportation perspective. Since we want to make this more

precise, we would like to find the corresponding bound on information for the case of AdS3,

so that we can compare it with our result.

Something else that we are interested in calculating is the backreaction of the signal

on the geometry, instead of treating it as a probe. Hence, we would like to compute a

correlation function of the form 〈[φR(−t), φL(t)]〉V , which describes whether we will see the

signal appearing on the left boundary (if, of course, we send it from the right one).

Moreover, it would be interesting to see what happens when we couple the two boundaries

of AdS3 at all times (which is similar to what Maldacena does in [10] but in one more

dimension). This would give us a static, time-independent wormhole that remains open

forever.

We hope that understanding as much as possible about how such set-ups work will

teach more about black holes, such as whether their horizon is smooth and what happens

behind it. This knowledge could potentially help us solve long standing problems such as

the information paradox and even allow us to take a step forward towards the understanding

of quantum gravity.

60

Acknowledgements

First and foremost, I would like to thank my supervisor Ben Freivogel for his constant

guidance and availability, and for making this whole process fun. Thank you for teaching me

how to think and encouraging me to express my ideas. Secondly, I would like to thank Diego

Hofman for asking me interesting questions and inspiring me to work harder. I also wish to

thank Alejandra Castro for her help with my PhD applications and for being one of the most

inspirational teachers I ever had. I am also grateful to Damian Galante, for always being

available and willing to help me and for his encouraging words. Thanks to Antonio Rotundo

and Beatrix Muhlmann for listening to my pre-presentations, for interesting discussions and

emotional support.

I also owe a big thanks to my friend Evita Verheijden. Thank you for our physics and

non-physics discussions and for always being there for me. Of course, thanks to my fellow

master students and friends Lotte ter Haar, Abel Jansma, Rebekka Koch, Stratos Pateloudis,

Nikos Petropoulos, Lakshmi Swaminathan, Jorran de Wit, and Teun Zwart. As Abel said,

you made coming to the office the highlight of the day.

I would like to offer my special thanks to Vassilis Anagiannis. Thank you for our infinite

physics discussions, for helping me built my physics intuition, for your unending and constant

support in every aspect of my life and for truly believing in me. To my parents: thank you

for always helping me follow my dreams and for making my studies possible.

62

Appendix

A Electromagnetic Casimir effect in 3+1

We have followed the reasoning of [43]. The Lagrangian of the electromagnetic field, in the

absence of an external current is:

L = − 1

4πF µνFµν , (A.1)

where F µν = (∂µAν − ∂νAµ) the field strength and Aµ = (A0, Ai) the 4-potential. The

equations of motion derived from the Lagrangian are:

∂µFµν = 0, (A.2)

which correspond to the Maxwell equations without sources. The stress-energy tensor that

is obtained from the action is:

T µν = − δLδ(∂µAλ)

∂νAλ + ηµνL = F µλ∂νAλ −1

4ηµνF abFab, (A.3)

which is neither gauge invariant, nor symmetric. In order to fix this, we add to the stress

tensor we derived a term of the form ∂λ(F λµAν

), and get:

T µνsymm = F µλF νλ −

1

4ηµνF abFab, (A.4)

where we have used the equations of motion and that F µν is antisymmetric. In case we want

T symmµν we contact twice with the metric and obtain:

T symmµν = F λ

µ Fνλ −1

4ηµνF

abFab. (A.5)

For our purposes, we are going to use the Coulomb gauge where A0 = 0 and ∂iAi = 0. In

this gauge the equations of motion take the following form:

∂µ∂µAj = 0→

(−∂2

t + ∂2j

)Ai = 0, (A.6)

which is the Klein-Gordon equation for the components of Ai. We can write down the

positive and negative frequency solutions, where we have separated the time variable:

Ai(+)

J =1√2ωJ

e−iωJ tAiJ(r), Ai(−)

J =(Ai

(+)

J

)∗, (A.7)

and AiJ(r)(+)

J satisfies the equation:

− ∂2iAiJ(r) = ω2

JAiJ(r). (A.8)

The functions AiJ(r) are orthonormal and satisfy the equation:∫V

dr ηijAiJ(r)∗AjJ ′(r) = δJJ ′ . (A.9)

