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University of Amsterdam
MSc Physics
Track: Theoretical Physics
Master Thesis
Entanglement Entropy, Differential Entropy and the Dynamics
of Spacetime Geometry
In AdS/CFT
by
Alex Kieft
5836956
60 ECTS
September 2013 - August 2014
Supervisor:
Prof. dr. J. de Boer
Examiner:
dr. A. Castro
In memory of Floris Tromp
2
Abstract
A convenient measure for the amount of entanglement in a quantum mechanical
system is the entanglement entropy. In conformal field theories, the entanglement
entropy can be calculated using the replica trick, which we apply to several two-
dimensional systems. Whenever the CFT has a holographic dual, the entanglement
entropy is proportional to a minimal surface area ending on the asymptotic boundary
of the dual AdS spacetime − a statement which is encoded in the Ryu-Takayanagi
(RT) formula S = A/4G. We derive a new result for the holographic entanglement
entropy in a three-dimensional conical deficit spacetime using the RT formula and
show that it matches the associated CFT2 result for a certain excited state. Whereas
the entanglement entropy is related to minimal surfaces ending on the boundary of
AdS, the “differential entropy” is related to more general bulk surfaces, i.e. those
which do not nesessarily extend all the way to the boundary. We review the defini-
tion of differential entropy and the closely related “residual entropy” within quantum
mechanical systems and offer illustrative examples, before reviewing its application
within AdS3/CFT2. Elaborating upon the philosophy that geometry comes from en-
tanglement, it has recently been shown that the linearized vacuum Einstein equations
are equivalent to a “first law” of entanglement entropy. We review this derivation
and compare it with Jacobson’s derivation of the full non-linear Einstein equations
from the first law of thermodynamics. Based on the similarities between these two
approaches, we present some preliminary ideas towards a direct holographic deriva-
tion of Jacobson’s original argument using the “first law of differential entropy”,
although we raise questions to the viability of such an attempt.
3
Acknowledgements
I would like to express my deep gratitude to my advisor Jan de Boer for guiding
me throughout this project. I would also like to thank my examiner dr. Alejandra
Castro for her comments on the preliminary version of this thesis, and dr. Diego
Hofman for valuable conversations and for bringing useful literature to my attention.
I am greatly endebted to Benjamin Mosk, who has been a big help with his careful
explanations on differential geometry. Furthermore, I am very greatful to my fellow
students and friends Jochem Knuttel, Eva Llabres and Gerben Oling for lots of valu-
able conversations. Special thanks goes to Manus Visser for innumerable invaluable
discussions and a very fruitful collaboration over the past two years. I also wish to
thank the rest of my fellow master students for their pleasant company.
CONTENTS 4
Contents
1 Introduction 5
2 Background 9
2.1 Anti-de Sitter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Entanglement entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Entanglement Entropy in CFT2 16
3.1 Replica trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Entanglement entropy of the ground state . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Entanglement entropy for excited states . . . . . . . . . . . . . . . . . . . . . . . 22
4 Holographic Entanglement Entropy 26
4.1 Heuristic derivation of RT formula . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Some properties of holographic entanglement entropy . . . . . . . . . . . . . . . . 28
4.3 Examples of HEE in AdS3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4 Holographic entanglement entropy in a conical metric . . . . . . . . . . . . . . . 34
5 Hole-ography 37
5.1 Spherical Rindler-AdS space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Residual entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 Holographic holes in AdS3 and differential entropy . . . . . . . . . . . . . . . . . 42
6 Linearized Gravity from the First Law of Entanglement Entropy 48
6.1 The ‘first law’ of entanglement entropy . . . . . . . . . . . . . . . . . . . . . . . . 48
6.2 Holographic interpretation of the first law . . . . . . . . . . . . . . . . . . . . . . 51
6.3 First law of black hole thermodynamics and the Iyer-Wald theorem . . . . . . . . 54
6.4 Linearized gravity from the first law . . . . . . . . . . . . . . . . . . . . . . . . . 55
7 First Law of Differential Entropy 60
7.1 Jacobson’s derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.2 Local Rindler horizons and differential entropy . . . . . . . . . . . . . . . . . . . 62
8 Conclusion 67
Appendix A Scalar curvature of conical spaces 69
Appendix B Iyer-Wald formalism 71
1 INTRODUCTION 5
1 Introduction
How to formulate a consistent theory of quantum gravity is one of the major outstanding prob-
lems in contemporary theoretical physics. Our ordinary theory of gravity is Einstein’s general
relativity, which is a classical field theory. On the other hand, the other fundamental forces of
nature are all formulated in terms of local quantum field theories. Attempts to treat general
relativity on an equal footing with the other forces − by quantizing it on the nose − quickly
run into trouble because general relativity appears to be non-renormalizable. As such, infinitely
many counter-terms are required to obtain finite answers for the one-loop (i.e. quantum) correc-
tions, rendering the theory unpractical if not unphysical. Since the direct quantization of general
relativity appears to be a fruitless effort, a radically different approach is needed to arrive at a
consistent description of gravity at the quantum level.
As is well known, the leading candidate for a consistent theory of quantum gravity is string
theory, most notably because it naturally encompasses a spin-two particle representing the
graviton. In contrast to the point particles which occur in ordinary quantum field theories,
the fundamental constituents of string theory are one-dimensional objects of finite extent. As
such, string theory is UV-finite and hence does not suffer the same fate as general relativity
upon quantization: it is perfectly renormalizable! Another success of string theory is that it has
allowed for a precise comparison between the microscopic and macroscopic entropy of a black
hole, as was famously demonstrated by [1]. The macroscopic entropy of a black hole is given by
the celebrated Bekenstein-Hawking formula [2, 3],
S =kBc
3
~A4G
, (1.1)
where A is the area of the horizon and G, kB, c and ~ are the usual fundamental constants of
nature.1 The Bekenstein-Hawking formula presents remarkable connection between quantum
theory, gravity and statistical mechanics. It is therefore generally believed that black hole
horizons provide a window into the nature of quantum gravity.
Certainly, the most important lesson that the Bekenstein-Hawking formula has had to offer is
that the information about the black hole’s interior is fully encoded on the surface that forms the
event horizon. This realization sparked the idea of holography [4, 5]. The holographic principle
states that the degrees of freedom of quantum gravity are, as opposed to an ordinary local
quantum field theory, not localized within a given spacetime region, but are instead localized on
the boundary surface of that spacetime region. More precisely, a certain gravitational theory in
a d + 1-dimensional spacetime M can be fully described by a non-gravitational quantum field
theory on the d-dimensional boundary spacetime ∂M. Currently, the most concrete and well-
understood implementation of the holographic principle is the AdS/CFT correspondence [6]. It
states that there exists an exact equivalence between certain gravitational theories on a d + 1-
dimensional anti de-Sitter space and certain conformal field theories living on the d-dimensional
asymptotic boundary of AdS. Although it is generally believed that we do not live in an anti-
de Sitter universe, the AdS/CFT correspondence forms an excellent theoretical playground to
1From now on, we use natural units ~ = c = kB = 1.
1 INTRODUCTION 6
study the properties of holographic theories. It is hoped that these lessons will one day help us
understand more realistic settings like de Sitter space, which is considered as a toy model for
our own universe.
Although AdS/CFT was originally formulated as a duality between a certain realization of
string theory on AdSd+1 and a supersymmetric conformal gauge theory [6], the correspondence
is more general. In particular, AdS/CFT is an example of a strong-weak duality: when the
gravitational theory is strongly coupled, the field theory is weakly coupled, and vice versa.
Focusing on the weak coupling regime of string theory, the theory becomes approximately equal
to semi-classical (super-)gravity. Thus, semi-classical general relativity on d + 1-dimensional
AdS is dual to a strongly coupled d-dimensional CFT without supersymmetry. In the present
thesis, we will be interested in this regime of the AdS/CFT correspondence. Despite its relative
simplicity (compared to string theory realizations of the correspondence), its study allows us to
gain deep insights in the fundamental structure of spacetime.
One of the deepest conceptual implications of the holographic duality is the emergence of
spacetime. Emergence here means that smooth classical spacetime does not exist at a fun-
damental level, but rather that it arises as a thermodynamic phenomenon in a coarse-grained
description. The idea of emergence can be made precise within the context of AdS/CFT. Here
the d-dimensional CFT describes the behaviour of the fundamental degrees of freedom of the
quantum gravity theory. When we ‘zoom out’, meaning that we consider the theory at larger
distance scales or, equivalently, at lower energy scales, another spacetime dimension arises in
such a way that the full spacetime is now d+1-dimensional AdS. The question that arises, then,
is how the extra dimension emerges from field theory data. Recently, it has been suggested that
entanglement plays a central role in the emergence of spacetime. Ryu and Takayanagi have
made this connection precise by their proposal that the area of a certain minimal surface in AdS
is proportional to the entanglement entropy SA in the boundary CFT [7],
SA =A(γA)
4G, (1.2)
where γA is a surface anchored at the boundary, that stretches into AdS in such a way that
its shape minimizes the area functional. Aside from these technicalities, the Ryu-Takayanagi
formula indeed conveys a deep relationship between gravitional spacetime and entanglement, as
we will try to illuminate by the following thought experiment.
Consider splitting the entire space supporting the CFT into two parts, denoted by A and
its complement B. The entanglement entropy SA then quantifies the amount of entanglement
between these two field theory regions. According to the Ryu-Takayanagi formula, there exists
a certain minimal surface γA in the dual AdS spacetime that divides the manifold into two parts
A and B and whose area A(γA) is directly related to the value of SA. Suppose now that we
change the amount of entanglement between the regions A and B, yielding a different value
for the entanglement entropy. The geometric quantity A(γA) in the dual spacetime therefore
changes accordingly by virtue of (1.2), which serves as a heuristic confirmation that the emergent
spacetime geometry indeed comes from entanglement [8, 9, 10]. We can stretch this thought
experiment a bit further by decreasing the value of SA all the way down to zero, such that the
1 INTRODUCTION 7
Figure 1: Both figures represent a timeslice of AdSd+1. The conformal field theory lives on the boundary,which is split into two parts A and B. The holographic entanglement entropy is given by the area of theminimal surface shown in orange, which splits gravitational spacetime into two pieces A and B. Whenthe entanglement between A and B vanishes, the associated spacetime regions become disconnected. Thedrawback of this illustration is that the boundary geometry seems to change, which is not true. Figurebased on [8, 9].
surface in the dual gravitational theory has vanishing size, as illustrated in figure 1. Hence,
the associated regions of spacetime A and B become disconnected. We can therefore, again
very heuristically, interpret entanglement as the “glue” of spacetime. Extended to more general
settings, the idea is that smooth classical spacetime emerges from the entanglement between the
microscopic degrees of freedom of quantum gravity.
Now, if geometry comes from entanglement, what are the corresponding dynamical laws gov-
erning fluctuations in the geometry? Of course, we already know that fluctuations are governed
by the Einstein equations at the classical level. Thus, the question becomes how the Einstein
equations are encoded in the dual CFT. In an attempt to answer this question, it was shown
by [11] that the entanglement entropy obeys a first law-like relation δSA = δE, where δE is a
small perturbation of the energy of the quantum state. Using the Ryu-Takayanagi formula (1.2)
for the holographic entanglement entropy, the first law holographically translates into a relation
between energy and geometry, which schematically is the content of the Einstein equations.
The details, however, are quite different, because the holographic energy does not correspond
to a local bulk matter field. Nevertheless, it has been shown that the entanglement first law
is indeed equivalent to the gravitational field equations [12, 13], although only the linearized
vacuum Einstein equations have been derived holographically. A holographic derivation of the
full non-linear Einstein equations remains elusive. The main obstruction is that the holographic
approach uses global rather than local Rindler horizons [13]. Nearly twenty years ago, before
the era of holography, Jacobson showed that applying the first law of thermodynamics to a
local Rindler horizon, the full Einstein equations emerge as a consequence of the entropy/area
relation S = A/4G [14]. Inspired by his success, we would like to reconstruct his argument from
a holographic viewpoint using local Rindler horizons.
The fact that the holographic derivation of the Einstein equations makes use of global rather
than local Rindler horizons is closely related to the observation that the RT formula picks out a
very special class of surfaces, namely minimal surfaces that extend to the asymptotic boundary.
An interesting question is whether a similar holographic correspondence exists between entropy
and more general surface areas. In particular, one might ask which field theoretical quantity
is associated to the area of a spherical surface centered inside global AdS, i.e. a “hole”. This
1 INTRODUCTION 8
question was addressed by [15] within the context of AdS3. The authors argue that the relevant
field theory observable is the “differential entropy”, which is an alternating sum of entanglement
entropies around the full field theory (we will make this more precise later). Although this is
an interesting topic its own right, we will primarily be interested in the dynamics around such
bulk surfaces. Certainly, these bulk surfaces are much more local objects than the minimal
RT surfaces which are used to evaluate the entanglement entropy. The differential entropy thus
appears as a suitable candidate to replicate Jacobson’s argument in a holographic manner. How-
ever, it appears that the Einstein equations are already needed to compute of the variation of
the hole’s area, which means that we are regressing into circular reasoning when attempting to
holographically derive the Einstein equations.
The thesis is outlined as follows. We start with a brief introduction to anti-de Sitter space
and entanglement entropy in chapter 2. The calculation of the entanglement entropy in simple
two-dimensional conformal field theories is reviewed in chapter 3. In chapter 4, we present an
introduction to the Ryu-Takayanagi (RT) formula (1.2) and discuss various examples in AdS3,
displaying exact agreement with the CFT2 results from chapter 3. In addition, we present a novel
result at the end of chapter 4, finding agreement with the dual CFT calculation for an excited
state characterized by twist operators. Chapter 5 is dedicated to the residual entropy and the
differential entropy, where we present new independent results for the residual entropy of a three-
qubit system. In chapter 6, we review the derivation of the linearized Einstein equations from the
first law of entanglement entropy. In chapter 7, we recall Jacobson’s derivation of the Einstein
equations. We then give some initial steps towards a holographic derivation by proposing a first
law of differential entropy. Finally, we shall discuss the aforementioned circularity when trying
to derive the Einstein equations from the first law of differential entropy.
2 BACKGROUND 9
2 Background
In the current chapter we will present some relevant background material. Section 2.1 provides
a review of anti-de Sitter space and its use in the AdS/CFT correspondence. The focus lies on
three dimensions, but most observations and derivations straightforwardly generalize to higher
dimensions. In section 2.2, we will review the definition of entanglement entropy and discuss
some of its properties.
2.1 Anti-de Sitter space
Anti-de Sitter space (AdS) is a solution to the Einstein equations in the presence of negative
cosmological constant. In particular, it is a maximally symmetric spacetime, which means that
every point in spacetime is equivalent to every other point. As an immediate consequece, such
spacetimes must have constant curvature, but the conditions on the curvature tensors are more
stringent than that. The curvature tensors for maximally symmetric spaces are then given by
Rµνρσ =1
L2(gµσgνρ − gµρgνσ) (2.1)
Rµν = − d
L2gµν (2.2)
R = −d(d+ 1)
L2, (2.3)
where L is a constant with dimension length and is known as the AdS-radius and d is the number
of spatial dimensions of the AdSd+1 spacetime. One can easily verify that this spacetime is a
solution to the vacuum Einstein equations sourced by a cosmological constant
Rµν −1
2Rgµν + Λgµν = 0. (2.4)
Taking the trace of this equation and comparing with (2.3) fixes the value of the cosmological
constant as
Λ = −d(d− 1)
2L2. (2.5)
2.1.1 Three-dimensional gravity
The most striking feature of three-dimensional gravity is that is has no local degrees of freedom,
which follows from a simple counting argument. Notice that the metric has six components,
three of which are fixed by diffeomorphism (i.e. gauge) invariance, whereas the other three are
fixed by the Einstein field equations. Therefore, the metric has no free parameters and thus
no local degrees of freedom.Therefore, the curvature of a three-dimensional spacetime has to
be constant. More precisely, every three-dimensional spacetimes is maximally symmetric, which
means any three-dimensional spacetime with negative cosmological constant is locally equivalent
to AdS3. The theory is therefore easy to handle, but not completely trivial. Different global
identifications give rise to a variety of spacetimes, including one with black hole-like features.
In this section, we will review some properties of AdS3.
2 BACKGROUND 10
2.1.2 Global coordinates
It is convenient to represent anti-de Sitter space as a hypersurface in a higher dimensional em-
bedding space. This is how one typically deals with maximally symmetric spaces. For instance,
the 2-dimensional Euclidean sphere can be defined as the solution to the constraint ~x2 = R2 in
flat 3-dimensional Euclidean space R3, whose coordinates are known as embedding coordinates.
Introducing the usual parametrization in terms of angular coordinates θ, φ for the constraint
equation ~x2 = R2 allows one to calculate the induced metric on the sphere.
We apply the same procedure to find the AdS3 metric. A negatively curved object with
constant curvature is known as a hyperboloid. Since we are looking for a negatively curved space
in Lorentzian signature, the embedding space is R2,2, with metric
ds2 = −dT 21 − dT 2
2 + dX21 + dX2
2 . (2.6)
The constraint equation for the Lorentzian hyperboloid is
− T 21 − T 2
2 +X21 +X2
2 = −L2, (2.7)
which can be parametrized with the following set of coordinates
T1 = L cosh ρ sin τ
T2 = L cosh ρ cos τ
X1 = L sinh ρ cosφ (2.8)
X2 = L sinh ρ sinφ.
To find the induced metric, we take the differentials of (2.8) and plug these into metric (2.6),
yielding
ds2 = L2(− cosh2 ρ dτ2 + dρ2 + sinh2 ρ dφ2
). (2.9)
What we have obtained here is AdS3 in global coordinates. It is quite special that one
can cover a whole spacetime with a single coordinate system (generically one needs multiple
coordinate patches to cover the whole space).
Note that there is a subtlety concerning τ . As can be seen from (2.8), τ is periodic on
the hyperboloid, which would produce unphysical features like closed timelike curves. This
issue is easily remedied by reminding ourselves that the embedding structure is just a helpful
tool to arrive at metric (2.9), which ultimately defines anti-de Sitter space with non-periodic
τ . However, the embedding coordinates have other computational benefits, such as finding
geodesics in AdS3, as we shall do in section 4.3.
Next, consider the coordinate transformation ρ = arcsinh tan ξ with 0 ≤ ξ ≤ π/2. In these
coordinates the metric reads
ds2 =L2
cos2 ξ
(−dτ2 + dξ2 + sin2 ξdφ2
). (2.10)
2 BACKGROUND 11
Figure 2: Global AdS3 compactified as a solid cylinder, where the physical distance to the boundary isinfinite. Each timeslice corresponds to a hyperbolic disk with radial coordiate ρ.
Setting aside the overall conformal factor L2/ cos2 ξ, (2.10) corresponds to the metric of a solid
cylinder, leading to the pictorial reprentation of AdS in figure 2. The dual field theory resides
on the conformal boundary of this cylinder (at ξ = π/2), but it should be remembered that this
boundary is actually infinitely far away.
Another convenient way to represent AdS in global coordinates is to introduce the coordinates
r = L sinh ρ, t = Lτ and φ = φL :
ds2 = −(
1 +r2
L2
)dt2 +
(1 +
r2
L2
)−1
dr2 + r2dφ2, (2.11)
where we have dropped the tilde on φ. In this thesis we shall mostly make use of (2.11), although
we will also make use of metric (2.9).
2.1.3 Poincare coordinates
There exists another convenient coordinate system for AdS. Going back to the hyperboloid (2.7),
we parametrize it differently as
T1 = t/z
X1 = x/z
X2 + T2 = −L/z (2.12)
X2 − T2 =L
z
(−t2 + x2 + z2
),
for which the metric reads
ds2 =L2
z2
(−dt2 + dz2 + dx2
). (2.13)
The asymptotic boundary is at z → 0, which is simply two-dimensional Minkowski space.
Poincare coordinates are therefore highly convenient for AdS/CFT calculations. The price we
2 BACKGROUND 12
pay, however, is that these coordinates cover only half of the full AdS space.2 To see this, note
that the third and fourth coordinate in (2.12) are essentially light-like coordinates. In particular,
from the third line it then follows that z itself is light-like, which splits the space into two charts:
z > 0 translates into −X2 < T2, corresponding to one half of the hyperboloid, whereas the other
half z < 0 gives the second Poincare chart, i.e. −X2 > T2. When we refer to the Poincare patch
of AdS, we refer to the region of AdS covered by one of these charts (z > 0 is typically chosen),
where z is now chosen to be spacelike. The locus that is not covered is z = 0, which corresponds
to the boundary of AdS.
