Considerations Regarding AdS Backreaction and Holography1117777/FULLTEXT01.pdfframework. This led to...

Master Thesis

Considerations Regarding AdS2Backreaction and Holography

Julius Engelsoy

Theoretical Particle Physics, Department of Theoretical Physics,School of Engineering Sciences,

KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden

Stockholm, Sweden 2016

Typeset in LATEX

TRITA-FYS 2016:62ISSN 0280-316XISRN KTH/FYS/--16:62--SE

c© Julius Engelsoy, September 2016Printed in Sweden by Universitetsservice US AB, Stockholm September 2016

Abstract

This thesis investigates the ADS2 dilaton gravity model studied by Almheiri andPolchinski in [1] and proposes a dual boundary theory. From the bulk model wederive a general equation of motion for the reparametrized boundary time and in-vestigate pulse solutions. We include quantum effects and calculate an expressionfor the boundary stress–energy tensor whose “classical” part turns out to be pro-portional to a Schwarzian derivative of the preferred coordinate transformation. Westudy black hole evaporation which is shown to decay exponentially. The boundarytheory can be reduced to a Schwarzian derivative action suggesting a connectionto the Sachdev–Ye–Kitaev model [2]. We investigate the canonical structure of thetheory and verify that it reproduces both the reparametrization equation and theboundary energy as obtained from the bulk system. Finally, we compute the matterPoisson bracket and find that the system is maximally chaotic as conjectured byMaldacena, Shenker, and Stanford [3].

Key words: AdS2, AdS/CFT Correspondence, Dilaton Gravity, Backreaction,CFT, Holography, Black Holes, Schwarzian Derivative Action, SYK Model.

Sammanfattning

Detta examensarbete undersoker en AdS2-dilatongravitationsmodell som studeratsav Almheiri and Polchinski i [1] och foreslar en dual randteori. Fran bulkmodellenharleder vi en generell rorelseekvation for den reparametriserade randtiden ochberaknar randens stressenergitensor vars ”klassiska” del visar sig vara porportionellmot en Schwarzisk derivata av den priviligierade koordinattransformationen. Vistuderar forangning av svarta hal vilken vi visar avtar exponentiellt. Randteorinkan reduceras till en verkan bestaende av en Schwarzisk derivata vilket antyderen koppling till Sachdev–Ye–Kitaev-modellen [2]. Vi undersoker teorins kanoniskastruktur och verifierar att den reproducerar bade reparametrisationsekvationen ochrandens stressenergitensor sasom de beraknas i bulkteorin. Till sist beraknar vimateriafaltets Poissonparentes och finner att systemet ar maximalt kaotiskt somformodat av Maldacena, Shenker och Stanford [3].

Nyckelord: AdS2, AdS/CFT-Korrespondens, Dilatongravitation, Bakatreaktion,CFT, Holografi, Svarta Hal, Verkan med Schwarzisk Derivata, SYK-Modellen.

iii

Preface

The research underlying this thesis was conducted during the spring semester of2016 at Princeton University under the supervision and guidance of Prof. HermanVerlinde and in collaboration with Dr. Thomas G. Mertens. The outcome of thatresearch was the paper “An investigation of AdS2 backreaction and holography”published in JHEP 07 (2016) 139 and with arXiv identifier arXiv:1606.03438 [4].This thesis can be viewed as containing some of the basic considerations that wentinto the more advanced investigation of the paper.

Acknowledgements

I would like to express my deepest gratitude toward my supervisor at PrincetonUniversity, Prof. Herman Verlinde, for welcoming me to Princeton, guiding methrough the research process, and for his clear vision and fruitful ideas. I wouldalso like to thank Dr. Thomas G. Mertens for teaching me everything worth know-ing about being a young researcher and for all the very helpful discussions anddeep insights that have contributed greatly to the completion of this thesis. Lastly,I would like to thank my Swedish supervisor Prof. Tommy Ohlsson for his greatadvice throughout this process.

v

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Preface v

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Contents vii

1 Introduction 1

2 The Almheiri–Polchinski model 5

2.1 Coordinate patches in AdS2 . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 The Poincare patch . . . . . . . . . . . . . . . . . . . . . . 8

3 The boundary equations of motion 11

3.1 Example: The black hole frame and infalling pulses . . . . . . . . . 13

3.1.1 Black hole frame set-up . . . . . . . . . . . . . . . . . . . . 14

3.1.2 Infalling pulse in the black hole frame . . . . . . . . . . . . 15

3.2 Example: The global frame and infalling pulses . . . . . . . . . . . 17

3.2.1 Global frame set-up . . . . . . . . . . . . . . . . . . . . . . 17

3.2.2 An infalling pulse in the global frame . . . . . . . . . . . . 18

4 Quantum effects of the matter fields 21

5 The boundary stress tensor 23

5.1 The stress tensor in the frame respecting the equation of motion . 25

5.1.1 Black hole formation and evaporation . . . . . . . . . . . . 28

5.1.2 The eternal heat bath . . . . . . . . . . . . . . . . . . . . . 29

5.1.3 The evaporating black hole . . . . . . . . . . . . . . . . . . 31

5.2 The quantum contribution to the boundary stress tensor . . . . . . 34

vii

viii Contents

6 A boundary theory 376.1 Canonical structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.2 SL(2,R) symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.3 Bulk dynamics from the boundary . . . . . . . . . . . . . . . . . . 416.4 External constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 426.5 The matter Poisson bracket . . . . . . . . . . . . . . . . . . . . . . 42

6.5.1 Solutions to the equations of motion . . . . . . . . . . . . . 426.5.2 The Poisson bracket . . . . . . . . . . . . . . . . . . . . . . 44

7 Conclusion 45

A A subtlety in the boundary equation of motion 47

B Multiple-pulse dynamics 49B.1 Consistency check using infalling pulses . . . . . . . . . . . . . . . 49B.2 Other frames obtained by time delays . . . . . . . . . . . . . . . . 50

C Poisson bracket computation 53

Bibliography 58

Chapter 1

Introduction

The last century has seen some of the greatest advances in physics ever witnessedto date. There were two major discoveries made in the early 1900s leading theway for subsequent developments. The first was the realization made by Planck in1900 that an assumption of energy quantization was needed in order to describethe black body spectrum which eventually gave rise to the development of quantummechanics [5]. The other was the development of the special, and later, the generaltheory of relativity made by Einstein in 1905 through 1915 which he and otherscontinued in the 1920s and onward (see e.g., [6, 7]).

Only a few months after Einstein published his first paper on general relativitycontaining the famous Einstein field equations in 1915, the physicist and astronomerSchwarzschild solved the equations for a spherical, non-rotating mass while servingin the German army in the trenches of the Russian front during World War I.1 Thissolution turned out to be the basis of the first quantitative description of a blackhole which now assumes a special role in the modern field of quantum gravity.

In the 1920s, quantum mechanics was developed by physicists such as Dirac,Heisenberg, and Schrodinger, and in 1927, Dirac published a paper widely regardedas representing the inception of quantum field theory (QFT) [8]. For the firsttime special relativity could be incorporated in quantum mechanics, and hence,relativistic particles such as photons could be described in a quantum mechanicalframework. This led to the development of quantum electrodynamics (QED), andlater, the entire standard model (SM) in theoretical particle physics.

However, the problem of quantization in curved backgrounds remained a puzzle;an all-encompassing theory containing both quantum mechanics and gravitationwas still not in reach. The issue arises when varying the action with respect tothe metric introduces a new time-slicing of the spacetime in turn affecting thetime-ordering of quantum operators. Another way the problem manifests itself isthrough the nonrenormalizability of a QFT involving gravitons—the quanta of the

1Schwarzschild died the following year in complications caused by a hereditary autoimmunedisease.

1

2 Chapter 1. Introduction

gravitational field. For later reference, because of the two spacetime indices of themetric, gravitons are spin-2 particles.

In the late 1960s, phenomenological studies of hadron collisions gave rise toa theory in which different particle excitations were described by a model basedon quantized relativistic strings. The model had some initial success in describ-ing hadron collisions but certain peculiarities in the implications of the model ledphysicists to consider other applications. One such peculiarity was the inevitabilityof the theory’s prediction of spin-2 particles. This led researchers to consider thepossibility that this “string theory” in fact was describing quantum gravity. So far,string theory has had some important successes in quantum gravity but the lack ofempirical predictiveness still haunts the field.

Around the same time that string theory was developed, i.e., in the 1970s,Bekenstein and Hawking made major contributions to the understanding of thequantum nature of black holes. They arrived at expressions describing the entropy2

and temperature of black holes. Hawking also showed that black holes evaporatewith a thermal spectrum giving rise to the famous information paradox [10]. Itwas evident that a further understanding of this phenomenon required a theory ofquantum gravity and Hawking’s findings reminded researchers of the urgency ofdeveloping such a unified theory.

In 1993, ’t Hooft presented some speculative work on quantum gravity in whichhe concluded that the number of degrees of freedom in a region of spacetime sur-rounding a black hole is proportional to the area of the horizon [11]. This notionwas subsequently promoted by Susskind and is now called the holographic principle[12]. The remaining challenge was to incorporate this idea into a tractable theoryof quantum gravity.

The breakthrough came in late 1997 when Maldacena conjectured a dualitybetween theories containing strings in an anti-de Sitter spacetime (AdS) and lesser-dimensional quantum mechanical conformal field theories (CFT)—a manifestationof holography [13]. Shortly thereafter, the so-called “dictionary” between corre-sponding quantities in the respective dual theories was worked out to a large extentby Gubser, Klebanov, and Polyakov [14] and Witten [15, 16], most notably betweenasymptotically AdS spacetimes containing Schwarzchild black holes and finite tem-perature thermal states in the dual CFT [16].

AdS2 (AdS in 2 spacetime dimensions) has since proven to be problematic in thecontext of understanding holography. Some reasons being that any finite energyperturbation causes a runaway backreaction [17, 18] and that it is still unclearwhether one should think of CFT1 (CFT in 1 dimension) as conformal quantummechanics or as a chiral sector of CFT2 [19].

Recently, Almheiri and Polchinski [1] have considered a model that in the IRregime reduces to pure AdS2, but in the UV regime gets adjusted by a non-trivialdilaton profile. This regulates the infinite backreaction, allowing one to set up a

2This expression for the entropy of a black hole was actually reproduced by Strominger andVafa in [9] from a string theoretical perspective and counts as one of the successes of string theory.

3

more meaningful holographic dictionary. The model was first proposed some timeago by Jackiw [20] and Teitelboim [21].

One of the conclusions that Almheiri and Polchinski arrived at is that the timecoordinate on the boundary needs to be reparametrized dynamically as time pro-gresses. Only then, for instance, do the correlators match with holographic expecta-tions. Another interesting aspect is the application of the holographic stress tensorrecipe to this spacetime, in particular when including one-loop quantum effects inthe bulk.

Of course, studying quantum field theory in AdS2 purely from the bulk per-spective has been done extensively in earlier works. If one restricts oneself to onlymassless scalar fields propagating in the bulk, then conformal invariance of thismatter action in 2d provides powerful techniques for getting concise quantitativeresults in a relatively simple way [22–24]. Different quantum vacua and their rolein Hawking emission and Unruh heat baths were studied for instance in [25].

