UNIVERSITY OF AMSTERDAM Master Thesis Building traversable wormholes from Casimir energy and non-local couplings by Theodora Nikolakopoulou 11410310 Main Supervisor: dr. Ben Freivogel Second Supervisor: dr. Diego Hofman July 16, 2018
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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