Another way to think about the Poincare patch is as the asymptotic limit, r →∞, of global
AdS (2.11). This means that we can neglect the contant O(r0) term, giving
ds2 = − r2
L2dt2 +
L2
r2dr2 + r2dφ2.
Making the coordinate redefinition z = L2/r and x = Lφ gives us again the Poincare metric
(2.13). Hence, the Poincare metric can be thought of as the near-boundary metric of global AdS.
2.1.4 BTZ black holes
Einstein’s equations in three dimensions yield a family of solutions with black hole-like features
[17], known as BTZ black holes. Since there is no local curvature in three dimensions, the BTZ
black hole has no curvature singularity like its higher dimensional relatives. On the other hand,
the BTZ black hole has an horizon, which justifies its status as a black hole. Just like higher
dimensional black holes, its metric is completely determined by the black hole’s mass, angular
momentum (but no electric charge).
The metric of the neutral, non-rotating BTZ black hole is given by [17],
ds2 = −(r2 − r2
+
L2
)dt2 +
(r2 − r2
+
L2
)−1
dr2 + r2dφ2. (2.14)
Here r+ is the radial coordinate of the horizon, which is related to the mass M as and inverse
temperature β as
M =r2
+
8GL2and β =
2πL2
r+. (2.15)
2.2 Entanglement entropy
In classical physics the concept of entropy quantifies the uncertainty we have about the exact
state of a physical system at hand. Classical entropy is always related to ignorance about the
state of the system, not to some fundamental randomness of nature. In quantum mechanics,
the situation is radically different, where positive entropies may arise due to the probabilistic
nature of quantum mechanics, i.e. even without an objective lack of information. The entropy
2This is only true if we do not decompactify τ , i.e. keep τ periodic. If we decompactify τ , the Poincare patchfits infinitely many times in the full AdS space. This is understood by looking at the Penrose diagrams of thespaces, which is discussed in e.g. [38].
2 BACKGROUND 13
we are talking about is known as the entanglement entropy, whose definition we will review in
the current section.
Consider a quantum mechanical system described by a single state vector |Ψ〉, which is called
a pure state. In contrast, a mixed state cannot be described by a single state vector. The density
matrix of a pure state is defined as
ρ = |Ψ〉 〈Ψ| , (2.16)
which has an associated Von Neumann entropy, defined as
S ≡ −Tr ρ log ρ, (2.17)
which vanishes for pure states (we consider a simple example shortly hereafter). In contrast, the
density matrix of a mixed state can be expressed as ρ =∑
i pi |ψi〉 〈ψi|, where pi is the weight
of each state, yielding a non-zero von Neumann entropy.
Focusing on a pure state, suppose that we factorize the total Hilbert space as a tensor product
of two subspaces
H = HA ⊗HB.
Now, if there exist vectors |ΨA〉 ∈ HA and |ΨB〉 ∈ HB such that |Ψ〉 = |ΨA〉 ⊗ |ΨB〉, then
|Ψ〉 is untentangled; otherwise, it is entangled. A convenient way of quantifying the amount of
entanglement between A and B is given by first writing down the reduced density matrix ρA of
subsystem A,
ρA = TrB ρ, (2.18)
where the trace is over the Hilbert space of subset B. The entanglement entropy is then defined
as the von Neumann entropy of ρA,
SA = −TrA ρA log ρA, (2.19)
which vanishes (is positive) in the absence (presence) of entanglement. Another way of writing
the entanglement entropy is in terms of the eigenvalues of ρA, namely SA = −∑
i λi log λi, which
follows from the cyclicity of the trace.
As an intuitive example, consider a system consisting of two qubits. The first entry in the
two-qubit state vector denotes the state of qubit A, the second that of B. If we consider the
following state, then the reduced density matrix of qubit A is easily constructed by tracing over
B:
|Ψ〉 = cos θ |00〉+ sin θ |11〉 , ⇒ ρA =
(cos2 θ 0
0 sin2 θ
).
For θ = 0, we obtain the unentangled state |00〉, whose reduced density matrix is equivalent to
that of a pure state, with one eigenvalue equal to 1 and the other 0, yielding SA = 0. On the
other hand, the state with coefficients cos θ = sin θ = 1/√
2 is the maximally entangled state,
whose density matrix is diag12 ,
12, yielding the maximum value for the entanglement entropy,
SA = log 2.
2 BACKGROUND 14
Figure 3: A timeslice of the d-dimensional spacetime which harbors the quantum field theory. TheHilbert space is factorized as a tensor product of the interior A and the exterior B of an imaginarysphere. The entanglement entropy is proportiotional to the area of the boundary ∂A, shown in red.
2.2.1 Properties of entanglement entropy
We now discuss some general properties of entanglement entropy, which will be relevant when
considering the holographic entanglement entropy from chapter 4 onwards.
Strong subadditivity
The von Neumann entropy satisfies a relation known as strong subadditivity. For two overlapping
subsets A and B, the following inequality holds [18],
S(A ∪B) ≤ S(A) + S(B)− S(A ∩B). (2.20)
The proof is not elementary, so we will not review it here. However, in chapter 4 we shall see
that (2.20) is easily proven in a holographic context.
Area law for QFT ground state
Given the previous discussion on entanglement entropy, it is natural to ask about its relation
to thermodynamic entropy. The latter is usually an extensive quantity, meaning that it scales
with the volume of a system. In constrast, a simple argument convinces us the entanglement
entropy cannot be extensive. Consider a quantum field theory, which we split into the interior
A and exterior B of an imaginary sphere, as in figure 3. The Hilbert space then factorizes as a
tensor product of A and B, i.e.
Htot = HA ⊗HB, (2.21)
so that the state in the interior A is described by the the reduced density matrix ρA = TrB ρ, and
similarly ρB = TrA ρ for the exterior B. Now, it can easily be shown that if the total system is in a
pure state, then these two density matrices have identical eigenvalues, plus some additional zeroes
for the density matrix of the larger Hilbert space. Consequently, the entanglement entropy of is
equal for both regions, SA = SB, which suggests that the entanglement entropy can depend only
on properties that are shared by the two regions, and since Vol(A) 6= Vol(B), the entanglement
entropy cannot be extensive. On the other hand, the regions do have a shared boundary, i.e.
∂A = ∂B. Combining this observation with the locality of the quantum theory, the conjecture
2 BACKGROUND 15
is that the entanglement entropy of a pure state in a local QFT is proportional to the area3 of
the boundary surface [19]
SA =Area(∂A)
δd−1+ ..., (2.22)
where ∂A is the boundary of region A (e.g. the interior of the sphere), δ is a cut-off to regulate
the answer and the dots represent subleading terms which come from correlations across larger
scales. The area law is naturally explained by the fact that in a local quantum field theory, short-
distance correlations are the most dominant. Thus, the degrees of freedom which are localized
close to surface on one side are entangled predominantly with those close to the surface on the
other side. Since local quantum field field theories are UV divergent, there are in fact infinitely
many modes that contribute to the entanglement across the surface, rendering the entanglement
entropy infinite. When the theory is regularized on a lattice with spacing δ, the area law (2.22)
can be interpreted as ‘the number of entanglement lines being cut by the surface ∂A’.
Note that the area law does not hold if the system is in a mixed state. Mixed states are
characterized by an extensive contribution to the entanglement entropy, i.e. they contain a term
proportional to the volume of the region.
3It should be noted that the area law is slightly violated in two-dimensional CFTs, where SA ∼ log `δ, with `
the length of the interval. This will become clear in the next chapter.
3 ENTANGLEMENT ENTROPY IN CFT2 16
3 Entanglement Entropy in CFT2
The main goal of this chapter is to calculate the entanglement entropy of a single interval in two-
dimensional conformal field theories. The most direct calculation of the entanglement entropy
for a system with a finite number of degrees of freedom is to explicitly compute the reduced
density matrix (exactly or numerically) and then calculate SA = −Tr ρA log ρA = −∑λi log λi,
where λi are the eigenvalues of ρA. However, a direct reconstruction of the full reduced density
matrix of a generic interacting QFT is a notoriously difficult task. We therefore review the
approach of [22, 23], known as the “replica trick” (earlier work found in [24]), to obtain the
entanglement entropy of the ground state of a two-dimensional conformal field theory in section
3.2, including generalization to finite size and finite temperature. In section 3.3, we review the
calculation of the entanglement entropy for excited states.
3.1 Replica trick
The crucial observation that makes the replica trick possible is that the reduced density matrix
ρA is positive semi-definite, meaning that its eigenvalues satisfy λi ∈ [0, 1] and∑λi = 1.
Therefore, the convergence and analyticity of the quantity Tr ρnA =∑
i λni holds for n ≥ 1, so
that derivative with respect to n exists and is analytic for any n > 1. Therefore, one may infer
that the entanglement entropy SA = −∑λi log λi is equivalent to
SA = − limn→1
∂
∂nTr ρnA. (3.1)
so that the computation of SA now amounts to calculating Tr ρnA. For a general n ≥ 1, this will
only make our lives harder instead of easier. However, for positive integral n, the quantity Tr ρnAcan in some cases be computed by a suitable path integral. One then analytically continues the
result to a general (complex) value of n, such that the limit n→ 1+ is well-defined, from which
the entanglement entropy follows using (3.1).4
It turns out that the calculation of Tr ρnA is tantamount to computing the partition function
on an n-sheeted Riemann surface Rn, as will be discussed below based on [22, 23, 25].
3.1.1 Path integral formulation
Consider the ground state of a two-dimensional CFT, where the subset A consists of the degrees
of freedom within a single interval x ∈ [u, v] at τ = 0, with coordinates (τ, x) ∈ R2. For
simplicity, we consider a CFT that contains just one field φ(τ, x). The ground state wave
functional can then be found by path-integrating the Euclidean action S from τ = −∞ to τ = 0
[22],
Ψ(φ−(x)
)=
∫ φ(0,x)=φ−(x)
τ=−∞Dφ e−S(φ). (3.2)
4There are cases where the analytic continuation to non-integer n is ill-defined, a phenomenon known as“replica symmetry breaking” [20, 21].
3 ENTANGLEMENT ENTROPY IN CFT2 17
The total density matrix ρ is then defined as ρφ−,φ+ = Ψ (φ−(x)) Ψ† (φ+(x′)), where Ψ† can be
obtained by path-integrating from τ =∞ to τ = 0, giving
ρφ−,φ+ = Z−1
∫ τ=∞
τ=−∞Dφ e−S(φ)
∏x
δ(φ(0−, x)− φ−(x)
)∏x
δ(φ(0+, x)− φ+(x)
). (3.3)
where Z = Tr ρ is the partition function that ensures normalization.
Pictorially, there is a cut at τ = 0, coming from the fact that the fields φ−(x) and φ+(x)
are not yet identified at τ = 0. Taking the trace makes the identification, which means setting
φ−(x) = φ+(x) and integrating over these variables in the path integral (3.3), which clearly
reproduces Tr ρ = 1 due to the normalization factor Z−1. Now, to obtain the reduced density
matrix of region A, one traces over the degrees of freedom in its complement. In the path
integral, we sew together the fields at all points x /∈ A, leaving an open cut along the interval A
at τ = 0. The reduced density matrix is then given by expression (3.3), where x is now restricted
to the interval A
[ρA]φ−,φ+ = Z−1
∫ τ=∞
τ=−∞Dφ e−S(φ)
∏x∈A
δ(φ(0−, x)− φ−(x)
) ∏x∈A
δ(φ(0+, x)− φ+(x)
), (3.4)
where Z is the full partition function, so that tracing over A gives TrA ρA = 1.
Now we would like to construct ρnA, obtained by making n copies of the above
ρnA = [ρA]φ−1 ,φ+1
[ρA]φ−2 ,φ+2. . . [ρA]φ−n ,φ+
n, (3.5)
and sewing them together cyclically along the cuts for 1 ≤ j ≤ n,
φj(0+, x) = φj+1(0−, x). (3.6)
where the final identification φn(0+, x) = φ1(0−, x) comes from the trace Tr ρnA. As a result,
we obtain a structure known as an n-sheeted Riemann surface Rn, illustrated in figure4. The
quantity Tr ρnA is then obtained through the path-integral
Tr ρnA = Z−n∫
(τ,x)∈RnDφ e−S(φ) ≡ Zn
Zn, (3.7)
where Zn is the partition function on Rn.
3.1.2 Twist operators
Although it is sometimes possible to directly calculate the partition function (3.7) on Rn, such
a direct approach becomes difficult for Riemann surfaces with complicated topology [22]. We
therefore consider moving the problem to the complex plane C, where the structure of the
Riemann surface can be implemented through appropriate boundary conditions at the branch
points (u, v) (the boundary points between region A and its complement). However, there is a
subtlety involved in this procedure, as discussed in [25]. It is argued that the fields φ become
non-local on the plane, because whereas the space on which the fields live has grown by a factor
3 ENTANGLEMENT ENTROPY IN CFT2 18
Figure 4: Each sheet is associated to the two-dimensional plane and the cut arises because the fields arenot identified on each sheet separately. Instead, the sheets are sewed together cyclically, giving rise tothe n-sheeted Riemann surface shown in blue. Here n = 3 for simplicity. This figure is based on [22].
of n, the number of fields has remained the same. To remedy the problem, we instead consider
n fields φk, i.e. one on each sheet, where they are now local. The boundary conditions (3.6)
which encode the structure of the Riemann surface then read
φk(e2πi(w − u)) = φk+1(w − u), φk(e
2πi(w − v)) = φk−1(w − v), (3.8)
where w = x + iτ is the complex coordinate on the plane (the anti-holomorphic coordinate
ω = x − iτ is implicit everywhere). It is imporant to note that the coordinate w is multi-
valued on the n-sheeted Riemann surface, in the sense that each value of w actually represents
n different points, one for each sheet.
Equivalently, we can regard the twisted boundary conditions (3.8) as the insertion of two
twist operators Φ+(k)n and Φ
−(k)n at w = u and w = v, respectively, for each k-th sheet. Twist
operators can formally be defined through the path integral as [25],
〈Φ+n (u)Φ−n (v)〉C =
∫C(u,v)
Dφ exp
[−∫Cd2xL(φ)
], (3.9)
where∫C(u,v) denotes the restricted path integral with constraints (3.8). Since the complete
Riemann surface Rn consists of n sheets, we conclude that the full partition function Zn can be
written as
Zn =n∏k=1
〈Φ+(k)n (u)Φ−(k)
n (v)〉C, (3.10)
from which it follows that (3.7) can be written as
Tr ρnA =n∏k=1
〈Φ+(k)n (u)Φ−(k)
n (v)〉C. (3.11)
We conclude that the calculation of the entanglement entropy is now tantamount to the compu-
tation of a two-point function of the twist fields Φ+n and Φ−n . It should be stressed that the twist
operators are primary operators, which means that their two-point function is fully contrained
3 ENTANGLEMENT ENTROPY IN CFT2 19
by conformal symmetry to be of the form (see e.g. [26])
〈Φ+n (u, u)Φ−n (v, v)〉 ∝ 1
|u− v|2∆, (3.12)
where ∆ = h + h is the scaling dimension of the twist fields. Here we have written the anti-
holomorphic part explicitly for clarity, but we will omit it hereafter. The proportionality constant
is infinite in a continuum field theory, but may be regularized with a UV cutoff, as already
mentioned in section 2.2.1.
We conclude that in order to evaluate (3.11), we have to determine the scaling dimension ∆
of the twist fields. In order to do so, it is note (3.9) implies that a general correlation function
on the Riemann surface Rn can equivalently be written as a correlation function on the plane
C with appropriate boundary conditions:
〈O(k)(w) . . . 〉Rn =
∫C(u,v)Dφ O(w) . . . e−S(φ)∫
C(u,v)Dφ e−S(φ)=〈Φ+(k)
n (u)Φ−(k)n (v)O(w) . . . 〉C
〈Φ+n (u)Φ−n (v)〉C
, (3.13)
where O(k)(w) . . . represents any number of operators on the k-th sheet and φ is (again) short-
hand notation for n different fields.
3.1.3 Scaling dimension of twist fields
We will now determine the scaling dimension ∆ of the twist fields, by first making the mapping
from Rn to C explicit5 [22],
Rn → C : w → z =
(w − uw − v
) 1n
, (3.14)
which maps the branch points (u, v) to (0,∞). Whereas w is multi-valued on the entire Riemann
surface because it takes the same value for identical points on each sheet, the mapping (3.14)
produces a coordinate z which is single-valued on the entire plane. Now, under any conformal
mapping, the (holomorphic) stress tensor transforms accordingly as (see e.g. [26]),
T (w) =
(dz
dw
)2
T (z) +c
12z, w, (3.15)
where c is the central charge and
z, w =z′′′z′ − (3/2)(z′′)2
(z′)2(3.16)
is the Schwarzian derivative.
Now, the crucial observation is that the expectation value of the holomorphic stress tensor
vanishes on the plane due to translational and rotational symmetry, 〈T (z)〉C = 0. Therefore, we
5The power 1/n makes sure that the argument of z (i.e. the complex angle in z = |z|eiθ) has periodicity 2πinstead of 2πn.
3 ENTANGLEMENT ENTROPY IN CFT2 20
can calculate its expectation value on the Riemann surface using (3.14) and (3.15), yielding
〈T (w)〉Rn =c
12z, w =
c(1− n−2)
24
(u− v)2
(w − u)2(w − v)2. (3.17)
In the previous section, we noted that correlators on the Riemann surface may equivalently be
written as a correlation function on the plane involving the twist fields Φ±n . Therefore, the right
hand side of (3.17) is equal to the left hand side of (3.13), i.e.
〈T (w)Φ+n (u)Φ−n (v)〉C =
c(1− n−2)
24
(u− v)2
(w − u)2(w − v)2〈Φ+
n (u)Φ−n (v)〉C. (3.18)
We can evaluate the left hand side of (3.18) using the conformal Ward identity (see e.g. [26]),
〈T (w)Φ+n (u)Φ−n (v)〉 =
(h
(w − u)2+
∂uw − u
+h
(w − v)2+
∂vw − v
)〈Φ+
n (u)Φ−n (v)〉, (3.19)
where the derivatives may be calculated using (3.12), giving
〈T (w)Φ+n (u)Φ−n (v)〉C =
h(u− v)2
(w − u)2(w − v)2〈Φ+
n (u)Φ−n (v)〉C. (3.20)
By combining (3.18) and (3.20) and using h = h for the twist fields [25], we conclude that the
scaling dimension is given by
∆ =c(1− n−2)
12. (3.21)
3.2 Entanglement entropy of the ground state
Now that we know the value of ∆, it is easy to compute Tr ρnA, since this is simply given by n
powers of the two-point function 〈Φ+n (u)Φ−n (v)〉. From (3.11), (3.12) and (3.21) we obtain
Tr ρnA =
(`
µ
)− c6
(n−1/n)
, (3.22)
where ` = |u − v| is the size of interval A and µ is the UV cutoff. Plugging (3.22) into (3.1)
gives the famous result of [24] for the ground state entanglement entropy on an infinite line6
SA =c
3log
`
µ, (3.23)
where the presence of the UV cutoff µ reflects that the entanglement entropy is physically
divergent in a local quantum field theory.
3.2.1 Generalizations to finite size or finite temperature
We have learned that Tr ρnA transforms as the two-point function of primary operators Φ±n .
The upshot is that we can easily compute the two-point function of primary operators in other
6Here we neglect a non-universal constant term which does not depend on `.
3 ENTANGLEMENT ENTROPY IN CFT2 21
geometriesM, as long asM is related to the plane by a conformal mapping C→M : w → z(w),
under which the two-point function transforms as
〈Φ+n (z1)Φ−n (z2)〉 =
∣∣∣∣dw(z1)
dz
∣∣∣∣h ∣∣∣∣dw(z2)
dz
∣∣∣∣h 〈Φ+n (w1)Φ−n (w2)〉C. (3.24)
This observation allows us to quickly calculate the entanglement entropy in different geometries.
Consider a system of finite size Lcy, which is a cylinder when Euclidean time is included.
