University of Amsterdam

Home , BTZ black hole, Ricci decomposition

MSc Physics Theoretical Physics

Master Thesis

The BTZ Black Hole

In Metric and Chern-Simons Formulation

Eline van der Mast 0588105

July 2014 54 ECTS 02/2013 - 07/2014

Supervisor: Examiner: A. Castro, Dr. J. de Boer, Prof.

Institute for Theoretical Physics Amsterdam

ABSTRACT

In this thesis, we review the BTZ black hole within the metric formulation of general relativity and Chern-Simons theory. We introduce the necessary concepts of general relativity and discuss three-dimensional gravity, specifically Anti-de Sitter vacuum space. Then, we review the identification process to get to the BTZ black hole solution, as well as an explicit quotienting formulation. The thermodynamic and geometric properties of the black hole are then discussed. The Brown-York stress tensor procedure is explained to give these properties physical meaning, with a correction by Balasubramanian and Kraus. In the second part of this thesis we introduce Chern-Simons theory and show its equivalence to three-dimensional gravity. The BTZ black hole is then shown to be defined through holonomies and thermodynamics. We find that the identifications in the metric formulation are equivalent to the holonomies in the Chern-Simons formulation; and the thermodynamics are the same. CONTENTS

1. Introduction ...... 6

2. Introduction to Gravity ...... 8 2.1 General Relativity ...... 8 2.1.1 Diﬀerential Geometry ...... 9 2.1.2 The Riemann Tensor ...... 9 2.2 Gravity in Three Dimensions ...... 11 2.2.1 Degrees of Freedom ...... 11

3. Local and Global Aspects of Anti-de Sitter Space ...... 13 3.1 Global Anti-de Sitter Space ...... 13 3.2 Symmetries of AdS3 ...... 15 3.2.1 Killing Vectors and Group Generators ...... 15 3.3 Quotient Spaces of AdS3 ...... 17 3.4 The Identiﬁcations ...... 18 3.4.1 The Nature of the BTZ Singularity ...... 20 3.5 The Euclidean BTZ Black Hole ...... 21 3.6 An Explicit Quotient Formulation of the BTZ Black Hole ...... 23

4. The BTZ Black Hole ...... 26 4.1 Mass and Angular Momentum ...... 26 4.1.1 ADM Formalism and Asymptotic Symmetries ...... 27 4.1.2 A Quasilocal Stress Tensor for Gravity ...... 27 4.1.3 The Decomposition of the Spacetime ...... 28 4.1.4 Using the Hamilton-Jacobi Formulation ...... 29 4.2 A Boundary Stress Tensor for AdS3 ...... 30 4.2.1 An Action Principle ...... 31 4.2.2 Finding a Finite Stress Tensor ...... 33 4.2.3 A Stress Tensor for the BTZ Black Hole ...... 34 4.3 Geometric Aspects ...... 35 4.3.1 Curvature Tensors ...... 35 4.3.2 Event Horizons Using Killing Vectors ...... 36 Contents 5

4.4 Black Hole Thermodynamics ...... 37 4.5 Euclidean Time Periodicity and Temperature ...... 39

5. Chern Simons Theory as (2+1)-dimensional Gravity ...... 41 5.1 Introduction to Chern-Simons Theory ...... 41 5.2 From (2+1)-Dimensional Gravity to Chern-Simons Theory . . . . . 42 5.2.1 Vielbein Formalism ...... 42 5.3 Chern-Simons as a Theory for (2+1)-Dimensional Gravity . . . . . 44

6. The BTZ Black Hole in Chern-Simons Theory ...... 49 6.1 From the BTZ Metric to a Chern-Simons Connection ...... 49 6.2 Holonomies ...... 51 6.3 The BTZ as a Holonomy Around the φ-cycle ...... 51 6.4 The Holonomy Around the tE-cycle ...... 52

7. Conclusion and Outlook ...... 54 7.1 Possibilities for Further Research ...... 55 7.1.1 More General Treatment of the BTZ Black Hole ...... 55 7.2 Final Remarks ...... 57

8. Acknowledgements ...... 58 1. INTRODUCTION

Gravity is described by Einstein’s theory of general relativity. Differential geometry is the framework used to describe spacetime as a manifold and to introduce gravity as a result of the manifold’s curvature. Our physical universe is described by a four-dimensional manifold, but this does not mean that we are confined to studying only four-dimensional spacetimes. The theory is easily adapted to higher and lower-dimensional models, some of which are much easier to work with than four dimensions. These models can shed a different light on problems in other dimensions and give a more general understanding of how gravity works. In this thesis, we will focus on three-dimensional gravity, which has been shown to be much less trivial than a first look seems to imply. More specifically, we will work in Anti-de Sitter space, a vacuum space with a negative cosmological constant, in three dimensions. As a space with constant negative curvature and thus no local degrees of freedom, it appears to be a trivial space and not of much use for any interesting physics. However, upon closer inspection, it turns out to contain a black hole solution. This black hole, called the BTZ black hole (after its discoverers Bañados,Teitelboim and Zanelli), appears when looking from a global perspective. Black holes are probably the most exciting objects in the theory of relativity. Astrophysically, they are very massive stars in the final stage of their collapse. For a large enough star, gravity is so large upon collapse that it breaks down exclusion principles to become extremely dense. This creates a gravitional pull so strong that within a certain radius of the collapsed star, it attracts everything irrevocably. Mathematically, we model these black holes as singularities where the curvature of the spacetime is infinite, and as spacetimes possessing a causal structure that won’t allow light curves to leave once they have come within a certain radius of the black hole. To find such an object in a theory of constant negative curvature is unexpected to say the least. While the BTZ black hole is a solution of constant negative curvature locally equivalent everywhere to Anti-de Sitter space, it can be found by taking a quotient space of Anti-de Sitter space. This creates a different solution globally as it changes the topology. Even though this is a toy model and the BTZ black hole is created through mathematical manipulations and not through modeling the collapse of a massive star in the spacetime, it still turns out to behave like a real, 1. Introduction 7 physical black hole. It can be attributed values for mass and angular momentum, and it is consistent with the laws of black hole thermodynamics. In this thesis, will first consider the three-dimensional Anti-de Sitter black hole in a metric formulation within general relativity. Beginning with a brief introduction to the formulation of gravity within general relativity, we will discuss Anti-de Sitter space and the process necessary to reach the black hole. We then discuss the geometric and thermodynamic properties of the black hole. In the second part of this thesis, we will consider the black hole within Chern- Simons theory. Chern-Simons theory is a topological quantum field theory that is found to be able to model three-dimensional gravity. It also supports the idea of a black hole, through its thermodynamics and holonomies. We start this section with a brief introduction to Chern-Simons theory, and then specify it as a model for three-dimensional gravity. We then describe the BTZ black hole within this formulation. While the BTZ black hole was discovered in 1992, and is as such not new, it is still used as a simple example for more complicated and general theories being worked on today. Besides this for its current relevance, it is also a good way to learn more about general relativity after an introductory course. Last but not least, the study of the BTZ black hole provides a nice framework to become more acquainted with certain concepts in theoretical physics; like the AdS/CFT correspondence, asymptotic symmetries, group theoretical concepts, and Chern-Simons theory. This thesis attempts to introduce these concepts (some more in-depth than others) when necessary in order to describe the black hole. It is definitely not an exhaustive discussion of the black hole, as much more can be said about it than is done here. However, it discusses some important results and introduces some concepts relevant to further research. 2. INTRODUCTION TO GRAVITY

Since 1687, gravity was seen as an attractive force between two objects due to their mass as described by Isaac Newton - up until quite recently. In 1916, Albert Einstein published his theory of general relativity, which describes gravity as the result of the curvature of spacetime, in turn influenced by the presence of matter and energy. Spacetime is described by a manifold; a topological space everywhere locally equivalent to flat space. The description of gravity is written in the math- ematics of differential geometry, which employs calculus and algebra to describe geometry. It is the best theory of gravity so far, albeit purely classical. It repro- duces Newton’s laws at small distances, and has explained and predicted many empirical results that without relativity wouldn’t have been understood. While it takes some getting used to intuitively, its equations are remarkably elegant and simple. This chapter will start with a brief introduction to the most relevant concepts of general relativity that will be used throughout this thesis. We will explain how curvature is described using the Einstein equations, the Riemann tensor and its contractions and symmetries. Then we will discuss curvature in three dimensions specifically, and show why there are no local degrees of freedom in this case.

2.1 General Relativity

Within the theory of general relativity, gravity is described as the result of the curvature of spacetime; which in turn is caused by energy or matter present in the spacetime. The Einstein ﬁeld equations are 1 8πG R − g R + g Λ = T , (2.1) µν 2 µν µν c4 µν and give the relation between the curvature of spacetime, matter and energy. Here, Rµν is the Ricci tensor, a contraction of the Riemann tensor (to be discussed in section (2.1.1)), and R is the Ricci scalar. Λ is the cosmological constant, which gives a measure of the inherent energy of spacetime in vacuum. Tµν is the stress-energy tensor. It contains all information about energy and matter in the spacetime. 2. Introduction to Gravity 9

General relativity describes spacetime and gravity in the language of differential geometry. Instead of the Newtonian concept of gravity as an attractive force between massive objects in a flat Euclidean background, gravity is described as the result of the curvature of spacetime itself. We use manifolds as a fundamental mathematical structure to describe the dynamics of spacetime. A manifold is a space that can locally always be described by a flat Euclidean space. To be able to discuss the physics of a manifold, we use the metric tensor, which is a coordinate- based description of a patch on the manifold.

2.1.1 Differential Geometry The line element 2 µ ν ds = gµνdx ⊗ dx , (2.2) defines the local geometry and causality of the spacetime in question, with gµν the matrix-valued metric tensor. The most basic spacetime is flat space, also known as Minkowski space, which is usually denoted by ηµν. In four-dimensional flat space of Lorentzian signature, it is defined as

ds2 = −dt2 + dx2 + dy2 + dz2.

A Lorentzian signature has one negative sign and all other signs positive (or vice versa, depending on conventions). A Euclidean signature has all signs positive.

2.1.2 The Riemann Tensor The Riemann tensor gives every point on the manifold a tensor value. This value indicates how much each point in the spacetime differs from flat space. When we transport a vector in flat space around a loop, we know the vector always comes back pointing in the same direction. In curved space this is not necessarily so. The Riemann tensor tells us how much the transported vector’s direction differs from the initial vector. Following the conventions of [19], it is given by

ρ ρ ρ ρ λ ρ λ R σµν = ∂µΓνσ − ∂νΓµσ + ΓµλΓνσ − ΓνλΓµσ. (2.3) Γ is the Christoffel connection, which is defined using the metric tensor, to be 1 Γσ = gσρ(∂ g + ∂ g − ∂ g ). (2.4) µν 2 µ νρ ν ρµ ρ µν The Christoffel connection is the connection with which we can define covariant derivatives on a curved manifold. The trace of the Riemann tensor is known as the Ricci tensor:

λ µ ρ Rµν ≡ R µλν = g ρR σµν, (2.5) 2. Introduction to Gravity 10 and the trace of the Ricci tensor is known as the Ricci scalar:

λ R ≡ R λ. (2.6)

The Ricci Decomposition We can decompose the Riemann tensor into three diﬀerent tensors, known as the Ricci decomposition. This splits the Riemann tensor into a scalar part, a semi- traceless part, and a fully traceless part, as follows:

Rρσµν = Cρσµν + 2(gρ[µRν]ρ − gσ[µRν]ρ) − gρ[µgν]σR. (2.7)

The tensor Cρσµν is the traceless part, called the Weyl tensor. The Ricci curvature tensor and scalar hold all the trace information of the Riemann tensor. The trace of a matrix tells us how volumes distort while moving throughout spacetime, whereas the Weyl tensor shows how shapes are distorted as a consequence of gravitational tidal forces. The Ricci tensor gives the relation between spacetime and matter as seen in the Einstein field equations. These indicate how the stress-energy tensor Tµν influences the curvature of spacetime. The Weyl tensor then gives the part of the Riemann tensor that describes how spacetime is curved without a local, non-gravitational source field present - thus indicating the propagating degrees of freedom, which are gravitational waves.

Symmetries of the Riemann Tensor The number of degrees of freedom in the Riemann curvature tensor depends on the dimensions d of the spacetime it describes. It has four indices, and is (anti)symmetric in certain indices or pairs of indices. Following the reasoning of [19], we can count the degrees of freedom of Riemann using its symmetries. The Riemann tensor is antisymmetric in both its ﬁrst two and its second two indices, and symmetric under the exchange of these two pairs. We can see it as a symmetric matrix RAB where we ﬁll in antisymmetric matrix values for both A and B. 1 As a symmetric matrix has 2 d(d + 1) independent components and an an- 1 tisymmetric matrix has 2 d(d − 1) independent components, we get a total of 1 1 1 2 2 d(d − 1) 2 d(d − 1) + 1 independent components. Lastly, the Riemann tensor has the property that its antisymmetric part vanishes. Since we can decompose a tensor into the sum of its fully antisymmetric and symmetric parts, we can simply subtract the independent components of a fully antisymmetric tensor with four 1 indices, which is 4! d(d − 1)(d − 2)(d − 3). We thus end up with 2. Introduction to Gravity 11

1 d2(d2 − 1) (2.8) 12

d(d+1) degrees of freedom in d dimensions. The Ricci tensor has 2 independent degrees of freedom, as it is a symmetric tensor with two indices. The Weyl tensor has, by deﬁnition, all the same symmetries as the Riemann tensor.