64

We introduce a vector εi(λ)J that labels the number of independent solutions, namely the

polarization vector, which is perpendicular to the generalized wave vector J . The polarization

vectors satisfy the following orthonormality condition:

ηijεi(λ)

J εj(λ′)J = δλλ′ , λ, λ′ = 1, 2. (A.10)

In Minkowski space and plane boundaries the polarization vectors have the following form:

εi(1)

J =1

k⊥(k2, k1, 0) , εi

(2)

J =1

kk⊥

(k1k3, k2k3,−k2

⊥), (A.11)

where ~k = (k1, k2, k3). If the boundaries are, for example, in the z direction then ~k⊥ =

(k1, k2, 0). The vector function AiJ(r) can be expanded in terms of the different polarizations:

AiJ(r) =∑λ

Ai(λ)

J (r) =∑λ

A(λ)J (r)εi

(λ)

J , (A.12)

with A(λ)J (r) = ηijAiJ(r)εj

(λ′)J . So, the mode expansion of Ai(x) is:

Ai(x) =∑J

∑(λ)

1√2ωJ

εi(λ)

J

[e−iωJ tA(λ)

J (r)a(λ)J + eiωJ t

(A(λ)J (r)

)∗ (a

(λ)J

)†]. (A.13)

where(a(λ))†

and a(λ) are the creator and annihilator of the photon in the polarization state

λ with generalized momentum J . The commutation relations are:[a

(λ)J ,(a

(λ′)J ′

)†]= δJJ ′δλλ′ ,

[a

(λ)J , a

(λ′)J ′

]=

[(a

(λ)J

)†,(a

(λ′)J ′

)†]= 0, (A.14)

and the vacuum state of the photon is defined as:

a(λ)J |0〉 = 0 (A.15)

Now, using (A.5) we may write the vacuum energy density as:

〈0|T00(x)|0〉 =1

2

⟨0|∂tAi∂tAi + ∂jA

i∂jAi − ∂jAi∂iAj|0

⟩=

1

2

⟨0|∂tAi∂tAi + ∂j

(Ai∂jA

i)− Ai∂2

jAi − ∂i

(Aj∂jA

i)

+ Ai∂i∂jAj|0⟩

1

2

⟨0|∂tAi∂tAi − Ai∂2

jAi + ∂j

(Ai∂jA

i)− ∂i

(Aj∂jA

i)|0⟩,

(A.16)

where we have used the fact that ∂jAj = 0, due to the Coulomb gauge. We first calculate

the first term:⟨0|∂tAi∂tAi|0

⟩=∑λλ′

∑JJ ′

1

2√ωJωJ ′

⟨0|ωJωJ ′εi

(λ)

J εi(λ′)J ′ AiJ(r)

(AiJ ′(r)

)∗a

(λ)J

(a

(λ′)J ′

)†|0⟩

=

∑λλ′

∑JJ ′

1

2√ωJωJ ′

⟨0|ωJωJ ′εi

(λ)

J εi(λ′)J ′ AiJ(r)

(AiJ ′(r)

)∗δλλ′δJJ ′|0

⟩=

1

2

∑λ

∑J

ωJAiJ(r)(AiJ(r)

)∗.

(A.17)

65

The second term is calculated in the same way and the result is:⟨0|Ai∂2

jAi|0⟩

=∑λ

∑J

1

2ωJAiJ(r)∂2

j

(AiJ(r)

)∗= −1

2

∑λ

∑J

ωJAiJ(r)(AiJ(r)

)∗, (A.18)

where we have used (A.8). We are not going to compute the last two terms. The reason

is going to become clear in the next steps. We continue by integrating (A.16) over the

appropriate volume, i.e.:

E0 =

∫V

dr 〈0|T00(x)|0〉 =

1

2

∑λ

∑J

∫V

dr(ωJAiJ(r)

(AiJ(r)

)∗+ ∂j

(Ai∂jA

i)− ∂i

(Aj∂jA

i))

=(A.19)

In order to compute (A.19) we need to take into account the boundary conditions of the

problem. Here, we are only interested in the set-up of two infinite parallel planes. Let’s

assume that the planes are at z = 0 and z = a. From classical electrodynamics the electric

and magnetic fields should satisfy the following boundary conditions: the tangential compo-

nents Ex and Ey of the electric field must and the normal component of the magnetic field

Bz must be zero on the planes. These boundary conditions are of Dirichlet type. Keeping

in mind that:

F 0i = Ei, F ij = εijkBk, (A.20)

and that we work in the Coulomb gauge, we acquire the following equations for Ai:

Ax(t, x, y, 0) = 0, Ay(t, x, y, 0) = 0, ∂zAz(t, x, y, 0) = 0,

Ax(t, x, y, a) = 0, Ay(t, x, y, a) = 0, ∂zAz(t, x, y, a) = 0.