The cylinder is related to the plane via
w = e2πiz/Lcy , (3.25)
so the two-point function on the cylinder can be calculated from (3.24), giving
〈Φ+n (z1)Φ−n (z2)〉cyl =
∣∣w′(z1)w′(z2)∣∣∆ 〈Φ+
n (u)Φ−n (v)〉C
=
∣∣∣∣ L2πie2πi(z1+z2)/Lcy
∣∣∣∣∆∣∣∣∣e2πiz1/Lcy − e2πiz2/Lcy
µ
∣∣∣∣−2∆
=
(Lcy
πµsin
π`
Lcy
)−2∆
, (3.26)
where now ` = z2 − z1 is the size of the interval on the cylinder. To get the entanglement
entropy, we use the result for the scaling dimension (3.21) and subsequently take the derivative
with respect to n and set n = 1 [22], yielding
SA =c
3log
(Lcy
πµsin
π`
Lcy
). (3.27)
In the limit where the subsystem is much smaller than the total system, i.e. ` L, we recover
the result on the line (3.23).
Similarly, we can compute the entanglement entropy of a thermal state on an infinitely long
system, represented by a cylinder with compactified (Euclidean) time direction τ ∼ τ+β, where
β is the inverse temperature T−1. We map the plane to this cylinder via
w = e2πz/β. (3.28)
Observing that the current cylinder (3.28) is related to the cylinder (3.25) through Lcy → iβ,
we can substitute this in (3.27) to obtain,
SA =c
3log
(β
πµsinh
π`
β
). (3.29)
In the low temperature limit ` β, (3.29) again reduces to the entanglement entropy of the
ground state on a line (3.23), as expected. In the high temperature limit, ` β, we find
S ∼ πc3β `, which is extensive (linear in the volume `). In this limit, SA agrees with the ordinary
thermal entropy of a CFT because SA is now dominated by thermal contributions [22].
3 ENTANGLEMENT ENTROPY IN CFT2 22
3.3 Entanglement entropy for excited states
The entanglement entropy for excited states on a system of length Lcy was first calculated by
[27]. The excited state is denoted by∣∣Υ〉 and it is defined through the insertion of an operator
in the infinite past τ → −∞ acting on the vacuum7∣∣0〉,
∣∣Υ〉 = limτ→−∞
Υ(τ, x)∣∣0〉. (3.30)
The wave functional is, just like the ground state, given by a path integral (cq. (3.2)), but with
an extra factor due to the operator Υ(x,−∞)
Ψ(φ,Υ) =
∫ φ(0,x)=φ−(x)
τ=−∞Dφ e−S(φ)Υ(−∞, x). (3.31)
The associated density matrix reads
ρΥ = CZ−11
∫Dφ e−S(φ)
∏x
δ(φ(0−, x)− φ−(x)
)∏x
δ(φ(0+, x)− φ+(x)
)Υ(−∞, x)Υ†(∞, x),
(3.32)
where the fields are not yet identified at τ = 0, Υ†(∞, x) is the conjugate operator inserted in
the infinite future and Z is the partition function for the degrees of freedom φ in the absence
of Υ. To fix the normalization constant C, we take the trace and demand that Tr ρΥ = 1. The
trace produces a factor Z from φ and a two-point function due to the two inserted operators
Υ, Υ†, yielding
C =1
〈Υ(−∞, x)Υ†(∞, x)〉. (3.33)
The reduced density matrix ρΥ(A) is then obtained by tracing over the degrees of freedom in
B, giving an expression identical to (3.32), but with the cut restricted to x ∈ A.
We now again employ the replica trick. This entails making n copies of the above and sewing
them together cyclically along the cuts, yielding an n-sheeted Riemann surface Rn, which now
includes 2n operator insertions Υk and Υ†k, one for each sheet 1 ≤ k ≤ n. To calculate the
entanglement entropy, we are interested in the trace Tr ρnΥ(A). Similar to the above, we get
a contribution Zn from the field degrees of freedom, times by a 2n-point function from the
operators,
Tr ρnΥ(A) =ZnZn1〈∏nk=1 Υk(−∞, x)Υ†k(∞, x)〉Rn〈Υ1(−∞, x)Υ†1(∞, x)〉nR1
, (3.34)
where the subscripts 1 and n refer to the single- and multi-sheeted model, respectively. Observing
that the factor Zn/Zn1 is precisely Tr ρngs(A) of the ground state (see eq. (3.7)), we can define
7This is known as the state-operator correspondence. It crucially depends on the conformal invariance of thetheory, which allows us to employ radial quantization. The infinite past corresponds to the limit z, z → 0, wherez = exp (2π(τ + ix)/Lcy). Since this point is the origin of the z-plane, we can associate a local operator to it. Foran ordinary QFT (one without conformal symmetry), we cannot define a local operator in the infinite past sincewe cannot regard it as a single point in space.
3 ENTANGLEMENT ENTROPY IN CFT2 23
the ratio between the excited state and the ground state by
FΥ =Tr ρnΥ(A)
Tr ρngs(A)=〈∏nk=1 Υk(−∞, x)Υ†k(∞, x)〉Rn〈Υ1(−∞, x)Υ†1(∞, x)〉nR1
. (3.35)
The entanglement entropy may then be calculated as follows. First, note that the Renyi entropy
is given by S(n)A = 1
1−n log Tr ρnA, from which we can define the quantity
FΥ = exp[(1− n)
(S
exc,(n)A − Sgs,(n)
A
)]. (3.36)
Taking the derivative with respect to n and analytically continuing to n → 1 then gives the
entanglement entropy
SexcA = Sgs
A −∂FΥ
∂n
∣∣∣∣n=1
. (3.37)
3.3.1 2n-point function
Our aim is now to calculate the 2n-point function in (3.35). As remarked in the previous
section, the correlation functions are difficult to calculate on Rn. The solution again lies in
shifting the system to the complex plane. The complicated topology of the Riemann surface is
then encoded in the transformation factor of the correlation functions. Let the entanglement
interval be A = [u, v] and introduce complex coordinates w = x + iτ . Instead of immediately
mapping to the single complex plane, we first map Rn to n copies of the plane via w → ζ, with
ζ =
(eiπ(w−u)/Lcy − e−iπ(w−u)/Lcy
eiπ(w−v)/Lcy − e−iπ(w−v)/Lcy
), (3.38)
The branch points w = (u, v) get mapped to ζ = (0,−∞), whereas the points in the infinite past
w = w−∞ and future w = w+∞, corresponding to the coordinates of Υ and Υ†, respectively, get
mapped to
w±∞ → ζ±∞ = e±iπ∆φ, (3.39)
as can easily be verified using (3.38). Here ∆φ = |v − u|/Lcy is angular size of the interval.
Next, we map these n copies of the plane to a single plane C via ζ → z = ζ1/n, i.e.
z =
(eiπ(w−u)/Lcy − e−iπ(w−u)/Lcy
eiπ(w−v)/Lcy − e−iπ(w−v)/Lcy
)1/n
. (3.40)
Similar to the discussion of the previous section, the complicated topology of the Riemann
surface now gets encoded by imposing twisted boundary conditions on the fields. On the plane,
these conditions may be written as Υk(z) = Υk+1(e2πik/nz), for 1 ≤ k ≤ n. It follows that the
two operator insertion points ζ±∞ get mapped to the points
z−k,n = exp
(iπ
n(∆φ+ 2k)
), z+
k,n = exp
(iπ
n(−∆φ+ 2k)
), (3.41)
3 ENTANGLEMENT ENTROPY IN CFT2 24
where z−k,n (z+k,n) are the points coming from the infinite past (future).
Under the mapping (3.40), the fields Υ(w) scale as
Υ(w) =
(dz
dw
)hΥ(z), (3.42)
where the derivative is given by
dz
dw=z
n
4π
Lcysin(π∆φ)
(eiπLcy
(w−u) − e−iπLcy
(w−u))−1(
eiπLcy
(w−v) − e−iπLcy
(w−v))−1
, (3.43)
as can be verified either by hand or using mathematica. Evaluated at the coordinates (3.41),
this corresponds to(dz
dw
)z=z−k,n
=z−k,nn
Λ and
(dz′
dw
)z=z+
k,n
=z+k,n
nΛ, (3.44)
where
Λ =4π
Lcysin(π∆φ)e−2π|w|/Lcyeiπ(u+v)/Lcy . (3.45)
For the quantity FΥ, the Λ’s cancel in the ratio (3.35), giving
FΥ = n−2n(h+h)
∏nk=1(z−k,n)h(z+
k,n)h((z−1,1)h(z+
1,1)h)n 〈∏n
k=1 Υk(z−k,n)Υ†k(z
+k,n)〉C
〈Υ1(z−1,1)Υ†1(z+1,1)〉nC
, (3.46)
which is difficult to evaluate due to the explicit dependence on the coordinates z±k,n. We can
get rid of the powers of zk,n by transforming back to the cylinder, which is realized by z = eit.
The coordinates of the operators are then given by t±k,n = πn(∓∆φ+ 2k), as can be read off from
(3.41). Importantly, the operators transform accodingly as
Υ(t) = (iz)hΥ(z). (3.47)
Inserting (3.47) transformation in (3.46), the powers of zk,n cancel as promised, yielding
F(n)Υ = n−2n(h+h)
〈∏nk=1 Υk(t
−k,n)Υ†k(t
+k,n)〉cy
〈Υ1(t−1,1)Υ†(t+1,1)〉ncy
. (3.48)
which is a 2n-point function on a cylinder of circumference Lcy.
3.3.2 Small interval limit
While it is in principle possible to evaluate (3.48) exactly for certain models [27, 28, 29, 30],
such an endeavor lies outside the scope of this thesis. We are interested in its universal results
by considering the limit l L, or equivalently ∆φ 1, so that we can approximate the terms
3 ENTANGLEMENT ENTROPY IN CFT2 25
ΥΥ† appearing in (3.48) by an operator product expansion (OPE). The result is cited from [27],
F(n)Υ ' 1 +
h+ h
3
(1
n− n
)(π∆φ)2 +O
(∆φ3
), (3.49)
where we have included the anti-holomorphic weight h for completeness. Using equation (3.37),
we obtain the following universal result for the small interval limit of the entanglement entropy
of an excited state
∆SA '2π2
3(h+ h)
(`
Lcy
)2
. (3.50)
In the next chapter, we will see that when the excited state is a twist field, expression (3.50)
may be matched to the holographic entanglement entropy in a conical defect spacetime.
4 HOLOGRAPHIC ENTANGLEMENT ENTROPY 26
4 Holographic Entanglement Entropy
When we divide the total space of a field theory into two parts A and B, the entanglement
entropy SA measures how much information about the entanglement is hidden in B from an
observer in A. Given the AdS/CFT conjecture, we would like to know what the corresponding
division is in the dual bulk theory. In particular, one might ask whether it is possible to pinpoint
a well-defined division surface γA separating two bulk regions A′ and B′. Rather remarkably, it
turns out that such a surface exists, which asymptotically coincides with the surface that divides
the field theory regions A and B, i.e. ∂γA = ∂A, because the field theory lives on the boundary
of AdS. This, however, still gives infinitely many choices for the surface. The proposal of Ryu
and Takayanagi is that we should choose the minimal surface area. The entanglement entropy
SA is then holographically computed by the Ryu-Takayanagi (RT) formula [7, 31, 32]
SA = min∂γ=∂A
[Area(γA)
4G
], (4.1)
where G is Newton’s constant. Formula (4.1) was historically motivated by the analogy with
black holes, whose horizons can also be thought of as a surface beyond which information is
hidden.
4.1 Heuristic derivation of RT formula
The main idea behind the holographic computation of entanglement entropy is the holographic
analog of the replica trick outlined in the previous chapter. That is to say, we make n copies of
the gravitational spacetime and sew them together cyclically, defining an n-sheeted geometry Sn.
We then evaluate the gravitational partition function on this space and apply formula (3.1). The
reason we expect this procedure to yield the entanglement entropy is due to the fundamental
principle of AdS/CFT, namely the bulk to boundary relation, which entails an equivalence of
the partition functions,
ZCFT = ZAdS. (4.2)
The strong coupling regime of the CFTd is dual to classical GR plus matter terms8 on AdSd+1.
Hence, the right hand side of (4.2) is just the classical partition function of the Einstein-Hilbert
action
ZAdS = e−SEH , (4.3)
where
SEH = − 1
16πG
∫Mdd+1x
√g(R+ 2Λ), (4.4)
where we omitted the matter terms in the action.
The boundary Riemann surface Rn is characterized by the presence of a deficit angle 9
8Here we focus on the case without supersymmetry; with supersymmetry the CFT is dual to SUGRA plusmatter terms on AdSd+1.
9To see this, note that one would have to go around a branch point n times to return to the first sheet. Thismeans that the period of the angular coordinate on the complex plane (i.e. θ in the parametrization x = ρ cos θand τ = r sin θ) is 2πn. Normally, the periodicity is 2π, so the deficit angle is 2π − 2πn = 2π(1− n).
4 HOLOGRAPHIC ENTANGLEMENT ENTROPY 27
Figure 5: A timeslice of the near-boundary region of AdSd+1. When the field theory is bipartitionedinto region A and its complement B, the entanglement entropy is proportional by the area of the minimalsurface A that coincides with ∂A on the boundary. This figure is based on [31].
δ = 2π(1− n) on the surface ∂A. If we were sufficiently dilligent, we could find a back-reacted
(d + 1)-dimensional geometry Sn whose metric that approaches that of Rn at the boundary
r → ∞. In three dimensions, we can safely assume that the back-reacted Sn is given by an
n-sheeted AdS3 geometry, because locally every solution of the Einstein equations is still AdS3.
Similar to the n-sheeted Riemann surface on the boundary, the n-sheeted AdS3 geometry is
characterized by a deficit angle δ = 2π(1 − n) along some one-dimensional surface γA. It
should be noted that the conical deficit metric does not satisfy the Einstein equations, but
approximately does so in the limit where n → 1 [33]. Inspired by the AdS3 result, [7] makes
the natural assumption that this result generalizes to general d, where Sn is now an n-sheeted
AdSd+1 geometry and γA is a codimension two surface.
As reviewed in appendix A, the Ricci scalar for conical spaces behaves like a delta function
R = 4π(1− n)δ(γA) +R(0), (4.5)
where the delta function is localized on the entire codimension two surface, i.e. δ(γA) = ∞for x ∈ γA and δ(γA) = 0 otherwise, and R(0) is the Ricci scalar of pure AdSd+1. Plugging
(4.5) in expression (4.4) for the gravitational action and evaluating the integral yields a term
proportional to the area of γA,
SEH = −(1− n)Area(γA)
4G+ . . . , (4.6)
where the dots refer to terms linear in n coming from the part of AdS away from the deficit
angle. By virtue of the equivalence of the partition functions, we can compute the entanglement
4 HOLOGRAPHIC ENTANGLEMENT ENTROPY 28
entropy via10
SA = − ∂
∂n[logZn − n logZ1]
∣∣n=1
, (4.7)
where Z is the gravitational partition function defined in (4.3). Plugging (4.6) into (4.7) and
observing that the contributions from pure AdS cancel because these are linear in n, we find
SA = − ∂
∂n
[(1− n)Area(γA)
4G
]∣∣∣∣n=1
=Area(γA)
4G. (4.8)
That γA is the minimal surface can heuristically be understood from expression (4.6). By
invoking the action principle in the gravity theory, expression (4.6) tells us that γA should be
the minimal area surface.
4.2 Some properties of holographic entanglement entropy
Here we review some properties of the holographic entanglement entropy. In particular, we show
that strong subaddivity is satisfied and that the CFT area law is reproduced, both of which were
already discussed in the previous chapter. Notice that these are not the only properties of the
entanglement entropy, but merely the ones that will be useful in the remainder of the thesis.
For a full account of its properties, see [31, 32].
4.2.1 Strong subadditivity
As discussed in section 2.2.1, the entanglement entropy satisfies an inequality known as strong
subadditivity. It is given by
S(I1 ∪ I2) + S(I1 ∩ I2) ≤ S(I1) + S(I2), (4.9)
where I1 and I2 are two overlapping regions in a QFT. We will now show that the holographic
entanglement entropy given by (4.1) satisfies this inequality in AdS3. The arguments straight-
forwardly apply to higher dimensions, but AdS3 is particularly suitable because its minimal
surfaces are geodesics.
Proof: Consider for simplicity a timeslice of AdS3 in Poincare coordinates. Figure 6a shows
the boundary intervals I1 and I2, as well as the corresponding blue bulk geodesics used to
calculate the entanglement entropies, which intersect in the bulk at point p. Also shown are the
two green geodesics, associated to the entanglement entropies S(I1 ∪ I2) (big) and S(I1 ∩ I2)
(small). In the current state of affairs, it is not obvious that the inequality (4.9) is satisfied.
To proceed, we re-arrange the original blue geodesics at the intersection point p to form two
different curves, shown in red and yellow in figure 6b. This re-arrangement leaves the sum of
the areas unchanged. Thus, we can express the right hand side of (4.9) in terms of the areas
of these new curves. The crucial observation is that the red (yellow) curve is homologous to
10In the previous chapter, we expressed this equivalently as
SA = − ∂
∂nTr ρnA
∣∣n=1
, Tr ρnA ≡ZnZn1
.
Using Tr ρA = 1, we find ∂n log Tr ρnA|n=1 = (∂n Tr ρnA)/(Tr ρnA)|n=1 = ∂n Tr ρnA|n=1, showing the equivalency.
4 HOLOGRAPHIC ENTANGLEMENT ENTROPY 29
Figure 6: (a) The geodesics associated to the boundary intervals I1 and I2 are shown in blue, whereasthe geodesics associated to I1 ∪ I2 and I1 ∩ I2 are shown in green. In figure (b), the blue geodesics arerecombined at the intersection point p into the red and yellow geodesic, which in general are not minimalsurfaces. Since the big (small) green arc is a geodesic, its length is smaller than that of the red (yellow)curve, which proves the inequality (4.9).
(correspond to the same boundary interval as) the big (small) green geodesic. However, the
latter are geodesics due to the minimality condition in the RT prescription (4.1), which means
have the smallest area within their respective homology classes. Thus, the sum of the two green
surface areas is smaller than the combined area of the red and yellow curve. Going back to
the original configuration with blue geodesics, it follows that the inequality (4.9) is satisfied, as
desired.
4.2.2 Relation to CFT area law
It is not hard to convince oneself that the RT formula gives rise to the area law (2.22) in
the boundary field theory. First, note that since the AdS metric is divergent at z = 0, we
have to introduce a cutoff z = µ. The near-boundary region z = µ then gives the dominant
divergent contributions to the area of the minimal surface, so from the RT formula (4.1) we find
S ∝ Area(∂A)/µ, in accordance with the area law.
The correspondence between the RT formula (4.1) and the area law (2.22) is related to
the observation that areas and volumes are proportional on large enough scales, which is the
fundamental property of anti-de Sitter space which makes holography possible. In order to see
that area ∝ volume on large scales, consider a very large sphere (i.e. with radius R → ∞) in
AdSd+1 and compute its area and volume:
Area ∼ Rd−1
Volume ∼∫ L R′d−1dR′√
1 +R′2/L2
R→∞−−−−→ LRd−1
d− 1,
from which we conclude that indeed
Area(γA) ∝ Area(∂A)/µ. (4.10)
4 HOLOGRAPHIC ENTANGLEMENT ENTROPY 30
Figure 7: The figures shown in (a), (b) and (c) represent a timeslice of Poincare, global and BTZcoordinates, respectively. The boundary intervals are shown in green and the associated geodesic isshown in orange. The cutoff used to regulate the results is represented by the small gap between thegeodesic and the boundary.
4.3 Examples of HEE in AdS3
We will now explicitly consider some examples of the RT prescription (4.1) in three-dimensional
gravity theories, namely in Poincare, global and BTZ coordinates. Here minimal surfaces are
spatial geodesics, whose lengths are relatively easy to compute in an AdS3 metric. These ex-
amples are also instructive because the results can be compared to the CFT2 expressions from
the previous chapter. The agreement is perfect, which historically provided one of the first
compelling evidences for the validity of (4.1).
4.3.1 Poincare coordinates
We shall start with the calculation in Poincare coordinates (2.13), whose metric is reproduced
here for convenience:
ds2 =L2
z2
(−dt2 + dz2 + dx2
), (4.11)
where L is the radius of AdS3. The boundary is at z = 0, where the metric diverges. Hence,
the length of a spatial geodesic ending on the boundary is actually infinite, which is consistent
with the field theory result. To get a finite answer, we place the boundary at a finite distance
z = µ. This setup is represented in figure 7a.