2.2 Gravity in Three Dimensions

Our universe is described by a four-dimensional spacetime: three spatial and one time coordinate. However, sometimes it is convenient to look at other dimensions as they can be easier to work with, while still containing enough structure to give us the information we’re looking for. Because of this, three-dimensional gravity is an extensive ﬁeld of research. It is less complicated and therefore poses simpler problems with simpler solutions, that do have their uses in higher dimensions. While three-dimensional gravity has no degrees of freedom [16] - we will see why below - it still has some remarkable properties, such as being able to contain a black hole analogous to the rotating Kerr black hole in four dimensions.

2.2.1 Degrees of Freedom From (2.8) we see that in three dimensions, the Riemann tensor has six degrees of freedom. The Ricci tensor also has six (it is a symmetric three by three matrix). The Weyl tensor vanishes in three dimensions. It is fully traceless by construction:

ρ C µρν = 0,

1 which gives us 2 d(d + 1) restraints on its components. It has the same number of independent components as the Riemann tensor, meaning that we end up with 1 1 d2(d2 − 1) − d(d + 1) 12 2 independent components [34]. So for d = 3, the Weyl tensor has 0 independent components: the Weyl tensor vanishes in three dimensions, showing there are no propagating degrees of freedom in the theory. Let’s consider the remaining six independent components we have in the Rie- mann tensor. In the vacuum Einstein equations with non-vanishing cosmological constant, (2.1) with Tµν = 0, we ﬁnd that R = 6Λ. Plugging this back into the Einstein equation, we ﬁnd that it reduces to

Rµν = 2Λgµν. (2.9) 2. Introduction to Gravity 12

This means we have six independent equations - both the Ricci and the metric tensor are symmetric tensors with two indices - that constrain the Riemann tensor. As we had six independent components, they are now all fully constrained by Einstein’s equations. If we use the result Rµν = 2Λgµν in the Ricci decomposition (2.7), we get the simple expression R R = (g g − g g ). (2.10) µνρσ 6 µρ νσ νρ µσ So, the Riemann tensor is completely deﬁned in terms of the scalar curvature R, in turn determined by Λ and the metric. Thus, there are no degrees of freedom in three-dimensional gravity. The expression (2.10) is the same as the three- dimensional version of the more general R R = (g g − g g ), µνρσ n(n − 1) µρ νσ νρ µσ which is the relation of the Riemann curvature to the Ricci scalar in what is called a maximally symmetric spacetime. A maximally symmetric spacetime has the same amount of symmetries as Euclidean space of the same dimension and is thus locally equivalent to it everywhere. 3. LOCAL AND GLOBAL ASPECTS OF ANTI-DE SITTER SPACE

In discussing vacuum solutions to the Einstein equations in three dimensions, there are still different possibilities for Λ, the cosmological constant. Einstein added this (assumed positive) constant to his equation to achieve a static universe. While it so happens that, much to Einstein’s embarassment, our universe was found to be expanding, we still have need for a cosmological constant. The conventional matter we have in the universe does not suffice to explain its measured accelerated expansion. However, just as we, for informative purposes, study three-dimensional gravity while we live in a four-dimensional universe, we also consider spacetimes that have a negative cosmological constant for the same reason. There are two different ways we can look at a manifold: locally and globally. When we look at a manifold from a local perspective, we look at its geometry as given by the metric tensor. When looking from a global perspective, we look at its topology. A manifold’s topology is defined by global properties that are invariant under bending and stretching; so that a sphere and a box are equivalent topologically, but not a sphere and a donut. In this chapter, we will first consider the geometric aspects of Anti-de Sitter space by discussing the metric solution of global Anti-de Sitter space (AdS3), and show its constant negative curvature. We also discuss its symmetries. In the second section, we discuss a quotient space of AdS3, achieved by doing a patch-wise coordinate transformation and then identifying along a symmetry of the spacetime. This results in a three-dimensional black hole, which is suprising in a spacetime with constant curvature and no local degrees of freedom.

3.1 Global Anti-de Sitter Space

Vacuum spacetime solutions with a positive cosmological constant are called de Sitter spaces, and solutions with a negative cosmological constant are called Anti- de Sitter spaces. For positive Λ, the corresponding manifolds are spherical; for negative Λ, they are hyperbolic (or the higher-dimensional analogues). The cosmological constant is a way of deﬁning the “vacuum energy” of spacetime; or the 3. Local and Global Aspects of Anti-de Sitter Space 14 energy that does not come from conventional matter. We will from now on focus on Anti-de Sitter space in three dimensions. Three-dimensional Anti-de Sitter space (AdS3) can be visualized as an embedding in four-dimensional ﬂat space with a (2,2) signature through the embedding equation (following the convention of [7]):

−v2 − u2 + x2 + y2 = −`2 (3.1) Here, −`2 = Λ. l is the radius of curvature, a measure of the curvature of the spacetime, which is constant. The corresponding embedded metric is

ds2 = −du2 − dv2 + dx2 + dy2. (3.2)

To ﬁnd the metric of AdS3 itself without reference to an embedding in a higher- dimensional space, we set

u = ` cosh µ sin λ (3.3) p v = ` cosh µ cos λ, and ` sinh µ = x2 + y2.

If we then go to polar coordinates,

x = ` sinh µ cos θ, y = ` sinh µ sin θ, we get the following metric

ds2 = `2(− cosh2 µdλ2 + dµ2 + sinh2 µdθ2). (3.4)

λ is now the timelike coordinate (it has negative signature), but due to the transformation (3.3), it is an angle: λ ∼ λ + 2π. This means there are closed timelike curves in the spacetime. To ﬁx this, we unidentify λ with λ + 2π, and set t λ = (3.5) ` r = ` sinh µ.

Doing this, we get what is called the universal covering space of AdS3: r2 1 ds2 = −( + 1)dt2 + dr2 + r2dθ2. (3.6) `2 r2 `2 + 1 The universal covering space is the space one gets when a coordinate is unrolled. For example, the covering space of S1 is R. The metric (3.6) is conventionally referred to as Anti-De Sitter space. The coordinates used in it are global coordinates and cover the whole manifold. 3. Local and Global Aspects of Anti-de Sitter Space 15

PoincaréCoordinates Another coordinate system often used are Poincarécoordinates; or hyperbolic coordinates, denoted H3. We get to these from the embedded metric using ` y v z = , β = , γ = − , u + x u + x u + x which covers the upper-half plane chart (hence often referred to as the Poincaré patch) with line element

`2 ds2 = dz2 + dβ2 − dγ2 . (3.7) z2

3.2 Symmetries of AdS3

AdS3 is a maximally symmetric spacetime with isometry group SO(2, 2). “SO” stands for special orthogonal, where special means its matrix representations have determinant equals 1. The (2, 2) stands for the signature (− − ++), as in the embedded AdS3 metric (3.2). It is the group of all orientation- and distance- preserving linear transformations; or all proper rotations. In group theory, we can see these transformations as the group of all orthogonal matrices, meaning they consist of all scalar product-preserving transformations between vectors in Euclidean space (thus giving all possible isometries of Euclidean space). The group multiplication, which deﬁnes the representation’s action on the space, is simply matrix multiplication. It is isomorphic to SL(2, R)L×SL(2, R)R, the special linear group in two dimensions. The group SL(2, R) acts on the complex upper plane and has representation

a b , with ad − bc = 1. (3.8) c d The group action is given by ax + b x → . (3.9) cx + d

3.2.1 Killing Vectors and Group Generators In general relativity, we call an isometry of the manifold a Killing vector. They define symmetries of the spacetime metric with which we can associate conserved currents using Noether’s theorem. Informally speaking, we can say that if we move in the direction of a Killing vector on the manifold, the metric stays the same. Formally, Killing vectors are infinitesmal generators of a manifold’s isometries. As both Killing vectors and the generators of the symmetry group of a space 3. Local and Global Aspects of Anti-de Sitter Space 16 are infinitesmal generators of isometries of a manifold, they are equal to each other up to linear combinations. A Lie algebra can, for our purposes, be seen as the commutation relations between the symmetries’ infinitesmal generators. The relationship between the symmetries of a manifold and its generators is as follows.

Killing Vectors and Group Generators The isometries can be represented as matrices that obey a group multiplication (for instance as given by (3.8) for SL(2, R)). We ﬁnd the inﬁnitesmal generator ξ of the group of proper rotations with representation R by considering an expansion around the angle of rotation, say φ (following the logic of [25]):

dR(φ) R(φ) = 1 − iφξ + O(φ2); or − iξ = | . (3.10) dφ φ=0 The representation matrix R is then the exponent of the generator:

R(φ) = e−iφξ. (3.11)

The Killing vectors of AdS3 are linear combinations of the generators of SL(2, R)L× SL(2, R)R. The generators of SL(2, R) can be explicitly represented as (following [2])

1 0 0 0 1 2 0 L1 = ,L−1 = ,L0 = 1 . (3.12) −1 0 0 0 0 − 2 They obey the commutation relation (called the Lie algebra), which we denote with sl(2, R), [Li,Lj] = (i − j)Li+j, (3.13) and are closely related to the Pauli matrices. The Killing vectors of SO(2, 2) ∼= SL(2, R)L × SL(2, R)R are [7] ∂ ∂ J = x − x . (3.14) µν ν ∂xµ µ ∂xν In terms of the original embedding coordinates (x, y, u, v), they are explicitly [7]

Jvu = v∂u − u∂v Jvx = x∂v + v∂x

Jvy = y∂v + v∂y Jux = x∂u + u∂x

Juy = y∂u + u∂y Jxy = y∂x − x∂y. (3.15)

The relations between (6.6) and (3.15) are given by [32, 6] 3. Local and Global Aspects of Anti-de Sitter Space 17

1 L = (J + J ) 0 2 ux vy 1 L = (J − J − J + J ) −1 2 xy uv xv yu 1 L = (J − J + J − J ) (3.16) 1 2 xy uv xv yu for the generators of SL(2, R)L, and

1 L˜ = (J − J ) 0 2 ux vy 1 L˜ = (J + J − J − J ) −1 2 uv xy xv yu 1 L˜ = (J + J + J + J ). (3.17) 1 2 xv yu uv xy for the generators of SL(2, R)R. The sl(2, R) matrices are the generators of SL(2, R), with the group action given by (3.9). Canonically these are written as (6.6). The Killing vectors Jµν are the generators of the symmetries written speciﬁcally in the spacetime coordinate’s basis.

3.3 Quotient Spaces of AdS3

So far, we have only discussed the geometry, or the local aspects, of a AdS3. But a manifold is not only defined by its geometry, but also by its topology. For instance, a flat surface locally looks the same as a cylinder, but is globally inequivalent to it because two of its edges have been identified, or “glued together”. In the same way, we can have spacetimes with the same metric locally, but with an identification globally. As we shall see, this can drastically change the physics of a spacetime. Identifying points with each other in a topological space is called quotienting the space, or forming a quotient space. Points in the space are glued together, or identified - which is why a quotient space is sometimes called an identification space. See figure (3.1) for a simple example. If the identification is done with points on Killing orbits (an orbit along which the metric is symmetric), the quotient space will inherit the structure from the original space. A Killing vector gives us a subgroup in the symmetry group of the manifold. By exponentiating the Killing vector and using it to transform the manifold (identifying along the Killing vector’s direction), we get a manifold that is still locally isometric to AdS3. But in 1992 it was discovered, by Bañados, 3. Local and Global Aspects of Anti-de Sitter Space 18

Fig. 3.1: The two red edges of the surface are identiﬁed to create a diﬀerent topology

Teitelboim and Zanelli [8], that if we parametrize the metric patch-wise in a certain way and then identify a coordinate, we get a black hole metric. In this section we will show how these identiﬁcations are done, following [7].

3.4 The Identiﬁcations

By taking P → enξP, n = 0, ±2π, ±4π, ... (3.18) where ξ is a Killing vector, we get an identiﬁcation subgroup. Because the Killing orbits we identify along in AdS3 are isometries, the quotient space remains a manifold with a well-deﬁned metric with constant negative curvature. This way, it remains a solution to Einstein’s equations. To ensure preservation of causality, ξ must be spacelike:

ξ · ξ > 0, (3.19) to exclude closed timelike lines. Some patches of AdS3 do contain parts where the identification Killing vectors used are null or timelike. These patches need to be excluded from our solution, which does not seem logical before the identifications are made. However, they will make sense after the identifications are done, because we will see that the Killing horizon (defined as ξ · ξ = 0) coincides with the singularity in the quotient space and becomes the black hole horizon. We do the identification along the following combination of Jµν (a linear combination of a Killing vector is itself a Killing vector): r r ξ = + J − − J , (3.20) ` ux ` vy in which the r± can be seen as parametrizing constants for now. We will attribute them a physical interpretation later on, in section (4.1.2). As said earlier, ξ · ξ 3. Local and Global Aspects of Anti-de Sitter Space 19 must be larger than 0 to exclude closed timelike lines in the quotient space. We thus find that r2 r2 ξ · ξ = + (u2 − x2) + − (v2 − y2) `2 `2 r2 − r2 = + − (u2 − x2) + r2 , `2 −

2 2 r−` 2 2 which means that the region − 2 2 < u − x < 0 is where ξ · ξ > 0. We can r+−r− divide this region into three diﬀerent types of subregions separated by the null surfaces y2 − x2 = 0 (or v2 − y2 = `2 − (u2 − x2) = 0).