(A.21)

Using (A.21) and also assuming that the photon field falls off at ±∞, the total derivative

terms are both zero when integrated. Thus:

E0(a) =1

2

∑λ

∑J

∫ ∞−∞

dx

∫ ∞−∞

dy

∫ a

0

dz ωJAiJ(r)(AiJ(r)

)∗=

1

2

∑λ

∑J

ωJS,

(A.22)

where we have used (A.9). In the case at hand the set of solutions, is:

Aik⊥,n(r) =

bx cos k1x sin k2y sin k3z

by sin k1x cos k2y sin k3z

bz sin k1x sin k2y cos k3z

. (A.23)

and the coefficients of the expansion of AiJ(r) in terms of the polarization coefficients are:

A(1)k⊥,n

(r) =bxk⊥

(∂x − ∂y) cos k1x cos k2y sin k3z

A(2)k⊥,n

(r) = − 1

kk⊥

(bx∂x∂z + bx∂y∂z + bzk

2⊥)

sin k1x sin k2y cos k3z

(A.24)

66

Using the boundary conditions (A.21) we obtain:

sin k3a = nπ → k3 =nπ

a, n = 0, 1, 2, . . . (A.25)

8 and thus the oscillator frequencies are given by:

ωJ = ωn =

√k2⊥ +

n2π2

a2. (A.26)

So, we may write (A.22) as:

E0(a)/S =1

2

∫ ∞−∞

dk1

2π

∫ ∞−∞

dk2

2π

∑λ

∞∑n=0

√k2⊥ +

n2π2

a2=

1

2

∫ ∞0

k⊥dk⊥2π

∑λ

∞∑n=0

√k2⊥ +

n2π2

a2.

(A.27)

We must note that A(1)k⊥,0

(r) = 0 and hence for n = 0 only one polarization survives. Then

(A.27) becomes:

E0(a)/S =1

2

∫ ∞0

k⊥dk⊥2π

(2∞∑n=1

√k2⊥ +

n2π2

a2+ k⊥

), (A.28)

where the 2 in front of the sum arises because for n ≥ 1 we have two different polarizations.

If we want to compute the energy per unit area using Abel-Plana regularization we need the

sum to start from n = 0 . In order to achieve that, we add and subtract 2k⊥ and get:

E0(a)/S =

∫ ∞0

k⊥dk⊥2π

(∞∑n=0

√k2⊥ +

n2π2

a2− k⊥

2

). (A.29)

Now, the respective vacuum energy per unit area in Minkowski space is:

E0M(a)/S =a

2

∑λ

∫ ∞−∞

dk1

2π

∫ ∞−∞

dk2

2π

∫ ∞−∞

dk3

2πωk =

a

∫ ∞−∞

dk1

2π

∫ ∞−∞

dk2

2π

∫ ∞−∞

dk3

2π

√k2

1 + k22 + k2

3,

(A.30)

where we have taken into account that in Minkowski space the photon has two polarizations.

If we make a coordinate change we can rewrite equation (A.30) as:

E0M(a)/S =a

∫ ∞0

k⊥dk⊥2π

∫ ∞−∞

dk3

2π

√k2⊥ + k2

3 =

a

π

∫ ∞0

k⊥dk⊥2π

∫ ∞0

dk3

√k2⊥ + k2

3,

(A.31)

8If we hadn’t taken into account the two polarizations with the sum over λ, n would have been n =

0,±1,±2, · · ·

67

where we have used that the integrand of the second integral is even. So, now we can subtract

(A.31) from (A.29) and find the Casimir energy per unit area:

E(a) =

∫ ∞0

k⊥dk⊥2π

(∞∑n=0

√k2⊥ +

n2π2

a2− k⊥

2− a

π

∫ ∞0

dk3

√k2⊥ + k2

3

)(A.32)