A different motivation stems from studies of moving mirror models in the early1990s as 2d analogues of black hole evaporation [26, 27]. Equations of motionwere found that describe the dynamical trajectory of the mirror upon allowingbackreaction of the quantum fields in the asymptically flat spacetime. We here seekto analyze the question whether there is a way in which the holographic boundaryitself can be viewed in a similar way.

In this work, we aim to further investigate the explicit Almheiri–Polchinskimodel, exploiting the results on bulk quantum field theory for 2d massless scalarsand combining it with the more recent holographic perspective and in particularthe time evolution of the boundary observer.

Our goal is twofold. We will first investigate the boundary time reparametriza-tion equation and the holographic stress tensor. Both of these are important aspectsof 2d holographic dilaton gravity. The main advantage of this model, however, isthat backreaction effects can be fully resolved. We will analyze several situationsof interest, starting out with classical pulses and then introducing quantum effects.In particular, we build up towards applying these results to the evaporation of a2d black hole, which we will be able to describe fully analytically in this model,yielding results in unison with a back-of-the-envelope evaporation estimate.

The second goal will be to reproduce the bulk dynamics by a 1d system thatsupposedly lives on the boundary. This 1d system is interesting in its own right,as it represents the canonical formulation of a theory with a Schwarzian derivativeaction. Such an action was suggested by Kitaev [2] as providing an effective actionfor the so-called Sachdev–Ye–Kitaev (SYK) model, first proposed to be connectedto AdS2 holography in [28]. It has been studied further in [29–33]. An alterna-tive perspective on this system is viewing it as an effective model reproducing thenear-boundary dynamics of bulk the system with some resemblance to the actionsconstructed for the moving-mirror models in the past.

An interesting way to investigate the correspondence between the bulk andboundary systems is to study the connection between shockwaves in the bulk and

4 Chapter 1. Introduction

chaotic behavior in the boundary theory. The key constituents of such an inves-tigation are presented in this thesis but for the full analysis the reader is referredto [4]. The correspondence has been studied extensively in recent years [3, 34–38].As for the Almheiri–Polchinski model itself and more general dual systems to AdS2

dilaton gravity, there has been considerable interest lately [39, 40].This thesis is structured as follows. Chapter 2 contains a short review on the

Almheiri–Polchinski model itself and the different frames of AdS2 space of interestfor our purpose. Chapter 3 presents the derivation of the reparametrization equa-tion of the boundary time coordinate in this model, which in turn provides thepivotal starting point for nearly everything that follows. This expression is illus-trated with several classical situations. Chapter 4 briefly introduces the quantumaspects to the story, mainly from the perspective of quantum field theory in curvedspacetimes. Then Chapter 5 augments these aspects with the computation of theexpectation value of the holographic stress tensor. The results are then applied toan interesting example of relevance in the black hole information paradox context:we model a 2d evaporating black hole and compute the decrease of energy in thesystem as it evaporates through Hawking emission. Finally, Chapter 6 presents aboundary theory that agrees with both the equation of motion of the boundarytrajectory and the holographic boundary energy; it can be viewed as an effectivedescription of the bulk theory. We compute the Poisson bracket of the matterfields corresponding to quantifying the butterfly effect in the context of Hawkingradiation and find that it is maximally chaotic. We end with a brief conclusion inChapter 7. Some more detailed computations are contained in the appendices.

Chapter 2

The Almheiri–Polchinskimodel

The model we consider is a 2d dilaton gravity model proposed in [1]. It has theadvantage that its dynamics is classically solvable. The drawback is that it doesnot arise in the IR region of some physical higher-dimensional geometry but is areduction from a certain conformal Lifschitz spacetime.

The considerations that lead to the model being classically solvable are thefollowing. In conformal gauge (ds2 = −e2ω(σ+,σ−)dσ+dσ− where σ± = t ± z), wehave, in general,

R = 8e−2ω∂+∂−ω , (2.1)

where the subscript ± on the derivate denotes a derivate with respect to σ±. Thisimplies that in order to obtain a constantly AdS2 spacetime we must have theequation of motion

8e−2ω∂+∂−ω = − 2

L2, (2.2)

where L is the AdS length or

8∂+∂−ω + Ce2ω = 0 , (2.3)

where C = 2/L2. Starting with the action

S =1

16πG

∫d2x√−g[Φ2R− U0(Φ)− Ω(Φ)

2(∇f)2

], (2.4)

we find, by varying with respect to the dilaton,

8∂+∂−ω − ∂Φ2U0(Φ)e2ω = −2∂Φ2Ω(Φ)∂+f∂−f . (2.5)

Thus, in order to preserve AdS2, we must have Ω(Φ) = constant and

∂Φ2U0(Φ) = −C , (2.6)

5

6 Chapter 2. The Almheiri–Polchinski model

where C > 0, implyingU0(Φ) = −CΦ2 +D . (2.7)

The action of the model evaluated in [1] and the one we consider in this thesisis one in which C = D = 2 and Ω(Φ) = 1:

S =1

16πG

∫d2x√−g[Φ2R− 2 + 2Φ2 − 1

2(∇f)2

]. (2.8)

In conformal gauge the equations of motion are (excluding the one for f which onlystates that f is chiral)

4∂+∂−ω + e2ω = 0 , (2.9)

−e2ω∂+

(e−2ω∂+Φ2

)= 8πGTσ++ , (2.10)

−e2ω∂−(e−2ω∂−Φ2

)= 8πGTσ−− , (2.11)

2∂+∂−Φ2 + e2ω(Φ2 − 1

)= 0 , (2.12)

where

Tσ±± =(∂±f)2

16πG. (2.13)

The superscript σ on Tσ±± implies that the tensor is evaluated in the σ frame. Wecan immediately see that there is no backreaction on the metric because of thedecoupled equation (2.9). The general solution to the metric equation is

e2ω(σ+,σ−) =4∂+X

+(σ+)∂−X−(σ−)

(X+(σ+)−X−(σ−))2, (2.14)

where X± are general monotonic chiral functions. Substituting for the above ex-pression for the metric in (2.10) and (2.11) and solving for Φ2 yields, after somealgebra,

Φ2 =1

X+(σ+)−X−(σ−)

∫ X+(σ+)

−∞ds(s−X−(σ−))(s−X+(σ+))8πGT x++(s)

− 1

X+(σ+)−X−(σ−)

∫ X−(σ−)

−∞ds(s−X+(σ+))(s−X−(σ−))8πGT x−−(s)

+a+ bX+(σ+) + cX−(σ−) + dX+(σ+)X−(σ−)

X+(σ+)−X−(σ−), (2.15)

whereas (2.12) yields the condition

b− c = 2 . (2.16)

It turns out that for physical reasons that will become apparent later the constanta needs to be nonzero and positive. Furthermore, as will be discussed in detail

2.1. Coordinate patches in AdS2 7

later, any negative constant −µ (µ > 0) multiplying X+(σ+)X−(σ−) turns out tobe related to the total energy E in the spacetime (or, equivalently the boundaryenergy) through

E =µ

8πG, (2.17)

which, as it turns out, can be ascribed to a black hole embedded in the spacetime.From the form of the action (2.8), it is clear that the location at which Φ2 = 0 is astrong coupling singularity.

In compliance with the above considerations we will consider solutions of theform

Φ2 =

1 +1

X+(σ+)−X−(σ−)

a+

∫ X+(σ+)

−∞ds(s−X−(σ−))(s−X+(σ+))8πGT x++(s)

−∫ X−(σ−)

−∞ds(s−X+(σ+))(s−X−(σ−))8πGT x−−(s)

−∫ ∞−∞

ds(s−X−(σ−))(s−X+(σ+))8πGT x++(s)

, (2.18)

where this final term is added for reasons that are explained in Appendix A. Theparameter a is very important here: it represents the regularization away fromAdS2, allowing finite energy excitations. As discussed extensively in [1] and aswill be visible throughout this work, letting a → 0 leads to singular behavior ofmost quantities. When a black hole is formed by sending in a pulse of energyT x−−(s) = µ

8πGδ(s), the dilaton takes the form

Φ2 = 1 +a− µX+(σ+)X−(σ−)

X+(σ+)−X−(σ−). (2.19)

2.1 Coordinate patches in AdS2

“Geometrically” we will always be in AdS2, even after allowing backreaction, whichwas the rationale of the model in the first place. This makes knowledge of therelevant coordinate frames even more important. There are three coordinate framesthat enjoy a privileged position among the possible frames: the Poincare patch, theglobal frame and the black hole frame (see Figure 2.1). In this section, we willquickly review the most important one: the Poincare patch. We will review theothers as we go along.

8 Chapter 2. The Almheiri–Polchinski model

zy

ty

Global

Black hole

Poincare

Figure 2.1: The different coordinate frames of AdS2. The global frame is the entirevertical strip (uncolored). The labels on the axes denote the global coordinates zy and ty.The Poincare patch is the triangular region and will always be colored in yellow/beige.The black hole frame is a smaller triangular region and will always be colored in green.

2.1.1 The Poincare patch

The Poincare patch will henceforth be denoted by coordinates x+ and x− wherex± = tx ± zx and is obtained by letting

X±(x±) = x± , (2.20)

in (2.14). Then the line element becomes

ds2 = − 4

(x+ − x−)2dx+dx− (2.21)

where x± ∈ R and x+ > x−. The point x+ − x− = +∞ is a horizon since atthat point, g00 = 0. The AdS2 boundary is where x+ − x− = 2zx = 0. The properdistance to the horizon is

∫ +∞z0

1zx

dzx =∞ which showcases why the Poincare patchcan be viewed as the near-horizon region of an extremal black hole geometry. Inthe Poincare patch, in the presence of a black hole, the dilaton takes the form

Φ2 = 1 +a− µx+x−

x+ − x−= 1 +

a− µ(t2x − z2x)

2zx, (2.22)

2.1. Coordinate patches in AdS2 9

which is time-dependent but would evidently be time-independent without the blackhole term.

Chapter 3

The boundary equations ofmotion

It was argued in [1] that the holographic boundary time needs to be reparametrizedwhen considering spacetimes whose dilaton has different asymptotics than thePoincare case. The most convincing reason for this is that the holographic cor-relators turn out to correctly describe the thermal properties of black holes onlywhen this reparametrization is done [1]. We will have a slightly different viewon this, where we imagine starting in the Poincare patch and then changing theasymptotic coordinates in response to matter being thrown in (or taken out). Thisis more in line with the view of [41], where the very definition of an asymptoticobserver in 2d dilaton gravity is given by defining him/her as following a constantvalue of the dilaton. Hence, we assign special significance to the asymptotics of thePoincare dilaton (Equation (2.22) with µ = 0) merely because this is our initial con-figuration.1 In this chapter, we deduce the general form of this reparametrizationequation.

Starting in the Poincare frame, the preferred frame (henceforth denoted bycoordinates σ± = t± z) changes as matter is being thrown in. Given that

• the coordinate transformation is holomorphic, i.e., it preserves the chiralstructure;2

• it preserves the asymptotic form of the metric with respect to the Poincareform; and

• it preserves the asymptotic form of the dilaton with respect to the Poincareform;

1One possible justification for starting in the Poincare frame is that the spacetime (dilatonincluded) turns out to have zero boundary energy.

2This statement must be true near the boundary in any case, but it is an external requirementif one wants a unique extension throughout the entire bulk.