We can classify these regions into three types:

2 2 2 2 I. u − x > ` , with r− < ξ · ξ < ∞

2 2 2 2 2 II.0 < u − x < l , with r− < ξ · ξ < r+

2 2 r−` 2 2 III. − 2 2 < u − x < 0, with 0 < ξ · ξ < r2 r −+r− Per region, we can introduce a certain parametrization with (t, r, φ)

I. r+ < r u = pA(r) cosh χ(t, φ) x = pA(r) sinh χ(t, φ) y = pB(r) cosh τ(t, φ) v = pB(r) sinh τ(t, φ)

II. r− < r < r+ u = pA(r) cosh χ(t, φ) x = pA(r) sinh χ(t, φ) y = −p−B(r) cosh τ(t, φ) v = −p−B(r) sinh τ(t, φ)

III.0 < r < r− u = p−A(r) cosh χ(t, φ) x = p−A(r) sinh χ(t, φ) y = −p−B(r) cosh τ(t, φ) v = −p−B(r) sinh τ(t, φ) 3. Local and Global Aspects of Anti-de Sitter Space 20

Where 2 2 2 2 2 r − r− 2 r − r+ A(r) = ` 2 2 ,B(r) = ` 2 2 r+ − r− r+ − r− 1 r t 1 −r t τ = + − r φ , χ = − + r φ . ` ` − ` ` + On each of the three patches, this brings us to the metric

2 2 2 2 2 2 2 2 2 (r − r+)(r − r−) 2 ` r dr 2 2 r+r− ds = − 2 2 dt + 2 2 2 2 dr + r dφ − 2 dt . ` r (r − r+)(r − r−) `r (3.21) Here, 0 < r < ∞, −∞ < t < ∞, and −∞ < φ < ∞. We now have a spacetime metric that takes the form

−N 2(r)dt2 + N −2(r) + r2(dφ + N φ(r)dt)2, (3.22) which is the shape of a general rotating black hole solution. It has singularities at r = r+ and r = r−. However, for it to be interpreted as a black hole, the φ-coordinate must be angular, otherwise it will just be a boosted portion of AdS3. Thus, we identify φ ∼ φ + 2π. (3.23) The exact nature of the black hole - and why we can call it a genuine black hole while still locally equivalent to AdS3 - will be explained in section (3.4.1). The identiﬁcation along ∂φ - an isometry of the metric (3.21) - is exactly the identiﬁcation along (3.20) in the new coordinate system:

r √ √ r √ √ ξ = + ( A sinh χ∂ + A cosh χ∂ ) − − ( B sinh τ∂ + B cosh τ∂ ) ` u x ` y v = ∂φ

Thus, we have created a black hole from a space of constant scalar curvature by identifying along a symmetry and parametrizing the metric on distinct patches.

3.4.1 The Nature of the BTZ Singularity A black hole is a singularity in the spacetime, but as a quotient space, the BTZ black hole solution is still locally equivalent to AdS3. One way to identify a true singularity rather than a coordinate singularity is by doing coordinate transformations and seeing if the spacetime is smooth in the new coordinates where it was singular in the previous ones. Another way is by looking at curvature invariants: scalars that are coordinate-independent. They show if there are places in the 3. Local and Global Aspects of Anti-de Sitter Space 21 spacetime that have a true singularity in the curvature. One of these invariants is the Kretschmann scalar: µνρσ K = R Rµνρσ. (3.24) For example, for the Schwarzschild black hole (the most simple four-dimensional black hole), with the line element

r dr2 ds2 = 1 − s dt2 − − r2(dθ2 + sin2θdφ2), (3.25) r rs 1 − r

48G2M 2 the Kretschmann scalar is given by K = c4r6 . This shows that for r → 0 we have a genuine curvature singularity, while the apparent singularity at r = rs is not a genuine curvature singularity. The Kretschmann scalar for the BTZ black hole is [10] , 12 K = , (3.26) `4 which shows there is no curvature singularity. This makes sense, as it is locally equivalent everywhere to a space of constant negative curvature. The question is now what we mean when we talk about a black hole, if we are not talking about a curvature singularity. The nature of the singularity in the BTZ black hole is in its causality, not in its curvature [7]. The three regions deﬁned in the previous section are separated by null surfaces (where a vector switches sign) at r = r±. A causal curve that goes through one of these surfaces can never return. This is called a causality singularity, and is thus equivalent to a black hole, which is also deﬁned by the irreversibility of world lines that enter through its event horizons.

3.5 The Euclidean BTZ Black Hole

While so far the signature of AdS3 and the BTZ black hole we have been working in has been Lorentzian, (− + +), we can also do a Wick transformation and work in Euclidean signature: (+ + +). This geometry provides a useful and less laborious way of ﬁnding the black hole’s thermodynamics (as wel shall see in the next chapters). We go from (3.21) to the BTZ metric in Euclidean signature by setting

t → it (3.27)

M → ME (3.28)

J → iJE (3.29)

r− → ir− (3.30) 3. Local and Global Aspects of Anti-de Sitter Space 22

We then get the Euclidean black hole metric

2 2 2 2 2 2 2 (r − r+)(r + r−) 2 ` r 2 2 ir+(r−) 2 dsE = 2 2 dt + 2 2 2 2 dr + r (dφ + 2 dt−) . ` r (r − r+)(r + r−) `r (3.31) In this solution, r+ < r < ∞. The metric ends at r+; it is not deﬁned beyond r+; r− is purely imaginary. The coordinate singularity is now at r = r+. But we want the geometry to be smooth here, because there is nothing special that happens there locally. We will denote Euclidean time tE. Following [33], we examine the near-horizon geometry r = r+ by doing the following transformations

2 2 0 0 02 r − r+ tE = r+tE + r−φ, φ = r+φ − r−tE r = 2 2 . (3.32) r+ + r− Which leads us to the metric

r02 `2dr02 ds2 = dt02 + + (1 + r02)dφ02. (3.33) E `2 E 1 + r02 0 If we now take the near-horizon limit, r → 0, or r → r+, the non-angular part looks like a plane: 02 02 02 r dtE + dr . 0 0 0 It is non-singular provided we identify tE as an angle. So, we identify tE ∼ tE +2π. Going back to our original coordinates, this means

0 0 ∆tE = 2π = r+∆tE + r−∆φ and ∆φ = 0 = r+∆φ − r−∆tE, leading to the identiﬁcation of our original coordinates: 2 2π` r+ 2π`r− (tE, r, φ) ∼ tE + 2 2 , r, φ + 2 2 . (3.34) r+ + r− r+ + r−

So upon imposing smoothness at the horizon, the Euclidean time tE has a periodicity we will call β, and a periodicity in the φ-direction we call Φ:1

2 2π` r− 2π`r+ Φ = 2 2 , β = 2 2 . (3.35) r+ + r− r+ + r−

As there are now two identiﬁed cycles, in the φ and the tE directions, and one radial component, the Euclidean black hole spacetime has a solid torus topology, as we can see in ﬁgure (3.2).

1 It is important to note that when working in Lorentzian formulation, the periodicities Φ and 1 β are ∝ 2 2 , due to r− ← ir−. r+−r− 3. Local and Global Aspects of Anti-de Sitter Space 23

Fig. 3.2: The Euclidean BTZ black hole has a solid torus topology, with a non- contractible cycle in the φ-direction, and a contractible cycle in the Euclidean timecycle. 0 < r < ∞ is suppressed here.

3.6 An Explicit Quotient Formulation of the BTZ Black Hole

As we have seen, the BTZ black hole is created as a quotient space of AdS3. We can see the AdS3 to BTZ quotienting in a matrix formulation as well, as in [29] in explicit group theoretical formulation. Helpful discussions of this procedure can also be found in [15, 16, 17, 18, 20]. In Euclidean Poincar´ecoordinates, the metric of BTZ is

`2 (dz2 + dwdw¯). (3.36) dz2 Group theoretically, the Euclidean BTZ black hole is described as the quotient space H3/Z, which means it is the hyperbolic group in three dimensions with the integers quotiented out. We can write Euclidean AdS3 as a matrix element of SL(2, C)

ww¯ w z + z z M = w¯ 1 , (3.37) z z where the metric is given by `2 ds2 = T r(M −1dM M −1dM), (3.38) 2 which returns (3.36). Taking a quotient space in this formulation now explicitly means that

M = gMg† (3.39) where (g, g†) ∈ SL(2, C) for speciﬁc (g, g†). We now have explicitly the creation of a quotient space using the symmetry group of a space and its generators. Explicitly, the element g is given by 3. Local and Global Aspects of Anti-de Sitter Space 24

π(r++ir−) ! π(r+−ir−) ! e ` 0 † e ` 0 g = −π(r++ir−) ; g = −π(r+−ir−) . (3.40) 0 e ` , 0 e `

The eigenvalues of these matrices are π(r + ir ) λ = ± + − (3.41) ` for g, and for g† we ﬁnd π(r − ir ) λ(†) = ± + − . (3.42) ` The coordinates (w, z) are given explicitly by

1 2 2 2 r+ − r− r+ ir− z = 2 2 exp φ + tE (3.43) r − r− ` ` 1 2 2 2 r − r+ r+ + r− w = 2 2 exp (φ + itE) , (3.44) r − r− ` so that our matrix M becomes r r  2 2 2 2 r+φ+r−t 2 2 i(−r−φ+r+t ) r−+r+ r −r+ E r −r+ E √ ` ` r2+r2 + 2 2 e r2 +r2 e  − r−+r+ − +  M =  r r  . (3.45) 2 2 (r−φ−r+t ) 2 2 (r+φ+r−t )  r −r+ i E r +r+ − E  2 2 e ` 2 2 e ` r−+r+ r−+r+

If we now do (3.39), in the (t, r, φ) coordinates, we find r r  2 2 2 2 r+(φ+2π)+r−t 2 2 −ir−(φ+2π)+r+t ) r−+r+ r −r+ E r −r+ E √ ` ` r2+r2 + 2 2 e r2 +r2 e †  − r−+r+ − +  gMg =  r r  2 2 ir−(φ+2π)−r+t ) 2 2 (r+(φ+2π)+r−t )  r −r+ E r +r+ − E  2 2 e ` 2 2 e ` r−+r+ r−+r+ (3.46) so we can see that taking the quotient gMg† = M means identifying φ ∼ φ + 2π; exactly as we saw before in section (3.4). In this section we saw the quotient space as an identification along the Killing vector in the φ direction. If we write the identification Killing vector (3.20) in terms of the SL(2, R) generators using (3.16) and (3.17) we get (setting r− → ir− to go to Euclidean space): r ir r ir ξ = + J − − J = + (L + L˜ ) − − (L − L˜ ). (3.47) φ ` ux ` vy ` 0 0 ` 0 0 3. Local and Global Aspects of Anti-de Sitter Space 25

This leads to r − ir r + ir ξ = + − L + + − L˜ . (3.48) ` 0 ` 0 In terms of the generators of SL(2, R),

1 0 0 0 1 2 0 L1 = ,L−1 = ,L0 = 1 , (3.49) −1 0 0 0 0 − 2 ˜ in which representation L0 = L0, we can write our identiﬁcation matrices as π(r − ir ) π(r + ir ) g = exp − − L ; g† = exp − − L . (3.50) ` 0 ` 0

We see that the Killing vector ξ used in (3.4) is related to the identiﬁcation matrices g, g†: eπξφ = g†g, (3.51) recalling equation (3.18). This explicit quotient formulation is equivalent to the identiﬁcation procedure in section (3.4). 4. THE BTZ BLACK HOLE

Now that we have a black hole solution, we can use it to find the properties that are attributed to black holes. For instance, black holes have a mass, and they have an angular momentum for a spinning solution. We will show how these are found for the BTZ black hole by using the approach of [12] and [5], which is to find a quasilocal stress tensor for asymptotically AdS spacetimes, defined at the boundary. To do this, we must first introduce some concepts such as foliating a spacetime and the holographic principle. We will also determine the black hole’s Killing vectors and use these to find its surface gravity and determine its thermodynamics. The four laws of black hole thermodynamics will also be discussed here. We will also see that the periodicity found in the Euclidean time cycle in section (3.5) is the inverse of the black hole temperature.