We will now show that spatial geodesics with both endpoints on the boundary at z = µ are
semi-circles. To construct the geodesic, we minimize the distance functional
Length(γA) =
∫ds√
det(gabxaxb) = L
∫dz
z
√1 +
(dx
dz
)2
, (4.12)
leading to the Euler-Lagrange equation
L
z
x′√1 + (x′)2
= constant, (4.13)
where the prime denotes differentiation with respect to z. To fix the constant, note that the
4 HOLOGRAPHIC ENTANGLEMENT ENTROPY 31
geodesic has a turning point at some maximal distance z = z∗, where the derivative diverges, i.e.
x′(z∗)→∞. Plugging this into (4.13) informs us that the constant is equal to L/z∗. Therefore,
the Euler-Lagrange equation may be written as the following differential equation
dx
dz=
z√z2∗ − z2
. (4.14)
Integrating this expression, we find that geodesics must satisfy x2 +z2 = z2∗ , justifying the claim
about semi-circles.
For a boundary interval of size `, with coordinates x ∈ [−`/2, `/2], we can parametrize the
semi-circles as
(x, z) =`
2(cos s, sin s), (ε ≤ s ≤ π − ε) (4.15)
where ε = 2µ/l is the parametrized UV cutoff, which consistently reproduces lims→0 z = µ.
Using the parametrization (4.15), the length of this geodesic can obtained as
Length(γA) = 2L
∫ π/2
ε
ds
sin s= 2L log
`
µ, (4.16)
from which the entanglement entropy can be found by dividing by 4G. We wish to relate (4.16)
to the CFT result (3.23). Note that the cutoffs can simply be identified, since the AdS/CFT
correspondence says that the UV of the CFT is dual to the IR of the gravitational theory. Using
the standard AdS/CFT dictionary c = 3L/2G [31], we finally obtain
Svac =c
3log
`
µ. (4.17)
4.3.2 Global coordinates
We will now calculate the length of a spatial geodesic ending on the boundary of global AdS3,
as in figure 7b, with metric
ds2 = L2(− cosh2 ρ dτ2 + dρ2 + sinh2 ρ dφ2
). (4.18)
In order to evaluate the entanglement entropy, we use the embedding construction of AdS3
(following [34]), which we reviewed in section 2.1. That is, we embed the hyperboloid in the
embedding space R2,2, with metric gαβ = diag(−1,−1,+1,+1) and embedding coordinates
T1 = L cosh ρ sin τ, X1 = L sinh ρ cosφ
T2 = L cosh ρ cos τ, X2 = L sinh ρ sinφ, (4.19)
For convenience, we combine these coordinates into a single vector Xα = (T1, T2, X1, X2), whose
inner product is defined with respect to the embedding metric. Hence, the hyperboloid is given
by X2 = −T 21 − T 2
2 +X21 +X2
2 = −L2.
4 HOLOGRAPHIC ENTANGLEMENT ENTROPY 32
To find a spacelike geodesic, one looks for solutions which minimize the following functional
δI =1
2
∫ds gαβ
dXα
ds
dXβ
ds=
∫ds L(X, X), (4.20)
where gαβ is the embedding metric. To ensure we stay on the hyperboloid, it is helpful to
introduce a Lagrange multiplier, so that the Lagrangian reads
L =1
2X2 + λ(X2 + L2). (4.21)
The Euler-Lagrange equations for this Lagrangian are Xα = 2λXα. To solve for λ, we make use
of the fact that X2 = −L2, so that
d2X2
ds2= X ·X + X · X = 0. (4.22)
Combining (4.22) with the equations of motion, it follows that λ = X2/2L. Therefore, the
equations of motion take the rather simple form
L2Xα = X2Xα. (4.23)
For spacelike geodesics, we have X2 = 1, so the solution to (4.23) is
Xα(s) = mαes/L + nαe−s/L, m2 = n2 = 0, 2m · n = −L2 (4.24)
where mα and nα are constant vectors chosen which enure X2 = −L2. We can then compute
the proper distance between two geodesic points Xα(s1) and Xα(s2) via
X(s1) ·X(s2) = m · n(e(s2−s1)/L + e−(s2−s1)/L
)= −L2 cosh(∆s/L), (4.25)
where ∆s is the proper distance between the two points on the geodesic. In particular, if these
are its endpoints, the proper distance equals the geodesic length, i.e. ∆s = Length(γA).
We now have all the means to evaluate the length of a spatial geodesics whose endpoints lie on
the boundary of global AdS3. These points are given by (τ, ρ, φ) = (0, ρ0,−`/2L) and (τ, ρ, φ) =
(0, ρ0, `/2L), where ` is the size of the boundary interval and 2πL = Lcy is the circumference11
of the boundary cylinder and the radial coordinate ρ0 is a large distance cutoff. In the embed-
ding coordinates, we denote these points by Xα(s1) and Xα(s2), respectively. Inserting these
coordinates in the left hand side of (4.25), we obtain the following expression for the geodesic
distance,
cosh(∆s/L) = 1 + 2 sinh2 ρ0 sin2 `
2L. (4.26)
Assuming that the IR cutoff ρ0 is large, eρ0 1, we can then approximate the geodesic length
11This means that the boundary field theory lives on a circle of circumference Lcy = 2πL. It may seem oddthat the AdS radius L is directly related to the circumference of the boundary. However, taking a differentcompactification amounts to a simple rescaling of L, which is irrelevant.
4 HOLOGRAPHIC ENTANGLEMENT ENTROPY 33
as
Length(γA) = ∆s ≈ L arccosh
(1
2e2ρ0 sin2 `
2L
)≈ 2L log
(eρ0 sin
`
2L
), (4.27)
which when divided by 4G gives the entanglement entropy. In order to match the CFT result
(3.27), we relate the cutoffs via eρ0 = 2L/µ and use the standard AdS/CFT dictionary c =
3L/2G, giving
Svac =c
3log
(2L
µsin
`
2L
), (4.28)
which agrees with (3.27) (where Lcy = 2πL).
Alternatively, (4.28) can be obtained from the map between global and Poincare coordinates
given in [35],
1
z= cosh ρ cos τ + sinh ρ cosφ
t = z cosh ρ sin τ (4.29)
x = z sinh ρ sinφ
which in particular allows us to map the boundary points from Poincare to global coordinates.
To wit, we map (t, z, x) = (0, µ, `/2) to (τ, ρ, φ) = (0, ρ0, `/2L), yielding the following relation
`
2= µ sinh ρ0 sin(`/2L) = L sin(`/2L), (4.30)
where we again used eρ0 = 2L/µ for the cutoff. Substituting relation (4.30) into the Poincare
result (4.17) reproduces the global result (4.28).
4.3.3 BTZ coordinates
We now consider the BTZ black hole, whose metric is written as
ds2 = −(r2 − r2
+
L2
)dt2 +
(r2 − r2
+
L2
)−1
dr2 + r2dφ2. (4.31)
Changing coordinates to r = r+ cosh ρ, t = L2
r+τ, φ = L
r+φ, we may write its metric equivalently
as
ds2 = L2(− sinh ρ2dτ2 + dρ2 + cosh2 ρdφ2). (4.32)
The strategy which will give us the entanglement entropy is to exploit the high degree of sim-
ilarity with the global metric (4.18). First, move to Euclidean signature via τ = iτE . Now, in
order to obtain a smooth geometry, the Euclidean time is compactified as τE ∼ τE + β, where
β = 2πL2/r+ (4.33)
4 HOLOGRAPHIC ENTANGLEMENT ENTROPY 34
is the inverse temperature of the black hole. To find the geodesic, it is useful to construct
the embedding coordinates explicitly. This can be done by interchanging the time and angular
coordinate, φ = τ ′E and τE = φ′ in the global embedding coordiates (4.19). The computation of
the geodesic line is then similar to what we did for global AdS in section 4.3.2, although there
are some subtleties involved because we have swapped the temporal and spatial coordinate.
Performing the computation results in an expression similar to (4.26)
cosh(∆s/L) = 1 + 2 cosh2 ρ0 sin2 π`
β. (4.34)
Assuming again that the cutoff eρ0 = β/πµ 1 is large, it is straightforward to arrive at the
following result for the entanglement entropy [31],
SBTZ =c
3log
(β
πµsinh
π`
β
), (4.35)
which agrees perfectly with the CFT result (3.29).
4.4 Holographic entanglement entropy in a conical metric
Consider a point particle at rest at the origin of AdS3. Since there are no local gravitational
degrees of freedom in three-dimensional gravity, all curvature will be concentrated at the point
r = 0. The metric therefore becomes conical, described by the metric [43]
ds2 = −(γ2 +
r2
L2
)dt2 +
(γ2 +
r2
L2
)−1
dr2 + r2dφ2, (4.36)
where L is the AdS radius. The parameter γ exists in the range 1 ≥ γ ≥ 0, corresponding to
empty AdS and a massless BTZ black hole, respectively. It is convenient to introduce rescaled
coordinates
t = tγ, r = r/γ, φ = φγ, (4.37)
so that (4.36) now reads
ds2 = −(
1 +r2
L2
)dt2 +
(1 +
r2
L2
)−1
dr2 + r2dφ2, (0 ≤ φ ≤ 2γπ), (4.38)
which looks exactly like the global metric of vacuum AdS (2.11), but has a different periodicity
for the angular coordinate, as is depicted in figure 8. Since the identification occurs for angles
smaller than 2π, we say that this metric has an angular deficit of δ = (1−γ)2π, known simply as
the deficit angle. Note that metric (4.38) is locally equivalent to empty AdS3, but with different
global identifications. By this virtue, we may simply take the result for the entanglement entropy
in global vacuum AdS (4.28), and rescale the relevant quantities according to (4.37). The ratio
`/L is the angular size of the boundary interval and thus rescales as an angular parameter.
A non-trivial rescaling of the regulator µ comes from rescaling the radial coordinate r. In
particular, the gravitational cutoff is related to the UV cutoff through r0 ∼ 1/µ. We fix the
4 HOLOGRAPHIC ENTANGLEMENT ENTROPY 35
Figure 8: Figure (a) shows a point particle at rest in (a timeslice of) global AdS represented by metric(4.36), where the angular coordinate has normal periodicity, φ ∼ φ + 2pi. Figure (b) refers to the samephysical situation, but here the presence of the point particle is represented as global AdS with differentperiodicity, φ ∼ φ+ 2πγ (metric (4.38)).
cutoff r0 in the metric with proper periodicity, i.e. (4.36), which is the same r0 as in vacuum
AdS, and then rescale it accordingly. Rescaling the two mentioned quantities according to (4.37)
then yields the following expression for the entanglement entropy
Scon =c
3log
2L
γµsin
γ`
2L. (4.39)
An alternative derivation of this result comes from the analytic continuation of the BTZ ge-
ometry. A quick glance reveals that the BTZ metric (4.31) is related to the conical metric (4.36)
via the identification γ2 = −r2+/L
2. This identification bascially allows us to use the results
obtained from calculations in the BTZ geometry. In particular, the holographic entanglement
entropy in this geometry is given by (4.35), which under the analytic continuation r+/L = iγ
and using (4.33) reproduces the result (4.39).
Comparison with field theory calculations
In section 3.3, we reviewed the derivation of the entanglement entropy of an excited state∣∣Υ〉
on a circle with length Lcy = 2πL. In the limit ` L, it was found (cf. eq. (3.50)) that the
entanglement entropy increases relative to that of the ground state by an amount
∆SA = Sexc − Sgs '1
6(h+ h)
(`
L
)2
, (4.40)
where ∆ ≡ h+ h is the scaling dimension of the primary operator [27].
In the case where the excited state is dual to a conical geometry, we expect to be able to
reproduce this result holographically. From (4.28) and (4.39), we obtain the (finite) expression
∆SgravA = Scon − Svac =c
3log
sin(γ`/2L)
γ sin(`/2L), (4.41)
4 HOLOGRAPHIC ENTANGLEMENT ENTROPY 36
which has the following ` L expansion
∆SgravA ' c
72(1− γ2)
(`
L
)2
+O(`/L)4. (4.42)
Comparing equations (4.40) and (4.42) gives a relation between the scaling dimension of the
CFT operator and the deficit angle in the dual spacetime, i.e.
∆ =c
12(1− γ2), (4.43)
which can be recognized as the scaling dimension of a twist operator, cf. expression (3.21), with
n = γ−1.
The duality between the conical metric and the field theory with twist operators (known as
an orbifold CFT) was already noted in e.g. [36]. The fact that we reproduced (4.43) purely
from the entanglement entropy may be interpreted as an independent check of the validity of
this duality.
5 HOLE-OGRAPHY 37
5 Hole-ography
The Ryu-Takayanagi formula (4.1) provides elegant support for the conjecture that smooth
classical geometry emerges from the entanglement between nature’s fundamental degrees of
freedom. One important caveat, however, is that the RT prescription picks out a very special
class of surfaces, namely minimal surfaces which are anchored at the boundary and dip into the
bulk geometry. In contrast, it is generally believed that the Bekenstein-Hawking entropy applies
to any surface, − i.e. non-minimal surfaces, and/or closed surfaces that do not extend to the
asymptotic boundary. It would therefore be interesting to study the reconstruction of arbitrary
bulk surfaces in a holographic setup and, in particular, whether the finite entropy SBH = A/4Gassociated to a spherical bulk surface corresponds to any field theoretical quantity. It was argued
by [15] that the relevant quantity is the “differential entropy”.
In this chapter, we will review the original derivation of [15] within the context of AdS3/CFT2.
The authors refer to the spherical bulk surface as a “hole” centered inside AdS, based on a con-
struction involving a spherical family of Rindler observers, which we will review in section 5.1.
It is argued that the Rindler observers collectively connect to a domain on the boundary field
theory that covers all of space but not all of time, giving rise to a quantity called “residual
entropy”. We will explore the concept of residual entropy in section 5.2. In section 5.3, we
will show that the residual entropy is bounded by the differential entropy and that the latter
reproduces the hole’s area.
5.1 Spherical Rindler-AdS space
In order to understand the holographic derivation of [15], it is useful to know how to compute
the (non-holographic) gravitational entropy of a spherical hole with radial size R0 in Minkowski
space, which was done by the same authors in earlier work [37]. First, recall that a non-zero
gravitational entropy can typically be assigned to a Rindler observer, i.e. an accelerating observer
who is out of causal contact with a part of spacetime. Now, a spherical hole is the bifurcation
surface that seperates a region of spacetime which is out of causal contact from a spherically
symmetric family of Rindler observers.
We can lift the spherical Rindler construction to a holographic setting by introducing a
small negative cosmological constant. The Rindler observers are then causally connected from
a spherical hole inside AdS, whose radius is denoted by R0. Within AdS/CFT, it is conjectured
that the radial coordinate of AdS should be associated to the energy scale in the dual field theory.
Hence, the exterior (interior) of the hole corresponds to the UV (IR) of the dual field theory.
Given the relation between geometry and entanglement as discovered in [7], it was proposed
that areas of a closed bulk surfaces should therefore be associated to a UV/IR entanglement
in the dual field theory [39]. The achievement of [15] is that they make this conjecture precise
within AdS3/CFT2.12 It is argued that the relevant field theoretical division between UV and
IR observables is in terms of the timescales over which local observers can make measurement.
The crucial observation that validates the above statement is that Rindler trajectories reach
12Strictly speaking, the concept differs a bit different from entanglement entropy, because the Hilbert space ofquantum gravity does not factorize in the inside and outside of the hole [15].
5 HOLE-OGRAPHY 38
Figure 9: A spherically symmetric union of Rindler observers gives rise to a causally disconnected holeinside AdS. The physics of each invidual accelerating observer is holographically encoded in a single causaldiamond on the boundary field theory. The union of the diamonds comprises a field theory domain thatcovers all of space but not all of time. Figure taken from [15]
the asymptotic boundary of AdS in finite global time.13 To see this, recall that a Rindler trajec-
tory asymptotes to future- and past-directed light rays, which in this case are those propagating
outwards from the edge of the hole. From the global AdS3 metric (2.11), it is evident that a
light ray (for which ds2 = 0) propagating from the hole’s edge to the asymptotic boundary, we
have
T0 =
∫ ∞R0
dR
1 + r2
L2
= L cot−1 R0
L. (5.1)
Causality alone then suggests that observations carried out by this Rindler observer should be
holographically encoded within a time interval T ∈ [−T0, T0] in the boundary field theory. In
two dimensions, the field theory region dual to a single Rindler observer is a diamond. The
holographic equivalent of spherical Rindler space is therefore a spherically symmetric union of
overlapping causal diamonds, which collectively cover the entire spatial dimension of the field
theory, which is illustrated in figure 9. It should be stressed that the finite time interval implies
that the spatial interval which a boundary observer has access to is limited to a length 2T0 due
to causality.
We thus have a situation where the local boundary observers collectively cover every inch
of the system, but invidually see just a fraction of it. Suppose that each observer can see the
reduced density matrix on its interval. Can the observers then collectively reconstruct the global
state of the system? The key insight of [15] is they generally cannot determine the precise state
with 100% certainty. This is because the observers are oblivious to non-local correlations, i.e.
entanglement, over a distance larger than the size of their interval. Part of the success of [15] is
that they prosose a formula for this collective uncertainty, denoted by the residual entropy. We
shall elaborate on this notion in section 5.2.
5.2 Residual entropy
In order to give a precise definition to the residual entropy, we consider a simple quantum
mechanical spin chain consisting of N sites. Suppose we know the reduced density matrices for
13This statement is only valid if the acceleration exceeds a certain value, which we will assume to be the case.
5 HOLE-OGRAPHY 39
all N sequences of L consecutive sites. Now, if these are consistent with more than one global
density matrix, then residual entropy is nonzero. A quantitative defintion can then be given as
follows. Having found the set of all density matrices which have common reduced states ρi,one can calculate the von Neumann entropy S(ρ) for each such density matrix. The residual
entropy is then defined as the von Neumann entropy of the state with maximal von Neumann
entropy [15], i.e.
Sres ≡ maxS(ρ) : ∀i(Tri ρ = ρi)
. (5.2)
Let us make the above more precise. Considering a periodic spin chain with N spins, we
denote the subset of L consecutive spins i, ..., i+L−1 by Ai, whereas its complement is denoted
by Aci . To find the density matrix for the full system with maximal entropy, we wish to extremize
the entropy functional
F = −Tr (ρ log ρ) , (5.3)
subject to the constraint that ρ projects onto the a priori given reduced density matrices ρi,i.e.
TrHAci
(ρ) = ρi, (5.4)
which can be incorporated into the extremization of (5.3) using appropriate Lagrange multipliers
Λi,
F = −Tr (ρ log ρ) +
N∑i=1
TrHAi
(Λi(TrHAc
i(ρ)− ρi)
). (5.5)
Varying this functional with respect to δρ yields
δF = −Tr (δρ log ρ) +N∑i=1
TrHAi
(Λi(TrHAc
i(δρ))
). (5.6)
If this is to vanish for all variations δρ, it must obey 14
ρ ≡ e−Heff = exp
(N∑i=1
Λi ⊗ IAci
). (5.7)
Let us interpret expression (5.7). The Λi’s are matrices of the same size as the a priori
given reduced density matrices ρi. The matrices Λi can be thought of as describing correlations
between the spin sites within region Ai, whose size is given by the range L. For example, if L = 2,
the matrices Λi contains interactions between nearest neighbours only. Hence, the exponential
in expression (5.7) has the structure of an effective Hamiltonian Heff. It should be stressed that
14This can be verified by plugging this relation back into (5.6) and using the fact that tracing over the fullHilbert space is the same as tracing first over a subspace and successively over its complement:
Tr
(∑i
Λi ⊗ IAciδρ
)= TrHAi
(TrHAc
i
(N∑i=1
Λi ⊗ IAciδρ
))
=
N∑i=1
TrHAi
(Λi TrHAc
i(δρ)
),
where we’ve used the fact that Tr is a linear operator to take the sum up front.
5 HOLE-OGRAPHY 40
this Hamiltonian could be a complicated non-local operator.15
The general procedure for computing residual entropy is then as follows. First, we write down
the most general Hamiltonian Heff with interactions over the same distance as those accessible
to individual observers. Next, we demand that the reduced density matrices obtained from
ρ = e−Heff are precisely the a priori given matrices ρi. This uniquely fixes ρ. Finally, the von
Neumann entropy corresponding to ρ gives the residual entropy, summarized by the formula
Sres ≡ −Tr ρ log ρ, ρ = e−Heff = exp
(N∑i=1
Λi ⊗ IAci
). (5.8)
5.2.1 Example: qubits
We have reviewed the definition of the residual entropy for generic quantum mechanical systems.
However, to actually compute the residual entropy is extremely difficult in most cases, because
a generic density matrix is typically extremely large. Let us therefore consider the easiest ex-
ample, namely uninteracting qubits.