Introducing a new variable t =ak3

πwe arrive at:

E(a) =π

a

∫ ∞0

k⊥dk⊥2π

(∞∑n=0

√k2⊥a

2

π2+ n2 −

∫ ∞0

dt

√k2⊥a

2

π2+ t2 − k⊥a

2π

)(A.33)

Next, we use the Abel-Plana formula, which is:

∞∑n=0

F (n)−∫ ∞

0

F (t)dt =1

2F (0) + i

∫ ∞0

dt

e2πt − 1[F (it)− F (−it)] (A.34)

where F (z) is an analytic function in the right half plane. We can define x = k⊥aπ

and so, in

our case we get:

∞∑n=0

√x2 + n2 −

∫ ∞0

dt√x2 + t2 =

x

2+ i

∫ ∞0

dt

e2πt − 1[F (it)− F (−it)] (A.35)

where F (t) =√x2 + t2. We can consider a more general function F (z) = eb log(x2+z2). The

branch points are at z1,2 = ±ix. By going around the branch points one can prove that:

F (it)− F (−it) = 2ieb log(t2−x2) sin bπ θ(t− x), (A.36)

and thus, for b = 1/2 we get:

F (it)− F (−it) = 2i√t2 − x2θ(t− x). (A.37)

Using the above, (A.35) becomes:

∞∑n=0

√x2 + n2 −

∫ ∞0

dt√x2 + t2 =

x

2− 2

∫ ∞x

dt

√t2 − x2

e2πt − 1. (A.38)

We can now substitute (A.38) in (A.33) and get:

E(a)/S = −π2

a3

∫ ∞0

xdx

∫ ∞x

dt

√t2 − x2

e2πt − 1. (A.39)

We change the order of integration:

E(a)/S = − π2

3a3

∫ ∞0

dt

∫ t

0

xdx

√t2 − x2

e2πt − 1= −π

2

a3

∫ ∞0

dtt3

e2πt − 1. (A.40)

68

We make a change of variable to y = 2πt and get:

E(a)/S = − π2

3a3

1

(2π)4

∫ ∞0

dyy3

ey − 1= − π2

3a3

1

(2π)4

π4

15= − π2

720a3. (A.41)

One can find the expectation value of the whole stress-energy tensor as follows. It has been

shown by [44] that the Green’s function for the set-up of two infinite planes at z = 0, a

depends only on a unit four-vector zµ = (0, 0, 0, 1). Thus, the stress-energy tensor must only

depend on zµ as well. This, together with the tracelessness of the stress-energy tensor and

the symmetry of the problem require that:

〈0|Tµν(z)|0〉 =

(1

4ηµν − zµzν

)f(z). (A.42)

The function f(z) must, in fact, be constant c to make the stress-energy tensor free of

divergence. So, since 〈0|Tµν(z)|0〉 does not depend on z, we can find 〈0|T00(z)|0〉 by just

dividing (A.41) with a. In order to find the vacuum expectation values of the rest of the

components we solve for c:

〈0|T00(z)|0〉 = − π2

720a4=

(1

4η00 − z0z0

)c→ c =

π2

180a4. (A.43)

Consequently:

〈0|T11(z)|0〉 =π2

720a4, 〈0|T22(z)|0〉 =

π2

720a4, 〈0|T33(z)|0〉 = − 3π2

720a4. (A.44)

Another way of arriving at the same results is performed in [45], where they compute ex-

plicitly the electromagnetic field correlators.

B Second order stress tensor in 1+1 flat spacetime

In this section we are going to calculate the energy density to second order in g.

〈Ψ| : Tuu(u) : |Ψ〉 =

g2 〈φLφR : ∂uφ∂uφ : φLφR〉 −g2

2〈φLφRφLφR : ∂uφ∂uφ :〉 − g2

2〈: ∂uφ∂uφ : φLφRφLφR〉 ,

(B.1)

where we have omitted terms of first order in g and terms of higher order than g2. We begin

by manipulating the first term of (B.1):

φLφR : ∂uφ∂uφ : φLφR =(

: φLφR : + : φLφR :)

: ∂uφ∂uφ :(

: φLφR : + : φLφR :)

=

: φLφR :: ∂uφ∂uφ :: φLφR : + : φLφR :: ∂uφ∂uφ :: φLφR : + : φLφR :: ∂uφ∂uφ :: φLφR : +

: φLφR :: ∂uφ∂uφ :: φLφR : .