4.1 Mass and Angular Momentum

The interpretation of the BTZ solution as a black hole requires a mass and, for a spinning solution, angular momentum. But these are not obvious from the point of view of the black hole as a quotient space of a vacuum AdS3 solution: how do we interpret the values of the parameters r± above as physically meaningful in a black hole context? One way to arrive at a black hole solution in AdS3 is through the way we saw in section (3.4): by going through different sets of parametrizations and then making an identification along a certain Killing vector, which we can interpret as an angular coordinate. Because the identification is along a Killing vector - a symmetry - the metric we arrive at is well-defined and locally a solution of a spacetime with constant negative curvature. But after identifying the angle it also has an interpretation as a black hole analogous to a well-known four-dimensional black hole solution: the Kerr metric, which describes the spacetime geometry around a rotating black hole with no charge. The inner and outer horizons are, however, just constants if we do not give them any further physical interpretation. We want to find the mass and angular momentum of the solution by means of a more fundamental method than just by analogy of known solutions. There are 4. The BTZ Black Hole 27 different ways of reaching the same values for the mass and angular momentum. We will briefly discuss a conventional method based on the ADM formalism of a spacetime and a corresponding action principle. However, we will use the stress-tensor procedure of Brown and York [12] and its correction by Balasubramanian and Kraus [5] to define the conserved quantities. We will discuss this procedure and its background.

4.1.1 ADM Formalism and Asymptotic Symmetries To find the conserved charges of a theory one needs to define a Hamiltonian. The conventional way of doing this is through what is called the ADM-formalism, after creators Arnowitt, Deser and Miser [3, 4]. It is also referred to as the Hamiltonian formulation of gravity. In this method, one defines an action principle in a spacetime foliated into 1 spacelike hypersurfaces of constant time Σt. Then one considers the asymptotic symmetries of the metric, which are the symmetries that don’t change the metric in its asymptotic form (defined more specifically in section (4.2)). The ADM energy is defined as the variation of the metric tensor from its asymptotic form. The asymptotic symmetries are generated by the conserved Noether charges. These symmetries form their own group, which for AdS3 is two copies of the Virasoro algebra; of which SO(2, 2) is a subgroup. When varied, a boundary term must be added to cancel the lapse and shift functions at infinity; which are asymptotic displacements. These are conjugate to the mass and angular momentum. The two Killing vectors of the BTZ black hole, ∂t and ∂φ are the generators of these conserved quantities. They are conjugate to the two non-zero elements of the asymptotic symmetry group (the zero-modes of the Virasoro algebra). We will use these concepts loosely but continue now in more detail with the stress-tensor procedure.

4.1.2 A Quasilocal Stress Tensor for Gravity A fundamental way to find the mass and momentum of a system is by working with an action principle and finding conserved quantities using symmetries. To find conserved quantities we need a way to define the energy of the spacetime: a stress tensor. But how do we find a local stress tensor in a theory that is everywhere locally flat? A method of finding a stress tensor for a vacuum spacetime has been introduced by Brown and York [12], who defined a quasilocal stress tensor by

1 In this formulation, three-dimensional gravity is often referred to as (2+1)-dimensional gravity because it is decomposed into one time and two spatial dimensions. Three-dimensional and (2 + 1) dimensional gravity can be used interchangeably in this thesis, as with three dimensions we also mean one time and two spatial dimensions. 4. The BTZ Black Hole 28 looking at the energy of the spacetime as analogous to the energy of a system found in the Hamilton-Jacobi formalism. This idea uses the Hamiltonian formulation of gravity, in which a manifold is decomposed into hypersurfaces and a boundary. In this formalism, the Hamiltonian H of a system is minus the rate of change of the action over time. Brown and York defined a stress tensor for gravitational fields as minus the change over time in the action, at the boundary of the manifold. The boundary of the manifold thus serves as the analogue to the endpoints in time in the Hamilton-Jacobi formulation. The energy of a particle is found by varying the action over time, say on an interval (t1, t2), and like this the gravitational action is varied using the metric. As the metric carries more information than just the time interval, we don’t just get the energy, but the stress-energy tensor of the system. This approach had some problems that were solved by Balasubramanian and Kraus [5] for asymptotically Anti de Sitter spacetimes, in which the AdS/CFT correspondence (more on this below) was applied. Starting with explaining the Brown and York procedure, we then continue to discuss the approach of Bala- subramanian and Kraus for asymptotically AdS spacetimes specifically, and then go on to find a stress tensor for the BTZ solution to find its mass and angular momentum.

4.1.3 The Decomposition of the Spacetime Brown and York deﬁne a stress tensor on the boundary of a manifold analogously to the way energy is deﬁned in the Hamilton-Jacobi formalism, using an action principle. To do this, [12] starts with a decomposition of the manifold called an ADM decomposition. In this decomposition, a d-dimensional manifold M is foliated by spacelike surfaces Σt of constant t. Two functions, the lapse and shift functions, N and N µ respectively, connect these spacelike surfaces. A metric in this decomposition takes the following general form:

2 2 2 µ µ ν ν ds = −N dt + gij(dx + N dr)(dx + N dr) (4.1) In this decomposition, the dynamical variables are the (d − 1)-dimensional metric (called the induced metric, denoted gij) on the spacelike slices and its conjugate momentum πij. Every surface Σt has a boundary line Bt. All boundary lines together are the surface denoted (d−1)B, which is timelike. In our case, d = 3 and thus the boundary is two-dimensional. The boundary of M, ∂M, is then 2 deﬁned by B and the spacelike slices t1 and t2. The boundary metric is the 2 metric on B, which we denote γµν. The metric on a spacelike slice is denoted gij. The action can now be written as a function of the proper time between a slice B and its neighbour. 4. The BTZ Black Hole 29

4.1.4 Using the Hamilton-Jacobi Formulation In the Hamilton-Jacobi formulation, the momentum of a system with a classical action Scl at the endpoint t2 is deﬁned as

∂Scl P |t2 = , (4.2) ∂x2 and the Hamiltonian as

∂Scl H|t2 = − . (4.3) ∂t2

Analogously to these equations, in a foliated spacetime with a boundary metric γµν, and a hypersurface Σt metric gij we can deﬁne the components of the angular momentum of the spacetime as

ij δScl P |t2 = . (4.4) δgij For the equation analogous to the Hamiltonian in the foliated spacetime, the time between the slices Σt is determined by the induced metric, γµν. But the induced metric doesn’t just give information about the time interval, but about all the spacetime intervals. Thus, if the action is varied with respect to the boundary metric, the Hamiltonian as found in the Hamilton-Jacobi formulation becomes generalized to the energy-momentum tensor at the boundary: 2 δS T µν = √ cl . (4.5) −γ δγµν The above expression holds close similarity to the stress-energy tensor in the Ein- stein equation, m µν 2 δS TEinstein = √ . −g δgµν It is important to note that even though T µν is only defined by the induced metric on the boundary, it gives an expression for the whole of the spacetime, including both gravitational and matter contributions. However, as the boundary at constant r is moved to infinity (r → ∞), the stress tensor (4.5) will diverge. To get a useful value for the stress tensor, it needs to be finite. By adding a boundary term, the equations of motion in the bulk won’t be affected. Brown and York tried to fix the divergence by subtracting a boundary term with the same intrinsic metric as the stress tensor metric, embedded in a reference spacetime. However, this did not result in a well-defined stress tensor. For asymptotically Anti de Sitter spacetimes, there is an attractive solution to this problem in what is known as the AdS/CFT correspondence. 4. The BTZ Black Hole 30

AdS/CFT Correspondence Proposed in 1998 by Maldacena [28], the AdS/CFT correspondence has become a huge field of research. It is a conjecture that proposes a duality between gravity in d dimensions and string theory, or more specifically quantum field theory, in (d − 1) dimensions. Specifically, it states that (quantum) gravity in Anti de Sitter, in our specific case in three dimensions, is dual to a conformal field theory in two dimensions. A conformal field theory is a quantum field theory invariant under conformal (angle-preserving) transformations. So far it is the best realization of the holographic principle, which states that the information in the bulk of a volume of space can be found on the boundary of the volume. Going back to the problem of a stress tensor in AdS; instead of working with reference spacetimes, we can see the boundary stress tensor of the bulk spacetime defined in (4.5) as the (expectation value of the) stress tensor of the dual conformal field theory as described by the AdS/CFT conjecture. The divergences can be removed by adding local counterterms and are then the ultraviolet divergences well-known in quantum field theories. The added terms are boundary curvature invariants of the stress tensor. We look for the terms in the action we need to get a well-defined stress tensor.

4.2 A Boundary Stress Tensor for AdS3

To now specifically find a boundary stress tensor for AdS3, we follow the procedure of [5, 26]. The solution to the problem with the Brown and York procedure is found for asymptotically Anti-de Sitter spacetimes. An asymptotically AdS spacetime is defined as follows.

Asymptotically AdS Spacetimes

As we have seen, AdS3 is invariant under the isometry group SO(2, 2) with six Killing vectors. Simply said, an asymptotic AdS3 spacetime is one that asymptotically looks like AdS3 and is invariant under the AdS3 symmetry group. Another way of deﬁning asymptotic AdS3 is through a Feﬀerman-Graham expansion. First, we write the metric in the form

2 2 k i j ds = dρ + gij(x , ρ)dx dx , i, j = 1, 2; (4.6)

in which we can write gij in the Feﬀerman-Graham expansion [2]:

2ρ/` (0) µ (2) µ (4) k gij = e gij (x ) + gij (x ) + gij (x ) + ... (4.7) This expansion indicates what happens to the metric asymptotically - by which we 2 dr2 mean for large ρ; which is the radial coordinate in this case - defined dρ = r2 in 4. The BTZ Black Hole 31 relation to the radial coordinate r we use. So, when we talk about an asymptotic 2 2ρ/` (0) µ spacetime, we mean as r → ∞. The leading part of the metric is dρ +e gij (x ), (2) and the subleading parts the terms gij and higher. An asymptotically AdS3 space is defined by the leading part of its metric being invariant under the same isometry group as AdS3, SO(2, 2), but the subleading terms can differ. The diffeomorphisms that leave the leading part of the metric intact but change the subleading parts are then part of the asymptotic symmetry group. These have nonvanishing boundary charges, which for the BTZ black hole, as an asymptotically AdS3 metric, are the mass and angular momentum. We will now continue with the boundary stress procedure, which defines first a stress tensor for AdS3 and then incorporates for asymptotic solutions as well.

4.2.1 An Action Principle We start with the Einstein-Hilbert action with an added negative cosmological 1 constant Λ = − `2 , Z 1 d+1 √ 2 S = − d x g(R − 2 ). (4.8) 16πG M ` We want to rewrite the gravitational action in terms of the foliation we discussed. Following the reasoning of [30], we foliate the manifold in a radial direction r instead of foliating the spacetime in spacelike hypersurfaces of constant t. This way we get an ADM-like decomposition [5]:

2 2 2 µ µ ν ν ds = N dr + γµν(dx + N dr)(dx + N dr). (4.9)

The decomposition is now done by foliating the spacetime in the radial direction, so that a hypersurface in the manifold of constant r is normal ton ˆr, and is referred to as ∂Mr. The radial direction is then the normal vector to the slices. The metric on ∂Mr is the boundary metric γµν at r. We can then define the boundary of the whole manifold for r → ∞, and denote it ∂M. Now the metric is decomposed into the form (4.9) but with the radial component taking the place of the timelike direction, we can discern between two forms. The first fundamental form in the foliation is the induced metric of the hypersurfaces at constant r. The second fundamental form is the extrinsic curvature: the curvature as seen from an embedded point of view. A vector is decomposed into a tangential part to the hypersurface as defined by the induced metric γµν, and a normal part as defined by the extrinsic curvature Θµν. We define them on the boundary

α β α β γµν ≡ gαβeµeν , Θµν ≡ ∇βnαeµeν , (4.10) 4. The BTZ Black Hole 32

α ∂xα α µ with eµ = ∂yµ , where x are the coordinates on the spacetime and y denote the coordinates on boundary lines of the hypersurfaces. nα is the unit normal pointing outward in the radial direction. The trace of the extrinsic curvature is given by µν γµνΘ = Θ. We can, for simplicity, write 1 Θ = ∂ γ . (4.11) µν 2 r µν The reason we can do the above for the extrinsic curvature is that [30] it is symmetric: Θµν = Θνµ, (4.12) so we can rewrite 1 1 Θ = (∇ n + ∇ n ) = (L γ ), (4.13) µν 2 ν µ µ ν 2 r µν in which Lr is the Lie derivative along r - which, for a vector field, is simply the commutator of the two fields, as we see in the equality above. As the Lie derivative is the change of a tensor field along the flow of another vector field, we can write it as (4.11). To now rewrite our action, we use the decomposition of the Ricci scalar [30] in terms of this foliation:

2 2 µν ρ σ ρ σ R = R + Θ − Θ Θµν + 2∇ρ(∇σn n − n ∇σn ). (4.14) 2R is the Ricci scalar in two dimensions:

2 µν λ R = γ Rµλν. (4.15) Now, using Gauss’ theorem, Z I α√ 4 α ∇αA −gd x = A dΣα (4.16) V ∂V √ √ and −g = nµ γ, we can write the last term in the decomposed Ricci scalar as I I ν µ ν µ µ√ 2 −2 (∇µn n − n ∇µn dΣρ) = −2 ∇µn γd x ∂V ∂V I √ = −2 d2x γ Θ. (4.17) ∂V This way we get the following gravitational (Einstein-Hilbert) action:

Z Z 1 2 √ (2) 2 µν 1 2 √ SEH = d xdr γ R + Θ − Θ Θµν − 2Λ − d x γ Θ. 16πG 8πG ∂M (4.18) 4. The BTZ Black Hole 33

The last term spoils the variational principle, in which the variation of the metric δγµν is constant but not its normal derivative, δ∂rγµν. To ﬁx this, we add what is called the Gibbons-Hawking term. This saves the variational principle and doesn’t aﬀect the equations of motion in the bulk: Z 1 2 √ SGH = d x γΘ. (4.19) 8πG ∂M After adding this term, we get for the action Z 1 2 √ 2 2 µν SEH+GH = d xdη γ R + Θ − Θ Θµν − 2Λ . (4.20) 16πG M

Varying the action with respect to γµν, integrating by parts the terms that include δ∂rγµν and keeping only the resulting boundary terms (the rest will be bulk terms that don’t contribute to the stress tensor at the boundary), we end up with Z 1 2 √ µν µν δS = − d x γ (Θ − Θγ ) δγµν. (4.21) 16πG ∂M Using (4.5), we ﬁnd for the stress tensor thus far 1 T µν = (Θµν − Θγµν) . (4.22) 8πG

4.2.2 Finding a Finite Stress Tensor Now, this stress tensor is undefined as r → ∞. In [5] a method is used in which µν T is seen as a conformal field theory on the boundary of AdS3. This way, the divergences in the stress tensor become ultraviolet divergences well-known in quantum field theories, that can be canceled by adding local counterterms. These local terms only depend on the boundary’s intrinsic geometry and so an embedding reference spacetime becomes redundant. For every dimensionally different Anti de Sitter spacetime, curvature invariants are added to the action to make it finite. For AdS , with S = R L , they have found 3 ct ∂Mr ct 1√ L = − −γ. ct ` This means that when we add this term to the action, we get a stress tensor that looks like

1 1 T µν = Θµν − Θγµν − γµν . (4.23) 8πG `

If we now use the Poincar´ecoordinates of AdS3: 4. The BTZ Black Hole 34

`2 r2 ds2 = dr2 + (−dt2 + dx2), (4.24) r2 `2 r2 where the metric at the boundary is given by −γtt = γxx = `2 and the radial normal µ r µ,r vector isn ˆ = ` δ , we get the following values for the stress tensor components:

r2 1 8πGT = − − γtt tt `3 ` r2 1 8πGT = − γxx xx `3 ` 1 8πGT = − γtx. tx `

For AdS3, the added term in the action cancels the divergences, and actually yields 1 µν Tµν = 0. Thus, because of the added term − ` γ , the stress tensor is ﬁnite even as ∂Mr → ∂M.

4.2.3 A Stress Tensor for the BTZ Black Hole

Now, the BTZ black hole is an asymptotically AdS3 solution, so the method described above can be used to ﬁnd the quasilocal stress tensor. To do this, the BTZ metric for large r is used:

−r2 + r2 + r2 r2 + r2 + r2 r r r2 + − dt2 + + − `2dr2 − + − dtdφ + dφ2, (4.25) `2 r4 ` `2 with which the stress tensor components Ttt and Ttφ can be found [5]: r2 + r2 T = + − (4.26) tt 2π`3 r r T = − + − . (4.27) tφ π`2 Then we integrate setting x → `φ, over the interval 0 to 2π, for the mass and angular momentum: Z 2π 2 2 r+ + r− M = 3 `dφ (4.28) 0 2π` Z 2π r+r− J = − 2 `dφ. (4.29) 0 π`

Basically, we see that using this technique, we find M = M and Pφ = J. Depending 1 on the values of M and J, global AdS3 can be reproduced for J = 0 and M = − 8G , 4. The BTZ Black Hole 35 and the Poincarépatch for M = 0 and J = 0. The definitions of mass don’t coincide due to the different time directions of the coordinates, by which different energies are thus defined.

4.3 Geometric Aspects

Every metric gives rise to certain curvature tensors, such as the Riemann tensor, and the Ricci tensor and scalar. We will compute these in the following section. After, we will calculate the event horizons and the angular momentum using the black hole’s Killing vectors.

4.3.1 Curvature Tensors Now that we have a black hole metric with its parameters deﬁned, we can look at its symmetries and determine its properties from the metric. Starting with our black hole metric as found in (3.21):

2 2 2 2 2 2 2 2 2 (r − r+)(r − r−) 2 ` r dr 2 r+r− ds = − 2 2 dt + 2 2 2 2 + r dφ − 2 dt , ` r (r − r+)(r − r−) `r

2 2 r++r− 2r+r− in which we now know M = 8G`2 and J = 4G` . We can write our metric in terms of the mass and angular momentum speciﬁcally [8]:

r2 16G2J 2 dr2 4GJ 2 8GM − − dt2 + + r2 dφ − dt . `2 r2 r2 16G2J2 r2 −8GM + `2 + r2 (4.30) With this metric, we can calculate the curvature tensors and Christoﬀel connections using the deﬁnitions (2.3), (2.4), (2.5), and (2.6).

2(−M`2 + r2) 8r2 J 2r2 R = ,R = − ,R = ,R = − , tt `4 rr 4r4 + `2(J 2 − 4Mr2) φt `2 φφ `2 (4.31) out of which we can ﬁnd the Ricci scalar by contracting: 6 R = − . (4.32) `2 1 As the Ricci scalar is dependent only on the cosmological constant (Λ = − `2 ) we can see the solution (3.21) is locally the same as AdS3. 4. The BTZ Black Hole 36

4.3.2 Event Horizons Using Killing Vectors To calculate certain properties of the black hole, we need to know what its symmetries, given by its Killing vectors, are. If a metric is independent of a certain coordinate, we have a Killing vector in this direction. It is not always the case that it is so easy to find the symmetries of a metric as this, but in the current case we can rely on there being only the two Killing vectors that are directly obvious µ µ µ in this way [7, 21]. We write their components as χ = (∂α) = δα, where α is the direction of the isometry (and in this case the coordinate the metric is independent of). The Killing vectors in this metric are ∂t and ∂φ. As with a four-dimensional rotating (Kerr) black hole, it is their combination that gives us the Killing vector µ field: χ = A∂t +B∂φ, where A and B are constants. Now, analogously to the Kerr metric, we want to find what is called the Killing horizon: a null hypersurface at which the norm of the Killing vector is null for certain values of r. For the metric (4.30), these values for r coincide with the event horizons. As the Killing vector goes null at the hypersurface, it switches from timelike to spacelike. As black hole horizons, this means the Killing vectors cross the inner and outer event horizons into the black hole. To find for which values of r this happens, we compute

µ ν µ χ χµ = gµνχ χ = 0. (4.33) We know that the Killing vector is timelike before it hits the horizon; so we have the extra condition that µ χ χµ < 0 for r → ∞. (4.34) Using the form (4.30) of the metric, we get for the norm

r2 χµχ = −(−8GM + )A2 − 8GJAB + r2B2. (4.35) µ `2

A To satisfy the condition (4.34) we must demand that ` > |B|. We are free to set A = 1 for simplicity as there are no further restrictions on it, and thus we end up with r2 χµχ = −(−8GM + ) − 8GJB + r2B2. (4.36) µ `2 The value for B is then found by setting the norm to zero as above. In terms of metric coeﬃcients, we then get q −g ± g2 − g g tφ tφ φφ tt B± = . (4.37) g φφ r± 4. The BTZ Black Hole 37

To ﬁnd the event horizons, we look for which values of r grr has a pole. This is when

r2 16G2J 2 −8GM + + = 0, `2 r2 so the horizons in terms of M and J are v u q u 2 J2 8GM ± 8G M − `2 r = `t . (4.38) ± 2 Now we can plug in these values in the expression for B we found earlier, which gives

4GJ r− 4GJ B+ = 2 = ,B− = 2 . (4.39) r+ `r+ r−

q 2 When plugging in either of the values for r±, the part gtφ − gφφgtt disappears, so that we end up with the equivalence of

gtφ B± = − . g φφ r±

dφ By analogy, the angular momentum associated to the Kerr metric is Ω = dt = gtφ − . We rename B± = Ω±. We end up with the Killing vector ﬁeld gφφ

µ χ = ∂t + Ω±∂φ. (4.40)

The factor Ω+ is actually the angular velocity of the black hole, and if we compare it to the values of the periodicities β and Φ of the Euclidean black hole found in 2 2 section (3.5) but in Lorentzian signature so r− → −r−;

2 2π` r+ 2π`r− β = 2 2 and Φ = 2 2 , r+ − r− r+ − r− we ﬁnd the relation βΩ+ = Φ. So, the periodicity in tE multiplied by the angular velocity gives us the periodicity of the φ-cycle.

4.4 Black Hole Thermodynamics

Black holes obey rules analogous to the laws of thermodynamics [9]. While at ﬁrst the laws that black holes obey seemed to be coincidentally analogous to the laws of thermodynamics, it later became clear that we really can interpret black holes 4. The BTZ Black Hole 38 as thermodynamical systems. In 1974 Hawking [23] discovered that black holes radiate due to quantum processes; conﬁrming that they are really thermodynamic systems. As with the laws of thermodynamics, there are four laws in total, to be discussed separately below.

The Zeroth Law The Zeroth Law of black hole thermodynamics states that a black hole has constant surface gravity (denoted by κ) in a stationary metric. It is uniform on the entire event horizon. The zeroth law of thermodynamics states that temperature is constant in a system in thermal equilibrium, indicating that the surface gravity of a black hole functions as temperature. Surface gravity is the acceleration needed to keep a test particle at the horizon as evaluated at inﬁnity. Mathematically, there are three equivalent deﬁnitions [30]:

β α β α 2 1 α;β 1. (−χ χβ);α = 2κχα , 2. χ;βχ = κχ , 3. κ = − χ χα;β . (4.41) r+ r+ 2 r+ Using any of these gives the same value for the BTZ black hole’s surface gravity:

q 2 8G − J + M 2 2 2 l2 r+ − r− κ = = 2 (4.42) r+ ` r+ The relationship between a black hole’s surface gravity and its temperature (known as the Hawking temperature), is

2 2 κ r+ − r− TH = = 2 (4.43) 2π 2π` r+

The First Law The First Law gives the relationship between changes in mass, angular momentum and surface area. It is analogous to the ﬁrst law of thermodynamics: dE = T dS − pdV, and is given by κ dM = dA + Ω dJ. (4.44) 8π +

A is the area of the event horizon r+. To ﬁnd it, we use a covariant surface integral: Z A = p|g|dx. (4.45) . 4. The BTZ Black Hole 39

For BTZ, the metric on the horizon (the outer horizon, so for r+) is found by setting dr = 0, r = r+ and dt = 0. So we end up with only the gφφ component at r+ and integrating over φ from 2π to 0. Z 2π q 2 A = r+dφ = 2πr+. (4.46) 0

r q 2 8GM+8G M 2− J l2 dA Now we can ﬁll in for (4.44): A = 2πr+ = 2πl 2 into dA = dM dM+ dA dJ dJ. After some manipulations, we end up with q 2 J2 1 G M − l2 J dA = dM − 2 dJ; π r+ 2r+ exactly (4.44).

The Second Law This law is also called the area theorem, and it states that the area of a black hole never decreases. As a black hole does in fact radiate due to quantum processes, this law only holds in a classical sense. It is comparable to a degree to the second law of thermodynamics, which is a statement about entropy never decreasing in an isolated system. This leads to a relation between a black hole’s area and its (Bekenstein-Hawking) entropy: A S = (4.47) BH 4G Speciﬁcally for the BTZ black hole, this gives for its entropy r π S = + . (4.48) BTZ 2G

The Third Law This law states that in ﬁnite time, the surface gravity of a black hole can never be reduced to zero. Analogously, the entropy of a thermodynamic system will be a constant at absolute zero temperature.

4.5 Euclidean Time Periodicity and Temperature

The value of the Euclidean time periodicity is actually the same as the inverse temperature T −1 of a thermodynamical ensemble. If we look at the path integral 4. The BTZ Black Hole 40

of a field, we see [22] that the amplitude of moving from configuration φ1(t1) to φ2(t2) is given by Z hφ2, t2|φ1, t1i = d[φ]exp(iI[φ]), (4.49) with hφ2, t2|φ1, t1i = hφ2, t2|exp(iH(t2 − t1))|φ1, t1i, (4.50) with H the Hamiltonian. If we now set t2 − t1 = −iβ and the field configuration stays the same, we get Z Tr exp(−βH) = d[φ]exp(iI[φ]). (4.51)

Now we see that Tr exp(−βH) is the partition function for a canonical ensemble with temperature T = β−1. Thus we can equate the periodicity of imaginary time in the Euclidean signature of a black hole with the inverse of the temperature. Looking back at equations (4.42), and (4.43), we ﬁnd

2 2 −1 κ r+ − r− β = = TH = 2 . (4.52) 2π 2π` r+ 5. CHERN SIMONS THEORY AS (2+1)-DIMENSIONAL GRAVITY

General relativity has given us a model of spacetime that quantifies gravity that thus far has not been superseded by any other theory of gravity. However, it has the problem that it does not incorporate quantum mechanics. It is a purely classical theory that gives no information about quantum processes in gravity. A lot of attempts to quantize gravity have been made, and so far none have given a definitive solution. One attempt at finding a quantum (toy) model of gravity has been made by interpreting (2 + 1)-dimensional gravity as a Chern-Simons theory, originated by Witten [36] and Achucarro and Townsend [1]. Formulating gravity in this form not only has the possibility of quantizing the theory, but also sheds light on other aspects of gravity. In this chapter, we start with a short introduction to Chern-Simons theory, after which we will show how and why it is possible to formulate (2 + 1)-dimensional gravity as a Chern-Simons theory. This will also require a discussion of the vielbein formalism, a different way of specifying the curvature of a manifold.