Two qubits, single-particle reduced states (N = 2, L = 1)
Consider two a priori given reduced states ρ1 and ρ2. From equation (5.7) we know that the den-
sity matrix which maximizes the entropy functional is given in terms of matrix valued Lagrange
multipliers Λi:
ρ = exp (Λ1 ⊗ I2 + I1 ⊗ Λ2) , (5.9)
where the labels on the identity matrices denote the Hilbert space, not the size. The Lagrange
multipliers Λ1 and Λ2 are determined by the condition that the reduced states of ρ are the same
as the a priori given ρ1 and ρ2. Solving for the Lagrange multipliers we obtain [40]
ρ = ρ1 ⊗ ρ2. (5.10)
In other words, among all states that have the same reduced states ρ1 and ρ2, the one with
maximal entropy is the product state (5.10).16 Since the residual entropy is defined as the von
Neumann entropy of the state with largest entropy via (5.8), we conclude that the residual
entropy (5.8) is the von Neumann entropy associated to the product state (5.10), which can be
written as
Sres = S(ρ) = S(ρ1) + S(ρ2). (5.11)
It is straightforward to check that for general N and L = 1, the residual entropy is equal to
15Locality of a Hamiltonian is a very restrictive condition. Since the effective Hamiltonian Heff is quite general,we do not necessarily expect it to be local.
16To see that this state has maximal entropy, recall that the mutual information for any state, I(1, 2) =S(ρ1)+S(ρ2)−S(ρ12), is always positive. Note further that the von Neumann entropy of the product state (5.10)is simply the sum of the entropies of its reduced states, S(ρ) = S(ρ1) + S(ρ2). We then see that the positivityof mutual information implies that among the class of states ρ12 with the same reduced states ρ1 and ρ2, theproduct state (5.10) is the one with largest entropy.
5 HOLE-OGRAPHY 41
the von Neumann entropy of the product state ρ1 ⊗ · · · ⊗ ρN , yielding
Sres =
N∑i=1
S(ρi). (5.12)
We will argue below that this is in sharp contrast to the case where the size of the reduced states
is larger than a single qubit, i.e. L > 1.
Three qubits, two-particle reduced states (N = 3, L = 2)
Let us now consider the more interesting case where N = 3 and L = 2. That is, we wish to
calculate the residual entropy for a system of three qubits knowing its three two-party reduced
states. The same question was posed and answered elegantly in [40]. Starting from a pure state
|η〉, the authors analysed whether there are other density matrices which have the same reduced
states as |η〉 〈η|. If there are, then the system is not uniquely determined by its reduced states,
which we can quantify with the residual entropy formula. Surprisingly, it was found that for
almost every state |η〉, the system is uniquely determined by its reduced states. Thus, in general,
the residual entropy for a generic three-particle pure state vanishes.
The exception to the above is the following state
|ψ〉 =1√2
(|000〉 − |111〉) , (5.13)
known as the GHZ state. Its three reduced density matrices are given by
ρred =1
2(|00〉 〈00|+ |11〉 〈11|) , (5.14)
which are identical due to translation invariance.
To see that a system described by the reduced states (5.14) has non-vanishing residual
entropy, consider the mixed state
ρ =1
2(|000〉 〈000|+ |111〉 〈111|) , (5.15)
which has the same two-particle reduced density matrices (5.14) as the GHZ state |ψ〉. Note
that normalized linear combinations of |ψ〉 〈ψ| and ρ also have same the reduced states (5.14).
Hence, the reduced states (5.14) do not uniquely determine the quantum state of the full system.
To quantify this uncertainty, we use the proposed notion of residual entropy (5.8). The mixed
state (5.15) has the largest von Neumann entropy, given by S(ρ) = log 2, which immediately
gives the residual entropy,
Sres = log 2. (5.16)
Let us interpret the presence of residual entropy for the GHZ state (5.13). The two-particle
reduced states (5.14) encode correlations between any pair of qubits, but fail to capture corre-
lations that occur between all three qubits concurrently. The distinguishing feature of the GHZ
state is precisely that it exhibits correlations between all three qubits, known as three-partite
5 HOLE-OGRAPHY 42
entanglement. Since the reduced states (5.14) do not capture the information associated to
this three-partite entanglement, we cannot reconstruct the global three-particle state uniquely
from the reduced ones. The uncertainty in reconstructing the global state is quantified by the
residual entropy (5.16). In constrast, a generic three-particle state does not have three-partite
entanglement (the GHZ state is, up to transformations of the local bases, the only state with
genuine three-partite entanglement). Therefore, the two-particle reduced states collectively en-
code all possible correlations of such a generic state. We conclude that there is no uncertainty in
reconstructing the global state from the reduced states and hence the residual entropy vanishes
for a generic state, in accordance with the observeration of [40].
In summary, the concept of residual entropy was introduced by [15] to quantify the collective
ignorance that local observers have in reconstructing the global quantum state. The general
procedure for finding the residual entropy in a generic quantum mechanical system was outlined
in the beginning of this section and subsequently some examples were given to make the proposal
more concrete. We now turn to the application in holographic CFTs.
5.3 Holographic holes in AdS3 and differential entropy
As described in the introductory part of this chapter, the concept of residual entropy depends
crucially on causality and observers. In particular, the time it takes for a light ray to reach
the asymptotic boundary defines a time interval on the boundary CFT, which is reminiscent of
causal holographic information [41]. However, as pointed out by [16], this construction is specific
to AdS3/CFT2. For holographic holes in higher dimensions, a more geometric construction is
needed to reproduce their area. To this end, another notion of entropy was introduced, namely
“differential entropy”. Although we shall not consider higher dimensions, it is helpful to review
both derivations to elucidate some conceptual differences.
5.3.1 Differential entropy
In the introduction to this chapter, we argued that due to holographic causality considerations,
an individual boundary observer can make measurements during a time 2T0. Within this amount
of time, the observer can perform measurements on an interval of length 2T0, or equivalently,
an interval of angular size α0 = 2T0/L.17 Thus, the state over which these measurements are
performed is described by the reduced density matrix obtained by tracing over the complement
of this interval. If the full system is in the ground state, corresponding to empty AdS, the
entanglement entropy is given by (3.27),
SA(l0) =c
3log
(2L
µsin(α0/2)
)(5.17)
where L is the radius of the cylinder and µ is a UV cutoff, which is related holographically to
the divergence of spatial geodesics as they approach the asymptotic boundary. In constrast, the
circular bulk curve which we are trying to reproduce holographically has finite area, so we’re
17In the original paper [15], α0 is half the opening angle of the interval, whereas here it corresponds to the fullopening angle.
5 HOLE-OGRAPHY 43
Figure 10: The angular size of a single interval Ik on the boundary field theory is denoted by α0, whosesize is determined by the timescale T0 via α0 = 2T/L. The interval size of the overlap between twoneighbouring intervals Ik ∩ Ik+1 is α0−∆θ, where ∆θ = 2π/n is the angle of separation. Figure adaptedfrom [15].
seeking a CFT quantity that is UV-finite.
Consider adding a second observer, whose causal diamond overlaps with the first. In general,
when we have overlapping systems, strong subaddivitity can be used to put an upper bound on
the entropy of the union of these systems S(A ∪B) ≤ S(A) + S(B)− S(A ∩B). Therefore, the
residual entropy associated to these two observers cannot exceed this bound either, i.e.
S(2)res ≤ S(I1) + S(I2)− S(I1 ∩ I2). (5.18)
Now consider a family of n observers, which are not yet periodically identified. Iterating
formula (5.18) n− 1 times bounds their collective ignorance S(n)res via
S(n)res ≤
n∑k=1
S(Ik)−n−1∑k
S(Ik ∩ Ik+1). (5.19)
Now let these intervals collectively cover the entire circle supporting the field theory, which means
that we periodically identify In+1 ≡ I1. The identification introduces one more overlapping
interval In ∪ I1, so the second sum in (5.19) obtains an additional term, yielding
S(n)res ≤
n∑k=1
[S(Ik)− S(Ik ∩ Ik+1)
]≡ S(n)
DE, (5.20)
where quantity S(n)DE is known as the differential entropy [15, 16], where the superscript refers to
the number of intervals. Importantly, SDE is a UV finite quantity, since the terms in brackets
have the same logarithmic UV divergence, which therefore cancel.18 This is a first indication
that the differential entropy is a good candidate to reproduce the area of the holographic hole.
Indeed, we will see that in a certain continuum limit, the differential entropy reproduces the
area of the holographic hole.
18Note that this only works in 1 + 1 dimensions. The divergences are associated to the boundary area of theinterval, i.e. S ∼ area(δIk). In 1+1 dimensions, this is simply a point. Therefore, the individual intervals Ik havethe same UV divergence the intervals of overlaps Ik ∩ Ik+1. This does not hold for higher dimensions, becausethe boundary area of Ik is then greater than that of the overlap Ik ∩ Ik+1.
5 HOLE-OGRAPHY 44
To proceed, we first express the entanglement entropy in terms of the interval size. The size
of the interval of overlap Ik ∩ Ik+1 is obtained by subtracting the angular separation between
two neighbouring intervals, ∆θ = 2π/n, from the angular size α0 of a single interval. This is
illustrated in figure 10. Hence, the differential entropy as a function of interval size is given by
S(n)res ≤ S
(n)DE =
n∑k=1
[S(α0)− S(α0 −∆θ)
]. (5.21)
Now we take the continuum limit n → ∞, which holographically constitutes a smooth
circular curve in the bulk. We multiply and divide by ∆θ, so that the continuum limit can be
re-expressed as ∆θ → 0. The difference then becomes a derivative and the sum becomes an
integral, yielding
Sres ≤ SDE =
∫ 2π
0dφ
dS(α)
dα
∣∣∣∣α=α0
, (5.22)
where Sres and SDE are quantities defined as the continuum limits of their discrete versions S(n)res
and S(n)DE, respectively.
We are now in a position to show that (5.22) holographically reproduces the area (length)
of the spherical hole with radius R0. Substituting the expression for the entanglement entropy
on a cylinder (5.17) into (5.22), we obtain after integration
SDE =cπ
3cotα0. (5.23)
We can express this result in terms of gravitational quantities by using α0 = T0/L, the AdS/CFT
dictionary c = 3L/2G and, finally, the relation between the radius of the hole and the duration
T0 of the time strip (5.1), giving
SDE =2πR0
4G=A4G
, (5.24)
which is indeed reproduces the area of the hole.
In summary, the residual entropy is a measure for the collective ignorance of a family of
observers. Strong subadditivity was used to argue that this quantity should be bounded by the
residual entropy SDE defined in (5.20). By considering the continuum limit where the number of
observers goes to infinity, the differential entropy takes the value (5.23). Upon translating this
expressions’ field theoretical variables into gravitational variables, it precisely coincides with the
Bekenstein-Hawking entropy SBH = A/4G of the hole!
5.3.2 Holographic derivation using strong subadditivity
Let us try to understand the above from a holographic point of view, based on the holographic
version of strong subadditivity. In section 4, we saw that the RT prescription for holographic
entanglement entropy indeed satisfies the strong subadditivity bound,
S(I1 ∪ I2) ≤ S(I1) + S(I2)− S(I1 ∩ I2). (5.25)
5 HOLE-OGRAPHY 45
Figure 11: All figures are timeslices of AdS3. Figure (a) shows two overlapping boundary intervals andtheir minimal surfaces, which are recombined aroung the point p to make the yellow and red curves. Thered curve is referred to as the ‘outer envelope’. Figure (b) represents a closed union of evenly spacedoverlapping intervals. The outer envelope is the closed curve shown in red, whose area is bounded bythe differential entropy. In the continuum limit, the outer envelope becomes a smooth curve as shown in(c), where now the bound saturates so that the area of the circle divided by 4G matches the differentialentropy. All figures are taken from [16].
Figure 11a shows the relevant construction for the proof of the bound in global coordinates.
Notice that the red arc k1∪2 does not yet have an associated entropy. Since it is crucial for the
derivation of the hole’s area, this arc is referred to as the ‘outer envelope’ by [16]. Its entropy is
defined via the Bekenstein-Hawking entropy as
S(I1, I2) ≡ A(k1∪2)
4G, (5.26)
which is smaller than the bound on the right hand side of (5.25). To see this, note that i1∩2 and
k1∩2 are two homologous line segments, and since the former is a geodesic, it is the one with
smallest length within its homology class. By the same line of reasoning, one easily sees that
S(I1, I2) is smaller than S(I1 ∪ I2). Hence, we arrive at the following set of inequalities
S(I1 ∪ I2) ≤ S(I1, I2) ≤ S(I1) + S(I2)− S(I1 ∩ I2). (5.27)
Analogous to the field theory case, we iterate (5.25) n times around the circle, yielding
S(∪Ik) ≤ S(Ik) ≤n∑k=1
[S(Ik)− S(Ik ∩ Ik+1)
]= S
(n)DE, (5.28)
where we used the definition of differential entropy (5.20). Note that S(Ik) is the BH entropy
corresponding to the closed red surface in figure 11b.
As before, consider the continuum limit n → ∞. The quantity S(Ik) defined by (5.26) is
then the entropy associated to the smooth circular curve composed of the tips of the n boundary
5 HOLE-OGRAPHY 46
Figure 12: The holographic hole with constant profile z = z∗ in Poincare coordinates is shown in blue.It is formed from the tips of the minimal surfaces shown in orange which are anchored to their boundaryintervals of size `. The distance of separation between two intervals is denoted by ∆x. This figure isadapted from [16].
geodesics (figure 11c), whose area is denoted as
limn→∞
A(k1∪···∪n) = A. (5.29)
The crucial observation is that in this limit, the second inequality in (5.28) saturates. Hence,
we find that the (continuum) differential entropy indeed reproduces the area of the hole
SDE ≡ limn→∞
S(n)DE =
A4G
. (5.30)
Some remarks are in place. In contrast to the analysis of [15], the outer envelope construction
makes no reference to the causality arguments and Rindler observers. Instead, it is based
purely on minimal surfaces. Yet, for spherical entangling surfaces in the vacuum, these two
concepts seem to agree perfectly [41]. Since a single interval in CFT2 has spherical symmetry, it
makes sense that the construction of [15] with residual entropy works. However, when moving
to holographic holes in higher dimensions do subtleties appear which favour this geometric
construction using the outer envelope [16, 42].
5.3.3 Poincare coordinates
We will now briefly demonstrate that the holographic construction of [15] is valid also for
Poincare coordinates, whose metric in AdS3 reads
ds2 =L2
z2
(dz2 − dt2 + dx2
), (5.31)
where L is the AdS radius. Consider a bulk curve with constant profile z = z∗, built up from
a family of minimal surfaces extending to the boundary. In Poincare coordinates, these are
semi-circles, z2 + x2 = z2∗ , so the maximal distance z∗ is related to the size of the boundary
interval via ` = 2z∗.
Now we turn to the bulk curve z = z∗. Since the x-direction is non-compact, the area of
5 HOLE-OGRAPHY 47
the bulk curve diverges, which can be regulated by compactifying the x-direction with period
Lx l. The Bekenstein-Hawking entropy of this curve is then given by an integral over the
induced metric hab = gµν∂axµ∂bx
ν , where gµν is the Poincare metric (5.31). This yields
A(z = z∗)
4G=
1
4G
∫ Lx
0dx√h =
LLx4Gz∗
. (5.32)
This result is holographically reproduced by the formula for differential entropy (5.22), which
in terms of spatial (instead of angular) variables reads
SDE =
∫ Lx
0dx
dS(`)
d`
∣∣∣∣`=`0
. (5.33)
It is straightforward to check that when we substitute the entanglement entropy of a single
interval S(`) = (L/2G) log(`/µ) (cf. expression (3.23) with c = 3L/2G) into (5.33), it reproduces
the area of the bulk curve (5.32) using ` = 2z∗.
5.3.4 Differential entropy for excited states
It was pointed out by [15] that the differential entropy associated to a thermal state on the CFT
reproduces the area of spherical holes in the dual BTZ geometry. First, note that light rays
projected radially outward from an R0-sized hole reach the asymptotic boundary after a time
T0 = (L2/R+) coth−1(R0/R+), (5.34)
which can be seen by setting ds2 = 0 in the BTZ metric (4.31) and subsequently solving for the
resulting differential equation. The entanglement entropy of a single interval is given by (4.35),
which can be plugged into the differential entropy formula (5.22). Converting into gravitational
variables as before, the differential entropy indeed reproduces the area of the hole19
SDE = A/4G. (5.35)
The fact that the differential entropy formula applies to excited states serves as a first indication
that we might be able to set up a first law of differential entropy, as we shall attempt in section
7.
19However, there is a sublety involved here, as can be seen from expression (5.34). When the bulk circle coincideswith the BTZ horizon, i.e. R0 →+ R+, the time it takes light rays to reach the boundary diverges, T0 →∞. Thismeans that α0 2π, i.e. the subinterval winds the entire boundary multiple times. Hence, the entanglemententropies for the intervals become ill-defined, and yet, the differential entropy continues to reproduce the areaof the hole. This issue is addressed in the later paper [43], where the differential entropy is studied in a conicaldeficit spacetime and the winding can be studied in the covering space of the orbifold.
6 LINEARIZED GRAVITY FROM THE FIRST LAW OF ENTANGLEMENT ENTROPY 48
6 Linearized Gravity from the First Law of Entanglement En-
tropy
The conjecture that spacetime emerges from entanglement is made very explicit within the
context AdS/CFT by Ryu and Takayanagi’s proposal (4.1). The entanglement entropy of a given
field theory state provides a direct window in the structure of the emergent geometry via areas
of certain minimal surfaces. Of course, when we make the statement that spacetime emerges
from entanglement, we must also include dynamics of the spacetime geometry. Fluctuations in
geometry are governed by the Einstein equations, so we would like to know its counterpart in
terms of entanglement.
This central question permeates the work of [11, 12, 13]. The crucial concept is the “first
law of entanglement entropy”, which states that under small perturbations of the CFT state,
the entanglement entropy changes as δSA = δ〈HA〉. The operator HA is known as the modular
Hamiltonian, which we shall make more precise later on. Holographically, the variation of
the field theory state translates into to a small metric perturbation in the dual spacetime.
Using the Ryu-Takayanagi prescription for the holographic entanglement entropy (4.1), the first
law δS = δE translates into certain integral constraint relations on the metric perturbation.
Since the first law holds for any spherical entangling surface, we essentially have an infinite set
of integral relations, which can be converted into local equations equivalent to the linearized
Einstein equations. Hence, the linearized Einstein equations are equivalent to a CFT first law
of entanglement entropy.
The organization of this chapter is as follows. In section 6.1, we review the derivation of the
first law and its application to spherical entangling surfaces. In section 6.2, we consider the first
law in holographic CFTs. We will review a theorem due to Iyer and Wald in section 6.3, which
will subsequently be applied to the holographic setup to derive the linearized Einstein equations
in section 6.4.
6.1 The ‘first law’ of entanglement entropy
It was shown in [11] that entanglement entropy obeys a ‘first law’ analogous to the first law of
thermodynamics. We will review this derivation below.
The reduced density matrix ρA is by construction a hermitian and positive (semi)definite
matrix, meaning that its eigenvalues are non-negative and finite real numbers. Therefore, ρA
can always be written in the following form [11]
ρA =e−HA
Tr e−HA, (6.1)
where HA is the modular Hamiltonian, on which we shall comment below. For now, consider
an infinitesimal variation of the state of the system, represented by a variation of the reduced
density matrix
ρA → ρA + δρA. (6.2)
6 LINEARIZED GRAVITY FROM THE FIRST LAW OF ENTANGLEMENT ENTROPY 49
The corresponding first order variation of the entanglement entropy then reads
δSA = −δ (Tr ρA log ρA)
= −Tr (δρA log ρA)− Tr (ρAδ log ρA)
= −Tr (δρA log ρA)− Tr (δρA)
= Tr (δρAHA) + log(Tr e−HA) Tr (δρA)− Tr (δρA) ,
where the computation of the second term in the second line is a bit tricky, but can be done
using the series expansion of the logarithm and the cyclicity of the trace. In the fourth line we’ve
taken the log of expression (6.1). Since both the original and the perturbed density matrix have
unit trace we have Tr (δρA) = 0, so the second and third term vanish. Hence, the variation of
the entanglement entropy becomes simply
δSA = Tr (δρAHA)
= Tr(ρ(1)A HA)− Tr(ρ
(0)A HA),
which can be written as
δSA = δ〈HA〉, (6.3)
known as the first law of entanglement entropy. It got its name due to the similarity with the
ordinary first law of thermodynamics dE = TdS. Note, however, that we are now dealing with
systems which may be far from equilibrium (e.g. the ground state), so equation (6.3) presents a
quantum generalization of the ordinary first law.
