Investigation of a Scalar Wave Equation in Asymptotically Anti-de Sitter Spacetimes

INOM

EXAMENSARBETE TEKNISK FYSIK, AVANCERAD NIVÅ, 30 HP

STOCKHOLM SVERIGE 2020 ,

Investigation of a Scalar Wave

Equation in Asymptotically Anti-de Sitter Spacetimes

ASK ELLINGSEN

TRITA TRITA-SCI-GRU 2020:189

Investigation of a Scalar Wave Equation in Asymptotically Anti de-Sitter Spacetimes

Author:

Ask Ellingsen askel@kth.se

Department of Physics

KTH Royal Institute of Technology

Supervisor: Professor Bo Sundborg

June 30, 2020

TRITA-SCI-GRU 2020:189

Ask Ellingsen, 2020 c

Abstract

We study a scalar field in asymptotically Anti-de Sitter spacetime, coupled to gravity through an interaction term ξR, where R is the Ricci scalar and ξ is the coupling constant.

It is known that in such field theories, the radial part of the equation of motion for the scalar field can be put on the form of a Schr¨ odinger equation by using a special choice of coordinates. In this thesis we investigate the asymptotic behaviour of this equation and its solutions for asymptotically Anti-de Sitter background metrics that can be defined in terms of a single function f (r) dependent only on the radial coordinate r. We lay out a general method for finding and solving the asymptotic equation using a first principles application of the method of Frobenius. We also apply this method to both Anti-de Sitter space and some examples of asymptotically Anti-de Sitter spaces in various dimensions.

We find that in general, one of the asymptotic solutions picks up a logarithmic behaviour in a sequence of ‘critical’ cases related to the effective squared mass pa- rameter m ² = m ² ₀ + ξR, where m ₀ is the bare mass parameter of the scalar field.

Namely, the logarithmic behaviour only appears when m ² takes one of the critical values m ² = (N + 1)(N − 1)/4, with N a nonnegative integer. For N > 0, logarithmic behaviour of the solution may or may not appear, depending on other parameters than m. We find that the smallest critical mass, corresponding to N = 0, coincides with the well known Breitenlohner-Freedman bound.

Finally, we also discuss the issue of choosing boundary values for the equation of

motion for the scalar field in asymptotically Anti-de Sitter spaces.

Sammanfattning

Vi studerar ett skal¨ arf¨ alt i asymptotiskt-anti-de Sitter-rumtider, kopplat till gravitatio- nen genom en interaktionsterm ξR, d¨ ar R ¨ ar Ricciskal¨ aren och ξ ¨ ar kopplingskonstanten.

Det ¨ ar k¨ ant att i s˚ adana f¨ altteorier kan den radiella delen av skal¨ arf¨ altets r¨ orelseekvation skrivas p˚ a formen av en Schr¨ odingerekvation genom ett speciellt koordinatval. I denna studie unders¨ oker vi det asymptotiska beteendet f¨ or denna ekvation och dess l¨ osningar f¨ or asymptotiskt-anti-de Sitter-bakgrundsrumtider som kan definieras i termer av en enda funktion f (r) som beror enbart p˚ a den radiella koordinaten r. Vi l¨ agger fram en allm¨ an metod f¨ or att hitta och l¨ osa den asymptotiska ekvationen genom att anv¨ anda Frobe- nius metod. Vi applicerar ocks˚ a denna metod till b˚ ade anti-de Sitter-rummet och n˚ agra exempel av asymptotiskt-anti-de-Sitter-rum i olika dimensioner.

Vi finner att i allm¨ anhet plockar en av de asymptotiska l¨ osningarna upp ett logarit- miskt beteende i en f¨ oljd av ‘kritiska’ fall, relaterade till den effektiva masskvadratparam- etern m ² = m ² ₀ + ξR, d¨ ar m ₀ ¨ ar den nakna massparametern f¨ or skal¨ arf¨ altet. Det logarit- miska beteendet uppst˚ ar bara n¨ ar m ² tar ett av de kritiska v¨ ardena m ² = (N +1)(N −1)/4, d¨ ar N ¨ ar ett ickenegativt heltal. F¨ or N > 0 beror det ¨ aven p˚ a andra parametrar ¨ an m ² om logaritmiskt beteende uppst˚ ar. Vi finner att den minsta kritiska massan, motsvarande N = 0, sammanfaller med den v¨ alk¨ anda Breitenlohner-Freedman-gr¨ ansen.

Till sist diskuterar vi ¨ aven problemet med att v¨ alja randv¨ arden f¨ or skal¨ arf¨ altets

r¨ orelseekvation i asymptotiskt-anti-de Sitter-rum.

Acknowledgements

First and foremost, I would like to thank my supervisor, Professor Bo Sundborg, for simplifying the impossible. I’m also indebted to Professor Edwin Langmann, for showing me how to ask questions. Their patience and very helpful input has been crucial to this project.

I would also like to thank Professor Mats Wallin for jumping in as the second examiner at short notice.

I’m grateful to M˚ ans Andersson for a multitude of complaining-over-coffee-sessions, and to Sven Sandfeldt for pulling me through all the maths courses I thought it was a good idea to read in parallell with working on this thesis.

Finally, I’d like to thank my mum and dad for being my mum and dad.

Contents

1 Introduction 4

2 Theoretical Background 6

2.1 Anti-de Sitter Space . . . . 6

2.1.1 Construction of AdS space . . . . 6

2.1.2 Global coordinates . . . . 7

2.1.3 Poincar´ e coordinates . . . . 9

2.2 Asymptotically Anti-de Sitter Spaces . . . . 9

2.3 The AdS/CFT Correspondence . . . . 11

2.4 The AdS-Schwarzschild Solution . . . . 12

2.5 BTZ Black Holes . . . . 13

2.6 The Quantum Mechanical 1/x ² -Potential . . . . 14

2.7 The Breitenlohner-Freedman Bound . . . . 15

3 Free Scalars in AAdS Spacetimes 17 3.1 The Equations of Motion . . . . 17

4 Solving the Equations of Motion for the Scalar Field 20 4.1 The Massless BTZ Metric . . . . 20

4.1.1 The Tortoise Coordinate in BTZ Space . . . . 20

4.1.2 The Potential . . . . 21

4.1.3 Solving the Equation . . . . 23

4.2 AdS Space . . . . 23

4.2.1 The Tortoise Coordinate in AdS Space . . . . 23

4.2.2 The Potential . . . . 24

4.3 General Approach . . . . 27

4.3.1 The Tortoise Coordinate . . . . 27

4.3.2 The Method of Frobenius . . . . 28

4.3.3 Discussion of Critical Cases . . . . 30

4.3.4 BTZ Space Revisited . . . . 32

4.4 The AdS-Schwarzschild Metric . . . . 33

4.4.1 The Tortoise Coordinate in AdS-Schwarzschild Space . . . . 33

4.4.2 The Potential . . . . 37

4.4.3 Solving the equation . . . . 38

5 Discussion 40

5.1 Boundary Values . . . . 40

5.1.1 General Theory and Self-Adjoint Extensions . . . . 40

5.1.2 Quantum Mechanics and AdS Physics . . . . 41

5.1.3 A modern approach . . . . 42

5.1.4 Black Hole Physics . . . . 43

5.2 Are the Critical Cases Critical? . . . . 44

5.3 Backreaction . . . . 45

6 Conclusion 46 A Calculations 48 A.1 Asymptotic Expansion of the Tortoise Coordinate . . . . 48

A.2 Solving the Schr¨ odinger Equation for the 1/ρ ² -Potential Using Frobenius’ Method . . . . 49

B Asymptotic Solutions 50

B.1 AdS-Schwarzschild space . . . . 50

Conventions and Units

• We will be working in natural units, where

~ = c = 1.