5.1 Introduction to Chern-Simons Theory

Chern-Simons theory is a quantum field theory in (2+1) dimensions that computes only topological invariants. It can be defined on any manifold, and a metric does not need to be specified as it is a topological theory. This means it only has interesting topological features and no local geometric features. Thus, the physics does not depend on the local geometry. It is a gauge theory as well: given a gauge group G and a manifold M, the theory is defined by a principal G-bundle on the manifold. A principal G-bundle is a formalism defining the action of the group G on the manifold, and the projection of this action on the manifold. It is given by a connection that specifies parallel transport but is compatible with the group action. This connection takes the shape of a one-form A that takes values in the theory’s Lie algebra, denoted g. The Chern-Simons action is given by k Z 2 S [A] = Tr(A ∧ dA + A ∧ A ∧ A), (5.1) CS 4π 3 5. Chern Simons Theory as (2+1)-dimensional Gravity 42 where the Lagrangian 2 L = AdA + A ∧ A ∧ A (5.2) 3 is called the Chern-Simons three-form. k is what is called the Chern-Simons level, which is only relevant once the theory is quantized. We will focus on the classical Chern-Simons solutions. “Tr” stands for the trace, the sum of all diagonal elements of a matrix. The equations of motion from this action after varying it with respect to A is the vanishing field strength, also known as the curvature two-form:

F = dA + A ∧ A = 0. (5.3) F vanishes if there is no curvature; i.e., the connection A is ﬂat. This means that there is no local information to speak of in the connection, and we must see globally if there is anything that deﬁnes the connection. In Chern-Simons theory, the gauge- invariant observables are Wilson loops: holonomies around noncontractible cycles on the manifold M on which it lives.

5.2 From (2+1)-Dimensional Gravity to Chern-Simons Theory

The question is now how to connect gravity and Chern-Simons theory: how to formulate a theory of gravity without using a metric. The answer is specific to the dimensions of our spacetime. We have seen that (2+1)-dimensional gravity has no local degrees of freedom, and that possible solutions are all locally equivalent to a solution with constant negative curvature. The only possible difference in solutions is thus a global difference, such as with a global identification changing the topology of the manifold and creating a black hole. The same way, a Chern-Simons theory has no relevant local geometry, and is fully determined by its topology and global symmetry group. This way we can equate it to (2 + 1)-dimensional gravity. The first step is to write the geometry of the manifold in a different formalism. By discussing curvature in the form of the metric tensor gµν, the way we have discussed the curvature of a manifold so far is by taking coordinate-based tangent vectors as the basis vectors with which we define our curvature. Another way to describe curvature is to work in a basis that is not coordinate-dependent: the vielbein formalism.

5.2.1 Vielbein Formalism The curvature of a manifold in the metric tensor formulation has its basis in the partial derivatives at the tangent space of a point on the manifold. Instead of this, the vielbein formalism introduces a set of orthonormal basis vectors at each point on the manifold, so that their inner product, g(ea, eb), with g the usual metric, is 5. Chern Simons Theory as (2+1)-dimensional Gravity 43

ηab. Following the conventions and reasoning of [19], we will use the familiar Greek index notation for the coordinate-based metric formulation and Latin indices to label the components in the vielbein formalism. As in the metric formulation, we usually need to work patch-wise on the manifold to describe it fully, instead of one set of vielbeins suﬃcing for the whole manifold. We can write the metric tensor basis in terms of the partial coordinate derivates as

a ∂µ =e ˆ(µ) = eµ eˆ(a), (5.4) a where eµ, the vielbein components notation, is matrix-valued. Its inverse gives

µ a µ a µ a ea eν = δν , eµ eb = δb . (5.5)

The metric can be written in terms of these:

a b gµν = eµ eν ηab, (5.6) where due to the form of equation (5.6), the vielbein e is sometimes called the metric tensor’s square root. Now that we have defined a non-coordinate based formalism, how do we define ρ differentiation in it? The Christoffel connection Γµν is certainly coordinate-based, so we can’t just carry this connection into the new formalism. Where the Christof- fel connection was a correction based on the manifold’s deviation from flat space and had a term for each index, we define what is called the spin connection in the same way in the vielbein formalism:

a a a c c a ∇µX b = ∂µX b + ωµ cX b − ωµ bX c. (5.7) The spin connection is connected to the Christoﬀel connection by

a a λ ν λ a ωµ b = eν e bΓµλ − e b∂µeλ . (5.8)

Obviously, the spin connection has both Greek and Latin indices. Differentiating must be done with respect to a direction given by a coordinate, and the connection holds values in the vielbein formulation. Instead of the metric tensor, we now have a formulation of general relativity with as dynamical variables the vielbein (or, specifically in our three-dimensional case, the dreibein) e and the spin connection ω. Both e and ω are matrix-valued forms, (e a one-form and ω a two-form), which obey exterior algebra. Multiplication is done through the wedge product, denoted ∧; and differentiation by means of the exterior derivative, denoted d. In the metric formalism we have been using thus far, the exterior derivative is defined

dX ≡ ∂µXν − ∂νXµ, (5.9) 5. Chern Simons Theory as (2+1)-dimensional Gravity 44 and the wedge product of two one-forms is deﬁned by the two-form

(X ∧ Y ) ≡ XµYν − YµXν. (5.10)

In vielbein formulation, the Riemann tensor is written as

a a a c R = dω + ω ∧ ω ⇐⇒ Rµν b = (dω b + ω c ∧ ω b)µν, (5.11)

a a a a or alternatively Rµν b = ∂µων b − ∂νωµ b + [ωµ, ων] b. Lastly, as with the metric tensor, we can use vielbeins to raise and lower indices and contract tensors, like when ﬁnding the Ricci tensor:

b σ a Rµν = eν e a Rµσ b.

Chern-Simons Theory in Vielbein Formulation To give some background on the general idea of the concept of Chern-Simons theory as gravity, we will follow the reasoning of [36], by Witten, who, together with Achucarro and Townsend, was among the first to introduce Chern-Simons theory as a model for gravity. He formally introduces the concept of looking at the vielbein formalism as follows. One starts with a d-dimensional space-time manifold M of Lorentzian signature and its associated tangent bundle T (the group of all its tangent spaces). A d-dimensional vector bundle V with a structure group SO(d − 1, 1) is also introduced, meaning it has a metric ηab = (− + + + ...). A vector bundle is a definition of how one or more vector spaces are parametrized by another space - in our case the manifold M. It is assumed that V is of the same topological type as M, so that there will exist isomorphisms between the vector bundle V and the tangent bundle T , but not an obvious choice of specific isomorphism. The choice made for an isomorphism is a vielbein, which we will presume to be invertible. The associated spin connection can be seen as a SO(d − 1, 1)-valued connection on V . So, the vielbein gives us the transformation from the tangent space (which is how we formulate our metric) to the vector bundle.

5.3 Chern-Simons as a Theory for (2+1)-Dimensional Gravity

The general idea of the proof given for the equivalence in [36] goes as follows: starting with the Chern-Simons action (5.1), A is a gauge ﬁeld that takes values in the group’s Lie algebra. For equivalence, the Chern-Simons action needs to be invariant under the same gauge transformations as (2 + 1)-dimensional gravity. Witten [36] starts with three-dimensional vacuum space with a negative cosmological constant, which is invariant under SO(2, 2), and thus also SL(2, R)×SL(2, R). 5. Chern Simons Theory as (2+1)-dimensional Gravity 45

He constructs the Chern-Simons action to be invariant under this group, and then ends up with the Einstein-Hilbert action (5.13). We will see how the Chern-Simons action is equivalent to the Einstein-Hilbert action in three dimensions. To begin with, the Einstein-Hilbert action in three dimensions, is in metric notation given by 1 Z √ S = d3x −g(R − 2Λ), (5.12) EH 16πG and in terms of the vielbein formalism takes the shape [16] Z 1 a b c Λ a b c SEH = e ∧ 2dωa + abcω ∧ ω + abce ∧ e ∧ e . (5.13) 16πG M 3 Using the conventions in [16] we start with the speciﬁc case of the Chern-Simons action ¯ SCS = S[A] − S[A], (5.14) in which S[A] is equation (5.1),

1 Z 2 S[A] = Tr(A ∧ dA + A ∧ A ∧ A). 2 3 As A is a connection one-form that is valued in the gauge group’s Lie algebra of SL(2, R), we write it in component notation as 1 1 A = ω + e → A = (ωa + ea )L dxµ, (5.15) ` µ ` µ a 1 1 A¯ = ω − e → A¯ = (ωa − ea )L dxµ, (5.16) ` µ ` µ a in which the spin connection is 1 ω a = abcω . (5.17) µ 2 µbc ¯ 1 If we now ﬁll in for SCS = S[A] − S[A], we ﬁnd 2 4 S[A] − S[A¯] = eaL ∧ dωbL + ωaL ∧ debL − eaL ∧ ebL ∧ ecL (5.18) ` a b a b 3`3 a c c 2 + ωaL ∧ ωbL ∧ ecL + ωaL ∧ ebL ∧ ωcL + eaL ∧ ωbL ∧ ωcL . 3` a b c a b c a b c 1 The Greek indices are suppressed here for clarity as the Lie-algebra valued forms are more relevant. 5. Chern Simons Theory as (2+1)-dimensional Gravity 46

Now, using the following [16, 36]: 1 Aa ∧ Bb = [Aa,Bb], (5.19) 2 1 Tr(T T ) = δ , (5.20) a b 2 ab Z Z a b Tr A ∧ dA = Tr(LaLb) A ∧ dA , (5.21)

c [La,Lb] = abcL , (5.22) we ﬁnd for the ﬁrst term, using integration by parts [35] and the antisymmetry of the wedge product,

a b a b a a a a 2Tr(LaLb)(e ∧ dω + ω ∧ de ) = (e ∧ dω + ω ∧ de ) (5.23) = 2eadωa.

For the second term, we find 4 2 − eaL ∧ ebL ∧ ecL = Tr( [L ,L ],L )ea ∧ eb ∧ ec (5.24) 3`3 a b c 3`3 a b c 1 = ea ∧ eb ∧ ec, 3`3 abc and for the first part of the third term 4 2 Tr(eaL ∧ ωbL ∧ ωcL ) = ([L ,L ],L )ea ∧ ωb ∧ ωc (5.25) 3` a b c 3` a b c 1 = ea ∧ ωb ∧ ωc, (5.26) 3` abc alike for the last two parts of the third term. As all three parts of the last term 1 are the same; we end up with the above term without the factor 3 . Adding all these together and filling in gives for the Chern-Simons action (5.14) 1 Z ea 1 ∧ 2 ∧ dωa + ωb ∧ ωc − eb ∧ ec . (5.27) 4π ` abc 3`2 abc

−1 This is simply the Einstein-Hilbert action (5.13), for which `2 = Λ, the covariant √ ea integration variable g is ` (ﬁttingly, as the square root of the metric’s trace, as was mentioned in the previous section on vielbeins), and the Ricci tensor is as in (5.11) - remembering that in three-dimensional gravity, the Riemann tensor comes down to the Ricci tensor. Thus, the Chern-Simons action is equivalent to the Einstein-Hilbert action in three dimensions with algebra SL(2, R); after setting ` k = . (5.28) 4G 5. Chern Simons Theory as (2+1)-dimensional Gravity 47

We can also see that the equations (5.34) come down to the Einstein equations of motion as discovered by Achùcarroand Townsend [1]. To fill them in using the vielbein formalism and combining F and F¯, remembering (5.3) and defining F¯ using A¯, we use F = dA + A ∧ A, and equivalently F¯ = dA¯ + A¯ ∧ A¯ = 0. We will work in component notation and thus write the curvature two-form as

a a b c b c Fµν = La[∂µeν − ∂νeµ + abc(ωµ eν + eµ ων ) 1 + ∂ ω a − ∂ ω a + abc(ω ω + e e )]dxµdxν (5.29) µ ν ν µ µb νc `2 µb νc