11

12 Chapter 3. The boundary equations of motion

one can prove that the coordinate transformation is uniquely defined throughout theentire bulk. The first two constraints specify the coordinate transformation in termsof a single function X+(σ) = X−(σ) ≡ X(σ) of one variable. The last constraintwill then yield an integro–differential equation to be solved for this function. Eventhough one only specifies conditions on the near-boundary form of this function, onefinds a unique solution throughout the entire bulk. We now prove this statement.

Starting out in the Poincare metric with coordinates x±, we transform to anew frame σ± through the chiral transformation x± = X±(σ±). The generaltransformed metric is

ds2 = −4∂+X+(t+ z)∂−X

−(t− z)(X+(t+ z)−X−(t− z))2 (dt2 − dz2) . (3.1)

The condition that the metric keeps the Poincare form close to the boundary yields,after expanding the denominator in (3.1),

X+(σ) = X−(σ) ≡ X(σ) . (3.2)

This is equivalent to stating that the boundary remains at z = 0 in any preferredpatch. The condition that the dilaton has the same asymptotic form as before aperturbation in the Poincare patch, can be expressed, close to the boundary, i.e.,z = ε, as

Φ2 =

1

X(t+ ε)−X(t− ε)

[a+ 8πG

∫ X(t+ε)

−∞ds(s−X(t− ε))(s−X(t+ ε))T x++(s)

− 8πG

∫ X(t−ε)

−∞ds(s−X(t+ ε))(s−X(t− ε))T x−−(s)

−8πG

∫ +∞

−∞ds(s−X(t− ε))(s−X(t+ ε))T x++(s)

]' a

σ+ − σ−=

a

2ε, (3.3)

Series expanding the left-hand-side to lowest order in ε implies, after taking thelimit ε→ 0,

∂tX(t) = 1 +8πG

a

∫ X(t)

−∞ds(s−X(t))2

(T x++(s)− T x−−(s)

)− 8πG

a

∫ +∞

−∞ds(s−X(t))2T x++(s) . (3.4)

3.1. Example: The black hole frame and infalling pulses 13

For later reference we take the derivative with respect to t twice and rearrange toobtain

− 1

∂tX(t)∂t

(∂2tX(t)

∂tX(t)

)+

16πG

a

∫ X(t)

−∞(T x++(s)− T x−−(s)) ds

− 16πG

a

∫ ∞−∞

T x++(s) ds = 0 . (3.5)

This integro–differential equation for X(t) describes the reparametrization of thenear-boundary Poincare time X as a function of the new preferred time t, as matteris being thrown in (or taken out).

The boundary observer himself/herself specifies his/her trajectory by stayingat z = ε in his/her current preferred frame. This implies that his/her trajectoryis not at fixed zx when viewed in Poincare coordinates. To trace the trajectory ofthe boundary as viewed in the Poincare coordinate system we note that the timecoordinate is parametrized by

tx =1

2(X(t+ ε) +X(t− ε)) = X(t) +O(ε2) ' X(t) , (3.6)

whereas the parametrization of the spatial coordinate can be obtained through

zx =1

2(X(t+ ε)−X(t− ε)) = ε∂tX(t) +O(ε3) ' ε∂tX(t) . (3.7)

The “velocity” of the boundary as viewed from the Poincare coordinate patch canbe computed through

dzxdtx

= ε16πG

a

[∫ ∞tx

ds(s− tx)T x++(s) +

∫ tx

−∞ds(s− tx)T x−−(s)

], (3.8)

whereas the “acceleration” is

d2zxdt2x

= −ε16πG

a

[∫ ∞tx

ds T x++(s) +

∫ tx

−∞ds T x−−(s)

]. (3.9)

In the next two sections we analyze two specific applications of these expressions,thereby introducing the black hole frame and the global frame. Especially the blackhole frame will be important for us further on.

3.1 Example: The black hole frame and infallingpulses

The concrete physical situation that we want to analyze in more detail is the cre-ation of a black hole by an infalling matter pulse in the Poincare frame. This isdepicted in Figure 3.1.


zy

ty

tξ = +∞

Blackhole

Poincare

Figure 3.1: Creation of a black hole by sending in a pulse in the Poincare patch. Thedashed line represents the black hole horizon. Again, the axes denote global coordinateszy and ty. After the pulse, the more natural time coordinate to consider is the black holeframe time tξ.

We know from [1] that the black hole patch becomes the preferred frame bysending in the pulse

T x−−(x−) =µ

8πGδ(x−) . (3.10)

Substituting for this expression in (3.4) indeed reproduces the same equation ob-tained in [1] for t > 0, namely

∂tX(t) = 1− µ

aX(t)2 (3.11)

with solution X(t) =√a/µ tanh

√µ/a t.

3.1.1 Black hole frame set-up

From now on, let us denote the black hole coordinates by ξ± = tξ ± zξ. Thesecoordinates are obtained from (2.14) by taking

X±(ξ±) =

√a

µtanh

√µ

aξ± , (3.12)

where µ is related to the black hole energy E via E = µ/(8πG). This results in theline element

ds2 = − 4µ

a sinh2(√

µa (ξ+ − ξ−)

)dξ+dξ− . (3.13)

3.1. Example: The black hole frame and infalling pulses 15

The horizon is found at ξ+ − ξ− = +∞, which implies that either x+ =√a/µ or

x− = −√a/µ. In global coordinates, x± = tan y± = tan (ty ± zy),3 which implies

that the black hole horizon is located at

ty + zy = arctan

√a

µ, ty − zy = −arctan

√a

µ. (3.14)

Since arctan√a/µ < π/2, these diagonal lines are inside the Poincare patch. If

µ→ 0, these lines coincide with the Poincare horizon, which can also be seen fromthe coordinate transformation itself, since then ξ± → x±. The horizon at zξ = +∞is at a finite proper distance in the black hole frame.

In the black hole frame, the dilaton takes the form

Φ2 = 1 +√µa coth

(√µ

a(ξ+ − ξ+)

), (3.15)

which is time-independent. If need be, one can bring this into the conventionalSchwarzschild-form by redefining the radial coordinate as

ρ =√µa coth

(2

√µ

azξ

). (3.16)

3.1.2 Infalling pulse in the black hole frame

Next, we study the situation where we throw in a new pulse in an eternal black hole.This would merely increase the mass of the black hole (see Figure 3.2). We deducedthe above equations based on the assumption that we started in the Poincare frameand transformed to other patches by sending in pulses. As we will show in thesubsequent sections, these patches make up a closed family of patches under theaction of sending in pulses whose energies are defined in local coordinates. We statethe boundary equation of motion in the black hole frame (starting with any µ > 0):

∂tΞ(t) = 1 +8πG

µ

∫ Ξ(t)

−∞ds sinh2

(√µ

a(s− Ξ(t))

)(T ξ++(s)− T ξ−−(s))

− 8πG

µ

∫ +∞

−∞ds sinh2

(√µ

a(s− Ξ(t))

)T ξ++(s) . (3.17)

In the black hole patch given by a black hole with mass µ1/(8πG), the relevantcoordinate transformation for an infalling pulse at tξ = ∆ and with magnitudeµ2/(8πG) is given by

∂tΞ(t) = 1− µ2

µ1sinh2

(√µ1

a(Ξ(t)−∆)

), (3.18)

3This frame will be introduced more elaborately in the next chapter.


zy

ty

tξ1 = +∞tξ2 = +∞

Black hole

Figure 3.2: Increasing the mass of an eternal black hole (with coordinates ξ±1 ) by sendingin a pulse giving rise to a “new” black hole with coordinates ξ±2 .

for t > ∆, with solution√a

µ1tanh

(√µ1

a(Ξ(t)−∆)

)=

√a

µ1 + µ2tanh

(√µ1 + µ2

a(t−∆)

), (3.19)

where the clocks are synchronized at t = tξ = ∆. It is clear that this describes astatic black hole with just a larger mass. To further investigate the consequencesof creating a larger black hole, let us compute the Hawking temperature TH . Asdescribed in [1], the prescription involves transforming to Scwarzschild coordinates(3.16). Then, TH might be computed through

TH = − 1

4π∂ρ

√− gttgρρ

. (3.20)

Following this prescription, we find that upon sending in a pulse starting with anexisting black hole, TH jumps as

TH =

√µ1

π→ TH =

√µ1 + µ2

π. (3.21)

This example is studied deeper in Appendix B.1, where we imagine the originalblack hole being created dynamically by an initial pulse.

3.2. Example: The global frame and infalling pulses 17

3.2 Example: The global frame and infallingpulses

We can make contact with the global frame, by extracting energy from the space-time, by emitting a negative energy pulse with the cleverly chosen amplitude

T x−−(x−) = − a

8πGδ(x−) . (3.22)

Substituting for this expression in (3.4) yields, for t > 0,

∂tX(t) = 1 +X(t)2 , (3.23)

which integrates to X(t) = tan t. This transformation indeed defines the timecoordinate of the global frame.

3.2.1 Global frame set-up

The global coordinates will hereinafter be denoted by y± = ty ± zy and are definedby taking X±(y±) = tan y± in (2.14), thus arriving at the line element

ds2 = − 4

sin2(y+ − y−)dy+dy− , (3.24)

where, arriving from the Poincare patch, one has y± ∈ [−π/2, π/2] and y+ > y−,which implies that the Poincare horizon is located at

ty + zy =π

2, ty − zy = −π

2. (3.25)

To arrive at the full global solution, we extend the interval such that ty ∈ (−∞,+∞).In the global frame, the dilaton takes the form (obtained by setting µ = −a in(2.19))4

Φ2 = 1 +a+ a tan y+ tan y−

tan y+ − tan y−= 1 + a coth 2zy . (3.27)

4For completeness, we mention that directly transforming the general expression for the dilatonwith a black hole of mass µ (2.19) into global coordinates, yields

Φ2 = 1 +a− µ tan y+ tan y−

tan y+ − tan y−= 1 + a coth 2zy −

µ+ a

2

[cos 2zy − sin 2ty

sin 2zy

], (3.26)

which is time-dependent unless µ = −a. These global frames are not part of the family of framesthat can be obtained from one another by sending in matter from the Poincare patch.


3.2.2 An infalling pulse in the global frame

The boundary equation of motion, when starting in the global frame (starting withµ = −a as required by (3.23)), is given by

∂tY (t) = 1 +8πG

a

∫ Y (t)

−∞ds sin2(s− Y (t))(T y++(s)− T y−−(s))

− 8πG

a

∫ +∞

−∞ds sin2(s− Y (t))T y++(s) . (3.28)

As an example of the use of this equation, suppose we send in a pulse in the globalframe of the form

T y−−(y−) =µ

8πGδ(y−) . (3.29)

Then we obtain, for t > 0,

∂tY (t) = 1− µ

asin2 Y (t) , (3.30)

which has the solution

tanY (t) =

√a

a− µtan

(√a− µa

t

)≡ X(t) . (3.31)

This relates global time Y with the new time coordinate t. Clearly, no periodicityin imaginary time is generated, and hence, this does not represent a black hole; theinfalling pulse is too weak (see Figure 3.3). We can immediately identify tanY (t)as the Poincare time. Furthermore, if the pulse is sufficiently strong, i.e., µ > a, wehave

X(t) =

√a

µ− atanh

(√µ− aa

t

). (3.32)

This allows us to identify µ − a = µ, where µ is the corresponding parameterstarting in the Poincare patch, and t = tξ, where tξ is the the black hole patch timecoordinate. We thus conclude that we obtain the same preferred frame by eitherstarting in the global frame and sending in a pulse with magnitude (a+ µ)/(8πG)or starting in the Poincare frame and sending in a pulse with magnitude µ/(8πG).This illustrates the fact that the global frame can be seen as being lower in energyby −a/(8πG) than the Poincare patch. This will be discussed in further detail inChapter 5.