6.1.1 First law for ball shaped regions in CFTs
For a general quantum field theory, a general state, and a general region A, the modular Hamil-
tonian HA is not known independently, but is implicitly defined by ρA through (6.1). The first
law (6.3) can thus be regarded as a tautology, because both sides of the expression are derived
from the same quantity ρA. The reason that an expression for the modular Hamiltonian cannot
be derived independently is because it is typically expected to be some complicated non-local
operator.
However, in the special case where the subsystem of a CFT has the shape of a round ball,
the modular Hamiltonian can in fact be written as an integral of a local operator. The crucial
observation made in [46] is that the domain of dependence D for a ball-shaped region B is
related by a conformal transformation to the hyperbolic cylinder H = Hd−1 ×Rτ , where Hd−1
is the hyperbolic space.20 This maps the vacuum state on B to a thermal state on Hd−1 with
temperature T = 1/2πR [46]. Therefore, the density matrix (6.1) may be expressed as a thermal
20This procedure is very similar to the derivation of the Unruh effect. In fact, the Rindler wedge is relatedto the hyperbolic cylinder by a conformal transformation which is time independent and thus acts trivially onthe modular Hamiltonian [46]. That is, HR = HH ' 2πRHτ . This can be seen by writing the usual Rindler
coordinates ds2 = − z2
R2 dτ2 + dz2 + d~x2 as ds2 = Ω2
(−dτ2 + R2
z2[dz2 + d~x2]
), where Ω2 = z2/R2. By getting rid
of the conformal factor Ω, we are left with the hyperbolic space H. Since Ω is time independent, the Hamiltoniandoes not change.
6 LINEARIZED GRAVITY FROM THE FIRST LAW OF ENTANGLEMENT ENTROPY 50
Figure 13: On the left, we see the ball-shaped region B shown in turquoise and its causal developmentD shown as the two light cones, which is generated by the Killing vectors shown in green. The rightfigure shows the hyperbolic cylinder H, where the hyperbolic space is shown in turquoise and the greenlines are timelike Killing vectors. There exists a conformal tranformation between D and H which allowsus to construct the Killing vectors on D from the ones on H. This image is adapted from [13].
density matrix,
ρ =1
Ze−2πRHτ , (6.4)
where Z is the thermal partition function and Hτ is the operator associated to time translations
in H. In other words, Hτ is the ordinary Hamiltonian of the field theory on the hyperbolic
space. Comparing this to (6.1), it follows that the modular Hamiltonian HH can be written as
HH = 2πRHτ + logZ. This operator defines a local geometric flow on H. An equivalent way of
saying this is that there exists a Killing vector ζH = 2πR∂τ on this geometry.
With the expression for the modular Hamiltonian on the hyperbolic cylinder at hand, we can
map it back to the causal development D of the ball-shaped region B to obtain an expression
for the modular Hamiltonian HB. In other words, we are looking for the image ζB of the Killing
vector ζH under the conformal transformation which takes us from H to D [46]. The precise
expression for the mapping can be found in [46], leading to the following expression for the
(conformal) Killing vector [13]:
ζB =π
R
([R2 − (t− t0)2 − |~x− ~x0|2
]∂t + 2(t− t0)(xi − xi0)∂i
), (6.5)
where ~x0 are the coordinates of the center of the ball. It is straightforward to check that the
Killing vector (6.5) generates a flow that remains entirely in D, i.e. it acts as a null flow on
∂D and vanishes at the future and past tips of D and on the sphere ∂B. The corresponding
modular Hamiltonian is then simply the conserved charge associated to this local flow,
HB =
∫SdΣµTµνζ
νB, (6.6)
where dΣµ is the volume-form on the (d−1)-dimensional hypersurface S and Tµν is the conformal
stress tensor. Evaluating (6.6) on the spacelike hypersurface S = B, which entails setting t = t0
6 LINEARIZED GRAVITY FROM THE FIRST LAW OF ENTANGLEMENT ENTROPY 51
in (6.5), we obtain the result [46]
HB = 2π
∫Bdd−1x
R2 − |~x− ~x0|2
2RTtt(t0, ~x). (6.7)
In summary, the entanglement entropy satisfies a first law δSA = δ〈HA〉. In general, HA
might be a complicated non-local operator, in which case the first law does not have a lot of
predictive power. However, ball-shaped regions B have such a high degree of symmetry that it
harbors the Killing vector (6.5), which may be calculated as the image of the Killing vector on
the hyperbolic cylinder (where it is easily constructed). The Killing vector defines a conserved
quantity EB, which is the modular energy defined as δEB ≡ δ〈HB〉. Thus, the entanglement
first law (6.3) for ball-shaped regions B reads
δSB = 2π
∫Bdd−1x
R2 − |~x− ~x0|2
2Rδ〈Ttt(t0, ~x)〉 = δEB. (6.8)
6.2 Holographic interpretation of the first law
The entanglement first law (6.3) is a general result for quantum field theories. In conformal
field theories, the explicit expression (6.8) can be written down for ball-shaped regions when
considering perturbations around the ground state. In the current section, we will be interested
in understanding the first law for ball-shaped regions in holographic CFTs with a classical
gravity dual. Translating the entropy and energy into the dual gravitational observables δSgravB
and δEgravB , the first law (6.8) gives a prediction for the equivalence of these quantities, i.e.
δSgravB = δEgravB , (6.9)
in any spactime dual to a small perturbation above the CFT vacuum state.
The power of the equality (6.9) lies in the fact that it represents an infinite set of constraint
relations, because the first law can be applied to every ball-shaped region in any Lorentz frame
on the boundary geometry. The insight of [12, 13] is that this infinite set of relations can be
converted into local constraints on the spacetime metric, which are equivalent to the linearized
Einstein equations. In order to get there, we shall first specify the dual spacetime and find
explicit expressions for δSgravB and δEgravB .
6.2.1 Perturbed geometry
The CFT ground state on Minkowski space is dual to pure AdSd+1 in Poincare coordinates,
ds2 = g0abdx
adxb =L2
z2
(dz2 + ηµνdx
µdxν). (6.10)
Upon slightly varying the CFT density matrix, the dual geometry can be treated as a small
perturbation around pure AdS,
gab = g0ab + δgab. (6.11)
6 LINEARIZED GRAVITY FROM THE FIRST LAW OF ENTANGLEMENT ENTROPY 52
Note that the perturbed metric (6.11) has to be an asymptotically AdSd+1 spacetime, as it oth-
erwise would not qualify as a holographic CFT. The most general metric which is asymptotically
AdSd+1 is the (d+ 1)-dimensional Fefferman-Graham metric [47]
ds2 =L2
z2
(dz2 + ηµνdx
µdxν + zdHµν(x, z)dxµdxν), (6.12)
so by comparing to (6.11) we read off that
δgµν = L2zd−2Hµν , δgzµ = δgzz = 0. (6.13)
We will now calculate the gravitational quantities δSgravB and δEgravB associated to this metric
perturbation.
6.2.2 Variation of holographic entanglement entropy
The holographic entanglement entropy is computed by the RT formula21
SB =A(B)
4G=
1
4G
∫Bdd−1σ
√h (6.15)
where σ is the affine parameter along the extremal (d− 1)-dimensional surface B, and
h = det(hab) = det(g0cd)
dxc
dσadxd
dσb(6.16)
is the induced metric on the minimal surface B, where g0ab is the pure AdSd+1 metric (6.10).
Recall from section 4.3 that the minimal surface in Poincare-AdS3 is shaped as a semi-circle.
For ball-shaped regions in a d-dimensional CFT, a similar analysis shows that the homologous
minimal surface in AdSd+1 is a hemisphere. We parametrize the hemisphere in terms of the
boundary coordinates xi as z(x) =√R2 − ~x2, where R is the radius of the ball. With this
parametrization, the induced metric (6.16), together with its inverse22 and determinant, are
given by
h0ij =
L2
z2
(δij +
xixjz2
), hij0 =
z2
L2
(δij −
xixj
R2
), h0 =
(L2
z2
)d−1R2
z2, (6.17)
21The Ryu-Takayanagi formula is valid for holographic CFTs whose gravitational dual is governed by theEinstein-Hilbert action. There exist gravitational theories, e.g. those whose action contains higher powers of theRicci scalar. Since every power of Ricci scalar contains a single derivative of the metric, higher powers of R havea higher derivative dependence. These are therefore called higher derivative gravity theories. Now, there exists aformula to find the gravitational entropy for any theory of gravity. This is the Wald-functional [45]
SWald = −2π
∫Hddσ√gδLδRabcd
nabncd, (6.14)
where the integral is over bifurcate Killing horizon H, L is the gravitational Lagrangian and nab is the binormalto the horizon H. The Wald entropy functional can be used to calculate the entanglement entropy for ball-shapedregions in higher derivative holographic CFTs [13].
22The inverse of the induced metric can be found by writing down the most general symmetric matrix gij0 =aδij + bxixj and fixing the constants a and b by demanding g0
ikgkj0 = δji .
6 LINEARIZED GRAVITY FROM THE FIRST LAW OF ENTANGLEMENT ENTROPY 53
respectively, where the latter two will be used in a moment.
Now consider turning on the metric perturbation, so that the spacetime metric is given
by (6.12). The RT prescription dictates that we should find the minimal surface in this new
geometry and compute its length in order to find the entanglement entropy. In general, the shape
of the new minimal surface is expected to change accordingly, which complicates the calculation
of its length immensely. However, if the metric perturbation is small, we can expand the profile
of the minimal surface in terms of the perturbation z(x) = z0(x) + δz +O(δz2), where δz is of
order δg. The crucial observation of [11] is that the original surface was extremal, i.e. the first
order variation δz vanishes by construction. Hence, to linear order, the minimal surface area is
found by evaluating the area of the original surface B in the perturbed metric (6.12), yielding
the following expression for the change in area:
δA =
∫Bdd−1σ δ
√h =
1
2
∫Bdd−1σ δ
√h0h
ab0 δgab, (6.18)
where the inverse and determinant of the pull-back metric h0 are given by the equations in
(6.17). Using (6.13) and dividing by 4G, we are then able to write the first order variation of
the holographic entanglement entropy as [12]
δSB =RLd−3
8G
∫Bdd−1x
(δij − 1
R2(xi − xi0)(xj − xj0)
)Hij(z, t0, ~x). (6.19)
6.2.3 Holographic energy perturbation
We now wish to find the gravitational counterpart of the modular energy δEB. The AdS/CFT
correspondence entails that the gravitational observable dual to the expectation value of the
field theory stress tensor is proportional to the asymptotic metric, a fact which can be estab-
lished through a systematic procedure called holographic renormalization [48, 49]. The main
result obtained from holographic renormalization is that the asymptotic value of the bulk metric
perturbation (6.12) is related to the expectation value of the boundary stress tensor via23
δ〈Tµν(x)〉 =dLd−1
16πGHµν(z = 0, x). (6.20)
For later convenience, we note that tracelessness and conservation of the CFT stress tensor
imply that
Hµµ = 0 and ∂µH
µν = 0. (6.21)
With expression (6.20) for the holographic stress tensor at hand, we can write the change in
23There is, however, a major caveat in using the holographic stress tensor for our present purposes. To obtainthe holographic stress tensor (6.20) via the holographic renormalization method, the metric perturbation Hµνhas to satisfy the Einstein equations. Hence, we encounter a circular reasoning if we want to use the result (6.22)to derive the linearized Einstein equations from the first law. Luckily, a different route to derive the holographicstress tensor has been offered by [13]. The authors show that the R → 0 limit of the holographic first law (6.9)(combined with expressions (6.8) and (6.19) for the entanglement entropy), relates the asymptotic value metricto the boundary stress tensor in precisely the same way as (6.20). Since their derivation is straightforward butinvolves quite a bit of algebra, we will not reproduce it here, but we take their derivation of (6.20) for granted.
6 LINEARIZED GRAVITY FROM THE FIRST LAW OF ENTANGLEMENT ENTROPY 54
Figure 14: The causal development of the boundary ball-shaped region B is generated by the Killingvectors shown in green, representing expression (6.5). The light rays projected inwards which intersectthe minimal surface B constitute the AdS-Rindler horizon, which is generated by the gravational Killingvectors (6.23) shown in red. The dark-grey region bounded by B and B is the timeslice Σ. This figure isbased on [13].
energy (6.8) in terms of gravitational observables as
δEgravB =dLd−1
8G
∫Bdd−1x
R2 − |~x− ~x0|2
2RHtt(z = 0, t0, ~x), (6.22)
which expresses the gravitational energy as an integral of the asymptotic value of the metric
perturbation over region B, which lies at the asymptotic boundary of AdS.
6.3 First law of black hole thermodynamics and the Iyer-Wald theorem
We have obtained expressions for the gravitational quantities δSgravB and δEgravB , given by (6.19)
and (6.22), respectively. Now, δSgravB is expressed as an integral over the bulk surface B,
whereas δEgravB is an integral over the boundary interval B. Equating these two quantities
through the first law (6.9) gives non-local integral constraints on the metric perturbation. It is
not immediately obvious how these non-local integral relations will give rise to local constaints,
i.e. the Einstein equations, on the metric.
The crucial observation that allows us to proceed is that the minimal surface B resembles
the bifurcation surface of a black hole Killing horizon [46]. This can be understood as follows.
In the CFT, the vacuum density matrix for the ball-shaped region B was obtained via a con-
formal transformation to a thermal state of the CFT on hyperbolic space. Now, the AdS/CFT
correspondence tells us a thermal state is dual to a certain black hole. Embedded in Poincare
coordinates, the black hole dual to the thermal state on hyperbolic space is simply the wedge
shown in figure 14, which is known as a Rindler wedge. The null lines projected inwards from
the causal development D which intersect the minimal surface B form the AdS-Rindler horizon.
6 LINEARIZED GRAVITY FROM THE FIRST LAW OF ENTANGLEMENT ENTROPY 55
Importantly, the AdS-Rindler horizon is generated by a Killing vector, given by [13]
ξB = −2π
R(t− t0)[z∂z + (xi − xi0)∂i] +
π
R[R2 − z2 − (t− t0)2 − (~x− ~x0)2]∂t. (6.23)
It can be checked that (6.23) is indeed an exact Killing vector of the Poincare metric (6.10),
which remains inside the wedge and vanishes on B. The existence of this Killing vector justifies
the initial claim that B resembles the bifurcation surface of a black hole Killing horizon.
Why, then, is the existence of this Killing symmetry so crucial for the derivation of the
Einstein equations? The answer lies in a theorem due to Iyer and Wald [45], which states
that for stationary spacetimes with a bifurcate Killing horizon generated by a Killling vector
ξ, arbitrary on-shell metric perturbations satisfy a first law of thermodynamics δS = δE. In
other words, if the metric perturbation satisfies the Einstein equations, then the first law holds.
Essentially, we are looking for the conserve statement: if the first law holds, then the metric
perturbation must satisfy the Einstein equations. In the next section we shall demonstrate that
this converse statement is indeed valid, provided that the first law holds for any ball-shaped
region in any Lorentz frame (which it does).
6.4 Linearized gravity from the first law
As we are looking for the converse statement of the Iyer-Wald theorem, we shall first review
the original theorem. Mote that the formalism which substantiates the following statements is
reviewed in appendix B. We focus on the essence of the Iyer-Wald theorem, which is that for
Killing horizons generated by a Killing vector ξB, there exists a differential (d− 1)-form χ that
satisfies
(i) δSgravB =
∫Bχ and (ii) δEgravB =
∫Bχ. (6.24)
and satisfies the following relation off-shell
(iii) dχ ∼ ξaBδEgabε
b, (6.25)
where δEgab are the linearized Einstein equations which vanish on-shell, εb is the volume form on
the d-dimensional hypersurface Σ defined in (B.31), and ξB is the gravitational Killing vector
given by (6.23).
In appendix B we review the formalism which proves the existence of χ with the above three
propoerties and in the next section, we will explicitly construct the form χ that matches the
expressions (6.19) and (6.22) for δSgravB and δEgravB , respectively. For now, we take expressions
(6.24) and (6.25) for granted, because together with the first law (6.9) they will allow us to
derive the linearized Einstein equations. Using the two expressions in (6.24) we may write
δSgravB − δEgravB =
∫Bχ−
∫Bχ =
∫∂Σχ =
∫Σdχ, (6.26)
where the last equality is due to Stokes’ theorem and Σ is the region enclosed by B and B, as
can be seen in figure 15. It is now straightforward to verify the original Iyer-Wald statement
6 LINEARIZED GRAVITY FROM THE FIRST LAW OF ENTANGLEMENT ENTROPY 56
Figure 15: Timeslice of AdSd+1, where the region Σ is enclosed by the ball-shaped boundaryregion B and the minimal surface B.
from (6.25) and (6.26), namely that on-shell metric perturbations satisfy δSgrav = δEgrav.
As mentioned, our mission is to reverse the theorem of Iyer and Wald. Starting from δSgrav =
δEgrav, we see that the left hand side of equation (6.26) vanishes, i.e.∫
Σ dχ = 0. We then use
expression (6.25) to obtain ∫ΣξtBδE
gttε
t = 0. (6.27)
where we used that the spacelike interval Σ with volume element εt is defined at constant time
t = t0, which implies that only the time component of ξB is non-vanishing on Σ.
We now wish to isolate the linearized Einstein equations in the integrand. In order to do so,
we multiply by R and subsequently take the derivative with respect to R, yielding
0 =
∫Σ
(RξtB)′δEgttεt +
∫Σ
(RξtB)δEgtt(εt)′
= 2πR
∫ΣδEgttε
t +
∫B
(RξtB)δEgttr · εt. (6.28)
where we used expression (6.23) for the first term in the second line.24 The second term drops
out because ξtB vanishes on B, so (6.27) is equivalent to∫ΣδEgttε
t = 0, (6.29)
for every spacelike region Σ.
From the vanishing of the integral we cannot directly deduce that the Einstein equations
are satisfied locally, i.e. that δEgtt = 0 everywhere. However, as mentioned before, the crucial
observation that allows us to extract local information is that the first law holds for ball-shaped
regions of any size R and any position x0. Thus, we have an infinite set of integral relations
(6.29), one for every timeslice Σ(R, x0). It is straightforward to show that if the integral (6.29)
vanishes on every domain Σ, then the integrand itself must be zero (see appendix A of [13]).
24To see how to obtain the second term in the second line from the second term in the first line, note that Σ isthe interior of a hemisphere with radius R. Taking the derivative of the volume element with respect to R meansthat we take the difference between the volume of the R-sized hemisphere B’s interior and another hemispherewhich has infinitesimally larger radius. This difference formally defines an area element on the hemisphere B.
6 LINEARIZED GRAVITY FROM THE FIRST LAW OF ENTANGLEMENT ENTROPY 57
Therefore, the linearized Einstein equations are locally satisfied,
δEgtt = 0. (6.30)
Let us note for completeness that the other components of the Einstein equations can be
derived by boosting to a general frame of reference and demanding that δSgravB = δEgravB in
the general frame, leading to δEgµν = 0. In other words, δEgµν = 0 immediately follows from
the Lorentz invariance of the theory. The remaining equations δEgzµ = 0 and δEgzz = 0 can
be obtained through the initial value formulation of gravity. This formulation implies that if
δEgzµ = 0 and δEgzz = 0 at z = 0 and δEgµν = 0 everywhere, then δEgzµ = 0 and δEgzz = 0
everywhere. That the Einstein equations δEgzµ = 0 and δEgzz = 0 are satisfied at z = 0 follows
immediately from the tracelessness and conservation of the stress tensor, so from the initial value
formulation we dedude that all components of the Einstein equations are satisfied everywhere,
i.e.
δEgab = 0, (6.31)
where a = z, µ. We refer to [13] for a more detailed discussion on the above arguments.
In summary, we have reviwed the derivation of the linearized Einstein equations from the
first law (6.9), provided that a form χ exists that satisfies (6.24) and (6.25), which is reviewed
in appendix B.
6.4.1 Explicit construction of χ
So far, the above discussion has been quite abstract. In the current section, we shall explicitly
construct the form χ which satisfies (6.24) and (6.25). Note that the explicit construction is
not necessary to derive the linearized Einstein equations; the existence of χ with the desired
properties is guarenteed by the Iyer-Wald formalism reviewed in appendix B. The computations
in the current section are therefore not worked out in much detail, and the results are mostly
just taken from [13].