Unless stated otherwise, we will also choose units so that the curvature radius of spacetime is

L = 1,

where L is defined via the cosmological constant Λ (with Λ < 0) as the positive root of the equation

Λ = − d(d − 1) 2L ² .

More on this in section 2.1. This fixes all relevant units, and means that in the generic case we will have Newton’s constant

G 6= 1.

• The notation

T _µν

means either the µν:th component of a tensor T in some coordinate system under- stood by context, or the entire (coordinate-independent) tensor itself expressed in abstract index notation. We will use brackets,

(T _µν ),

to denote the matrix representation of the tensor T _µν in implicit coordinates.

• We use the ‘mostly plus’ metric signature, (−, +, +, . . .).

• We will be employing the Einstein summation convention, so for example,

g µν A ^ν =

d

X

ν=0

g µν A ^ν ,

where d is the spatial dimension. When working in d+1 bulk spacetime dimensions,

repeated greek indices starting from µ, ν, ρ, . . . are implicitly summed from 0 to d,

and latin indices i, j, k, . . . are implicitly summed over only the spatial coordinates,

from 1 to d. It will sometimes be useful to sum from 0 to d, but skip one special

(radial) spatial coordinate. We will then use greek indices starting from α, β, . . ..

Chapter 1 Introduction

In a 1982 paper [4], Breitenlohner and Freedman derived a stability bound on the squared mass parameter of a scalar field in Anti-de Sitter (AdS) space. Below this bound, the dynamics of the system are predicted to be wildly unstable, causing spontaneous collapse.

Therefore, the Breitenlohner-Freedman bound says that only fields with mass parameters above this critical value can be physical. However, the Breitenlohner-Freedman bound on the squared mass of the field is negative, and so allows for certain tachyonic fields in AdS space. Their result has since been reproduced using methods of varying degrees of rigour – notably by Ishibashi and Wald in [8].

The modern interest in AdS and Asymptotically AdS (AAdS ) spaces stems largely from the theory known as the AdS/CFT correspondence, proposed by Juan Maldacena in 1997, and expanded upon by Witten in 1998 [13]. This theory says that field theories involving gravity defined on an AAdS space are dual to conformal field theories (CFT:s) defined on the boundary of those spaces. It suggests that results derived about the confor- mal field theory defined on the boundary can be translated into results about the theory involving gravity defined on the interior, ¹ and vice versa. In Chapter 2 we introduce some of the concepts discussed here in more detail and give the relevant theoretical background for this thesis.

Maldacena’s result sparked a significant interest in the study of AAdS spaces, and the various field theories that can be defined on them. In Chapter 3, we introduce one of the simplest such field theories – that of a scalar field directly coupled to gravity through a simple interaction term proportional to the Ricci scalar.

We will focus on the behaviour of the scalar field, rather than the effect it has on gravity. In particular, we are interested in how it behaves toward spatial infinity, which is the ‘boundary’ on which the corresponding CFT would be defined. Treating gravity as a static background field, it turns out that the radial part of the equation of motion for the field reduces to a Schr¨ odinger equation. In AAdS spaces, the resulting ‘potentials’

have a general asymptotic behaviour. In Chapter 4 we will study this general behaviour, and apply our results to some special cases.

For certain, simple spaces, the potential is simple enough for the Schr¨ odinger equation to be explicitly solvable. We will study two such cases: the standard, vacuum AdS space (in arbitrary dimension), and a (2 + 1)-dimensional space in which there exists a massless, pointlike ‘black hole’, known as a (massless) BTZ black hole [1]. For more complicated

1

Known as the bulk in the literature.

potentials, finding exact solutions proves quite difficult. Instead we will take the approach of expanding the potential as an asymptotic series, and solving the resulting asymptotic equation using the method of Frobenius. This process immediately yields certain ‘critical’

values of the squared mass parameter m ² of the scalar field, at which one of the linearly independent solutions picks up a logarithmic behaviour. The critical masses will be modified depending on the curvature coupling. We show that the smallest such critical value appears to correspond to the Breitenlohner-Freedman bound. This leads us to conclude that the other critical masses might also be of importance – although at the point of writing, the specifics of this potential importance is unknown.

After laying out the general theory of this approach, we use it to calculate some

examples that are not easily explicitly solvable, and study the critical masses and the

solutions to the equation. Finally, in Chapter 5, we will discuss the problem of choosing

proper boundary values in spaces equipped with black hole metrics, and the potential

physical interpretation of the critical masses and the corresponding solutions.

Chapter 2

Theoretical Background

2.1 Anti-de Sitter Space

In this section we formally introduce Anti-de Sitter space and discuss different parametri- sations.

2.1.1 Construction of AdS space

The Einstein equations, that govern the curvature of spacetime, have the general form R _µν − 1

2 Rg _µν + Λg _µν = κT _µν , (2.1.1) where g _µν is the metric, R _µν is the Ricci tensor, R is the Ricci scalar, κ = 8πG is Einstein’s constant, Λ is the cosmological constant, and T _µν is the stress-energy tensor, which encodes the matter content of the universe. An Einstein metric is one where the Ricci tensor is proportional to the metric itself:

R _µν = kg _µν , (2.1.2)

for some constant k. This is equivalent to solving the Einstein equations for some T _µν proportional to g _µν . A vacuum solution to the Einstein equations is an Einstein metric g _µν that solves equation (2.1.1) with T = 0 (in particular, any solution to (2.1.1) with T = 0 is automatically an Einstein metric). A maximally symmetric solution is a solution with all possible symmetries. There are three fundamentally different maximally symmetric vacuum solutions, depending on the sign of Λ. The solution corresponding to Λ = 0 is the traditional Minkowski metric of special relativity. If Λ > 0, the corresponding solution is known as the de Sitter (dS) metric, after the Dutch mathematician and cosmologist Willem de Sitter. For obvious reasons, the solution corresponding to Λ < 0 is called the Anti-de Sitter (AdS) metric. The manifold equipped with the AdS metric is known as Anti-de Sitter space and denoted by AdS _n , where n is the dimension of the manifold.

We will show how AdS _d+1 can be constructed explicitly, following [2]. First, define R ^d,2 as the set R ^d+2 considered as a pseudo-Riemannian manifold with metric

ds ² = −dX ₀ ² − dX _d+1 ² +

d

X

i=1

dX _i ² . (2.1.3)

That is, R ^d,2 is simply R ^d+2 , but we consider d dimensions to be spacelike and 2 to be timelike. The Anti-de Sitter space AdS _d+1 can then be embedded into R ^d,2 as the solution set to the equation

− X ₀ ² − X _d+1 ² +

d

X

i=1

X _i ² = −L ² , (2.1.4)

where L is the radius of curvature, related to the cosmological constant by Λ = − d(d − 1)

2L ² . (2.1.5)

From now on, we choose units so that L = 1. We call the X _i embedding coordinates.

The metric on AdS _d+1 is the one inherited from R ^d,2 , and any explicit expression for it will depend on the particular choice of coordinates made. We will construct explicit expressions in some different coordinate systems.