3.2. Example: The global frame and infalling pulses 19

ty

Figure 3.3: The emission of a pulse with magnitude µ/(8πG) < a/(8πG) starting in theglobal patch. No black hole is created and the boundary conditions dictate the behaviorof the pulse.

Chapter 4

Quantum effects of thematter fields

In this chapter, we will briefly review how quantum effects modify the above equa-tions of motion and how they are incorporated in this framework.

As is well-known, the conformal matter field f has a stress tensor that is anoma-lous at one-loop, due to the Virasoro anomaly [1]. The general form of the ++ and−− components of the stress tensor, including the conformal anomaly, is then [1, 24]

〈Tσ±±(σ+, σ−)〉 =N

12π(∂2±ω − (∂±ω)2)+ : Tσ±±(σ±) : +Tσ,cl± (σ±) . (4.1)

The first term is the conformal anomaly, which, given the form of the metric (3.1)and (3.2), is proportional to a Schwarzian derivative:

∂2±ω − (∂±ω)2 =

1

2X(σ±), σ± =

1

2

[∂3±X(σ±)

∂±X(σ±)− 3

2

(∂2±X(σ±)

∂±X(σ±)

)2], (4.2)

whereas the second term is to be interpreted as the normal-ordered stress “tensor”with respect to the σ± vacuum. The third term is the classical stress–energy andwill not require further treatment here; we will omit it from this discussion. Thefirst two terms are separately non-tensor, but the total sum is a coordinate covari-ant quantity. We note that, a priori, the first term in (4.1) (and hence the fullstress tensor) can generally depend on both lightcone coordinates, depending onthe particular ω studied in the frame at hand. However, in AdS2, we find that thefull stress tensor remains chiral because of the result (4.2).

Both of the first two terms have merit on their own. The covariant tensor isthe one that should be inserted into Einstein’s equations and hence is responsiblefor backreaction. The normal-ordered stress tensor corresponds to the stress ten-sor that would be measured by local observers using detectors calibrated to their

21

22 Chapter 4. Quantum effects of the matter fields

vacuum. Of course, the above expressions should be read as being evaluated in agiven quantum state |ψ〉, but throughout the rest of this thesis we will just writeTσ±±(σ+, σ−) for brevity.

The trace of the stress tensor is also non-zero and given by

Tσ+− = − N

12π∂+∂−ω . (4.3)

Now, the equations of motion get slightly modified into

2∂+∂−Φ2 + e2ω(Φ2 − 1) = 16πGTσ+− , (4.4)

4∂+∂−ω + e2ω = 0 , (4.5)

−e2ω∂+(e−2ω∂+Φ2) = 8πGTσ++ , (4.6)

−e2ω∂−(e−2ω∂−Φ2) = 8πGTσ−− . (4.7)

Equations (4.3), (4.4), and (4.5) imply a modified version of the dilaton equation:

∂+∂−Φ2 + e2ω

(Φ2 − 1− GN

3

)= 0 . (4.8)

where simply shifting the dilaton by the constant +GN3 correctly incorporates the

effect of Tσ+−. We also see that the same restriction on the integration constantsapply as before with the replacement

b− c = 2 +2GN

3(4.9)

in (2.16).

Chapter 5

The boundary stress tensor

In this chapter, we will consider the holographic boundary stress tensor, includingthe one-loop quantum effects discussed in chapter 4. To start with, we split thematter field f into a classical part fcl and a quantum part. For the quantum part,it is well-known (see e.g., [1, 24]) that the conformal anomaly can be included intoan effective action, using an auxiliary field χ with action

Sχ = − N

24π

∫d2x√−g [∂µχ∂

µχ+ χR]− N

12π

∫dt√−γχK . (5.1)

The relevant total bulk action to be used is given by

S = SG + Sf + Sχ + Sct , (5.2)

where

SG =1

16πG

∫d2x√−g[Φ2R− 2 + 2Φ2

]+

1

8πG

∫dt√−γΦ2K , (5.3)

Sf =1

32πG

∫d2x√−g∂µfcl∂µfcl . (5.4)

Here we included the trace of the extrinsic curvature K =√gzz(∂z

√−γ)/

√−γ as

prescribed in [42, 43]. The equations of motion arising from this effective action,are the same as before with the replacements

Tσ+− =N

12π∂+∂−χ , (5.5)

Tσ±± = Tσ,cl±± +N

12π

(−∂2

+χ+ ∂+χ∂+χ+ 2∂+ω∂+χ), (5.6)

∂+∂−(χ+ ω) = 0 , (5.7)

where Tσ,cl±± = (∂±fcl)2/(16πG). The last equation can be solved by

χ = −ω + g+(σ+) + g−(σ−) . (5.8)

23

24 Chapter 5. The boundary stress tensor

Plugging these back into expressions (5.6) and comparing with (4.1), one finds thatthese two chiral functions g+ and g− are related to the normal-ordered stress tensorthrough

: Tσ±± := − N

12π

(∂2±g± −

(∂±g

±)2) . (5.9)

In conformal gauge, i.e., using R = 8e−2ω∂+∂−ω, one finds1

SG + Sχ =∫dzdt

[1

8πG

(−4∂(+Φ2∂−)ω + (Φ2 − 1)e2ω

)+N

6π

(∂+χ∂−χ+ 2∂(+χ∂−)ω

)],

(5.10)

where we performed an integration by parts whose boundary contribution cancelswith the boundary action.

The counterterm action Sct can be found by cancelling all z → 0 divergences.Since the entire family of patches we are considering is determined precisely suchthat they have the same asymptotic behavior for both metric and dilaton, the samecounterterm action can be used in all cases we will consider. Moreover, the classicalfield fcl will be assumed not to diverge at the AdS boundary:

fcl = O(1) , (5.11)

and does not contribute to the counterterm action.

In all cases, the divergent contribution at z = ε is cancelled by adding thecounterterm action

Sct =

∫dt√−γ(

1

8πG(1− Φ2)− N

24π

), (5.12)

which turns out to be unique. The prescription then proceeds by varying the on-shell bulk action with respect to the boundary metric γtt = ε2γtt(ε),

2 as

〈Ttt〉 = − 2√−γ

δS

δγtt. (5.13)

The details can be found in [1]. The only addition here is the classical matteraction, but this is decoupled from the metric and hence does not contribute to theboundary stress tensor.

1We have used the convention A(ab) = (Aab +Aba)/2.2The boundary metric is obtained by removing the conformal prefactor from the bulk metric

as: ds2 = γttdt2 = γttdt2/z2.

5.1. The stress tensor in the frame respecting the equation of motion 25

Utilizing standard arguments related to the Hamilton-Jacobi method, one ar-rives at

〈Ttt〉 = 2εeω∂zΨ− εe2ω

(1− Φ2

8πG− N

24π

), (5.14)

where

Ψ =Φ2

16πG− Nχ

24π. (5.15)

Hence all that is needed to compute the boundary stress tensor is the asymptoticexpansion of Φ2, eω and χ.

5.1 The stress tensor in the frame respecting theequation of motion

As we argued above in Chapter 3, the correct definition of the boundary amountsto keeping fixed the asymptotic metric and dilaton as matter is falling in. Thisrequires the boundary time coordinate and asymptotic trajectory to be redefineddynamically as time progresses.

Hence, we need to allow the asymptotic frame to “flow” instead of keeping itfixed to a predefined choice. Here we derive an elegant expression for the stresstensor using this “dynamical” prescription.

Following the general prescription, we expand the relevant quantities aroundz = 0, while including terms linear in ε. On a technical level, this merely requiresgoing to one order higher in ε for Φ2 than the case studied in Chapter 3. Using(2.14), (2.18) (but including the +GN/3 term), (3.4), and (5.8), one finds for thenear-boundary expansion in the general frame characterized by the chiral functionX(t),

e2ω =1

ε2+

2

3X, t+O(ε2) , (5.16)

χ = g+(t) + g−(t) + log ε+[∂tg

+(t)− ∂tg−(t)]ε+O(ε2) , (5.17)


and

Φ2 = 1 +GN

3+

1

X(t+ ε)−X(t− ε)×[

a+ 8πG

∫ X(t+ε)

−∞ds(s−X(t− ε))(s−X(t+ ε))T x++(s)

− 8πG

∫ X(t−ε)

−∞ds(s−X(t+ ε))(s−X(t− ε))T x−−(s)

−8πG

∫ +∞

−∞ds(s−X(t− ε))(s−X(t+ ε))T x++(s)

]=

a

2ε+ 1 +

GN

3− a

3X, tε+O(ε2) . (5.18)

Using equation (5.14), we obtain

〈Ttt〉 = − a

16πGX, t − N

12π

[∂tg

+(t)− ∂tg−(t)]. (5.19)

The significance of the term proportional to N will be discussed in Section 5.2 andwe will omit this term throughout the rest of the current discussion.3 Thus, we willbe using the equation

〈Ttt〉 = − a

16πGX, t (5.20)

instead. Furthermore, by starting in the global patch and using (3.28) one canprove that

e2ω =1

ε2+

2

3tanY, t , (5.21)

Φ2 =a

2ε+ 1 +

GN

3− aε

3tanY, t , (5.22)

leading to

〈Ttt〉 = − a

16πGtanY, t = − a

16πGX, t . (5.23)

Analogously, one obtains the same kind of formulae when one refers the procedureto the black hole patch. Using (3.17) yields

e2ω =1

ε2+

2

3

√a

µtanh

(√µ

aΞ

), t

, (5.24)

Φ2 =a

2ε+ 1 +

GN

3− aε

3

√a

µtanh

(√µ

aΞ

), t

, (5.25)

3If one also assumes that there is no quantum contribution in the bulk matter stress–energytensor contributing to the preferred coordinate transformation (3.4), this is equivalent to ignoringquantum matter effects.


again leading to

〈Ttt〉 = − a

16πG

√a

µtanh

(√µ

aΞ

), t

= − a

16πGX, t . (5.26)

Having arrived at the same expression for the boundary stress–energy tensorregardless of which coordinate frame one starts in, we might first conclude that theboundary energy in the Poincare frame is

〈Ttt〉 = − a

16πGX, t = − a

16πGt, t = 0 . (5.27)

We know from Section 3.1 that, upon sending in a positive-energy pulse startingin the Poincare frame, one obtains the black hole frame as the preferred frame.Plugging this coordinate transformation into (5.20) yields

〈Ttt〉 =µ

8πG. (5.28)

Furthermore, sending in a pulse

T x−−(x−) = − a

8πGδ(x−) , (5.29)

in the Poincare frame produces the global frame which yields the boundary stress–energy tensor

〈Ttt〉 = − a

8πG. (5.30)

As described in the work of Balasubramanian and Kraus [44] for the 3d case, this canbe interpreted as the global patch having lower vacuum energy than the Poincarepatch.