We have already constructed δSgravB and δEgravB explicitly, given by (6.19) and (6.22), re-
spectively. First, let us reproduce these in the following suggestive form:
δSgravB =Ld−1
8GR
∫Bdd−1x
(R2δij − (xi − xi0)(xj − xj0)
)Hij(z, t0, ~x)
?=
∫Bχ (6.32)
δEgravB =dLd−1
16GR
∫Bdd−1x
(R2 − |~x− ~x0|2
)H i
i(z = 0, t0, ~x)?=
∫Bχ, (6.33)
where the tracelessness of the stress tensor (6.21) allowed us to replace Htt by H ii = δijHij ,
where δij are the (ij)-components of the pure AdS metric.
We will now demonstrate that the form χ by (B.26) indeed satisfies the above two expressions.
The expression for χ is reproduced here for convenience as
χ = δQ[ξB]− ξB ·Θ(δφ), (6.34)
where ξB is the gravitational Killing vector given by (6.23), Θ is known as the symplectic
6 LINEARIZED GRAVITY FROM THE FIRST LAW OF ENTANGLEMENT ENTROPY 58
potential, Q is the Noether charge and φ = gµν , . . . are all dynamic fields appearing in the
Lagrangian of the theory (we are only interested in the metric gµν). The gravity theory we are
interested in is governed by the Einstein-Hilbert action
SEH =1
16πG
∫dd+1x
√−g(R+ 2Λ), (6.35)
where we stress that the equations of motion do not follow from the action principle, but are
derived from the first law (6.9) (as we have seen in the previous section).
The symplectic potential is related to the surface term appearing in the variation of the
action, which reads (see, e.g. [38]):
δSsurface =1
16πG
∫dd+1x
√−g∇a
(∇aδgbb −∇bδgab
), (6.36)
from which we may read off the symplectic potential Θ as the term in parentheses. Taking the
interior product with the Killing vector ξB, we get
ξB ·Θ =1
16πGξbBεab
(∇cδgac −∇aδgcc
), (6.37)
where εab is defined in (B.31).
An expression for Q was calculated25 in [45]:
Q = − 1
16πGεab∇aξbB. (6.39)
With expressions (6.38) and (6.39) at hand, we insert these in the definition for χ to obtain
χ = − 1
16πG
[δ(εab∇aξbB) + ξbBεab(∇cδgac −∇aδgcc)
], (6.40)
which defines χ everywhere. We are interested in its restriction to the spacelike hypersurface Σ,
yielding [13]
χ∣∣Σ
=zd
16πG
εtz
[(2πz
R+d
zξt + ξt∂z
)H i
i
]+ (6.41)
+ εti
[(2π(xi − xi0)
R+ ξt∂i
)Hj
j −(
2π(xj − xj0)
R+ ξt∂i
)H i
j
],
where ξtB = πR(R2−z2−|~x−~x0|2). Using expression (6.41), it is straightforward, though tedious,
to verify the second equality in equations (6.32) and (6.33), and also check that its derivative
satisfies
dχ∣∣Σ
= −2ξtδEgttεt, (6.42)
25One can show that Q may always be written as [50]
Q = Wcξc + Xcd∇[cξd], Xcd = − δL
δRabcdεab, (6.38)
where Wc is a covariant expression constructed from the metric and its derivatives. For Einstein gravity, we haveXcd∇[cξd] = − 1
16πGεab∇aξbB , which by itself gives (6.39). It is not clear to me why the first term vanishes.
6 LINEARIZED GRAVITY FROM THE FIRST LAW OF ENTANGLEMENT ENTROPY 59
where [13]
δEgtt = −zdL2−d
32πG
(∂zH
ii +
d+ 1
z∂zH
ii + ∂j∂
jH ii − ∂i∂jHij
)(6.43)
is the (tt)-component of the linearized Einstein equations. By the arguments discussed at the
end of section 6.3, we infer that δEgtt = 0. The other transverse components of the Einstein
equations δEgµν = 0 are satisfied when we boost to an arbitrary frame of reference. Finally, the
components δEgzν and δEgzν vanish by appealing to the initial value formulation and observing
that the stress tensor is traceless and conserved, which implies that δEgzν = δEgzν = 0 at z = 0.
Thus, for theories where the entanglement entropy is computed by the Ryu-Takayanagi for-
mula, we explicitly see that the first law for entanglement entropy is equivalent to the linearized
Einstein equations.
7 FIRST LAW OF DIFFERENTIAL ENTROPY 60
7 First Law of Differential Entropy
At the end of their paper [13], the authors discuss the relation of their derivation to the work
of Jacobson [14]. Jacobson considers a local Rindler horizon and investigates what happens as
a certain bulk energy flows through this horizon. Assuming that the entropy is proportional
to the horizon area, the first law of thermodynamics relates the energy flow to a fluctuation in
area. From this constraint between energy and geometry, it was shown that the back-reacted
metric must satisfy the Einstein equations. However, Jacobson’s approach lacks a microscopic
understanding of the entropy and the first law. It is, if you like, a phenomenological derivation.
In contrast, the work of [13] offers a precise microscopic understanding of both the energy and
the entropy and also offers a microscopic proof for the first law, which is a major improvement
compared to Jacobson.
On the other hand, the authors of [13] derive the Einstein equations only at the linearized
level, whereas Jacobson obtains the full non-linear Einstein equations sourced by a matter term.
That the holographic derivations is limited to the linearized level seems to be related to the
fact that they are considering global rather than local Rindler horizons [13]. In particular, the
holographic setup presents δSgravB and δEgravB as integrals of gravitational observables (the former
is over the bulk minimal surface B, the latter over the boundary region B), whereas Jacobson
expresses δSgravB and δEgravB directly in terms of local gravitational quantities. It seems that we
would come closer to Jacobson’s original derivation if we could somehow make the AdS-Rindler
horizons more local objects. In other words, we seek to isolate the local bulk tips of the global
AdS-Rindler horizons.
The holographic construction discussed in section 5 seems to precisely isolate these bulk tips.
Hence, it would be interesting to combine the first law and the concept of differential entropy
into a first law of differential entropy.26 In this final chapter, we will give some preliminary
ideas in this direction, but also discuss what appears to be a fundamental objection towards
such an approach. We start in section 7.1 by reviewing Jacobson’s original derivation in detail.
In section 7.2, we discuss the relationship between local Rindler horizons and differential entropy.
In the same section, we propose a formula for the first law of differential entropy and check its
validity with an example. Finally, we will briefly comment on its limitations in the holographic
derivation of Jacobson’s approach.
7.1 Jacobson’s derivation
Here we review Jacobson’s derivation of the Einstein equations from the first law of thermody-
namics in four-dimensional asymptotically flat spacetime. Invoking the equivalence principle,
Jacobson argues any small enough region of spacetime is approximately flat, which therefore
approximately has the usual Poincare symmetries of Minkowski space. Of particular interest
are the boost symmetries, generated by the boost Killing vector field ξa, which give rise to a lo-
26Note that we discussed differential entropy mostly within the context of AdS3/CFT2. Since gravity in threedimensions has no local curvature, the Einstein equations become rather trivial in this case. However, it wasshown by [16] that differential entropy applies to holographic holes in higher dimensions. Hence, the lessons welearn in three dimensions may be transferable to higher dimensions, where the non-trivial gravitational dynamicscome into play.
7 FIRST LAW OF DIFFERENTIAL ENTROPY 61
Figure 16: The left image illustrates the local Rindler horizon generated by the null geodesics ka. Theboost Killing vector ξa is shown in red, which is approximately proportional to the null geodesics nearP. The bifurcation surface is P0, and the spacelike surfaces P0 and P are generated by the congruenceof null geodesics ka, as illustrated under the magnifying glass. Figure (b) represents the focusing of thehorizon generators, which is governed by the Raychaudhuri equations. Figure (a) is based on [51].
cal Rindler horizon. The generators of this horizon are the null geodesics ka, which are affinely
parametrized by λ. Along every point on the horizon, we find a two-dimensional surface P.
These spacelike surfaces are generated by the congruence of null geodesics ka normal to it. The
bifurcation surface, i.e. the surface on which ξa vanishes and where λ = 0, is denoted by P0.
The described setup is represented in figure 16a.
Now consider a local energy flux passing through the Rindler horizon. The energy is defined
as the conserved charge associated to the boost Killing vector,
δE =
∫HTabξ
bdΣa, (7.1)
where Tab is the stress tensor of the matter fields and dΣa is the volume element is defined as
dΣa = kadλdA, where dA is the area element of the two-surfaces normal to ka. Approximating
the Killing vector as27 ξa = −κλka, where κ is the surface gravity, we can express the energy
flux as
δE = −κ∫HλTabk
akbdλdA. (7.2)
Next, Jacobson makes the assumption that the entropy is proportional to the horizon area,
δS = δA/4G. (7.3)
27Since ξa is normal to the bifurcation surface P0 it obeys the geodesic equation along the Killing horizon,ξa∇aξb = −κξb. Here κ is the surface gravity, which arises because ξa is not affinely parametrized. In contrast,ka is affinely parametrized by definition of λ, and thus satisfies ka∇akb = 0. If we now approximate the Killingvector as ξa = f(λ)ka in the vicinity of the horizon, it then follows that f(λ) = −κλ by combining these twogeodesic equations. Hence, ξa = −κλka.
7 FIRST LAW OF DIFFERENTIAL ENTROPY 62
where the change in area can generically be expressed as
δA =
∫HθdλdA, (7.4)
where θ is the expansion of the horizon generators. The expansion can be computed using the
Raychaudhuri equation [52]dθ
dλ= −1
2θ2 − σ2 −Rabkakb, (7.5)
where σ2 is the square of the shear and Rab is the Ricci tensor. Jacobson argues that both the
quadratic terms may be neglected, since they vanish at the bifurcation surface P0 and therefore
represent higher order corrections in the neighbourhoud of P0. Integrating the Raychaudhuri
equation then simply gives θ = −λRabkakb, from which it follows that expression (7.4) can be
written as
δA = −∫HλRabk
akbdλdA. (7.6)
From expressions (7.2), (7.3) and (7.6), we see that the first law
δE = TδS (7.7)
relates the presence of the energy flux to a change in geometry. In particular, this energy flux is
associated with a focusing of the horizon generators, as illustrated in figure 16b. We shall now
show that this focusing is in accordance with the Einstein equations.
Using the usual value of the Unruh temperature T = κ/2π [53], we see that the first law can
only be valid if Rabkakb = 8πGTabk
akb for all ka. It follows that Rab + fgab = 8πGTab, because
the second term on the left vanishes when contracted with two null vectors. Conservation of
the stress tensor ∇aTab = 0, together with the contracted Bianchi identity ∇aRab = 12∇
a(gabR),
enforce that f = −R/2 + Λ, for some constant Λ. Following Jacobson, we thus infer that the
first law is equivalent to the full non-linear Einstein equations
Rab −1
2Rgab + Λgab = 8πGTab, (7.8)
which can therefore be interpreted as an equation of state, i.e. a macroscopic relation that
governs how the system changes between two equilibrium states.
In summary, by assuming that local Rindler horizons enjoy a first law of the form (7.7), the
relation between entropy and area is tantamount to the full non-linear Einstein equations (7.8).
The flaw in this derivation is that it fails to give a micropscopic explanation of the entropy/area
relation (7.3). Since holography provides such an explanation in terms of entanglement via
the Ryu-Takayanagi formula, it seems worthwhile to pursue a holographic replica of Jacobson’s
approach.
7.2 Local Rindler horizons and differential entropy
In the previous section, we reviewed the thermodynamic derivation of the Einstein equations
from local Rindler horizons. In contrast, the holographic derivation discussed in section 6 uses
7 FIRST LAW OF DIFFERENTIAL ENTROPY 63
global Rindler horizons. In an attempt to remedy this discrepancy, we resort to the differential
entropy. Recall from section 5 that the differential entropy is dual to the area of a closed bulk
curve, which is formed through the union of the near-tip segments of the minimal surfaces
associated to a spherically symmetric union of Rindler horizons. In particular, the differential
entropy is given by expression (5.22), i.e.
SDE =
∫dφ
dS
dα, (7.9)
where S is the entanglement entropy of a single interval, α is the angular size of the interval
and φ is the angular coordinate of the global AdS3 metric (2.11). The crucial observation that
connects this to the construction of a local Rindler horizon is that we can the line element dA,
given bydA4G
= (dS/dα)dφ (7.10)
represents an infinitesimal bulk segment, which we may think of as the local tip of the spatial
surface associated to a Rindler horizon.
Notice that the hole itself does not correspond to the bifurcation surface of a Killing horizon
(in fact, it is not even a minimal surface28). Since both the Wald’s and Jacobson’s approach
require the existence of a Killing symmetry, it seems unlikely that we can apply the laws of
thermodynamics to the surface of the entire hole. The hope is that the infinitesimal segment d`
can approximately be regarded as a bifurcation Killing surface, although we have not been able
to formally verify this.
7.2.1 First law of differential entropy
We will now consider variations of the differential entropy (7.9), from which we can later on
easily isolate a local line element (7.10) by dismissing the integral in front. Recall first that the
entanglement first law was applied to vacuum perturbations on ball-shaped regions in Minkowski
space, yielding (6.8). The observation that connects the first law to variations of the differential
entropy is that an interval in two-dimensional CFTs is a line, which by definition is equivalent
to one-dimensional ball. Since the first law for ball-shaped regions is set up in Minkowski space,
it is convenient to use expression (5.33) for the differential entropy in planar coordinates, which
together with the first law (6.8) is reproduced here for convenience:
SDE =
∫ Lx
0dx0
dS
d`and δS = δE = 2π
∫|x|<`/2
dx(`/2)2 − (x− x0)2
`δ〈Ttt(x)〉, (7.11)
where Lx ` is an IR regulator, x0 is the central coordinate of each interval and ` = 2R
is the interval length. Combining these two yields the following functional as the first law of
28If the total system is in the vacuum state, then the minimal surface homologous to the entire boundary shrinksto a point of vanishing size (see [15, 16] for a discussion).
7 FIRST LAW OF DIFFERENTIAL ENTROPY 64
differential entropy29
δSDE = 2π
∫ Lx
0dx0
∫ x0+`/2
x0−`/2dx
(1
4+
(x− x0)2
`2
)δ〈Ttt(x)〉. (7.12)
7.2.2 Example calculation: BTZ as a perturbation above the vacuum
As a check, we would like to show that this entropy variation holographically matches the
variation of the hole’s area. In particular, let this variation be the difference between the area
in the BTZ geometry and that in the Poincare vacuum. It is important that the BTZ mass
parameter µ is small,
µ`2 1. (7.13)
because the first law used in (7.12) only holds for small perturbations above the vacuum. Since
the boundary metric is planar, it is convenient to express the BTZ metric as a Fefferman-Graham
expansion30,31
ds2 =1
z2
[dz2 −
(1− µ
4z2)2dt2 +
(1 +
µ
4z2)2dx2
], (7.15)
which upon comparison to the general FG metric (6.12) tells us that the first order metric
perturbation δgµν = z2Hµν is given by Htt = Hxx = µ/2. Using (6.20), we may relate the
metric perturbation to the boundary stress tensor as δ〈Ttt〉 = µ/8πG, which tells us that δ〈Ttt〉is contant along the field theory space. We can therefore evaluate both integrals in (7.12) to
obtain
δSDE =`Lxµ
24G. (7.16)
We would like to reproduce (7.16) from the first order area variation of a bulk surface with
constant profile z = z∗, which was shown to be dual to the differential entropy in chapter 5. The
crucial question that arises is what quantity we ought to keep fixed when doing the variation.
For the variation of a minimal RT surface, we kept the end-points of the surface fixed and varied
both its shape and the metric. In chapter 6, we furthermore argued that the shape is unchanged
to first order, so it is only the metric which changes. The fact that we keep the end-points fixed
makes sense because the size of the boundary interval does not change, only the energy within
the interval does. The same physical argument applies to the present case: we definitely ought
to keep the value of ` fixed. What quantity, then, does change due the metric perturbation? It
29By analogy with the ordinary first law, the quantity on the right could be referred to as the “diffential energy”δEDE . However, its interpretation is unclear. It is the energy present on the entire spatial part of field theorywithin a finite time interval. Therefore, it cannot be represented as the conserved charge of a single local geometricflow.
30In this section, we set the AdS radius to unity.31Starting from the BTZ metric
ds2 = −(r2 − µ)dt2 +dr2
r2 − µ + r2dφ2, (7.14)
we define a new coordinate z through the differential equation dz/z = −dr/√r2 − µ, whose solution yields
z = 2µ
(r −
√r2 − µ
), where the constant is chosen such that z ∼ 1/r for r → ∞. Inverting the relation from
z(r) into r(z) yields r = 1/z + (µ/4)z, which can be plugged in the above BTZ metric to obtain metric (7.15),where x is simply the non-compact boundary coordinate obtained by replacing x = φ.
7 FIRST LAW OF DIFFERENTIAL ENTROPY 65
is the value of z∗, i.e. the location of the tips of the geodesics that form the bulk curve. We
refer to figure 12 in chapter 5 for an illustration of the setup.
Our aim is therefore to compute the new value of z∗. The area for a single curve γ extending
from x = 0 to x = ` with profile x = x(z) in metric (7.15) is given by
A′(γ) = 2
∫ z∗
ε
dz
z
√1 + f(z)x2, f(z) ≡ 1 + (µ/2)z2. (7.17)
where the prime refers to the perturbed metric, ε is the IR cutoff and x = dx/dz. The shape of the
curve is determined by minimizing the area functional under the variation x(z)→ x(z) + δx(z),
δA(γ) = 2
∫ z∗
εdz
[d
dz
(1
z
x√1 + f(z)x2
)]δx, (7.18)
where we have partially integrated to express the integral as a variation of x rather than x.
If (7.18) is to vanish for all variations δx, then the term in parentheses must be a constant
independent of z. To fix the constant, we evaluate its value at the turning point z = z∗, where
the derivative x(z∗) is infinite, so the constant is given by 1/z∗√f∗, where f∗ ≡ f(z∗). With the
expression for the constant at hand, we can express the term in parentheses appearing in (7.18)
as
x(z) =
√z2
f∗z2∗ − f(z)z2
, (7.19)
which upon integration gives an implicit equation for z∗,
`
2=
∫ z∗
0dz
√z2
f∗z2∗ − f(z)z2
. (7.20)
Unfortunately, (7.20) cannot be solved exactly. However, the starting premise (7.13) justifies
an expansion of the integrand to first order in µ`2. This expansion allows us to evaluate the
integral, which yields a third order equation for z∗, which in turn can also be expanded to first
order in µ`2, yielding
z∗ =`
2− µ`3
96+O
(µ2`4
). (7.21)
We now compute the area of the hole in the perturbed metric, which we denote by A′, as
A′(z = z∗) =
∫ Lx
0dx
1
z∗
√1 + (µ/2)z2
∗ . (7.22)
Doing one final expansion to first order in µ`2 of (7.22) and subsequently substracting the
vacuum contribution, i.e. the zero’th order term, gives a precise match with the CFT result
(7.16),δA(z = z∗)
4G=`Lxµ
24G= δSDE. (7.23)
This agreement was of course to be expected. In section 5.3 we (following [15]) already
demonstrated that the differential entropy in BTZ coordinates correctly reproduces the bulk
7 FIRST LAW OF DIFFERENTIAL ENTROPY 66
curve’s area. The first order variation is then obtained through the Taylor expansion around
µ`2 of both sides of this equality, yielding (7.23). However, the above calculation serves as a
straightforward check for the first law of differential entropy (7.12).
7.2.3 Dfferential entropy and the Einstein equations
In the above, the metric perturbation Hµν was constructed from the difference between the BTZ
metric and the vacuum. As these are both solutions of the Einstein equations, the perturbation
must also be on-shell. Now what about arbitrary, not necessarily on-shell perturbations? One
might wonder whether the first law of differential entropy constraints the metric consistent with
the Einstein equations. There is a major against such an aim. In order to obtain the holographic
dual of δSDE, i.e. the variation of the hole’s area, it is not immediately clear what quantity we are
keeping fixed. As mentioned, the length of the boundary interval does not change for obvious
physical reasons. Thus, whenever we wish to compute the change in area of the hole, there
seems to be no other option but to first compute the maximal distance of the geodesic in the
perturbed metric, which gives the new radius of the hole. However, determining this distance
already requires the Einstein equations! At present, we see no way to circumvent falling into
circular reasoning when trying to derive the Einstein equations from the first law of differential
entropy.