2.1.2 Global coordinates

First, we define coordinates (ρ, t, Ω) on AdS _d+1 through the following parametrisations of the embedding coordinates in (2.1.4):

X ₀ = cos t cos ρ , X _d+1 = sin t

cos ρ ,

X _i = x _i tan ρ, i = 1, . . . , d.

(2.1.6)

where the x _i are the standard embedding coordinates of S ^d−1 in R ^d , satisfying the standard equation for the (d − 1)-sphere, ¹

d

X

i=1

x ² _i = 1. (2.1.7)

The coordinate ranges are ρ ∈ [0, π/2), t ∈ [0, 2π), and x _i ∈ [−1, 1]. One can then verify that Eq. (2.1.4) is satisfied. ² The metric is the pullback metric ³ inherited from R ^d,2 .

1

In particular, this constraint means there are only d − 1 independent x

, that may for example be parametrised by d − 1 angles θ

as

x

= cos

θ

, x

=

d−1

Y

j=1

sin

θ

, and x

= cos

θ

i−1

Y

j=1

sin

θ

, for 2 ≤ i ≤ d − 1.

2

A problem with these coordinates is that the ‘time’ coordinate t is cyclic, with period 2π. This means we can have closed timelike curves (CTC:s) that violate causality in AdS space. The standard way to formally solve this problem is to pass to the universal covering space – the (unique up to diffeomorphism) simply connected space that covers AdS – sometimes denoted CAdS. This space can be though of as an

‘unwound’ version of AdS space – like the real line R is an ‘unwound’ version of the circle S

. In this text we will take ‘AdS space’ to mean the universal cover, but the distinction will not be of huge importance.

3

Given two manifolds M and N , a metric g defined on N , and a diffeomorphism (a smooth bijective map with smooth inverse) from M to N , the pullback metric Φ

g on M given by Φ and g is defined by

Φ

g(ξ, η) = g (dΦ(ξ), dΦ(η)) ,

Using Eq. 2.1.6, we can calculate this to be ds ² = 1

cos ² ρ −dt ² + dρ ² + sin ² ρdΩ ² _d−1
, (2.1.8) where dΩ ² _d−1 is the standard metric on the (d − 1)-sphere S ^d−1 . ⁴ This version of global coordinates is noteworthy since the range of ρ is finite, with spatial infinity at ρ = π/2. In these coordinates, ‘infinity is a finite distance away’. This is a fundamental idea behind conformal compactification, to be discussed in Section 2.2. We will be seeing much of ρ and its cousins throughout this report, first related to defining functions, and later, as tortoise coordinates. A notable feature of the ‘radial’ coordinate ρ is that it suggests that the spatial range of the space is, in a sense, finite in these coordinates, and by formally increasing the range of the coordinate to ρ ∈ [0, π/2] we may ‘compactify’ the space – adding a ‘boundary at infinity’. We will return to this notion in the next section.

For now, note that ρ cannot represent an ordinary spherical radial coordinate, because the surface at fixed ρ and t does not have the area of a sphere of radius ρ. Introducing the coordinate r = tan ρ, we may rewrite the metric (2.1.8) as ⁵

ds ² = −(1 + r ² )dt ² + dr ²

1 + r ² + r ² dΩ ² _d−1 . (2.1.9) The coordinates (t, r, Ω) are the standard choice of coordinates on AdS _d+1 . The reason for this standard is that the r-coordinate now plays the role of an ‘ordinary’ – namely, a sphere of radius r has the expected area

A(rS ^d−1 ) = r ² Z

S

dΩ _d−1 .

To suggest the connection between the r-coordinate and our intuitive understanding of radial distance, we will be calling the r-coordinate the spatial radial coordinate.

If we introduce the function

f _AdS (r) = 1 + r ² , (2.1.10)

we may rewrite (2.1.9) as

ds ² = −f _AdS (r)dt ² + dr ²

f _AdS (r) + r ² dΩ ² _d−1 . (2.1.11) Metrics on this form will be our objects of interest in this report. Usually, we will take the opposite route to the one taken here. We will start with a metric on the form (2.1.11), and then introduce a coordinate ρ to rewrite it in a ‘compactified’ form similar to the one in (2.1.8).

where ξ and η are smooth vector fields on M . The metric inherited from an ambient space can in this way be defined as the pullback metric of the embedding map.

4

This metric may be defined as the pullback metric on S

of the standard metric ds

= δ

dx

dx

on R

.

5

To see this, note that cos

(arctan r) = 1/(1 + r

), and dρ = dr/(1 + r

).

2.1.3 Poincar´ e coordinates

Another common coordinate system in AdS space is Poincar´ e coordinates. These are constructed by first introducing the light-cone coordinates

u = X _d+1 − X _d , v = X _d+1 + X _d , (2.1.12) and parametrising the rest of the embedding coordinates as

x ^α = X α

u , 0 ≤ µ ≤ d − 1. (2.1.13)

The coordinate x ⁰ = t will play the role of a time coordinate. Inserting these parame- terisations into Eq. (2.1.4) yields an equation that we may solve for v. Doing so, and introducing

z = 1

u , (2.1.14)

we obtain the expressions

X _d = 1

2z 1 + z ² (η _αβ x ^α x ^β + 1) , (2.1.15) X _d+1 = 1

2z 1 + z ² (η _αβ x ^α x ^β − 1) , (2.1.16) X _µ = x ^α

z , 0 ≤ α ≤ d − 1, (2.1.17)

where η is the Minkowski metric on M ^d and the implicit sum is taken over 0 ≤ α, β ≤ d−1.

The coordinates (z, x ^α ) = (t, z, x ⁱ ) are the Poincar´ e coordinates. In these coordinates, the metric on AdS _d+1 takes the simple form

ds ² = 1

z ² (dz ² + η _αβ dx ^α dx ^β ). (2.1.18) This form of the metric will be useful in deriving the Breitenlohner-Freedman bound in Sec. 2.7, but beyond that, we will be sticking to the global coordinates of Sec. 2.1.2 throughout the text.

2.2 Asymptotically Anti-de Sitter Spaces

In this section we will engage in some relatively heavy mathematics. Many of the detailed definitions discussed here will not be central to the thesis, but are included for complete- ness. Our goal is to define asymptotically Anti-de Sitter (AAdS) spaces, which intuitively are spacetimes (Lorentzian manifolds) equipped with metrics that behave asymptotically like the Anti-de Sitter metric of equation (2.1.9) as r → ∞; and to make the connection to conformal field theories (CFT:s), to be discussed further in section 2.3. First, we will introduce conformal structures.

For a given manifold M, we can introduce an equivalence relation on the set of metrics

defined on that manifold by saying that g ∼ g ⁰ if g ⁰ = e ^w g, for some smooth function

w : M → R. Two metrics g and g ⁰ related in this way are said to be conformally

equivalent. Let [g] be the equivalence class (known as the conformal class) of some metric g. We then say that [g] defines a conformal structure on M. We can think of a conformal structure as a metric defined ‘up to local rescaling’. A conformal structure, like a metric, uniquely determines angles between tangent vectors ⁶ – and therefore between crossing curves on the manifold – but, unlike a metric, does not determine distances between points.

Next, we will define conformally compact manifolds. We follow the definition given in [9]. Let M be a manifold equipped with a metric g. Let M be a manifold with boundary, and suppose M can be (smoothly) embedded onto the interior Int M of M.