(B.2)

69

The last term of (B.2) will not contribute when we put it in a correlator. We start by

computing : φLφR :: ∂uφ∂uφ ::

: φLφR :: ∂uφ∂uφ : =: φLφR∂uφ∂uφ : +2 : φLφR∂uφ∂uφ : +2 : φLφR∂uφ∂uφ : +2 : φLφR∂uφ∂uφ :

(B.3)

Using (B.3) we can write that the second term of (B.2) is:

: φLφR :: ∂uφ∂uφ :: φLφR := : φLφR∂uφ∂uφφLφR : +2 : φLφR∂uφ∂uφφLφR : +

2 : φLφR∂uφ∂uφφLφR : +2 : φLφR∂uφ∂uφφLφR :

(B.4)

The first, second and third term of (B.4) do not contribute when we put them in a correlator.

Thus: ⟨: φLφR :: ∂uφ∂uφ :: φLφR :

⟩= 2 〈φLφR〉 〈φL∂uφ〉 〈φR∂uφ〉 , (B.5)

and also: ⟨φLφR : ∂uφ∂uφ :: φLφR

⟩= 2 〈φLφR〉 〈∂uφφL〉 〈∂uφφR〉 . (B.6)

Next, we will compute the first term of (B.2), using (B.3):

: φLφR :: ∂uφ∂uφ :: φLφR :=:φLφR∂uφ∂uφ :: φLφR : +2 : φLφR∂uφ∂uφ :: φLφR : +

2 : φLφR∂uφ∂uφ :: φLφR : +2 : φLφR∂uφ∂uφ :: φLφR :

(B.7)

The last term of (B.7) is not going to contribute when sandwiched between the vacuum. Also,

the first term is always going to be left with unpaired operators and thus in a correlator would

give zero. The second term is:

: φLφR∂uφ∂uφ :: φLφR := : φLφR∂uφ∂uφφLφR : + : φLφR∂uφ∂uφφLφR : + : φLφR∂uφ∂uφφLφR : +

: φLφR∂uφ∂uφφLφR : + : φLφR∂uφ∂uφφLφR : + : φLφR∂uφ∂uφφLφR : +

+ : φLφR∂uφ∂uφφLφR : .

(B.8)

In a correlator only the last two terms of (B.8) will contribute. Next, the third term of (B.7)

is:

: φLφR∂uφ∂uφ :: φLφR : =: φLφR∂uφ∂uφφLφR : + : φLφR∂uφ∂uφφLφR : + : φLφR∂uφ∂uφφLφR :

: φLφR∂uφ∂uφφLφR : + : φLφR∂uφ∂uφφLφR : + : φLφR∂uφ∂uφφLφR : +

+ : φLφR∂uφ∂uφφLφR :

(B.9)

70

Again only the last two terms contribute. So, finally:

〈φLφR : ∂uφ∂uφ : φLφR〉 =2 〈φLφR〉 〈φL∂uφ〉 〈φR∂uφ〉+ 2 〈φLφR〉 〈∂uφφL〉 〈∂uφφR〉+2 〈φL∂uφ〉 〈φRφL〉 〈∂uφφR〉+ 2 〈φL∂uφ〉 〈φRφR〉 〈∂uφφL〉+2 〈φLφL〉 〈φR∂uφ〉 〈∂uφφR〉+ 2 〈φLφR〉 〈φR∂uφ〉 〈∂uφφL〉

(B.10)

Next, we will compute the second term of (B.1), while keeping in my mind that:

φLφRφLφR = : φLφRφLφR : + : φLφRφLφR : + : φLφRφLφR : + : φLφRφLφR :

: φLφRφLφR : + : φLφRφLφR : + : φLφRφLφR : + : φLφRφLφR :

φLφRφLφR : + : φLφRφLφR :=

: φLφRφLφR : +2 : φLφRφLφR : + : φLφRφLφR : + : φLφRφLφR : +

: φLφRφLφR : + : φLφRφLφR : + : φLφRφLφR : +

: φLφRφLφR : + : φLφRφLφR : .