¯ a a b c b c Fµν = La[∂µeν − ∂νeµ + abc(ωµ eν + eµ ων ) 1 + ∂ ω a − ∂ ω a + abc(ω ω + e e )]dxµdxν. (5.30) µ ν ν µ µb νc `2 µb νc While the deﬁnition (5.10) seems to indicate that both ω ∧ ω and e ∧ e would vanish, we must remember that they are matrix-valued one-forms (also recalling (5.17)); valued in the gauge group’s Lie algebra. Thus, we get for these wedge products not zero but 1 e ∧ e = ea L dxµ ∧ eb L dxµdxν = ea eb [L ,L ]dxµ ∧ dxν (5.31) µ a ν b µ ν 2 a b

1 ω ∧ ω = ωaL dxµ ∧ ωb L dxµdxν = ωaωb [L ,L ]dxµ ∧ dxν. (5.32) µ a ν b µ ν 2 a b As for e ∧ ω, we ﬁnd 1 ea L dxµ ∧ ωb L dxν = ea ωb [L ,L ]dxµ ∧ dxν, (5.33) µ a ν b µ ν 2 a b c where [La,Lb] = abcL [16] as before. These give 1 F + F¯ = 2L [∂ ω a − ∂ ω a + abc(ω ω + e e )]dxµdxν = 0 (5.34) a µ ν ν µ µb νc `2 µb νc

¯ a a b c b c µ ν F − F = 2La[∂µeν − ∂νeµ + abc(ωµ eν + eµ ων )]dx dx = 0. (5.35) Recalling the formulations (5.8) and (5.11) of the Riemann tensor and torsion tensor in the vielbein formalism, we see that (5.34) is equivalent to the Einstein equations of motion in three dimensions, (2.9): Rµν = Λgµν. We see, just as when comparing the actions; the Ricci tensor (using its three-dimensional equivalence to the Riemann tensor) described on the left-hand side using the spin connection 5. Chern Simons Theory as (2+1)-dimensional Gravity 48

1 and the metric tensor and cosmological constant Λ = − `2 described by the part in which the vielbeins are squared. The meaning of (5.35) is that the torsion tensor vanishes. In metric formulation, the torsion tensor is given by

λ λ λ Tµν = Γµν − Γνµ. (5.36)

It indicates how a frame is twisted when following a curve. If it vanishes as in (5.35), it means the connection is symmetric in its lower two indices and is thus the Christoﬀel connection, which we introduced in section (2.1.2). With these equations compatible, Chern-Simons theory can be seen as a model for 2+1- dimensional gravity, allowing us to now deﬁne the BTZ black hole in it. 6. THE BTZ BLACK HOLE IN CHERN-SIMONS THEORY

We now have a Chern-Simons formulation compatible with 2+1-dimensional gravity, but how can we define a black hole in it? We have no local degrees of freedom, and how can we talk about a “causal structure”? The ways black holes are usually defined, as curvature singularities or singularities in the causal structure are unavailable to us in Chern-Simons theory. We thus define the black hole through its topological features and its degrees of freedom at infinity, which give us a way of determining the thermodynamics of the system. We will see that the topological degrees of freedom, given by the manifold’s holonomies, are equivalent to the metric formulation of identifying the φ-cycle, and that when we compute the thermodynamics (entropy) of the system using asymptotic symmetries, we find the same entropy as we found in the metric formulation. We will introduce the concept of holonomies and asymptotic symmetries, and then define the black hole in terms of these.

6.1 From the BTZ Metric to a Chern-Simons Connection

To explicitly go from the BTZ metric (3.31) to a connection we can work with in the Chern-Simons formulation, we begin by writing the metric in the Feﬀerman- Graham form (4.7), using w = φ + it/` [2] :

ds2 = dρ2 + 8πG` Ldw2 + L˜dw¯2 + `2e2ρ/` + (8πG)2LL˜e−2ρ/` dwdw,¯ (6.1)

2 2 2 2 2ρ e−2ρ 2 2 2 r++r− in which r = ` e + 4`2 (r+ − r−) + 2 . The Virasoro zero modes in Euclidean signature1 can be written as

(r + ir )2 (r − ir )2 L = + − and L˜ = + − (6.2) 32πG` 32πG` 1 We will work in Euclidean signature. This means, as in section (3.5), that the time-cycle t → itE, J → iJE and r− → ir−. In compact vielbein notation with no components, we can i then write the connection as A = ω + ` e. 6. The BTZ Black Hole in Chern-Simons Theory 50 and equivalently in terms of mass and angular momentum as: M` J L + L˜ = and L − L˜ = . (6.3) 8πG 8πG Now, using the deﬁnition gµν = 2Tr(eµeν), (6.4) we can ﬁnd the vielbein components for the metric (6.1):

1 πL −ρ 1 ρ 2 0 0 − k e 0 2 e eρ = 1 , ew = 1 ρ , ew¯ = πL −ρ . (6.5) 0 − 2 − 2 e 0 − k e 0 ¯ 1 ¯ R R And in terms of A, A, e = 2 (A−A). As we we are working in SL(2, )L×SL(2, )R with the explicit basis for the generators as given by (6.6):

1 0 0 0 1 2 0 L1 = ,L−1 = ,L0 = 1 . (6.6) −1 0 0 0 0 − 2

We can see right away eρ = L0, and write the connections A in terms of sl(2, R) [2]:

2πL A = (eρL − e−ρL )dw + L dρ (6.7) 1 k −1 0 2πL˜ A¯ = (eρL − e−ρL )dw¯ − L dρ. (6.8) −1 k 1 0 For large ρ, the BTZ charges L and L¯ are subleading. In terms of connections, we call a connection asymptotically AdS3 if it is like (6.7) for ρ → ∞. The connections that are asymptotically AdS3 can be written in the gauge forms [14]

A = b−1a(w)b + b−1db, A¯ = ba¯(w ¯)b−1 + bdb−1. (6.9) with sl(2, R)-valued b = eρL0 . Up to gauge transformations, we can then deﬁne the BTZ black hole connections we will be working with as

2π a(w) = L − LL dw (6.10) 1 k −1 2π a¯(w ¯) = L − L˜L dw.¯ (6.11) −1 k 1 6. The BTZ Black Hole in Chern-Simons Theory 51

6.2 Holonomies

As a topological theory, the observables of a Chern Simons theory are given by its Wilson loops. A Wilson loop is a gauge-invariant holonomy around the noncontractible cycles of the manifold. Explicitly, the holonomy of A at a point P along the cycle C is given by I P exp( A), (6.12) C The holonomy is an indication of how a connection fails (or succeeds) to be parallel transported along a curve; thus giving an indication of the curvature of the connection. However, our connection is flat, so that if we find a holonomy that is non-trivial, the holonomy comes from the topology. For contractible cycles, the holonomy matrices will be trivial, meaning they will be 1 or in the center of the group G of A. When an element is in the center of a group, it commutes with everything (the center of a group is defined as the set of elements that commute with every other element in the group). If the holonomy is non-trivial, it indicates that topologically there is a non-trivial feature in our theory - such as a singularity. This means the cycle we transport our connection along is non-contractible.

6.3 The BTZ as a Holonomy Around the φ-cycle

We now calculate the holonomy around the φ-cycle. We know it must return a non-trivial holonomy matrix as we know our manifold (a torus) has non-trivial topology in this cycle. Using (6.10) and (6.12): I Hφ = exp a(w), 0 < φ < 2π. (6.13) φ Now, I Z 2π 2π a(w) = (L1 − LL−1)dφ, (6.14) φ 0 k ` and remembering k = 4G , we get for the holonomy:

4π2 0 − L π(r+ + ir−) H = exp k , with eigenvalues λ = ± . (6.15) φ −2π 0 ` ¯ For Hφ, usinga ¯, we have I ¯ Hφ = exp a¯(w ¯), 0 < φ < 2π, (6.16) φ 6. The BTZ Black Hole in Chern-Simons Theory 52 and I Z 2π 2π ˜ a¯(w ¯) = (L−1 − LL1)dφ, (6.17) φ 0 k so we get for the holonomy matrix ¯ 0 2π π(r+ − ir−) Hφ = exp 4π2 ˜ , with eigenvalues λ = ± . (6.18) k L 0 ` These values show us that the holonomy around the φ-cycle is nontrivial; indicating a topological singularity. If we compare the eigenvalues of these matrices to the ones we used in the quotienting the BTZ black hole in section (3.6), we see that † the quotienting matrix g is equivalent to the holonomy Hφ and g is equivalent ¯ to Hφ. The quotienting of the BTZ black hole in the metric formulation is thus equivalent to deﬁning the BTZ black hole as a holonomy within Chern-Simons theory.

6.4 The Holonomy Around the tE-cycle

To ﬁnd the holonomy around the tE-cycle, we use the same connections, so it would seem we get the same holonomy. However, there is an important part we need to be careful not to miss here. First, we see the holonomy matrices return the same values as in the φ-cycle:

I Z 2π 2π HtE = exp a(w) = exp (L1 − LL−1) , (6.19) tE 0 k

` and remembering k = 4G , we get for 0 < tE < 2π the holonomy:

4π2 0 − L π(r+ + ir−) = exp k , with eigenvalues λ = ± . (6.20) −2π 0 ` Fora ¯, we have equivalently I Z 2π ¯ 2π ˜ HtE = exp a¯(w ¯) = exp (L−1 − LL1) , (6.21) tE 0 k so we get for the holonomy 0 2π π(r+ − ir−) H¯ = exp 2 , with eigenvalues λ = ± . (6.22) tE 4π ˜ k L 0 ` However, we know we are in a Euclidean formulation with two cycles, and that the torus topology has only one non-contractible cycle - the one in the φ-cycle. The 6. The BTZ Black Hole in Chern-Simons Theory 53

tE-cycle must therefore be contractible and return a trivial holonomy. Analogous to what we saw in section (3.5), we now integrate not from 0 to 2π, but to 2πβi q q for A, where β = k , and for A¯ from 0 to 2πβi¯ , with β¯ = k . This means 8πL 8πL˜ our holonomy matrices become

2 2 ¯ 0 −4iπ β L 0 −4iπ β L˜ H = exp k and H¯ = k ; (6.23) tE −2πβi 0 tE −2πβi¯ 0 with eigenvalues λ = ±iπ. This leaves our exponential matrices for both A and A¯ equivalent to

−1 0 H = H¯ = . (6.24) tE tE 0 −1 While not 1, the above matrix is in the center of SL(2, R), so it is a trivial holonomy through the periodicity of β and β¯: ` ` β = and β¯ = (6.25) (r+ + ir−) (r+ − ir−) While in the metric formulation we found β through the condition of regularity at the horizon, in the Chern-Simons formulation we found it by the demand that the holonomy around the Euclidean time-cycle must be trivial. We also saw the connection of this periodicity in the Euclidean signature with the inverse temperature of the black hole. If we take a combination of β and β¯ in the Chern-Simons formulation; we again ﬁnd a relation with the inverse temperature that is equivalent to the value of the inverse Hawking temperature found in the metric formulations after going back to Lorentzian signature [29]:

2 −1 ¯ 2π` r+ TH = π`(β + β) = 2 2 . (6.26) r+ − r− Not only the temperature [29] can be deciphered from the β, β¯ periodicities in this formulation, but also the entropy we found in section (4.4), by using instead of the area of the black hole 1 1 πr S = πk( + ) = + . (6.27) β β¯ 2G As we can see, within the Chern-Simons formulation we can ﬁnd our way to black hole thermodynamics, without the use of a metric. The values we found earlier in the metric formulation using black hole thermodynamics are returned to us, using only a connection and (non)-trivial holonomy cycles on a manifold. 7. CONCLUSION AND OUTLOOK

In this thesis we discussed the BTZ black hole from different angles. While three- dimensional gravity is seemingly trivial, we have seen that, at least in AdS3 gravity, this is not the case. It is possible to find a black hole in it as a quotient space: the BTZ black hole. To discuss this black hole, we started with an introduction to gravity and three-dimensional gravity in particular. Specifiying to three-dimensional gravity with a negative cosmological constant, we discussed the symmetries and properties of AdS3. Using these symmetries, we showed different ways to reach the BTZ black hole as a quotient space. First, by doing coordinate transformations in different patches and identifying an angular coordinate, and secondly by an explicit matrix quotient formulation. Both procedures ended in the identification of the φ-coordinate with 2π, resulting in the BTZ black hole metric. Once the BTZ black hole metric was found, we could attribute to it a mass and angular momentum defined through its inner and outer horizons. This was done by defining a stress-energy tensor at the boundary of the spacetime, using the Brown-York procedure [12] and then the correction given by Balasubramanian and Kraus [5]. Once the mass and angular momentum had been identified, the black hole laws of thermodynamics could be checked for the metric, and we could see that the BTZ black hole obeyed these like any other black hole. We also went through the Euclidean formulation of the black hole by doing a Wick rotation, whereupon we could identify the complex time coordinate with a periodicity in- versely proportional to the black hole’s temperature. We also looked at the formulation of three-dimensional gravity within Chern- Simons theory. After a brief introduction to Chern-Simons theory, we saw that the two approaches were equivalent due to their equivalent actions and resulting equations of motion, and then saw how we could define the BTZ black hole within this formulation. Looking at its holonomies, trivial and non-trivial, and its thermodynamics, we found the same black hole in a very different formulation. The holonomy matrices and the identification matrices of the metric formulation turned out to be equivalent. 7. Conclusion and Outlook 55

7.1 Possibilities for Further Research

Had this not been the end of the thesis, but the end of an introduction to another thesis or an article, what would we now introduce? What has been discussed in this thesis was groundbreaking - about twenty years ago. The research into this field has progressed much further however, in different directions. For example, the stress-tensor correction published in the article of Balasubramanian and Kraus [5] was among the first of a lot articles publishing successful results within the AdS/CFT correspondence. We introduced the AdS/CFT correspondence briefly, but we could have also used the BTZ black hole as part of the dictionary between AdS3 and CFT2 and gone into this duality in more depth. This could have also led to a broader description within string theory, whereas we limited ourselves to the BTZ black hole as a topological possibility in AdS3. There is a correspondence between the BTZ black hole and higher-dimensional black holes described within string theory [27], which we had seen in articles such as [29], but we restricted the discussion to the BTZ black hole. We could have also gone from the BTZ black hole into a discussion of higher spin gravity. This extends the gauge group we have been working with, SL(2, R)L× SL(2, R)R, to higher spin-N fields described by SL(N, R)L×SL(N, R)R. This leads to a more general definition of a black hole. Upon extending the gauge group to a higher spin group, the asymptotic symmetry group is also extended, so that the Virasoro algebra becomes a subgroup of the extended asymptotic symmetry group. A brief introduction into the role of asymptotic symmetries will be given below as an example of how this thesis could have been expanded. Another possibility is to look at different possibilities for the BTZ black hole, such as the extremal case for which J = M`, or equivalently r− = r+. This could then be discussed in the Chern-Simons formulation. However, as in the extremal case T = 0, a lot of our definitions do not hold, so further research could be defining the extremal black hole within Chern-Simons theory. After this is done, the extremal black hole formulation could be generalized into higher spin theories which hold no possibilities of a metric formulation.