Let Φ : M → Int M be the embedding diffeomorphism. We then identify M with Int M and g with the pushforward metric Φ ∗ g. ⁷ Now let ψ : M → R be a smooth, positive function such that the following two conditions hold:

1. ψ = 0 on ∂M;

2. dψ 6= 0 on ∂M. ⁸

Such a function ψ is called a defining function. If for any defining function ψ, the metric g = ψ ² g, a priori defined on M, extends continuously to a metric on the entire manifold with boundary M, we say that (M, g) is conformally compact. For a given point x on the boundary ∂M, we define γ _x : T _x ∂M × T _x ∂M → R by γ x = g _x | _T

_∂M×T

_∂M . In this way, any defining function ψ defines a metric γ on the boundary ∂M. The boundary

∂M equipped with the corresponding conformal structure [γ] is known as the conformal boundary (or conformal infinity) of M.

This definition makes sense since the conformal boundary is uniquely defined for a given metric g on the bulk space M – for note that if ψ ₁ and ψ ₂ are two defining functions, we can always write ψ ₁ = e ^w ψ ₂ , where w : M → R is some smooth function. Let g = ψ 1 ² g, and g ⁰ = ψ ₂ ² g, and let γ ₁ and γ ₂ be the corresponding induced metrics on the boundary.

We then have γ ₂ = e ^w|

γ ₁ , so they are in the same conformal class. Therefore any conformally compact manifold (M, g) has a unique conformal structure on the boundary.

The definition is quite technical, but the intuition is that a conformally compact man- ifold is a space that can be ‘compactified’, introducing a boundary that comes ‘finitely

6

Let p be a point in M, and consider two tangent vectors ξ and η in the tangent space T

M . The angle θ between them is then given by the relation g

(ξ, η) = kξk

kηk

cos(θ). If g

is conformally equivalent to g, the angle as measured by either metric will be the same, since

g

(ξ, η) = e

g

(ξ, η) = e

kξk

kηk

cos(θ) = kξk

kηk

cos(θ).

Thus, while the norms of ξ and η are potentially different as measured by the metrics g and g

, the angle between them is the same in both metrics.

7

If M and N are manifolds, g is a metric on M, and Φ : M → N is a diffeomorphism, then g and Φ define a metric on N by

Φ

g(ξ, η) = h Φ

g i

(ξ, η) = g dΦ

(ξ), dΦ

(η)
.

This metric is the pushforward metric defined by g and Φ. It is equivalent to the pullback metric (Φ

)

g defined by g and Φ

.

8

This condition means that for any point x ∈ ∂M there exists a tangent vector η ∈ T

∂M such that

dψ

(η) = η

∂

ψ(x) 6= 0. In other words, ψ has nonvanishing derivative on ∂M.

close’ to any internal point, and that then the metric on the bulk space uniquely deter- mines a conformal structure on the boundary.

Finally, we are ready to define AAdS spaces. We say that a pseudo-Riemannian manifold (M, g) is Asymptotically Anti-de Sitter (AAdS) if g is a conformally compact Einstein metric, for Λ < 0.

That an asymptotically AdS metric should solve the Einstein equations with a negative cosmological constant is natural, but it may be less clear why we go through the laborious process of introducing conformally compact spaces in order to define AAdS spaces. The intuition behind it is that the dominating behaviour of the AdS metric (2.1.9) as r → ∞ is due to the r ² -term of f AdS (r) in front of dt ² , and the factor r ² in front of dΩ ² (equation (2.1.10)). If ψ is a function that tends to zero to first order – but not to second order – at the boundary, its square ψ ² will tend to zero exactly ‘fast enough’ to cancel the divergence of g, but not so fast as to induce the trivial metric γ = 0 on the boundary.

In particular, the AdS metric (2.1.9) is conformally compact. It is shown in for example [12] that if we introduce the coordinate ρ = arctan r in (2.1.9), the function ψ(ρ) = cos ρ works as a defining function. But then ρ is precisely the original coordinate introduced in (2.1.6), and we may formally compactify AdS _d+1 by simply looking at the metric (2.1.8) and extending the domain of definition of ρ from the open interval [0, π/2) to the closed interval [0, π/2], ‘adding a hypersurface at infinity’.

The fact that AAdS-spaces induce conformal structures on the boundary is one of the fundamental ideas of the AdS/CFT-correspondence.

2.3 The AdS/CFT Correspondence

Our interest in AAdS spaces is, as mentioned in the introduction, largely due to the AdS/CFT correspondence. The correspondence does not play a large active role in this thesis, but we will still give a very brief overview of some of the main ideas by way of motivation. First, we must introduce conformal maps, which are maps between manifolds that preserve angles, but not necessarily distances. Technically, we define a conformal map between two manifolds (M, g) and (N , h) as a diffeomorphism Φ : M → N such that the pullback metric Φ ^∗ h is conformally equivalent to g.

A conformal field theory (CFT) on a manifold M is field theory that is invariant under conformal transformations M → M. They are important in for example certain areas of statistical physics and condensed matter theory. A CFT by definition does not depend on the particular metric that is chosen on M, only on its conformal class. Thus, it is natural to think of CFT:s as defined on manifolds equipped with conformal structures rather than metrics. ⁹ Since we concluded in section 2.2 that the metric of an AAdS space naturally defines a conformal structure on the boundary, we might not be that surprised to learn that AAdS spaces and CFT:s are related. However, the precise nature of that relation is a priori less clear.

The AdS/CFT correspondence states that a theory with gravity on an AAdS space is equivalent to a CFT without gravity on the conformal boundary of that space [13].

9

In practice, one often needs to specify a metric in order to define the theory, and one must then

show conformal invariance of that theory. Still it is useful to think of the theory as being defined on a

conformal manifold, and of the specific metric as giving one representation of the conformal class.

Specifically, certain problems that are untractable on one side of the equivalence may be solvable on the other, and the results can be transferred to the first side via the AdS/CFT correspondence. This motivates part of our interest in field theories on AAdS spaces. We will not give a quantitive statement of the correspondence rules, since we will not be using them, but rather we just note that there is reason to be interested in AAdS spaces – if for no other reason then at least for what they may say about conformal field theories on the boundary spaces, and in order to study the implications of the holographic principle – the general notion that information about the interior of a system may be stored on the boundary and vice versa – of which AdS/CFT is the most prominent and well-developed special case.

