University of AmsterdamUniversity of Amsterdam MSc Physics Track: Theoretical Physics Master Thesis...

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].


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


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


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.


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 `.


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].


3.3 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.


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)


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


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).


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


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.


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)


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).


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


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.


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.


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)


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


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)


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].


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)


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.


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


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)


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 .


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.


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.


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


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.


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


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.


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.


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.


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


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).


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.


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


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’


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)


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.


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.


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.


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.

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