7.1.1 More General Treatment of the BTZ Black Hole The discussion of the BTZ black hole was at times quite brief or simplified, es- pecially in Chern-Simons formulation. It was possible to do so because the BTZ black hole is a simple example of more general possibilities for gravity within Chern-Simons theory [16]. For example, the BTZ charges L and L˜ are zero-modes of a Virasoro algebra. The BTZ black hole can also be seen as the first excited mode of AdS3, and the Virasoro algebra as the algebra of an asymptotic symmetry group with non-vanishing charges at infinity that physically alter the state of the 7. Conclusion and Outlook 56 system. We will give a brief introduction to the concept of asymptotic symmetries, following [13], as an example of how this thesis could have been continued or developed differently from a more general background as starting point for further research.

Asymptotic Symmetries Besides the topological degrees of freedom we have in Chern-Simons theory, we also have degrees of freedom at infinity. While the global symmetry group of the theory is SL(2, R)L × SL(2, R)R, we can also define an asymptotic symmetry group. We have discussed this concept earlier in section (4.1.1) within the metric formulation. In the Chern-Simons gauge theory formulation, we speak of the same concepts as proper and improper gauge transformations. The global symmetry group of a theory is the group of trivial (“proper”) gauge transformations that do not change the physics of the system. At infinity, they come down to 1 and do not contribute a surface term in the action. There are also transformations one can do that leave the boundary conditions of the action principle intact, but are not trivial at infinity: they approach an element of the asymptotic symmetry group. These are improper gauge transformations, as they actually do change the physics of the system. These asymptotic symmetries form a Lie algebra and are generated by non-vanishing charges. The asymptotic symmetry group is the group of transformations that respect the boundary conditions at infinity but have a non-vanishing surface charge (and thus change the physics of the solution). They form a Lie algebra, of which the theory’s global symmetry group is a subgroup [24]. So we can define the asymptotic symmetry group as the quotient group of all allowed transformations, with the trivial proper gauge transformations quotiented out. To find the asymptotic symmetry group, one must define the boundary conditions and then find those transformations that leave these boundary conditions invariant. These transformations don’t change the theory asymptotically at the boundary (the AdS3 part), but they do change the subleading parts. For AdS3, the asymptotic symmetry group is two copies of the Virasoro algebra [11]: c [L ,L ] = (m − n)L + (m3 − m)δ . (7.1) m n m+n 12 m+n Here, c is the central charge (to be defined below). For the BTZ black hole, ¯ we have only the two zero modes, the constants L0 and L0. However, allowing for higher modes extends the possibilities. The connections A, A¯ can then also depend on these higher modes. In section (6.1) we found the connections of the BTZ black hole directly, but there is a more general way to do it: by using an 7. Conclusion and Outlook 57 action principle and finding the most general connections that obey the asymptotic boundary conditions (allowing for the non-vanishing charges at infinity), while still returning the desired equations of motion, F = F¯ = 0. As shown in [2, 13], the Chern-Simons action can be written in Hamiltonian form: ¯ SH = SH [A] − SH [A], (7.2) which includes a boundary term at infinity that ensures the field variations at infinity and cancels non-vanishing surface terms. The boundary conditions of the action principle must be determined to know the conditions the improper transformations must respect. These are given as Lagrange multipliers that go to 0 for proper gauge transformations and are related to the charge at infinity when they don’t. The boundary term given in the Hamiltonian action must ensure well- defined functional derivatives [31], which can then be expanded in Fourier modes. It can then be found that the Fourier modes obey the Virasoro algebra (7.1), with 3` central charge valued c = 6k = 2G . The BTZ black hole is as such just a small subset of all these possibilities. This approach can also be easily generalized to higher spin algebras; by valuing the connections in sl(n, R) algebras and generalizing the boundary conditions.

7.2 Final Remarks

We have seen an example of one the possibilities that are out there for further research, both for a master’s thesis in the form of a review and as a background for doing one’s own calculations. By looking at the BTZ black hole as part of a larger whole, it can serve as a simple model for other concepts. The range of research within the BTZ black hole can also be expanded, as we mentioned earlier, for example by looking at the extremal BTZ black hole. All in all, it is a very useful concept for a broad area of research, both accomplished already and yet to be done. 8. ACKNOWLEDGEMENTS

First of all, I would like to thank my supervisor, Dr. Alejandra Castro. I was her first master student, and I’m quite sure I will not turn out to have been her easiest one. Once I realized it was probably smarter to say I didn’t understand what was happening in the articles, I learned she’s wonderful at explaining and inspiring enthusiasm. I was usually nervous to go visit but always excited about my topic after I left. Then, I would like to thank my parents, who were always very supportive during my “gemiep” and helped to see it all in perspective, and were excited whenever I was. Also, the study buddies who have made it so much fun to be at Science Park, who made me laugh and let me do endless impressions with only a few accents at my disposal, and only very rarely asking me to please shut up: Sam, Nick, Jaco, Eva, Ludo. Another shout-out to the bètaboys from the bachelor days who went on to do other things but are always superfun; Paul, Tim, Tom, Willem, Thomas. A special thanks to Ido, for his patience when I reached the end of my math understanding, and the clarity of his explanations, and his friendship. Another special thanks to Marcus, who was always ready to explain any random question. He also made a point of ensuring me that they weren’t always dumb questions, and his complete irreverence for almost anyone put my insecurities in a nice perspective. Aagje, for all the fun times and the friendship and the joint frustration over QFT and Group Theory. Last but never least, Anna, because she wasn’t just a constant buddy for eight years and a comfort, but also because she’s so much fun to be around that every coffee/maximally extended lunchbreak/study session with wine was like a baby party in my head. BIBLIOGRAPHY

[1] A. Achucarro and P.K. Townsend. A Chern-Simons Action for Three- Dimensional anti-De Sitter Supergravity Theories. Phys.Lett., B180:89, 1986.

[2] Martin Ammon, Michael Gutperle, Per Kraus, and Eric Perlmutter. Black holes in three dimensional higher spin gravity: A review. J.Phys., A46:214001, 2013.

[3] R. Arnowitt, S. Deser, and C. W. Misner. Dynamical structure and deﬁnition of energy in general relativity. Phys. Rev., 116:1322–1330, Dec 1959.

[4] Richard L. Arnowitt, Stanley Deser, and Charles W. Misner. The Dynamics of general relativity. Gen.Rel.Grav., 40:1997–2027, 2008.

[5] Vijay Balasubramanian and Per Kraus. A Stress tensor for Anti-de Sitter gravity. Commun.Math.Phys., 208:413–428, 1999.

[6] Vijay Balasubramanian, Per Kraus, and Albion E. Lawrence. Bulk versus boundary dynamics in anti-de Sitter space-time. Phys.Rev., D59:046003, 1999.

[7] Maximo Banados, Marc Henneaux, Claudio Teitelboim, and Jorge Zanelli. Geometry of the (2+1) black hole. Phys.Rev., D48:1506–1525, 1993.

[8] Maximo Banados, Claudio Teitelboim, and Jorge Zanelli. The Black hole in three-dimensional space-time. Phys.Rev.Lett., 69:1849–1851, 1992.

[9] J.M. Bardeen, B. Carter, and S.W. Hawking. The four laws of black hole mechanics. Commun.Math.Phys., 31(2):161–170, 1973.

[10] Jens Boons. Explicit check of the validity of the btz solution without torsion. Master’s thesis, University of Cologne, 2014.

[11] J. D. Brown and Marc Henneaux. Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity. Communications in Mathematical Physics, 104(2):207–226, 1986. Bibliography 60

[12] J. David Brown and Jr. York, James W. Quasilocal energy and conserved charges derived from the gravitational action. Phys.Rev., D47:1407–1419, 1993.

[13] Claudio Bunster, Marc Henneaux, Alfredo Perez, David Tempo, and Ricardo Troncoso. Generalized Black Holes in Three-dimensional Spacetime. JHEP, 1405:031, 2014.

[14] Andrea Campoleoni, Stefan Fredenhagen, Stefan Pfenninger, and Stefan Theisen. Asymptotic symmetries of three-dimensional gravity coupled to higher-spin ﬁelds. JHEP, 1011:007, 2010.

[15] Steven Carlip. The (2+1)-Dimensional black hole. Class.Quant.Grav., 12:2853–2880, 1995.

[16] Steven Carlip. Conformal ﬁeld theory, (2+1)-dimensional gravity, and the BTZ black hole. Class.Quant.Grav., 22:R85–R124, 2005.

[17] Steven Carlip. Quantum gravity in 2+1 dimensions: The Case of a closed universe. Living Rev.Rel., 8:1, 2005.

[18] Steven Carlip and Claudio Teitelboim. Aspects of black hole quantum mechanics and thermodynamics in (2+1)-dimensions. Phys.Rev., D51:622–631, 1995.

[19] Sean M Carroll. Object Oriented Perl: A spacetime. Manning Publications Co., Connecticut, USA, 2000.

[20] Alejandra Castro and Alexander Maloney. The Wave Function of Quantum de Sitter. JHEP, 1211:096, 2012.

[21] G. de Berredo-Peixoto and M.O. Katanaev. Inside the BTZ black hole. Phys.Rev., D75:024004, 2007.

[22] G. W. Gibbons and S. W. Hawking. Action integrals and partition functions in quantum gravity. Phys. Rev. D, 15:2752–2756, May 1977.

[23] S.W. Hawking. Black hole explosions. Nature, 248:30–31, 1974.

[24] Kristian Holsheimer. Surface charges, holographic renormalization and lifshitz spacetime. Master’s thesis, University of Amsterdam, 2009.

[25] H.F. Jones. Groups, Represenations and Physics. Taylor and Francis, second edition, Abingdon, UK, 1998. Bibliography 61

[26] Per Kraus. Lectures on black holes and the AdS(3) / CFT(2) correspondence. Lect.Notes Phys., 755:193–247, 2008.

[27] Taejin Lee. The Entropy of the BTZ black hole and AdS / CFT correspondence. 1998.

[28] Juan Martin Maldacena. The Large N limit of superconformal ﬁeld theories and supergravity. Adv.Theor.Math.Phys., 2:231–252, 1998.

[29] Juan Martin Maldacena and Andrew Strominger. AdS(3) black holes and a stringy exclusion principle. JHEP, 9812:005, 1998.

[30] Eric Poisson. A Relativist’s Toolkit. Cambridge University Press, Cambridge, UK, 2004.

[31] Tullio Regge and Claudio Teitelboim. Role of Surface Integrals in the Hamil- tonian Formulation of General Relativity. Annals Phys., 88:286, 1974.

[32] Hai-Siong Tan. Aspects of Three-dimensional Spin-4 Gravity. JHEP, 1202:035, 2012.

[33] Balt van Rees. Wormholes in 2+1 dimensions. Master’s thesis, University of Amsterdam, 2006.

[34] Steven Weinberg. Gravitation and Cosmology. John Wiley and Sons, Inc, Somerset, NJ, 1972.

[35] Bastian Wemmenhove. Quantization of 2+1 dimensional gravity as a chern- simons theory. Master’s thesis, University of Amsterdam, 2002.

[36] Edward Witten. 2 + 1 dimensional gravity as an exactly soluble system. Nuclear Physics B, 311(1):46 – 78, 1988.