2.4 The AdS-Schwarzschild Solution

The AdS solution is not the only vacuum solution corresponding to a negative cosmo- logical constant. Rather, it is the unique maximally symmetric solution. ¹⁰ Distinctly different solutions with fewer symmetries do exist. By making the less restrictive as- sumption that spacetime is spherically symmetric and static (time-independent), we can derive a more general vacuum solution. This solution is called the AdS-Schwarzschild solution (to be compared with the classic Schwarzschild solution, which is the spherically symmetric, static vacuum solution for Λ = 0). The AdS-Schwarzschild solution models a spacetime in which there exists a spherically symmetric gravitating object, like a black hole or a star (we will be calling the spherically symmetric, gravitating object a ‘black hole’ throughout, although many results will be valid for any such object, such as a star or a planet). In d + 1 dimensions, with d 6= 2 (more on this in section 2.5), it can be written in coordinates as

g _S = −f _S (r)dt ² + 1

f S (r) dr ² + r ² dΩ ² _d−1 , (2.4.1) with

f S (r) = 1 + r ² − α

r ^d−2 , (2.4.2)

where α is a parameter related to the mass of the black hole. In spatial dimensions d 6= 2, we define a length parameter r ₀ by r ^d−2 ₀ = α, yielding

f _S (r) = 1 + r ² −  r ₀ r

 d−2

. (2.4.3)

In a Minkowski background, the r ² -term would disappear from this expression, and we would get back the Schwarzschild solution. In this case, the parameter r ₀ would be the Schwarzschild radius. We will however avoid this name, since the radius of the horizon of the black hole will generally not be equal to r ₀ . The case d = 2 has some peculiarities that will be considered in the next section. If r ₀ = 0, then there is no black hole, and the metric reduces to the AdS-metric (2.1.9). If r 0 is nonzero, then the equation f S (r) = 0

10

The appearance of an r-dependence of the metric (2.1.9) – which would suggest a non-symmetry,

picking out the point r = 0 as special – is an artefact of the arbitrarily chosen coordinate system. The

geometric information – the underlying curvature of spacetime – is actually the same for every point.

has a solution, call it r _h . The surface corresponding to r = r _h is called the event horizon of the black hole. Beyond this point, f _S (r) is negative, so the signs in front of the time and radial coordinates switch. Effectively, the roles of time and space are “reversed”

inside the event horizon – one of the many strange features of a black hole. We will not be getting into the topic of physics inside the event horizon any further in this report.

2.5 BTZ Black Holes

Three-dimensional spacetime is in many ways fundamentally different from its higher- dimensional counterparts. In Minkowski space, where Λ = 0, it can be shown that for any vacuum solution to the Einstein equations, the Riemann tensor vanishes – so the spacetime is completely flat. In particular, no black holes can exist in (2 + 1)-dimensional Minkowski space. However, it was shown in 1992 by Ba˜ nados, Teitelboim, and Zanelli that in a (2 + 1)-dimensional space with negative cosmological constant, there do exist black hole solutions to the Einstein-Maxwell equations [1]. These black hole solutions are called BTZ-black holes, after their discoverers.

The BTZ metric for a non-electrically charged black hole can be written [5]

ds ² = −N _⊥ ² dt ² + dr ²

N _⊥ ² + dr ² (dφ + N _φ dt) ² , (2.5.1) where the lapse and shift functions are

N ⊥ =



−M + r ² + J ² 4r ²

 1/2

, N _φ = − J

2r ² , (2.5.2)

with |J | ≤ M . The constants of integration M and J have interpretations as the mass and angular momentum of the black hole, respectively.

We will focus on non-rotating black holes, for which the solution once again takes the form (2.1.11), with the f -function given by

f (r) = −M + r ² . (2.5.3)

Here, the parameter M has an interpretation as the mass of the black hole. We may note that this metric can be put on the form (2.4.2) by letting α = 1+M . A horizon (a solution to f _BTZ (r) = 0) will appear at r _h = √

M . It is interesting to note that unlike the AdS- Schwarzschild solution (2.4.3), this solution does not reduce to the vacuum AdS-solution when M → 0. Rather, it reduces to the massless BTZ-solution

f _BTZ (r) = r ² , (2.5.4)

which has an event horizon at r = 0.

The simple form of the massless BTZ solution (2.5.4) makes it an attractive example

space for computational reasons. In particular, we will be interested in studying metrics

for which f has a series expansion with r ² as the leading term. For this reason the

massless BTZ metric, along with AdS space, will be our canonical, solvable examples.

2.6 The Quantum Mechanical 1/x ² -Potential

In this section we temporarily shift focus from gravitational physics and field theory to quantum mechanics. As we will see in Sec. 3, it namely turns out that the radial part of the equation of motion for the scalar field we wish to study in this article can, in the right choice of coordinates, be reduced to the form of a Schr¨ odinger equation, where the potential is asymptotically proportional to 1/ρ ² . It is therefore a good idea to familiarise ourselves with the Schr¨ odinger equation with the 1/x ² -potential. This is an equation that has been extensively studied in the theory of quantum mechanics, and is notable for its many pathological traits – being in a sense on the border between singular and regular potentials. We will follow the discussion laid out in [6].

Consider the Schr¨ odinger equation

− d ² ψ dx ² + A

x ² ψ(x) = κ ² ψ(x). (2.6.1)

This is the 1/x ² -potential, with A a parameter determining the strength of the potential, and κ ² the energy eigenvalue. We are interested (for reasons to be explained later) in the behaviour of the solutions as x → 0 ⁻ . Due to the invariance of the equation under the transformation x → −x however, we may as well consider x → 0 ⁺ .

An important feature of the 1/x ² -potential is that it either allows no bound states (solutions with κ ² < 0), or infinitely many, depending on A. For A larger than the critical value

A _BF = − 1 4 ,

the potential allows no bound states. ¹¹ For A > 0, this should be obvious, as the potential is repulsive. But for A in the range −1/4 < A < 0, a potential well appears. However, this well is still ‘too narrow’ to allow any bound states. An argument for this that crucially depends on the fact that we require normalisable solutions can be found in [6].

For A < −1/4 however, the well becomes ‘wide enough’ to allow bound states. In particular,

ψ(x) = √

x [C ₁ J _γ (κx) + C ₂ Y _γ (κx)] , (2.6.2) where J γ and Y γ are the Bessel functions of order γ = A + 1/4, is a solution. This gives some clue as to the criticality of A = −1/4. At this value, γ = 0. Above it, γ is real, and below, γ is pure imaginary. In particular, for imaginary γ, there exist normalisable solutions for pure imaginary κ, corresponding to negative energies κ ² < 0.

A peculiar feature of the 1/x ² -potential is that once we have one bound state solution, we automatically have a bound state solution for every negative energy. For if ψ(x) is a solution for the energy κ ² , then ψ β (x) = ψ(βx) is a solution for β ² κ ² . Namely,

− d ² ψ _β (x) dx ² + A

x ² ψ _β (x) = β ²



− d ² ψ(βx) dβx ² + A

β ² x ² ψ(βx)



= β ² κ ² ψ _β (x). (2.6.3) Note also that ψ _β is normalisable if ψ is, since

Z ∞ 0

|ψ _β (x)| ² dx = Z ∞

0

|ψ(βx)| ² d(βx)

β = 1

β Z ∞

0

|ψ(y)| ² dy < ∞. (2.6.4)

11

We use the subscript BF for ‘Breitenlohner-Freedman’. See Sec. 2.7.

But β is allowed to be any real number, so as soon as a bound state exists, there exist bound states with arbitrary negative energies. The conclusion is that the 1/x ² -potential for A < A _BF has no ground state. If it existed in reality then, any particle trapped in this potential would release an infinite amount of energy as it descended through the energies. We therefore conclude that the potential is unphysical for these values, and must be excluded from any reasonable theory. ¹²

2.7 The Breitenlohner-Freedman Bound

In a 1982 paper [4], Breitenlohner and Freedman derived a stability bound on the mass parameter of a scalar field in AdS space. In their honor, it is known as the Breitenlohner- Freedman (BF) bound. Below this value of the mass parameter, the dynamics of the spacetime become unstable. ¹³ The BF bound is related to to the 1/x ² -potential dis- cussed in Section 2.6. Breitenlohner and Freedman themselves used more sophisticated techniques and arguments in their derivation, but we will opt for deriving it through analogy with the quantum mechanical problem discussed previously. Some justification for this will be offered in Section 5.1.