We also calculated the coordinate transformations obtained by sending in apositive-energy pulse starting in the global patch, (3.31) and (3.32), which, afterplugging these into (5.23) and using µ = µ− a, also yields

〈Ttt〉 =µ

8πG, (5.31)

where now µ also can be negative.One can show that starting in any frame belonging to the family of frames with

Poincare asymptotics also yields

〈Ttt〉 =µ

8πG, (5.32)

where again µ can have any sign and is the energy difference between the current,preferred frame and the Poincare frame. These examples demonstrate that theboundary stress tensor is equal to the sum of stress–energy sent in to the spacetime,but with each respective stress–energy term evaluated in the instantaneous coordinate


frame from which it was sent in. The same can be shown to hold true for anymatter profile. Further treatment of the boundary stress–energy tensor in the caseof multiple consecutive pulses is presented in Appendices B.1 and B.2. The energyof the different coordinate frames we have discussed so far is illustrated in Figure 5.1.

E = 0

E = − a8πG

Poincare

Global

Black holes

E

Figure 5.1: Boundary energy of frames in the family of coordinate frames related toPoincare coordinates by having the same asymptotic form of the metric and dilaton. Ascan be seen here, the global frame is lower in energy than the Poincare frame.

5.1.1 Black hole formation and evaporation

We continue the study initiated in Section 3.1 of the creation of a black hole bysending in a classical pulse in the initial Poincare frame but this time we includequantum effects that will eventually lead to Hawking radiation.

In the Poincare patch, we always have

N

12π(∂2±ω − (∂±ω)2) =

N

24πx±, x± = 0 , (5.33)

implying that there is no ambiguity or anomaly in the stress tensor; the real tensorand the normal-ordered version coincide:

T x±± =: T x±± : . (5.34)

This is equivalent to computing quantum expectation values with respect to thePoincare vacuum. As discussed in connection with the result (4.2), the full (covari-ant) stress–energy tensor always depends on only one lightcone coordinate and has


the same functional form for both + and − indices. In general, this is not the casein non-AdS spacetimes (see e.g., [22]).

Now, we assume that no matter is outgoing before sending in the pulse, i.e.,T x++(x+) = 0. Since this is only a function of x+, the ++ component of the stress–energy tensor vanishes also after the pulse.4 We also assume that no ingoing matteris present before the pulse: T x−− = 0 for x− < 0. This identifies the quantum stateto be used in (4.1) as the Poincare vacuum |0〉P .

The AdS boundary can be reached by null rays in a finite affine time, andone should hence provide boundary conditions that specify what happens when alight ray hits the boundary z = 0. The most natural boundary condition is theperfect reflection: Tσ++(t) = Tσ−−(t) at any point along the boundary, since then themassive case and the massless case are smoothly connected. However, in principle,one is free to choose other boundary conditions. But since Tσ++ vanishes—unless theboundary actively emits radiation—the ingoing stress tensor Tσ−− will also vanish.

This then implies, through (4.1), that : Tσ++(σ+) : and : Tσ−−(σ−) : share thesame functional form. Hence, at the boundary,

: Tσ++(t) :=: Tσ−−(t) : , (5.35)

implying that the normal-ordered stress tensor satisfies perfect reflection in this sit-uation, regardless of what one chooses the boundary to do with ingoing radiation.The conclusion that Tσ++ = 0 seems inevitable, so there is no genuine Hawking ra-diation. However, it is still possible to enforce evaporation by making the boundaryperfectly absorbing for the normal-ordered stress tensor only:

: Tσ−−(t) := 0 . (5.36)

This can physically be realized by having local observers close to the boundarymeasure and absorb the ingoing flux with respect to their vacuum.

Hence, we have two possibilities to analyze: perfect reflection (Subsection 5.1.2)and absorption of ingoing quanta passing the boundary observer (Subsection 5.1.3).

5.1.2 The eternal heat bath

For the moment, we will focus on the earlier situation where both T x++ and T x−−vanish everywhere. Hence, these do not contribute to the dilaton solution. So, thesolution including a classical infalling pulse and the quantum matter has the form,in Poincare coordinates,

ds2 = − 4

(x+ − x−)2 dx+dx− , (5.37)

Φ2 = 1 +GN

3+a− µθ(x−)x+x−

x+ − x−. (5.38)

4The relevant frame is different after the pulse, but a chiral coordinate transformation isunable to create a non-zero T++ in any case.


As T x++ = 0, no Hawking radiation is present and the final black hole does notevaporate. The incoming radiation, T x−− = µδ(x−)/(8πG), represents only aningoing pulse and nothing else. However, an observer using a different time-slicingmight still experience a non-zero energy flow as measured from his/her vacuum.

As we have demonstrated previously, the boundary observer has a preferredframe for any given bulk perturbation (e.g., an infalling pulse). In this particularcase, the black hole frame is the new preferred frame. Static observers living in thisnew patch would measure a stress tensor

: T ξ±± := T ξ±± −N

12π(∂2±ω − (∂±ω)2) . (5.39)

Thus, in the black hole coordinate frame at hand, the total matter stress tensor issimply

: T ξ±± :=µN

12πa. (5.40)

Both the ++ and −− components of the normal-ordered stress tensor are equal.This represents a thermal heat bath of equal incoming and outgoing radiation. Thenet result is a static black hole that is equilibrated with its own radiation. Hence,one might say that there is Hawking radiation proportional to the black hole massreflected perfectly from the boundary. However, this radiation is the counterpartof the Unruh effect in AdS, not genuine Hawking radiation. The black hole is notevaporating and thermal radiation is only detected by fiducial observers, just likethe accelerated observers in flat space.

Let us finally mention that the observed spectrum is indeed thermal. Using the

coordinate transformation x± =√a/µ tanh

(√µ/a ξ±

), the vacuum expectation

value (VEV) of the particle number operator in the Poincare vacuum can be readilycomputed using CFT techniques [45]:

P 〈0|NωBH |0〉P = − 1

π

∫dξ+dξ′+e−iω(ξ+−ξ′+)

×

[∂ξ+X(ξ+)∂ξ′+X(ξ′+)

(X(ξ+)−X(ξ′+))2 −

1

(ξ+ − ξ′+)2

]

=1

e2πωκ−1 − 1, (5.41)

which characterizes a Planckian black body spectrum, with κ = 2√µ/a. This leads

to TH = κ/(2π) =√µ/a/π which agrees with the result found in [1].

Previously, in Subsection 3.1.2, we also analyzed the situation of sending in ad-ditional matter into an already present black hole. Computing the normal-orderedstress tensor shows that it immediately jumps to : T ξ±± := (µ1 + µ2)N/(12πa).In particular, no transient regime is present, and the infalling matter immediatelyequilibrates with the already present black hole, i.e., the scrambling time is zero.This property continues to hold when sending in continuous distributions of matter.


5.1.3 The evaporating black hole

We previously deduced that it is quite difficult to create an evaporating black holeas no outgoing Tσ++ can be produced at all. Moreover, the system has a tendencytoward perfect reflection at the AdS boundary and thus generates an eternal heatbath of virtual particles.

As is well-known for asymptotically flat black holes, draining some of the gas inthe thermal atmosphere is tantamount to simulating evaporation. So if we couldsomehow remove these virtual particles, we would allow the black hole to evaporate.So let us impose the perfect absorption boundary condition

: Tσ−−(t) := 0 , (5.42)

where we evaluate this expression in the instantaneous preferred patch. As aninitial analysis of this situation, we can plug in the black hole patch with fixedmass µ/(8πG) in (4.1). Then, one obtains

T ξ−−(ξ−) = − µN

12πa, (5.43)

meaning a negative flux of particles is thrown in to emulate evaporation. This isa bit strange, but is still in unison with the general mechanism of evaporation asnegative energy particles flowing into the interior.

However, (5.43) cannot be the full story. This is because the frame should bechanged dynamically throughout the entire evaporation process. This requires thesolution of a self-consistent set of equations. We will see that we will produce theevolution of the boundary energy without the need for a full solution.

Using (4.1) in combination with (5.42), the correct expression for the ingoingflux is

Tσ−−(t) =N

12π

(∂2−ω − (∂−ω)2

)=

N

24πX, t , (5.44)

where we used the general result (4.2). The boundary equation of motion (3.4) canbe rewritten in terms of the “flowing” stress tensor as

∂tX(t) = 1 +8πG

a

∫ t

−∞du

(X(u)−X(t))2

X ′(u)

(Tσ++(u)− Tσ−−(u)

)− 8πG

a

∫ +∞

−∞du

(X(u)−X(t))2

X ′(u)Tσ++(u) . (5.45)

As it stands, this equation represents a very difficult integro–differential equationto be solved for X(t). We are not able to do this in general, but we will obtain an


elegant formula for the change in holographic boundary energy. Taking respectivelyone and two time derivatives of the above equation, yields

X ′′ = −16πG

aX ′∫ t

−∞duX(u)−X(t)

X ′(u)

(Tσ++(u)− Tσ−−(u)

)− I ′ , (5.46)

X ′′′ =X ′′2

X ′+

16πG

aX ′2

∫ t

−∞du

1

X ′(u)

(Tσ++(u)− Tσ−−(u)

)+I ′X ′′

X ′− I ′′ , (5.47)

where

I =8πG

a

∫ +∞

−∞du

(X(u)−X(t))2

X ′(u)Tσ++(u) . (5.48)

The Schwarzian derivative can now be rewritten as

X, t = −1

2

(16πG

a

∫ t

−∞duX(u)−X(t)

X ′(u)

(Tσ++(u)− Tσ−−(u)

)+

I ′

X ′

)2

+16πG

aX ′∫ t

−∞du

1

X ′(u)

(Tσ++(u)− Tσ−−(u)

)+I ′X ′′

X ′2− I ′′

X ′. (5.49)

Taking a time derivative yields

∂t X, t = −16πG

a

(Tσ−−(t)− Tσ++(t)

). (5.50)

The part of the boundary energy given by the Schwarzian derivative term, i.e.,(5.20), is associated with the black hole itself (and not the quantum vacuum aroundit). We will henceforth call it EBH . Thus, we arrive at

dEBHdt

= − a

16πG∂t X, t = Tσ−−(t)− Tσ++(t) , (5.51)

meaning the change in boundary energy as measured in units of the preferredboundary time is merely the net flux of energy thrown into the spacetime. Thisexpression will be obtained (more straightforwardly) in the boundary theory furtheron in Chapter 6 as well.

Applying this formula for our case, we obtain, by plugging in (5.42) in (4.1) andusing (4.2),

Tσ−−(t) = E0δ(t) +N

24πX, t θ(t) . (5.52)

Now, using (5.51), we arrive at, for times t > 0,

dEBHdt

=N

24πX, t = −2NG

3aEBH , (5.53)

which, together with the initial value E0, is solved by an exponentially decayingenergy

EBH(t) = E0 exp

(−2NG

3at

), (5.54)


with evaporation rate 2NG/(3a). Reintroducing units of ~, the total boundaryenergy is

〈Ttt〉 = E0 exp

(−2NG~

3at

)− ~N

12π

[∂tg

+(t)− ∂tg−(t)], (5.55)

where the first term incorporates the decay of the black hole due to quantum effects,whereas the second term is the energy of the quantum matter fields themselves andis proportional to ~. We will come back to the second term in the next section,where we will argue that it vanishes in the case at hand, whereby we will excludeit in the following discussion.