(B.11)

Using (B.11), we can rewrite the second term of (B.1) as:

φLφRφLφR : ∂uφ∂uφ :=

: φLφRφLφR :: ∂uφ∂uφ : +2 : φLφRφLφR :: ∂uφ∂uφ : + : φLφRφLφR :: ∂uφ∂uφ : +

: φLφRφLφR :: ∂uφ∂uφ : +φLφRφLφR :: ∂uφ∂uφ : + : φLφRφLφR :: ∂uφ∂uφ : +

2 : φLφRφLφR :: ∂uφ∂uφ : + : φLφRφLφR :: ∂uφ∂uφ : + : φLφRφLφR :: ∂uφ∂uφ : .

(B.12)

The last three terms will not contribute to the expectation value. Also, the first term will

always have uncontracted operators and hence does not contribute either. The second term

of (B.12) is:

: φLφRφLφR :: ∂uφ∂uφ : =: φLφRφLφR∂uφ∂uφ : +2 : φLφRφLφR∂uφ∂uφ : +

2 : φLφRφLφR∂uφ∂uφ : +2 : φLφRφLφR∂uφ∂uφ :

(B.13)

The third one will be the same as the second. The fourth term of(B.12):

: φLφRφLφR :: ∂uφ∂uφ := : φLφRφLφR∂uφ∂uφ : +2 : φLφRφLφR∂uφ∂uφ : +

2 : φLφRφLφR∂uφ∂uφ : +2 : φLφRφLφR∂uφ∂uφ :,

(B.14)

the fifth term is:

: φLφRφLφR :: ∂uφ∂uφ := : φLφRφLφR∂uφ∂uφ : +4 : φLφRφLφR∂uφ∂uφ : +

2 : φLφRφLφR∂uφ∂uφ :,

(B.15)

71

and finally the sixth:

: φLφRφLφR :: ∂uφ∂uφ := : φLφRφLφR∂uφ∂uφ : +4 : φLφRφLφR∂uφ∂uφ : +

2 : φLφRφLφR∂uφ∂uφ : .

(B.16)

Using the above, we finally obtain that the second term of (B.1) is:

〈φLφRφLφR : ∂uφ∂uφ :〉 =6 〈φLφR〉 〈φL∂uφ〉 〈φR∂uφ〉+ 2 〈φRφL〉 〈φL∂uφ〉 〈φR∂uφ〉2 〈φLφL〉 〈φR∂uφ〉 〈φR∂uφ〉+ 2 〈φRφR〉 〈φL∂uφ〉 〈φL∂uφ〉 ,

(B.17)

and consequently the third term of (B.1) is:

〈: ∂uφ∂uφ : φLφRφLφR〉 =6 〈φLφR〉 〈∂uφφL〉 〈∂uφφR〉+ 2 〈φRφL〉 〈∂uφφL〉 〈∂uφφR〉2 〈φLφL〉 〈∂uφφR〉 〈∂uφφR〉+ 2 〈φRφR〉 〈∂uφφL〉 〈∂uφφL〉 .

(B.18)

Thus, 〈Tuu〉 to second order in g is:

〈Tuu〉 =

g2 (−2 〈φLφR〉 〈φL∂uφ〉 〈φR∂uφ〉 − 2 〈φLφR〉 〈∂uφφL〉 〈∂uφφR〉+ 2 〈φLφR〉 〈φL∂uφ〉 〈∂uφφR〉+2 〈φLφR〉 〈φR∂uφ〉 〈∂uφφL〉+ 2 〈φLφL〉 〈φR∂uφ〉 〈∂uφφR〉 − 〈φLφL〉 〈φR∂uφ〉 〈φR∂uφ〉− 〈φLφL〉 〈∂uφφR〉 〈∂uφφR〉+ 2 〈φRφR〉 〈φL∂uφ〉 〈∂uφφL〉 − 〈φRφR〉 〈φL∂uφ〉 〈φL∂uφ〉− 〈φRφR〉 〈∂uφφL〉 〈∂uφφL〉)

(B.19)

C Refinement of the expression of the stress tensor TUU

First we use equation (7a) from [46]:

F1 (a+ n; b, b′; c;x; y) =F1 (a; b, b′; c;x; y) +bx

c

n∑k=1

F1 (a+ k; b+ 1, b′; c+ 1;x; y) +

b′y

c

n∑k=1

F1 (a+ k; b, b′ + 1; c+ 1;x; y)