We will follow the article [10] by Moroz. The argument is, unlike the rest of this thesis, carried out in Poincar´ e coordinates. Recall that in these coordinates, the metric takes the form

g = 1

z ² dz ² + η _αβ dx ^α dx ^β
. (2.7.1) Breitenlohner and Freedman considered a scalar field Φ on a background AdS spacetime described by the action

S = − Z

d ^d+1 x p− det g g ^µν ∂ _µ Φ∂ _ν Φ + m ² Φ ^∗ Φ
. (2.7.2) The equations of motion corresponding to this action are

∂ _z ² Φ − d − 1

z ∂ z Φ − m ²

z ² Φ + η ^αβ ∂ α ∂ β Φ = 0, (2.7.3) where the sum over α, β is taken over all coordinates except z. We now perform a Fourier transform over all non-z coordinates, so that i∂ _α → q _α . Rewriting the field as

Φ = z

ψ, (2.7.4)

the equations of motion then become

− ∂ _z ² ψ + m ² + ^d

₄ ⁻¹

z ² ψ = −q ² ψ, (2.7.5)

12

If one wishes to model physical phenomena with this potential, one has to resort to renormalisation techniques, such as for example imposing a cutoff at x = ε, or looking for self-adjoint extensions of the Hamiltonian. We will not be getting into this issue here (but see [6] for a lucid review of these techniques, and section 5.1 for a discussion of self-adjoint extensions). Rather, the important point of this discussion is that there is a critical change in behaviour of the equation at the critical parameter value A

= −1/4.

13

The precise meaning of ‘unstable’ will be discussed in Sec. 5.1.

where q ² = η ^αβ q _α q _β .

We now recognise Eq. (2.7.5) as a Schr¨ odinger equation with the 1/z ² -potential, with strength

A = m ² + d ² − 1

4 . (2.7.6)

Instability occurs at the critical value A _BF = −1/4. Solving for m, we arrive at the famous Breitenlohner-Freedman bound

m ² _BF = − d ²

4 . (2.7.7)

In d = 3 spatial dimensions, the bound as originally derived by Breitenlohner and Freed- man is

m ² _BF = − 9

4 . (2.7.8)

We conclude that the requirement that the dynamics of the system should be stable

introduces a lower bound on the mass parameter m. The surprising part of the story is

that in AdS space we can define stable dynamics even for certain tachyonic fields.

Chapter 3

Free Scalars in AAdS Spacetimes

In this thesis, we will study a spacetime equipped with a metric that couples directly to a scalar field Φ through an interaction term in the action proportional to the Ricci scalar R.

Motivated by the AdS/CFT-correspondence, our primary focus will be on the behaviour of the scalar field approaching the boundary r → ∞. Furthermore, we will assume that there is a spherically symmetric gravitating object (a black hole) at the center of the space.

3.1 The Equations of Motion

Consider a (d + 1)-dimensional spacetime equipped with a standard Einstein-Hilbert action functional

S[Φ, g] = − Z

d ^d+1 xp− det(g)  1

2κ (R − 2Λ) + L _M



, (3.1.1)

where κ is Einstein’s constant, ¹ g is the metric, and the matter Lagrangian L M is given by

L _M = 1

2 ∂ _µ Φ ^∗ ∂ ^µ Φ + (m ² ₀ + ξR)Φ ^∗ Φ , (3.1.2) where Φ is a scalar field. The terms ∂ _µ Φ ^∗ ∂ ^µ Φ + m ² ₀ Φ ^∗ Φ are the standard kinetic energy terms for a free particle in a curved background. In addition, we also let the field Φ couple explicitly to the curvature via the Ricci scalar R, with coupling strength ξ.

The equation of motion for Φ corresponding to the action (3.1.1) can be calculated using the Euler-Lagrange equations. The result is

( + m ² 0 + ξR)Φ = 0, (3.1.3)

where  is the generalised d’Alambertian

 := 1

√ − det g ∂ _µ (p− det g∂ ^µ ). (3.1.4)

1

Which depends on Newton’s constant G.

We assume that the spacetime is static and spherically symmetric. Through symmetry arguments, it is possible to show that the metric then necessarily takes the form

g = ds ² = −A(r)dt ² + B(r)dr ² + r ² dΩ ² _d−1 , (3.1.5) where dΩ ² _d−1 is the standard metric on the (d−1)-sphere. Guided by the metrics discussed in section 2 we will in this thesis focus on metrics of the more specific form

ds ² = −f (r)dt ² + 1

f (r) dr ² + r ² dΩ ² _d−1 . (3.1.6) We will call the function f appearing in this expression the f -function of the metric, and reserve the symbol f for functions defining metrics in this way throughout the thesis.

Under this assumption we have

det g = −r ² , (3.1.7)

so the d’Alambertian reduces to

 = 1

r ∂ _µ (r∂ ^µ ). (3.1.8)

We may decompose any solution into eigenmodes Φ(t, r, Ω) = X

ω,k

e ^−iωt Y _k (Ω)φ _ω,k (r), (3.1.9)

involving the standard eigenfunctions {e ^−iωt } of the operator ∂ ² /∂t ² , and the spherical harmonics on the (d − 1)-sphere, {Y _k }, which satisfy

∇ ²

S

Y k = −k ² Y k , and Z

S

Y k Y k

= δ k,k

, (3.1.10) where the eigenvalue −k ² corresponding to the eigenfunction Y _k is given by the ‘angular momentum quantum number’ ` by k <sup>2</sup> = `(` + d − 2) [8].

What then remains is to find the solutions φ _ω,k to the radial part of the equation.

From now on we suppress the indices ω and k. Plugging one eigenmode of (3.1.9) into eq. (3.1.3), the radial equation takes the form

1

r ^d−1 ∂ r f r ^d−1 ∂ r φ
+  ω ²

f − `(` + d − 2)

r ² − m ² ₀ − ξR



φ = 0. (3.1.11) We now ‘rescale’ the scalar field φ by defining

u(r) := r

φ(r), (3.1.12)

and we also introduce the so called tortoise coordinate, ρ, via the differential relation dρ = dr

f (r) . (3.1.13)

The radial equation (3.1.11), rewritten in terms of u(r(ρ)) then takes the form of a Schr¨ odinger equation,

d ² u

dρ ² + [ω ² − V (r(ρ))]u = 0, (3.1.14)

with the potential V (r) given by V (r) =  `(` + d − 2)

r ² + (d − 1)(d − 3)f (r)

4r ² + (d − 1)f ⁰ (r)

2r + m ² ₀ + ξR(r)

 f (r).

(3.1.15) Note that the equation takes the form (3.1.14) (with the same potential V ) independently of the choice of integration constant for ρ. We only require the differential relation (3.1.13) to hold.

The equation of motion for the metric may also be derived from the action (3.1.1).

However, we will not do so, since we will be focused on the behaviour of the scalar field

on given background metrics in this thesis.

Chapter 4

Solving the Equations of Motion for the Scalar Field

In this chapter, we will attempt to solve the ‘Schr¨ odinger equation’ (3.1.14) on the dif- ferent background metrics discussed in section 2. Our approach will be to express the potential (3.1.15) in terms of the tortoise coordinate ρ defined by (3.1.13), and then solve the equation in terms of this coordinate. We are mainly interested in the behaviour of the solution close to the conformal boundary at r → ∞, and generally we will have to settle for asymptotic expansions of the solutions close to this limit. The massless BTZ case and the empty AdS space are the notable exceptions. We will also postpone discussion of boundary conditions to Sec. 5.1, focusing instead on the functional form of the solutions.