We note that the characteristic timescale associated with this evaporation pro-cess is

tevap =3a

2GN~. (5.56)

Using the analog of (3.21) together with (5.28) we find that the (quasi-static)Hawking temperature of the black hole as it evaporates equals

TH(t) =

√8GEBH(t)

πa=

√8GE0

πaexp

(−NG

3at

), (5.57)

decaying at half the rate. One can also compute the outgoing normal-ordered stresstensor, since according to (4.1),

: Tσ++(t) := − N

24πX, t =

2NG

3aE0 exp

(−2NG

3at

)= −Tσ−−(t) , (5.58)

with the initial value of : T++(t = 0) := µ0N/(12πa), matching indeed with theflux of energy found for a black hole of mass µ0/(8πG) as predicted by (5.40). Thedepletion of energy from the system can hence be viewed in two equivalent ways:

dEBHdt

= T t−−(t)− T t++(t)︸︷︷︸0

= : T t−−(t) :︸︷︷︸0

− : T t++(t) : . (5.59)

One can generalize this type of model toward non-perfect absorption. Supposewe only eliminate a fraction 1− α (0 ≤ α ≤ 1) of the energy hitting the boundary:

: T t−− := α : T t++ : . (5.60)

The boundary conditions now imply

T t−− = (1− α)N

24πX, t . (5.61)

which implies

EBH(t) = E0 exp

(−(1− α)

2NG

3at

)θ(t) , (5.62)

interpolating between fastest evaporation when α = 0 to no evaporation at all whenthere is perfect reflection, i.e., when α = 1.


The exponentially decaying profile could have been anticipated by a qualitativeargument based upon the Stefan–Boltzmann law:

dE

dt∼ −σSBT 2 ∼ −E , (5.63)

where the power of T is given by the Stefan–Boltzmann law in one spatial dimensionand we used the fact that T ∼ √µ ∼

√E.

5.2 The quantum contribution to the boundarystress tensor

Up to this point, we have omitted the quantum term in the boundary stress tensor(5.20). Here we explore this term in more detail. In (5.9), we saw that the chiralfunctions g+ and g− should be determined by solving

: Tσ±± := − N

12π

(∂2±g± −

(∂±g

±)2) . (5.64)

The solution, however, is not unique. If e.g., : T ξ±± := µN12πa , then the most general

solution is given by

g+(ξ+) =1

2ln

(4µ

a sinh2(√

µa (ξ+ + C)

))+D , (5.65)

g−(ξ−) =1

2ln

4µ

a sinh2(√

µa (C − ξ−)

)+ D , (5.66)

with arbitrary constants C, C, D, and D. The constants D and D are immaterialand we will henceforth omit them from the discussion. These functions are simplygiven by setting g+(ξ+) = ω(ξ+,−C) +D and g−(ξ−) = ω(C, ξ−) + D.

The boundary stress tensor has a term depending on these g± given by

− N

12π

(∂tg

+ − ∂tg−)

=

N

12π

(√µ

acoth

(√µ

a(tξ + C)

)+

√µ

acoth

(√µ

a(C − tξ)

)). (5.67)

As a particular case, let C = C → +∞. One obtains5

g+(ξ+)→ −√µ

aξ+, (5.68)

g−(ξ−)→√µ

aξ−, (5.69)

5Setting D = D =√µC − ln

(4√µ).

5.2. The quantum contribution to the boundary stress tensor 35

and one finds, for the “quantum part” of the boundary stress–energy:

− N

12π

(∂tg

+ − ∂tg−)

=N

6π

√µ

a, (5.70)

which is the result found in [1]. Our main point here is that this is not the onlypossibility, as also other (possibly time-dependent) energy profiles can be found.

For the eternal static black hole, the above choice C = C → +∞ is the one tomake. If not, one would find disagreement with the computation of the entropy[1]. However, for the black hole generated by an infalling pulse, this cannot bethe correct choice. One reason is energy conservation: sending in a pulse withamplitude µ0/(8πG) should generate a black hole with an associated boundaryenergy of equal magnitude, i.e., µ0/(8πG). The addition of (5.70) to the boundaryenergy would be inconsistent with this.

Let us now formulate a general approach to solving this ambiguity. We assumewe start out in the Poincare frame and throw in classical matter. Equation (5.64)represents a differential equation to be solved for g±. The functions g± are notscalar fields, but transform under change of coordinate frame as6

g±(x±) → g±(x±(σ±)) +1

2ln

(dx±

dσ±

), (5.71)

where, due to lack of letters suitable for denoting coordinates, the x± are herearbitrary coordinates. For pulses, the second term causes a discontinuity in thesecond derivative of g±, precisely corresponding to the jump in normal-orderedstress tensor as one crosses the pulse.

We first need to make a choice for g± in the initial Poincare patch. The mostgeneral solution there is

g+(x+) = − ln(C1x

+ + C2

), g−(x−) = − ln

(C1x

− + C2

), (5.72)

for possibly complex constants C1, C2, C1, and C2. The logic used to obtain g+

and g− after matter starts falling in, is different, even though the final result willturn out to be the same.

• g+ has been fully determined everywhere by the above choice. After thematter gets thrown in, the only change is that the frame is different. Thisimplies that g+ transforms as

g+(σ+) = − ln(C1X(σ+) + C2

)+

1

2ln

(dX

dσ+

). (5.73)

6The transformation rule might be deduced from how the metric (2.14) transforms. Thistransformation is unique up to the possible addition of a constant to g+ and subtraction of thatsame constant from g−. Such modifications have no influence whatsoever on anything one mightcompute.


• g− is only determined before the matter falls in. As soon as the matter startsfalling, the patch changes and g− needs to be determined by the differentialequation (5.64) using suitable initial conditions.7 As a second order ODE, werequire g−(σ− = 0) and ∂−g

−(σ− = 0) to have a unique solution. The initialvalues are determined by evaluating the initial profile in the transformed frameat σ− = 0. So we look for a solution of (5.64) that satisfies (using X(0) = 0):

g−(σ− = 0) = − ln(C2

)+

1

2ln

(dX

dσ−

∣∣∣∣σ−=0

), (5.74)

∂−g−(σ− = 0) = − C1

C2

dX

dσ−

∣∣∣∣σ−=0

+1

2

d2Xdσ−2

∣∣∣σ−=0

dXdσ−

∣∣σ−=0

. (5.75)

The unique solution is then simply given by

g−(σ−) = − ln(C1X(σ−) + C2

)+

1

2ln

(dX

dσ−

), (5.76)

which has the same functional form as g+.

If one demands the initial Poincare frame to have zero boundary energy, thenC2/C1 = C2/C1. In fact, just demanding a time-independent boundary energy inthe initial Poincare patch is already sufficient to reach the same conclusion. Thequantum contribution to the final boundary energy one would write down then isalso zero since ∂tg

+ = ∂tg−. This is the case of relevance to us, both when creating

a black hole with a classical pulse, and when we allow this black hole to evaporateafterward.

Note that the initial g+ and g− in the Poincare frame are part of the preparationof the initial state to consider and we are free to choose them in general.

7If : T−− : contains (only) a step-function, then g− and ∂−g− must be continuous and ∂2−g−

contains the relevant jump.

Chapter 6

A boundary theory

In this chapter, we will propose a 0+1d model that will capture at least the classicalaspects of the above bulk system. This model is the prospective boundary theorydual to the bulk gravitational system discussed up until now. We will require thatthe model fulfills two conditions:

• it should reproduce the reparametrization equation (3.5); and

• its energy should coincide with the holographic boundary stress tensor (5.20).

We will find that both properties are fulfilled for the following system. We startwith the action

S =a

16πG

∫dt

[1

2(∂tϕ)2 − λeϕ + λ∂tτ

+

∫ ∞0

dσ π+(∂t − eϕ∂σ)f+ −∫ ∞

0

dσ π−(∂t + eϕ∂σ)f−

], (6.1)

where f±, π± are functions of (t, σ). A priori the σ label is just an auxiliary la-bel, although we have in mind here the interpretation of the holographic radialcoordinate z. All other fields are functions of just the time t.

First of all, λ acts as a Lagrange multiplier field, forcing the system to bequadratic in the derivatives. Eliminating λ, yields for the subsystem (ϕ, τ , λ) theSchwarzian derivative action

S = − a

32πG

∫dt τ, t . (6.2)

An alternative perspective is provided by integrating out τ , which fixes λ to aconstant, and reduces this system to something reminiscent of a Liouville fieldtheory action:

S =a

16πG

∫dt

[1

2(∂tϕ)2 − λeϕ

]. (6.3)

37

38 Chapter 6. A boundary theory

6.1 Canonical structure

To start with, we investigate the underlying canonical (Hamiltonian) structure ofthis model. We obtain the equations of motion

δS

δϕ= 0 =⇒ ∂2

t ϕ = −eϕ[λ+

∫ ∞0

dσ π+∂σf+ −∫ ∞

0

dσ π−∂σf−

], (6.4)

δS

δλ= 0 =⇒ ∂tτ = eϕ , (6.5)

δS

δτ= 0 =⇒ ∂tλ = 0 , (6.6)

δS

δπ±= 0 =⇒ ∂tf± = ±eϕ∂σf± , (6.7)

δS

δf±= 0 =⇒ ∂tπ± = ±eϕ∂σπ± . (6.8)

The conjugate momenta are

πϕ =a

16πG∂tϕ , πλ = 0 , πτ =

a

16πGλ ,

ππ±(σ) = 0 , πf±(σ) =a

16πGπ±(σ) .

(6.9)

A characteristic feature of the chiral boson field f± is that the field itself and itscanonically conjugate momenta satisfy completely independent equations of motion.We note that all except the first momentum equation in (6.9) actually provideconstraint equations of the system in the Hamiltonian formalism. None of theconstraints commute with all other constraints, and hence, they are all second-class primary constraints in Dirac’s classification scheme [46]. Our goal is thus tocompute the Dirac brackets of all of the elementary fields. Working according toDirac’s prescription we first compute the matrix of (elementary) Poisson brackets.With

χ1 = πλ , χ2 = πτ −a

16πGλ ,

χ3(σ) = ππ± , χ4(σ) = πf± −a

16πGπ± ,

(6.10)

we find0 χ1, χ2 0 0

χ2, χ1 0 0 00 0 0 χ3, χ40 0 χ4, χ3 0

=

a

16πG

0 1 0 0−1 0 0 00 0 0 δ(σ1 − σ2)0 0 −δ(σ1 − σ2) 0

, (6.11)

6.1. Canonical structure 39

with inverse

16πG

a

0 −1 0 01 0 0 00 0 0 −δ(σ1 − σ2)0 0 δ(σ1 − σ2) 0

≡ (Cij) . (6.12)

We are now ready to compute the Dirac brackets according to

A,BDB = A,BPB − A,χiPBCijχj , BPB , (6.13)

where A and B are any functions of the canonical variables. One finds

λ, πλDB = 0 , τ, λDB =16πG

a, τ, πτDB = 1 , πλ, πτDB = 0 ,

(6.14)and similarly

π±(σ1) , ππ±(σ2)DB

= 0 , (6.15)

f±(σ1) , π±(σ2)DB =16πG

aδ(σ1 − σ2) , (6.16)

f±(σ1), πf±(σ2)DB

= δ(σ1 − σ2) , (6.17)ππ±(σ1), πf±(σ2)

DB

= 0 , (6.18)

consistent with the constraints. Alternatively, one can eliminate λ, πλ, π±(σ), andππ±(σ) before starting any computation and compute Poisson brackets with respectto only the remaining canonical variables. This agrees with the Dirac bracketformalism and is the approach we follow throughout the remainder of this thesis.