(C.1)

Using (C.1) we get:

F1

(3

2;1

2, 1 + ∆;

3

2−∆;

U0 − U2U0

,U0 − U

U0 (1 + UU0)

)= F1

(1

2;1

2, 1 + ∆;

3

2−∆;

U0 − U2U0

,U0 − U

U0 (1 + UU0)

)+

U0 − U4U0

(32−∆

)F1

(3

2;3

2, 1 + ∆;

5

2−∆;

U0 − U2U0

,U0 − U

U0 (1 + UU0)

)+

(U0 − U) (1 + ∆)

U0 (1 + UU0)(

32−∆

)F1

(3

2;1

2, 2 + ∆;

5

2−∆;

U0 − U2U0

,U0 − U

U0 (1 + UU0)

)(C.2)

72

which can be rewritten as:

1

2F1

(1

2;1

2, 1 + ∆;

3

2−∆;

U0 − U2U0

,U0 − U

U0 (1 + UU0)

)− 1

2F1

(3

2;1

2, 1 + ∆;

3

2−∆;

U0 − U2U0

,U − U0

U0 (1 + UU0)

)=

U − U0

4(2∆− 3) (1 + UU0)

[−4(∆ + 1)F1

(3

2;1

2, 2 + ∆;

5

2−∆;

U0 − U2U0

,U0 − U

U0 (1 + UU0)

)− (1 + UU0)F1

(3

2;3

2, 1 + ∆;

5

2−∆;

U0 − U2U0

,U0 − U

U0 (1 + UU0)

)](C.3)

By using (C.3) we can write (7.72) as:

〈TUU〉 =

− A(1− U/U0)∆+1/2 (1 + UU0)∆+1

[∆ F1

(1

2;1

2,∆ + 1;

3

2−∆;

U0 − U2U0

,U0 − U

U0 (1 + UU0)

)−1

2F1

(3

2;1

2, 1 + ∆;

3

2−∆;

U0 − U2U0

,U − U0

U0 (1 + UU0)

)](C.4)

Next, we use (6b) from [46]:

c F1 (a; b, b′; c;x; y)− (c− a)F1 (a; b, b′; c+ 1;x; y)− a F1 (a+ 1; b, b′; c+ 1;x; y) = 0. (C.5)

and (C.3) becomes:(1

2−∆

)F1

(1

2;1

2,∆ + 1;

1

2−∆;

U0 − U2U0

,U0 − U

U0 (1 + UU0)

)=

1

2F1

(3

2;1

2,∆ + 1;

3

2−∆;

U0 − U2U0

,U0 − U

U0 (1 + UU0)

)−∆ F1

(1

2;1

2,∆ + 1;

3

2−∆;

U0 − U2U0

,U0 − U

U0 (1 + UU0)

).

(C.6)

Consequently we may write 〈TUU〉 as:

〈TUU〉 =A

(1− U/U0)∆+1/2 (1 + UU0)∆+1

(1

2−∆

)F1

(1

2;1

2,∆ + 1;

1

2−∆;

U0 − U2U0

,U0 − U

U0 (1 + UU0)

)(C.7)

Next, we use the following property:

F1 (a; b, b′; c;x; y) = (1− x)−b(1− y)−b′F1

(c− a; b, b′; c;

x

x− 1;

y

y − 1

). (C.8)

and get:

F1

(1

2;1

2,∆ + 1;

1

2−∆;

U0 − U2U0

,U0 − U

U0 (1 + UU0)

)=(

U0 + U

2U0

)−1/2(U (U2

0 + 1)

U0 (1 + UU0)

)−(∆+1)

F1

(−∆;

1

2,∆ + 1;

1

2−∆;

U − U0

U + U0

,U − U0

U (1 + U20 )

).

(C.9)

73

So, the final expression for the stress tensor is:

〈TUU〉 =

AU−(∆+1)√

2 (1/2−∆)

(1− U/U0)∆+1/2 (1 + U/U0)1/2

(U2

0 + 1

U0

)−(∆+1)

F1

(−∆;

1

2,∆ + 1;

1

2−∆;

U − U0

U + U0

,U − U0

U (1 + U20 )

).

(C.10)

74

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