4.1 The Massless BTZ Metric

The massless BTZ metric serves as an excellent warm-up for what is to come due to the simple form of the f -function given by Eq. (2.5.4). Later we will be forced to look for asymptotic expansions of the function u, but in the BTZ case we can find explicit solutions. We will also observe some of the typical behaviour of spacetimes equipped with black hole metrics, such as the unboundedness of the tortoise coordinate close to the horizon.

4.1.1 The Tortoise Coordinate in BTZ Space

First, we will study how the tortoise coordinate ρ behaves in this metric. Recall that for the massless BTZ metric, the function f in (3.1.13) is given by f (r) = f _BTZ (r) = r ² . Integrating the defining equation (3.1.13) then yields

ρ = − 1

r + C, (4.1.1)

where C is an arbitrary constant signifying a translation of the coordinate system. We

will choose C = 0, so that ρ → 0 ⁻ when r → ∞. The important properties of the tortoise

coordinate are that it is bounded as we approach spatial infinity (r → ∞), but unbounded

as we approach the event horizon of the black hole (r = 0). It therefore forms a sort of

mirror representation of the spacetime with respect to the spatial coordinate r.

-5 -4 -3 -2 -1 ρ 0.5 1.0 1.5 2.0 2.5 r

Figure 4.1 – The spatial r-coordinate plotted as a function of the tortoise coordinate ρ.

We see that ρ tends to −∞ as r → 0, and r → ∞ as ρ → 0.

Equation (4.1.1) can readily be solved for r, yielding r = − 1

ρ . (4.1.2)

A plot of the spatial coordinate as a function of the tortoise coordinate can be found in Fig. 4.1.

4.1.2 The Potential

-5 -4 -3 -2 -1 ρ

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 - 1

ρ

(a) A < 0.

-5 -4 -3 -2 -1 ρ

0.5 1.0 1.5 2.0 2.5 3.0 1 ρ

(b) A > 0.

Figure 4.2 – The potential V (ρ) = A/ρ ² as a function of ρ for the massless BTZ metric, in units of A.

Since the BTZ metric is a vacuum solution of the Einstein equations, the Ricci scalar is a constant. To see this, take the trace of the Einstein equations (2.1.1) with T _µν = 0 and solve for R. This yields

R = 2Λ(d + 1)

d − 1 , (4.1.3)

for any vacuum solution whatsoever. In our units, where L = 1, we have Λ = −d(d−1)/2, so

R = −d(d + 1). (4.1.4)

In d = 2 spatial dimensions, we get

R = −6. (4.1.5)

Alternatively, we have through direct verification found a heuristic formula for the Ricci tensor for an arbitrary metric on the form (3.1.6) in terms of the f -function, namely

R(r) = (d − 1)(d − 2)(1 − f (r))

r ² − 2(d − 1)f ⁰ (r)

r − f ⁰⁰ (r). (4.1.6) We have not been able to prove this formula in general, but checked it at least for d = 1 through d = 6. For f (r) = r ² and d = 2 we see that it reproduces the result of (4.1.5).

Since R is constant, the term ξR in the action (3.1.1) then plays the role of an ‘extra’

mass term. For this reason, we introduce the effective mass

m ² := m ² ₀ + ξR. (4.1.7)

The tortoise coordinate and the known f -function f (r) = r ² can then be plugged into the expression (3.1.15) for the potential, which reduces to the simple form

V (r(ρ)) = ` ² +



m ² + 3 4

 1

ρ ² . (4.1.8)

The equation of motion (3.1.14) then becomes

− d ² u dρ ² + A

ρ ² u = κ ² u, (4.1.9)

where

A := m ² + 3

4 , and κ := √

ω ² − ` ² . (4.1.10)

Equation (4.1.9) is a Schr¨ odinger equation with a 1/x ² -potential, plotted in Fig. 4.2. The strength of the potential is determined by the mass m ² , whereas the angular momentum quantum number ` contributes to a shift of the energy level κ ² .

As discussed in Sec. 2.6 – 2.7, below the ‘Breitenlohner-Freedman bound’ A _BF = −1/4, corresponding to

m _BF = −1, (4.1.11)

the system becomes unstable. Fig. 4.2 offers a compelling graphical interpretation of what happens at this point, in terms of quantum mechanical energy eigenstates. In Fig. 4.2b, corresponding to A > 0, the potential is repulsive at the conformal boundary, confining the field inside the spacetime. At A = 0 it changes asymptotic behaviour, instead tending to negative infinity. However, the ‘well’ in Fig. 4.2a is still too ‘narrow’ to allow for any bound states at infinity for A > −1/4. Below the BF bound A = −1/4, the well becomes wide enough to allow bound states, and when this happens, it is suddenly steep enough to allow for an infinitude of negative energy bound states, making the system collapse.

It is interesting to note that the potential is confining even for tachyonic fields with

m ² > −3/4, and that there is an interval −1 < m ² < −3/4 where the potential is

nonrepulsive at the conformal boundary that is not excluded by the BF bound.

4.1.3 Solving the Equation

-5 -4 -3 -2 -1

0.2 0.4 0.6 0.8

|u( )|

(a) u(ρ) = √ ρJ ₀ (ρ)

-5 -4 -3 -2 -1

0.5 1.0 1.5

|u( )|

(b) u(ρ) = √ ρY ₀ (ρ)

Figure 4.3 – The magnitude of the two independent solutions at the BF bound γ = 0 (A = −1/4), with κ = 1.

Equation (4.1.9) has the general solution [6]

u(ρ) = √

ρ [C ₁ J _γ (κρ) + C ₂ Y _γ (κρ)] , (4.1.12) where J _γ and Y _γ are the Bessel functions of the first and second kind of order γ, where

γ :=

r A + 1

4 = √

m ² + 1, (4.1.13)

and C ₁ and C ₂ are constants. The magnitude of these solutions are plotted for γ = 0 and κ = 1 in Fig. 4.3. Note that the solution involving Y _γ , which would normally tend to infinity at ρ = 0, gets ‘killed off’ by the √

ρ term, making it a plausible physical solution.

Interestingly, even at the BF bound – where the potential in Fig. 4.2a looks non-confining – both linearly independent solutions tend to 0 at the conformal boundary (see Fig. 4.3).

Returning to the normal spatial radial coordinate r and the full scalar field φ, the solution becomes ¹

φ(r) = 1 r h

C ₁ J _γ 

− κ r



+ C ₂ Y _γ 

− κ r

i

. (4.1.14)

4.2 AdS Space

The standard AdS space is our second – more canonical, but slightly more cumbersome – explicitly solvable example. This is the case studied by Breitenlohner and Freedman.

In this section, we will follow the more modern and general approach found in [8].

4.2.1 The Tortoise Coordinate in AdS Space

We will begin by transforming to the preferred tortoise coordinate ρ, defined through Eq. (3.1.13), with f as in eq. (2.1.10). A simple calculation gives

ρ =

Z dr

1 + r ² = arctan(r) + C, (4.2.1)

1

We have removed an arbitrary overall constant of i from the result.