Naively, the constraints introduce an ambiguity in the Hamiltonian H but anice starting point is

H =8πG

aπ2ϕ + eϕ

[πτ +

∫ ∞0

dσ πf+∂σf+ −∫ ∞

0

dσ πf−∂σf−

]. (6.19)

In Dirac’s formalism the total, unambiguous Hamiltonian is HT = H + uiχi wherethe ui are to be determined using the fact that the constraints do not vary with time.Note that, when evaluated on the equations of motion, HT = H and no ambiguityis present at all. When computing the time-derivatives of the constraints through∂tχi = χi, HT , one finds the Lagrange multipliers ui to be1

u1 = 0 , u2 = 0 ,

u3(σ) = 0 , u4(σ) = ±eϕ∂σπ± ,(6.20)

1No secondary constraints arise in the process.


yielding the total Hamiltonian

HT =8πG

aπ2ϕ + eϕ

[πτ +

∫ ∞0


0

dσ πf−∂σf−

+

∫ ∞0

dσ ππ+∂σπ+ −

∫ ∞0

dσ ππ−∂σπ−

], (6.21)

whose Hamilton equations are consistent with the constraints. At this stage, onecan consistently remove the π± and ππ± degrees of freedom from the Hamiltonian.This implies that the Hamiltonian (6.19) is sufficient for our purposes. For brevity,let us henceforth denote

P± =

∫ ∞0

dσ πf±∂σf± , (6.22)

and

P0 = P+ − P− . (6.23)

6.2 SL(2,R) symmetry

Since AdS2 is isometric under SL(2,R) transformations, we expect an equiva-lent symmetry to be present in the boundary theory. It is well-known that theSchwarzian derivative found in the action (6.2) is invariant under SL(2,R) butwhat does the symmetry look like before eliminating any variables?

The SL(2,R) group has three generators L−1, L0, and L1, which satisfy thesl(2,R) algebra

[L0, L±1] = ∓i~L±1, [L1, L−1] = 2i~L0 . (6.24)

It turns out that the generators defined by

L0 = τ (πτ + P0) + πϕ , (6.25)

L1 =√

2

(1

2τ2(πτ + P0) + τπϕ −

a

16πGeϕ), (6.26)

L−1 =√

2 (πτ + P0) , (6.27)

indeed satisfy this algebra2 and commute with H given by (6.19). These conservedquantities may be obtained as Noether charges stemming from symmetries of thephase space Lagrangian. We will not expand on the meaning of these symmetrieshere but instead refer the reader to [4].

2One uses the following commutators:

[τπϕ, eϕπτ ] = i~ (eϕπϕ − eϕτπτ ) ,[

eϕ, πϕ2]

= i~ (eϕπϕ + πϕeϕ) .

6.3. Bulk dynamics from the boundary 41

6.3 Bulk dynamics from the boundary

Next we find out what the equations of motion entail and we determine initialconditions to match these to our earlier bulk description. We now substitute for(6.5) in (6.7) and (6.8) thus obtaining

f±(t, σ) = f±(τ(t)± σ) , (6.28)

π±(t, σ) = π±(τ(t)± σ) . (6.29)

The above result, in combination with the initial condition

π±(0, σ) = ±∂σf±(0, σ) , (6.30)

yields∫ ∞0

dσ π+∂σf+ −∫ ∞

0

dσ π−∂σf− =

16πG

a

∫ ∞0

(T τ++(τ(t) + σ) + T τ−−(τ(t)− σ)

)dσ , (6.31)

where we definedT τ±± =

a

16πG(∂±f±)2 (6.32)

as the left- and right-moving stress–energy tensor, associated with the fields f±.Now, solving for λ in the equations of motion and using (6.31), yields

λ = − 1

∂tτ∂t

(∂2t τ

∂tτ

)+

16πG

a

∫ τ(t)

−∞(T τ++(s)−T τ−−(s)) ds−16πG

a

∫ ∞−∞

T τ++(s) ds .

(6.33)

This equation of motion reduces to the boundary equation of motion we obtainedin the form of (3.5) by identifying τ(t) as X(t) and constraining the variable λthrough

λ = 0 . (6.34)

The Hamiltonian, when substituting for the equations of motion, has the form

H|EOM =8πG

aπ2ϕ + eϕ

[πτ +

∫ ∞0


0

dσ πf−∂σf−

]∣∣∣∣EOM

(6.35)

= − a

16πGτ, t , (6.36)

which, with τ(t) = X(t), matches the expression (5.20) derived for the boundaryenergy of the bulk system. We have hence shown that the conditions on the bound-ary theory stated earlier are fulfilled on a classical level. Lastly, one also obtains,evaluated on the equations of motion, that

dH

dt

∣∣∣∣EOM

= eϕ(P+ − P−) = T t−− − T t++ , (6.37)


where we have made use of P± = −τT τ±±, in agreement with (5.51).3

6.4 External constraints

To make contact with the bulk system we studied earlier, we imposed the twoconstraints

λ =16πG

aπτ = 0 , (6.38)

π±(0, σ) =16πG

aπf±(0, σ) = ±∂σf±(0, σ) . (6.39)

Since (3.5) is the derivative of the original equation of motion, we should also imposethe condition4

∂tτ → 1 as t→ −∞ . (6.40)

These should be viewed as initial conditions on the fields that are given such thatwe find equivalence with the bulk system we studied earlier.

Within the current canonical framework, the above constraints have to betreated in conjunction with the more fundamental second-class constraints to obtainthe correct dynamical evolution. It is clear that one should only impose the externalconstraints after computing Poisson brackets, since otherwise inconsistencies arise.

6.5 The matter Poisson bracket

Next, we would like to compute the Poisson bracket of the matter fields, evaluatedon their equations of motion with respect to the canonical variables. For that, weneed the explicit solutions to the equations of motion as functions of their initialconditions.

6.5.1 Solutions to the equations of motion

We will focus on the particular case where P0 is time-independent or at mostpiecewise constant. This is the case when all matter reflects perfectly, or when onesends pulse-shaped waveforms into the spacetime. This is physically reasonable butalso crucial to have any chance at an analytical solution of the equations. Now,taking the factor multiplying −eϕ in (6.4) to be constant (which is the case if

3It might seem strange at first that the Hamiltonian is not conserved in our dynamical systemin general, but this is due to our neglect of the surface term at σ = 0 in the previous manipulations.

4Provided the initial frame is the Poincare patch. If not, then one should impose a differentinitial condition as has been discussed in previous chapters. We will not go into this here.

6.5. The matter Poisson bracket 43

f+(t, σ = 0) = f−(t, σ = 0)), we find that the general solution to such a differentialequation can be expressed as

e−ϕ(t)2 =

√πτ + P+ − P−

Ecosh

(√8πGE

a(t− C)

), (6.41)

where E and C are integration constants, and thus, using (6.9), we find

πϕ(t) = − a

16πG

√32πGE

atanh

(√8πGE

a(t− C)

). (6.42)

We obtain5

E =8πG

aπ2ϕ + eϕ(πτ + P+ − P−) . (6.43)

We see that E can be identified with the Hamiltonian (6.19) and since it is aconstant of motion, we can choose to evaluate it at t = 0 to obtain

E =8πG

aπ2ϕ(0) + eϕ(0)(πτ + P+ − P−) . (6.44)

If we instead multiply (6.41) and (6.42) evaluated at t = 0 one can compute

C =

√a

8πGEarsinh

(√8πG

a (πτ + P+ − P−)πϕ(0)e−

ϕ(0)2

). (6.45)

Now, from (6.5), we have

τ(t) =

√a

8πGE

E

πτ + P+ − P−tanh

(√8πGE

a(t− C)

)+D , (6.46)

where D is an integration constant. Note that for any E > 0, this expressionexhibits periodicity in imaginary time. Setting t = 0, yields

D = τ(0) +

√a

8πGE

E

πτ + P+ − P−tanh

(√8πGE

aC

), (6.47)

Now, viewing the matter field as a function (with new notation x(t = 0) = x0

for brevity)

f±(t, σ) = f±

(f0±(σ), τ

[ϕ0, π0

ϕ, f0±(σ), π0

f±(σ), τ0, π0τ ; t], σ)

= f0±

(τ[ϕ0, π0

ϕ, f0±(σ), π0

f±(σ), τ0, π0τ ; t]± σ

), (6.48)

5Using the hyperbolic identity

tanh2 x+ cosh−2 x = 1 .


where the ± subscript of the arguments means that there is a dependence on boththe + and − fields, whereas the ± subscript of the function itself implies that wechoose the sign.

6.5.2 The Poisson bracket

The Poisson bracket is computed with respect to the canonical variables ϕ0, π0ϕ,

τ0, π0τ , f0

±(σ) and π0f±

(σ). The variables λ, πλ, π± and ππ± are eliminated beforecomputing the bracket in accordance with the Dirac bracket formalism. Let us nowcompute the Poisson bracket

f+(t1, 0), f+(t0, 0) (6.49)

in the regime where t1 t0. Without loss of generality, we focus on t0 = 0. Thecomputation is not that hard, but does require a meticulous treatment. The detailsare included in Appendix C. One finds

f+(t1, 0), f+(0, 0) ∼ e2√

8πGEa t1 for large t1. (6.50)

Since (6.46) has a periodicity in imaginary time corresponding to TH =√

8πGE/a/π,we may substitute for this in (6.50) to obtain

f+(t1, 0), f+(0, 0) ∼ e2πTHt1 , (6.51)

which is the conjectured maximum value of a Lyapunov exponent, and hence, theboundary theory is maximally chaotic, or, equivalently, has the minimum scram-bling time [3]. To see the link to shockwaves in the bulk, the reader is referred to[4].

Chapter 7

Conclusion

This thesis treated the bulk system studied by Almheiri and Polchinski as a step-ping stone to arrive at a proposal for a dual boundary theory. First we reviewedthe basics of the model. From it we derived a general equation of motion forthe reparametrized boundary time, or preferred coordinate frame, and investigatedpulse solutions. We then included quantum effects into our theory through an ef-fective action and calculated an expression for the boundary stress–energy tensorwhose “classical” part turned out to be proportional to a Schwarzian derivativeof the preferred coordinate transformation plus a term only containing quantumeffects. We studied black hole evaporation and it turned out that a quite specialsetup is needed in order to induce evaporation which decays exponentially in gen-eral. We also analyzed the quantum term of the boundary stress–energy tensor.The primary outcome of this work however was the proposed dual boundary the-ory. We investigated its canonical structure and verified that it reproduces boththe reparametrization equation and the boundary energy obtained from the bulksystem. Finally, we computed the matter Poisson brackets and arrived at the con-clusion that the system is maximally chaotic.

There are several open ended questions connected to the results presented inthis thesis. Some of them, such as the connection between bulk shockwaves andboundary chaos, are treated further in [4]. Others are still left unanswered.