8 CONCLUSION 67
8 Conclusion
We have focused on several aspects of the entanglement entropy. In quantum mechanical sys-
tems, it is defined as the von Neumann entropy associated to a reduced density matrix ρA. It is
extremely difficult to evaluate the entanglement in a generic quantum field theory. However, for
two-dimensional conformal field theories, the entanglement entropy may sometimes be computed
using the replica trick. This approach is characterized by the presence of twist operators, whose
correlation functions encode all the information needed to compute the entanglement entropy.
The replica trick was used to calculate the entanglement entropy of the ground state and a ther-
mally excited state. For a general low-energy excited state the method is similar, although the
operator insertions associated to the excited state introduce a 2n-point function to the partition
function. In the small interval limit, the 2n-point function can be approximated by its OPE,
leading to a universal result in terms of the scaling dimension of the operator generating the
excited state.
For holographic CFTs with a classical gravity dual, the RT prescription [7] states that
the entanglement entropy of a field theory region is equivalent to the area of a minimal surface
homologous to this field theory region. Thus, the RT formula may be used as a simple alternative
for the difficult field theory calculations. We have reviewed some simple calculations in AdS3 in
section 4.3, finding excellent agreement with the CFT results. In section 4.4, the entanglement
entropy was calculated in a conical metric. By comparing this to the result for an excited state
in the CFT, it follows that the scaling dimension of the operator in the dual field is precisely
that of a twist operator. This observation agrees with the fact that the field theory dual of
the conical metric is an orbifold CFT [36]. Our result can thus be interpreted either as an
independent check of this latter statement, or as a holographic check for the validity of the CFT
calculation of the entanglement entropy for excited states.
The RT formula also conveys a deep connection between geometry and entanglement. Adopt-
ing the view that the CFT completely describes quantum gravity at the UV, the radial bulk
dimension of AdS can be considered as an emergent one. The formula S = A/4G may then be
thought of as the definition of the geometric of classical spacetimes. In fancy language, we say
that spacetime emerges as a geometrization of entanglement. However, the RT formula singles
out a very special class of surfaces, namely minimal surfaces whose ends are anchored to the
boundary. The reconstruction of more general bulk curves from residual entropy due to [15]
definitely solidifies the emergent spacetime conjecture. Their work was reviewed in chapter 5,
where two novel notions of entropy were discussed: residual entropy and differential entropy.
Residual entropy measures the collective uncertainty associated to a family of observers. A
convenient working hypothesis is that all observers can determine the complete reduced density
matrix on the spatial region they have access to (which is unlikely to be true in practise). The
question of residual entropy then becomes a mathematical one: for a set of a priori given re-
duced density matrices rhoi, how many global density matrices consistently project onto the
set rhoi? A suitable definition of the residual entropy is the von Neumann entropy of the
global density matrix with maximal von Neumann entropy. In section 5.2, we presented some
examples of simple spin systems exhibiting residual entropy. However, for a generic quantum
8 CONCLUSION 68
field theory, it is extremely difficult to calculate the residual entropy. Even when restricted to
translationally invariant systems, such that both ρi and Λi are independent of i, the matching of
the matrix components of Λi to those of ρi is an intractable problem.32 Computing the residual
entropy in a holographic CFT thus seems like a hopeless task. However, the authors of [15]
make clever use of strong subadditivity to place an upper bound on the residual entropy. The
quantity that bounds it is called the differential entropy. In a particular continuum limit, this
quantity reproduces the area of a closed circular bulk curve, whose radius is determined by the
turning point of the geodesics associated to the single intervals.
In this thesis, we have also focused on the idea that if geometry emerges from entanglement,
then fluctuations in entanglement are governed by the Einstein equations. In chapter 6, we
reviewed some recent advances in this direction [11, 12], and in particular the work of [13]. In
particular, we have considered the entanglement first law applied to ball-shaped regions in a
holographic CFT, where the entanglement entropy is calculated via the RT formula. The first
law holographically translates in a non-local integral constraint on the dual metric perturbation.
Since the first law makes infinitely many predictions − one for every ball in any Lorentz frame in
the boundary geometry − these non-local constraints can be converted into the local linearized
Einstein equations. The drawback of this approach is that it only gives us the Einstein equations
at a linearized level. In contrast, the work of Jacobson [14] provides a thermodynamic derivation
of the full non-linear Einstein equations. To summarize his line of reasoning, Jacobson assumes
that a local Rindler horizon with Unruh temperature T enjoys an entropy/area relation S =
A/4G. A generic area variation of this horizon is governed by the expansion of the null geodesics
generating the horizon. In the presence of some local energy flux δE, the thermodynamic first
law δE = TδS can then only be valid if the back-reacted geometry satisfies the full non-linear
Einstein equations. The drawback is that this is a phenomenological derivation, because there
is no fundamental microscopic understanding of the entropy.
Combining the best of both worlds, we would ideally have microscopic explanation for the
full non-linear Einstein equations. Thus, in the final chapter, we presented some preliminary
ideas towards a holographic version of Jacobson’s approach. We argued that the differential
entropy can be used to isolate local segments of the AdS-Rindler horizon, which might be
fruitful because local Rindler horizons are essential in Jacobson’s line of reasoning. We derived
a first law of differential entropy, whose input is the entanglement first law and the continuum
limit of the differential entropy. A simple calculation in the BTZ geometry was used to verify
that for on-shell perturbations, the first law of differential entropy is indeed satisfied. However,
when attemping to invert the logic to derive the Einstein equations, we have stumbled upon a
major greater limitation, namely the lack of suitable boundary conditions for the variation of
a segment of the hole. We ought to keep the length of the boundary intervals fixed, but the
only way this boundary condition can be imposed while doing the variation of the hole’s area is
by using the equations of motion of the bulk metric and propagating inwards to determine the
new radius of the hole. Therefore, it seems unlikely that the approach outlines in chapter 7 is
suitable to holographically reconstruct Jacobson’s argument.
32We cannot even compute the residual numeraically for just a few degrees of freedom in a simple interactingspin chain.
A SCALAR CURVATURE OF CONICAL SPACES 69
A Scalar curvature of conical spaces
Here we calculate the scalar curvature for conical spaces, based on [54]. Consider a two-
dimensional cone C, with metric
ds2C = dρ2 + ρ2dφ2, 0 ≤ φ ≤ 2πα. (A.1)
It is convenient to embed the cone three-dimensional Euclidean space
ds2 = dx2 + dy2 + dz2, (A.2)
with embedding coordinates
x = αρ cos(φ/α), y = αρ sin(φ/α), z =√|1− α2|ρ, (??) (A.3)
that define the conical surface
z2 − |1− α2|
α2(x2 + y2) = 0, z ≥ 0, (A.4)
which is characterized by a singularity at z = 0. It is straightforward to verify that the induced
metric on this surface gives back (A.1).
To calculate the curvature of the cone, we consider a regularized surface C. The regularized
surface C is the same as the original cone away from the tip, but the tip is now smooth instead
of singular (see figure 17). It has the embedding coordinates x and y as before, but now
z =√|1− α2|f(ρ, a), (A.5)
where f(ρ, a) is a smooth function with regularization parameter a, such that through lima→0 f =
ρ we get back the original surface. For C, the function z has a minimum at ρ = 0, so the
regularized surface C should also have a minimum at this point. We thus infer the following two
conditions on f(ρ, a):
lima→0
f = ρ, ∂ρf |ρ=0 = 0, (A.6)
It is straightforward to verify that the induced metric on C is given by
ds2C = udρ2 + ρ2dφ2, u = α2 + (1− α2)(f ′)2, (A.7)
where the prime denotes differentiation with respect to ρ and u has the asymptotics
u|ρ=0 = α2, u|ρa = 1, (A.8)
which follow from the conditions on f .
We will now consider an explicit example of this regularization scheme. Let C be a hyper-
A SCALAR CURVATURE OF CONICAL SPACES 70
Figure 17:
boloid, defined by the surface
z2 − |1− α2|
α2(x2 + y2) = |1− α2|a2, z ≥ 0, (A.9)
in embedding space (A.2). Expression (A.9) corresponds to a choice f =√ρ2 + a2, which clearly
satisfies the conditions in (A.6). The induced metric on the regularized cone is given by (A.7),
where the function u is now explicitly given by
u =ρ2 + a2α2
ρ2 + a2. (A.10)
It can easily be checked that the Ricci scalar on C is given by RC = u′/ρu2. Taking into account
the volume element√hdρdφ =
√uρdρdφ, where h is the induced metric on C, we can compute
the integral of the Ricci scalar as∫Cdρdφ
√hRC =
∫ 2πα
0dφ
∫ ∞0
dρu′u−3/2
= 2πα
∫ 0
α2
u−3/2du
= 4π(1− α), (A.11)
where in the second line we used the asymptotics (A.8) to change the variable of integration
from ρ to u. Observing that the result does not depend on the regularization parameter a, and
because the original singular surface C is obtained via C = lima→0 C, we infer that the integral
curvature on C yields the same answer (A.11). Now, the defining feature of conical spaces is
that all curvature is concentrated at the singularity (its tip), from which it follows that the Ricci
scalar of a cone must be a delta function,
RC = 4π(1− α)δ(ρ). (A.12)
As a final comment, note that upon embedding the cone in a space with non-vanishing back-
ground curvature R(0), the Ricci scalars simply add to give R = 4π(1− α)δ(ρ) +R(0).
B IYER-WALD FORMALISM 71
B Iyer-Wald formalism
The formalism developed by Iyer and Wald applies to any gravitational theory, specified by a
gravitational Lagrangian density L. We shall review the general formalism. Afterwards, we
apply the formalism to Einstein gravity, and show that imposing the gravitational first law
indeed constrains the metric on-shell.
Covariant formulation of classical particle mechanics
As a stepping stone to the covariant formulation of field theories, we review phrasing ordi-
nary classical particle mechanics in covariant language. Most notions introduced here can then
straightforwardly be applied to field theories. Commencing, let L be the Lagrangian for particle
paths q(t), whose variation is given by
δL =
[∂L
∂q− d
dt
∂L
∂q
]δq +
d
dt
[∂L
∂qδq
]. (B.1)
The equations of motion are satisfied when the first term in brackets vanishes
E =∂L
∂q− d
dt
∂L
∂q= 0. (B.2)
The second term of (B.1) is a surface term and is known as the symplectic potential Θ,
Θ(q, q) ≡ ∂L
∂qδq = pδq, (B.3)
which is usually neglected to obtain the equations of motion. However, the symplectic potential
can also be used to obtain the Hamilton equations of motion. In order to do so, take another,
antisymmetrized variation of Θ and define the symplectic current Ω as
Ω(q, δ1q, δ2q) = δ1Θ(q, δ2q)− δ2Θ(q, δ1q) (B.4)
= δ1pδq − δ2pδ1q. (B.5)
For specific choices of the variations, the quantity (B.5) defines a variational expression for the
Hamiltonian. Choosing δ1 = δ and δ2 = ∂t defines the Hamiltonian as
δH = Ω(q, δq, q) (B.6)
= δpq − pδq, (B.7)
from which the equations of motion can be read off as
q =∂H
∂pp = −∂H
∂q, (B.8)
which are of course the familiar Hamilton equations of motion.
In summary, we see that the surface term obtained from Lagrangian variation still ‘knows’
B IYER-WALD FORMALISM 72
about the equations of motion and can even be used as a fundamental quantity to derive them.
Covariant formulation of classical field theories
We will now extend the above formalism to classical field theories. Consider a (d+1)-dimensional
spacetime, on which we define a gravitational Lagrangian L as a (d+1)-form. We use the symbol
φ to denote all the dynamical fields, including the metric. When we vary the Lagrangian with
respect to the fields, we obtain the equations of motion and an additional surface term,
δL = Eφδφ+ dΘ(δφ), (B.9)
where the equations of motion are Eφ = 0 and the boundary term Θ is called the symplectic
potential, which is a d-form. We can again consider an antisymmetrized variation of Θ, defining
the symplectic current density
ω(δ1φ, δ2φ) = δ1Θ(δ2φ)− δ1Θ(δ1φ), (B.10)
from which the symplectic current Ω is then obtained by integrating the symplectic current
density over a Cauchy surface
Ω(δ1φ, δ2φ) =
∫Cω(δ1φ, δ2φ). (B.11)
If one of the variations is with respect to the dynamical fields and the other with respect to a
vector field ξa, we define the Hamiltonian canonically via the differential equation
δH[ξ] = Ω(δφ, δξφ) =
∫Cω(δφ, δξφ), (B.12)
which can in principle be integrated to obtain the Hamiltonian.
In summary, we have again obtained a differential equation for Hamiltonian from specific,
antisymmetrized variations of the surface term Θ, now integrated over a spatial domain C.
Diffeomorphism covariant field theories
So far, the discussion has been quite general and applies to all classical field theories. General
relativity, and its extensions, are special cases of field theories: they are diffeomorphism covari-
ant, which implies that the Lagrangian depends solely on dynamical fields. In other words, it
has no fixed background structure. In general, diffeomorphisms are generated by vector fields
ξa. The variation of the Lagrangian under a diffeomorphism generated by ξa is then given by
Cartan’s magic formula
δξL = d(ξ · L) + ξ · dL (B.13)
= d(ξ · L), (B.14)
B IYER-WALD FORMALISM 73
where we used the fact that L is a top-form, so its exterior derivative vanishes.33 For diffeo-
morphism covariant theories, diffeomorphisms represent a local symmetry of the Lagrangian.
Therefore, Noether’s theorem guarentees that the vector field ξa which generates the diffeomor-
phisms induces a Noether current that is conserved on-shell. This Noether J[ξ] current is a
d-form, defined by [45]
J[ξ] = Θ[ξ]− ξ · L, (B.15)
where the dot denotes contraction of ξa with the first index appearing in the n-form L.
To show that J is conserved when the equations of motion are satisfied, we may use equations
(B.9), (B.14) and (B.15) in reversed order, yielding
dJ[ξ] = dΘ(δξφ)− d(ξ · L)
= dΘ(δξφ)− δξL
= −Eφδξφ, (B.16)
which implies that J is indeed conserved on-shell, as promised.
Equation (B.16) states that the Noether current J is a closed form. It was shown in [50] that
there exists a more stringent condition on J when the equations of motion are satisfied, namely
that it is also an exact form,
J[ξ] = dQ[ξ] (on-shell), (B.17)
where Q is the corresponding Noether charge, expressed as a (d− 1)-form. In [45], it was shown
that an off-shell definition of J exists, namely
J[ξ] = dQ[ξ] + ξaCa (off-shell), (B.18)
where Ca = 0 are constraint equations for the fields in the theory. For the metric, which is a
(0, 2)-tensor, these constraint equations are explicitly given by (cf. equation (5.12) of [13]):
Ca = 2Egabεb, (B.19)
where Egab are the metric equations of motion, i.e. the Einstein equations and εb is the d-
dimensional volume form defined in (B.31).
In summary, varying a diffeomorphism invariant Lagrangian with respect to diffeomorphisms
represents a symmetry of the theory. The corresponding Noether current J is conserved when
the equations of motion are satisfied, but can be defined off-shell in terms of the Noether charge
Q and the equations of motion Egab.
33On an n-dimensional spacetime, the highest non-vanishing form is an n-form. This is understood by notingthat forms are completely antisymmetric in their indices. Therefore, any k-form with k > n vanishes by thisantisymmetrization.
B IYER-WALD FORMALISM 74
Definition of energy
The gravitational energy may be defined as the conserved charge associated to the Noether
current (B.15). First, we use (B.15) to write
δJ[ξ] = δΘ(δξφ)− δ(ξ · L)
= δΘ(δξφ)− ξ · (dΘ(δφ) + Eφδφ)
= δΘ(δξφ)− δξΘ(δφ) + d(ξ ·Θ(δφ))
= ω(δξφ, δφ) + d(ξ ·Θ(δφ)). (B.20)
Let us explain the steps that we used. In the second line, we used (B.9) and furthermore assumed
that the AdS background metric around which we are considering small perturbation satisfies
the Einstein equations34, Eφ = 0, and finally that ξa is held fixed, i.e. δξa = 0. The third
line follows from Cartan’s magic formula δξu = ξ · du + d(ξ · u) and the forth line follows from
definition (B.10).
Using the definition of the Hamiltonian (B.12) and off-shell expression (B.18) for the Noether
current, we then find
δH[ξ] =
∫C
(δJ[ξ]− d(ξ ·Θ(δφ)
)= δ
∫CξaCa +
∫∂C
(δQ[ξ]− ξ ·Θ(δφ)
), (B.21)
where we used Stokes’ theorem in the second line.
Let us make the observation that when the equations of motion are satisfied, Ca = 0, the
Hamiltonian is a pure boundary term, which reflects the holographic character of gravitational
theories. We define the energy for an arbitrary perturbation of the background as this surface
contribution [13]:
δE[ξ] =
∫∂C
(δQ[ξ]− ξ ·Θ(δφ)
). (B.22)
In summary, we have given a definition of the energy associated to the vector field ξa. To
make contact with the setup from the previous section, we note that the boundary of the Cauchy
surface Σ = B∪B. Furthermore, the vector field ξ is in the Killing vector field ξaB which vanishes
on B, so that we get
δEgravB =
∫B
(δQ[ξB]− ξB ·Θ(δφ)
). (B.23)
Definition of entropy
Wald [55] has taught us that the entropy of a bifurcate Killing horizon can be constructed from
a local, geometrical quantity on the bifurcation surface. This geometrical quantity is precisely
the Noether charge Q we have considered earlier. Explicitly, it was shown that the following
34We wish to show that linearized perturbations to this background also satisfy Einstein’s equations, i.e. δEg =0.
B IYER-WALD FORMALISM 75
relation holds
SWald =2π
κ
∫H
Q[ξK ], (B.24)
where H is the bifurcate surface of the Killing horizon generated by Killing vector ξK and κ is
the surface gravity. We can normalize the Killing vector is such a way that the surface gravity
κ = 2π, in which case the prefactor is unity.
It is not immediately obvious that the first order variation of the Wald entropy is in fact
equal to the first order variation of the holographic entanglement entropy Sgrav. In general,
these quantities in fact differ for general gravitational theories. However, as argued in [13], these
two quantities coincide when the extrinsic curvature of the extremal surface vanishes. Since
extremal surface is the bifurcate surface of a Killing horizon, this condition is indeed satisfied to
linear order in the perturbations we are considering. Therefore, we have the following expression
for the first order variation of the holographic entanglement entropy
δSgravB =
∫BδQ[ξB]. (B.25)
Definition of χ
Recall that the form χ described in section 6.2 must satisfy the two properties in (6.24). From
expression (B.23) for the energy, we infer a plausible definition for χ
χ = δQ[ξB]− ξB ·Θ(δφ), (B.26)
from which δEgravB =∫B χ follows by definition. Does this forms also satisfy its relation to
the entropy? To see that it does, substite this definition in equation (B.25) and note that ξB
vanishes on B:
δSgravB =
∫BδQ[ξB] =
∫Bχ. (B.27)
We see that this choice of χ indeed satisfies the two properties in (6.24).
It remains to show that the form satisfies the off-shell relation (6.25). To see that it does,
note that for ξB a Killing vector of the background, the variation of the Noether current J
becomes δJ[ξB] = d(ξB ·Θ). This is because the Lie derivative δξφ vanishes for Killing vectors.
Therefore,
dχ = δ(dQ[ξB]− J[ξB]) = −ξaBδCa = −2ξaBδEgabε
b, (B.28)
where the second equality follows from the off-shell relation (B.18) for J and the last equality
follows from (B.19).
Thus, we have shown that the form χ satisfies all its requirement in full generality. In
particular, we made no explicit reference to any Lagrangian. We stress that it is simply the
existence of a form with these properties that guarentees that the linearized Einstein equations
are equivalent to the first law. In other words, it is possible to recover the linearized Einstein
field equations from the first law without knowing the explicit expressions for χ,L,Θ, or Q.
B IYER-WALD FORMALISM 76
Conventions on differential forms
On a (d+ 1)-dimensional spacetime, we define the volume form as
ε =1
(d+ 1)!εa1...ad+1
dxa1 ∧ · · · ∧ dxad+1 , (B.29)
where εa1...ad+1is the antisymmetric tensor, with sign convention
εzti1...id−1= +√−g. (B.30)
We also define lower dimensional volume forms
εa =1
d!εab2...bd+1
dxb2 ∧ · · · ∧ dxbd+1 , εab =1
(d− 1)!εabc3...bd+1
dxc3 ∧ · · · ∧ dxcd+1 , (B.31)
which give the volume elements on a d- and (d− 1)-dimensional hypersurface, respectively.
REFERENCES 77
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