0.0 0.5 1.0 1.5 ρ 0

2 4 6 8 10 12 14 r

Figure 4.4 – The black line is the spatial coordinate r as a function of the tortoise coordinate ρ. The gray, dashed line is the asymptote at ρ = π/2, where r goes to infinity.

so choosing C = 0 we get

r = tan(ρ). (4.2.2)

Note that the choice C = 0 can be made with impunity, since the defining relation (3.1.13) only defines ρ up to an additive constant. Choosing this constant corresponds to an unimportant shift of the coordinate range. ² With this choice, r = 0 corresponds to ρ = 0, and spatial infinity r → ∞ corresponds to ρ = π/2. We see that the tortoise coordinate effectively ‘pulls infinity to a finite distance’. A plot of the tortoise coordinate in AdS space is found in Fig. 4.4.

Interestingly, the tortoise coordinate in regular AdS space ³ given by Eq. (4.2.2) turns out to be exactly the original ρ-coordinate used to parametrise AdS space in the first place in Sec. 2.1, and later to compactify it in Sec. 2.2. This apparent coincidence is in fact what motivates our use of the symbol ρ for the tortoise coordinate.

4.2.2 The Potential

Next, we will find out how the potential (3.1.15) looks in the tortoise coordinate ρ.

Plugging in the known form of f and the general vacuum Ricci scalar (4.1.4), the potential takes the form

V (r(ρ)) = ν ² − 1/4

cos ² ρ + σ ² − 1/4

sin ² ρ , (4.2.3)

where

ν ² − 1

4 = (d − 1)(d + 1)

4 + m ² , σ ² − 1

4 = `(` + d − 2) + (d − 1)(d − 3)

4 .

(4.2.4)

2

Later, we will revisit this example in the context of the AdS-Schwarzschild metric, when it will be natural to choose C = −π/2.

3

Which, remember, was introduced in order to put the radial equation of motion on a specific form,

and is not explicitly linked to compactification.

0.5 1.0 1.5 ρ 20

40 60 80 100 V(r(ρ))

(a) ν = 1, σ = 1.

0.5 1.0 1.5 ρ

5 10 15 20 V(r(ρ))

(b) ν = 1, σ = 1/4.

0.5 1.0 1.5 ρ

-20 -10 10 20 30 V(r(ρ))

(c) ν = 1, σ = 0.

0.5 1.0 1.5 ρ

5 10 15 20 25 V(r(ρ))

(d) ν = 1/4, σ = 1.

0.5 1.0 1.5 ρ

-1.0 -0.5 0.5 1.0 V(r(ρ))

(e) ν = 1/4, σ = 1/4.

0.5 1.0 1.5 ρ

-8 -6 -4 -2 V(r(ρ))

(f ) ν = 1/4, σ = 0.

0.5 1.0 1.5 ρ

-20 -10 10 20 30 V(r(ρ))

(g) ν = 0, σ = 1.

0.5 1.0 1.5 ρ

-8 -6 -4 -2 V(r(ρ))

(h) ν = 0, σ = 1/4.

0.5 1.0 1.5 ρ

-30 -25 -20 -15 -10 -5 V(r(ρ))

(i) ν = 0, σ = 0.

Figure 4.5 – The P¨ oschl-Teller potential for some qualitatively different example values of ν and σ above or at the BF bound.

The naming of the constants follows Ishibashi and Wald ⁴ [8]. In the context of quantum mechanics, the potential (4.2.3) is known as the trigonometric P¨ oschl-Teller potential, ⁵ which is an analytically solvable potential with known eigenfunctions. A plot of this potential for a couple of different values of the parameters ν and σ can be found in fig.

4.5.

Focusing first on the parameter ν, which governs the asymptotic behaviour of the potential at infinity, we see that the potential is confining for ν ² > 1/4, but becomes concave below this value. Instability occurs at ν ² = 0. This might be expected from our discussion of the 1/x ² -potential in Sec 2.6, since we may Taylor expand cos ² ρ at ρ = π/2, yielding

cos ² ρ =  ρ − π

2

 2

− 1 3

 ρ − π

2

 4

+ O(ρ ⁶ ), (4.2.5)

meaning the potential should behave asymptotically like the 1/x ² -potential with strength A = ν ² − 1/4 in the limit ρ → π/2.

4

Our definition of ρ differs from the one found in [8], where they define their ‘tortoise coordinate’ by x = cot r, rather than our ρ = tan r – effectively inverting the coordinate range. Compensating for this, the results are the same.

5

As opposed to the hyperbolic one, where cos and sin are replaced by cosh and sinh.

Demanding stability, we obtain a criterion for m, namely m ² ≥ − (d − 1)(d + 1) + 1

4 = d ²

4 (4.2.6)

In particular, in 3 + 1 dimensions, we find the critical mass m ² _BF = − 9

4 . (4.2.7)

These results agree with what we found when we used Poincar´ e coordinates in Sec. 2.7.

We conclude that the approach of power series expansion is likely a possible way to derive the BF bound in other cases as well.

Next, consider the parameter σ. First note that nonzero σ cause a singularity at r = 0, which is either repulsive or attractive depending on the sign of σ ² . Again, there is a corresponding ‘Breitenlohner-Freedman’ critical case for σ = 0. However, we note that σ depends only on the dimension d and the ‘angular momentum quantum number’

`, and σ ² will always be nonnegative in dimensions d 6= 2.

Increasing ` creates a repulsive singularity at r = 0. We will call this an angular momentum barrier, as ` = 0 gives back the minimum value, and increasing it increases the strength of the repulsion. It is the wave equivalent of a centrifugal force.

In dimensions d = 3 and d = 1, the potential is “well-behaved” in the sense that

` = 0 corresponds to zero derivative of the potential at r = 0 for ` = 0, meaning there is no attractive or repulsive behaviour at our (arbitrarily) chosen center point. In two spatial dimensions however, we can reach the ‘BF bound’, σ = 0, when ` = 0. This case could probably be studied with more care, but we will find it enough to point out that the leading order term is proportional to 1/ρ ² , and therefore this potential should not allow for any bound states above the bound σ ² = 0.

It is noteworthy that this bound is exactly the smallest value that can be achieved for a physical system, but it is necessary that it be no smaller. Note that our description depends on an arbitrary choice of base point ρ = 0, and it is around this point that the potential becomes attractive. This may seem strange, but remember that an arbitrary field configuration Φ will be a sum over eigenmodes. If the sum should happen to termi- nate after the first term, then this special point is picked out by the state, rather than the background gravitational field. Our argument shows that nothing too drastical happens at this point for an arbitrary state.

We may think of the different behaviours of the ` = 0 mode for different dimensions as an effect of the phase space scaling with different powers close to a given point depending on d. For high powers of d, the amount of phase space available to the field close to a point goes to zero more quickly, creating a ‘dimensional barrier’. The field simply does not have enough space to be localised around the point, so it must go somewhere else.

Conversely, in low dimensions the phase space scales slowly with ρ, giving the field a relatively large amount of space to be localised in. It may be of interest that the break- even point where the potential is neither repulsive nor attractive is in d = 3 dimensions, the spatial dimension of the real world.

The general solution to the Schr¨ odinger equation with the trigonometric P¨ oschl-Teller potential is known to be [11]

u(ρ) = sin(ρ) ^σ+1/2 cos(ρ) ^ν+1/2 P _n ^(σ,ν) (cos(2ρ)), (4.2.8)