One such question is how exactly the boundary theory presented here is relatedto the action proposed by Kitaev [2]. Another one is the whether it is possible todevelop a full quantum theory of the bulk to see the microscopic correspondencebetween it and the boundary theory. Or if we should view it as an effective theorywhose underlying exact theory has to be investigated via the quantum theory of theboundary. One question is how to reproduce the quantum term of the boundaryenergy as given by (5.19) or whether a full quantum treatment of the boundarytheory yields this term “automatically.” This question is related to the yet unsolvedproblem of computing the exact commutators of the fields in the boundary theory;

45

46 Chapter 7. Conclusion

so far we have only done it to lowest order by using the correspondence with thePoisson bracket.

For more on the topic of this thesis the reader is referred to the more advancedtreatment given in [4]. Furthermore, since the topic has attracted quite some at-tention lately in works such as [39] and [40] we hope that further advances are justaround the corner.

Appendix A

A subtlety in the boundaryequation of motion

The final term in equation (2.18) needs to be added because of considerationsrelated to the boundary energy when outgoing pulses are present. If one were toignore this term the situation depicted in Figures A.1a and A.1b would arise.

ty

E = 0

E = 0

(a) Perfectly reflecting pulse.

ty

E = − µ8πG

E = 0

(b) Absorbed outgoing pulse.

Figure A.1: The figure on the left describes a perfectly reflecting pulse (T++ = T−−) towhich no energy at all is associated if we exclude the last term in the expression for Φ2.The figure on the right describes the perfect absorption of an outgoing pulse (T−− = 0)to which the naive formula only assigns an energy after it has been absorbed by theboundary.

47

48 Appendix A. A subtlety in the boundary equation of motion

If instead we add the final term (which basically makes sure outgoing energyis counted before the pulse hits the boundary instead of after), the outgoing pulsebecomes associated with a white hole horizon with a positive boundary energy (seeFigure A.2). After it hits the boundary, an energy µ

8πG is removed from the systemand one returns to the Poincare patch. It is somewhat strange to start out in adifferent patch, but this is required as it is merely the time-reversed situation ofthe infalling pulse studied before. This also allows for a more natural identificationwith the boundary model we propose in Section 6.

ty

E = 0

E = µ8πG

tξ = −∞

Figure A.2: White hole emitting a pulse that is absorbed on the boundary. This processis a natural transition from the black hole frame to the Poincare patch.

Appendix B

Multiple-pulse dynamics

B.1 Consistency check using infalling pulses

Let us now further analyze the situation discussed in the main text, namely thecreation of a black hole by sending in a pulse. Here, however, we will send in twoconsecutive pulses and to find the corresponding preferred coordinate transforma-tion and boundary energy. We let the pulses be separated by a Poincare time δand with amplitudes µ1/(8πG) and µ2/(8πG) as evaluated in the Poincare frame.Using (3.4) leads to the differential equation

∂tX = 1− µ1

aX2θ(X)− µ2

a(δ −X)2θ(X − δ) . (B.1)

For X > δ (i.e. after the second pulse has passed), one can easily check that X(t)

is given by (3.19) with δ =√a/µ1 tanh

(√µ1/a∆

)and µ2 = µ2 cosh2

(√µ1/a∆

).

This is a non-trivial consistency check but the outcome was expected. This finalrelation is obtained by contemplating the transformation properties of an infallingpulse:

T x−− =µ2

8πGδ(X− − δ) =⇒ T ξ−− = cosh−2

(√µ1

a∆

)µ2

8πG︸︷︷︸µ28πG

δ(ξ− −∆) . (B.2)

The boundary energy contained in the system after the second pulse is

〈Ttt〉 =µ1 + µ2 − µ1µ2δ

8πG=µ1 + µ2

8πG. (B.3)

The second equality shows that when describing the situation in terms of the rele-vant observer everything works out as expected.

49

50 Appendix B. Multiple-pulse dynamics

One might attempt to confirm the above results by instead starting from theflowing equation given by (5.45). Now, the stress–energy tensor “flows” with theboundary observer and we may use the form

T t−−(t) =µ1

8πGδ(t) +

µ2

8πGδ(t−∆) (B.4)

in order to check the result (3.19). Then we obtain the differential equation

∂tX(t) = 1− 1

a

∫ t

−∞ds

(X(u)−X(t))2

X ′(u)(µ1δ(u) + µ2δ(u−∆))

= 1− µ1

aθ(t)

(X(0)−X(t))2

X ′(0)− µ2

aθ(t−∆)

(X(∆)−X(t))2

X ′(∆). (B.5)

The solution X(t) can be found by studying (3.19) and is given by

X(t) =t , t < 0 ,√

aµ1

tanh(√

µ1

a t), 0 < t < ∆ ,√

aµ1

tanh

(artanh

(√µ1

µ1+µ2tanh

(√µ1+µ2

a (t−∆)

))+√

µ1

a ∆

), t > ∆ .

(B.6)

This function is everywhere continuously differentiable. One readily computes

X(0) = 0, X ′(0) = 1 ,

X(∆) = δ, X ′(∆) =µ2

µ2,

(B.7)

where we used ∆ =√a/µ1artanh

(√µ1/a δ

)and µ2 = µ2 cosh−2

(√µ1/a∆

).

B.2 Other frames obtained by time delays

Next to the three main frames (Poincare, global, and black hole) we have studiedso far, there are a host of new frames that can be obtained by sending in multiplepulses with time delays in between. From e.g., (B.3) one may deduce that theboundary energy is independent of the time separation between consecutive pulses;it only depends on the energies of the pulses. Since the classical boundary energyis a Schwarzian derivative, one might guess that the coordinate transformationscorresponding to a given boundary energy can be obtained through SL(2,R) trans-formations. This can easily be checked for the case of the two-pulse solution and itturns out that it holds for all types of situations involving consecutive pulses.

B.2. Other frames obtained by time delays 51

It is even possible to set up a transfer matrix framework to treat the most gen-eral multi-pulse solution. This prescription relates the coordinate transformationobtained by sending in all pulses at the same time to a coordinate transformationobtained by sending in pulses between which there are arbitrary time delays.

First, with

A =

(a11 a12

a21 a22

), (B.8)

we define the operation 〈 · 〉 as

A · f =a11f + a12

a21f + a22. (B.9)

Now, let

A(∆;µ) ≡

(cosh

(√µa∆) √

aµ sinh

(√µa∆)√

µa sinh

(√µa∆)

cosh(√

µa∆) )

∈ SL(2,R) . (B.10)

Furthermore, let n pulses with energy µ1/(8πG), . . . , µn/(8πG) separated bytimes ∆1 . . .∆n−1 fall into the spacetime. All of these quantities are to be evaluatedin the instantaneous preferred coordinate frame. The coordinate transformationobtained by sending in all the energy at once is given by

f(t) =

√a∑j µj

tanh

√∑j µj

at

. (B.11)

The actual solution X(t) (which includes the time delays) can now be obtained viaa concatenation of operations defined by (B.9):

X(t) = A(∆1;µ1) ·A(∆2 −∆1;µ1 + µ2) · · ·A

∆i −∆i−1;

i∑j=1

µi

· · ·×A

∆n−1 −∆n−2;

n−1∑j=1

µi

·A−∆n−1;

∑j

µj

· f(t) . (B.12)

Appendix C

Poisson bracket computation

The contribution to the bracket (6.49) with respect to the canonical variables(f0

+, π0f+

) is

∫dσ

[δf+(t1, σ1)

δf0+(σ)

δf+(t0, σ0)

δπ0f+

(σ)− δf+(t0, σ0)

δf0+(σ)

δf+(t1, σ1)

δπ0f+

(σ)

]

=

∫dσ

[δ(σ − τ(t1)− σ1) +

∂f+

∂τ

∣∣∣∣σ1,t1

∂τ

∂P0

∣∣∣∣t1

δP0

δf0+(σ)

∣∣∣∣t1

]

× ∂f+

∂τ

∣∣∣∣σ0,t0

∂τ

∂P0

∣∣∣∣t0

δP0

δπ0f+

(σ)

∣∣∣∣∣t0

− term with 1↔ 0 . (C.1)

In order to simplify our calculations we choose t0 = 0 because then we have∂τ/∂P0|t0=0 = 0. This is due to the fact that τ evaluated at this time is just

the independent variable τ0. Thus it is only the second term with 1↔ 0 that doesnot vanish. The part that remains is hence

−∫

dσδf+(0, σ0)

δf0+(σ)

δf+(t1, σ1)

δπ0f+

(σ)= − ∂f+

∂τ

∣∣∣∣σ1,t1

∂τ

∂P0

∣∣∣∣t1

δP0

δπ0f+

(τ0 + σ0)

∣∣∣∣∣t1

. (C.2)

From the above considerations we may also deduce that the contribution to thePoisson bracket from the canonical variables associated with f− vanishes.

The contribution from the other fields to the Poisson bracket isf0±

(τ[ϕ0, π0

ϕ, f0±(σ), π0

f±(σ), τ0, π0τ ; t1

]± σ1

), f0±(τ0 ± σ0

)ϕ0,π0

ϕ,τ0,π0

τ

= − ∂f+

∂τ

∣∣∣∣σ1,t1

∂f+

∂τ

∣∣∣∣σ0,t0=0

∂τ

∂π0τ

∣∣∣∣t1

. (C.3)

53

54 Appendix C. Poisson bracket computation

Collecting our results we have

f+(t1, σ1), f+(t0 = 0, σ0) = − ∂f+

∂τ

∣∣∣∣σ1,t1

∂τ

∂P0

∣∣∣∣t1

δP0

δπ0f+

(τ0 + σ0)

∣∣∣∣∣t1

− ∂f+

∂τ

∣∣∣∣σ1,t1

∂f+

∂τ

∣∣∣∣σ0,t0=0

∂τ

∂π0τ

∣∣∣∣t1

= − 1

τ

∣∣∣∣t1

f+

∣∣∣σ1,t1

∂τ

∂P0

∣∣∣∣t1

δP0

δπ0f+

(τ0 + σ0)

∣∣∣∣∣t1

− 1

τ

∣∣∣∣t1

1

τ

∣∣∣∣t0=0

f+

∣∣∣σ1,t1

f+

∣∣∣σ1,t0=0

∂τ

∂π0τ

∣∣∣∣t1

. (C.4)

Now, let us assume that fluctuations f± with respect to the boundary time t donot grow exponentially in t, i.e., f± ∼ O(1). We are now ready to compute thePoisson bracket given by (C.4) factor by factor. First, we note that

δP0

δπ0f+

(τ0 + σ0)

∣∣∣∣∣t1

=δ

δπ0f+

(τ0 + σ0)

∫ ∞0

dσ πf+(τ(t1) + σ)∂σf+(τ(t1) + σ)

= θ(τ0 + σ0 − τ(t1))∂τf+(τ0 + σ0)

= 0 , for τ0 = σ0 = 0 and t1 > 0 , (C.5)

implying that the first term in (C.4) vanishes. For the second term, we only careabout the large t1 behavior. In the end one ends up with:

∂τ

∂π0τ

∣∣∣∣t1

∼ O(1) for large t1 (C.6)

and

1

τ

∣∣∣∣t0=0

=µ

8πGEcosh2

(√8πGE

aC

)∼ O(1) , (C.7)

1

τ

∣∣∣∣t1

=µ

8πGEcosh2

(√8πGE

a(t1 − C)

)∼ e2√

8πGEa t1 for large t1 . (C.8)

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