MSc Physics
Theoretical Physics
Master Thesis
The BTZ Black Hole
In Metric and Chern-Simons Formulation
by
Eline van der Mast
0588105
July 2014
54 ECTS
02/2013 - 07/2014
Supervisor:
A. Castro, Dr.
Examiner:
J. de Boer, Prof.
In this thesis, we review the BTZ black hole within the metric formulation of gen-eral 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 thermo-dynamics are the same.
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
4.4 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
Gravity is described by Einstein’s theory of general relativity. Differential geom-etry 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 con-stant, 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 con-tain a black hole solution. This black hole, called the BTZ black hole (after its discoverers Ba˜nados, 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 exclu-sion 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 quo-tient 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 col-lapse of a massive star in the spacetime, it still turns out to behave like a real,
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 introduc-tion to the formulaintroduc-tion 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-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 corre-spondence, 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.
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
Rµν−
1
2gµνR + gµνΛ = 8πG
c4 Tµν, (2.1)
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.
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
ds2 = gµνdxµ⊗ dxν, (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
Rρσµν = ∂µΓρνσ − ∂νΓρµσ+ Γ ρ µλΓ λ νσ− Γ ρ νλΓ λ µσ. (2.3)
Γ is the Christoffel connection, which is defined using the metric tensor, to be Γσµν = 1
2g
σρ(∂
µgνρ+ ∂νgρµ− ∂ρgµν). (2.4)
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: Rµν ≡ Rλµλν = g
µ ρR
ρ
and the trace of the Ricci tensor is known as the Ricci scalar:
R ≡ Rλλ. (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 semi-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.
As a symmetric matrix has 12d(d + 1) independent components and an an-tisymmetric matrix has 1
2d(d − 1) independent components, we get a total of 1
2 1
2d(d − 1)
1
2d(d − 1) + 1 independent components. Lastly, the Riemann
ten-sor 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 sim-ply subtract the independent components of a fully antisymmetric tensor with four indices, which is 4!1d(d − 1)(d − 2)(d − 3). We thus end up with
1 12d
2(d2− 1) (2.8)
degrees of freedom in d dimensions. The Ricci tensor has d(d+1)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:
Cρµρν = 0,
which gives us 12d(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 12d
2(d2− 1) − 1
2d(d + 1)
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
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
Rµνρσ =
R
6(gµρgνσ − gνρgµσ). (2.10) 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 free-dom in dimensional gravity. The expression (2.10) is the same as the three-dimensional version of the more general
Rµνρσ =
R
n(n − 1)(gµρgνσ − gνρgµσ),
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.
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 topologi-cally, 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 sec-ond 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 cos-mological constant is a way of defining the “vacuum energy” of spacetime; or the
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
embed-ding in four-dimensional flat space with a (2,2) signature through the embedembed-ding 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)
v = ` cosh µ cos λ, and ` sinh µ =px2+ 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(− cosh2µdλ2 + dµ2+ sinh2µdθ2). (3.4) λ is now the timelike coordinate (it has negative signature), but due to the trans-formation (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
λ = t
` (3.5)
r = ` sinh µ.
Doing this, we get what is called the universal covering space of AdS3:
ds2 = −(r 2 `2 + 1)dt 2+ 1 r2 `2 + 1 dr2+ r2dθ2. (3.6) 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 re-ferred to as Anti-De Sitter space. The coordinates used in it are global coordinates and cover the whole manifold.
Poincar´e Coordinates
Another coordinate system often used are Poincar´e coordinates; or hyperbolic coordinates, denoted H3. We get to these from the embedded metric using
z = ` u + x, β = y u + x, γ = − v u + x,
which covers the upper-half plane chart (hence often referred to as the Poincar´e patch) with line element
ds2 = `
2
z2 dz
2+ dβ2− dγ2 . (3.7)
3.2
Symmetries of AdS
3AdS3 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 c d
, with ad − bc = 1. (3.8) The group action is given by
x → ax + b
cx + d. (3.9)
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
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]):
R(φ) = 1 − iφξ + O(φ2); or − iξ = dR(φ)
dφ |φ=0. (3.10) The representation matrix R is then the exponent of the generator:
R(φ) = e−iφξ. (3.11) The Killing vectors of AdS3are 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]) L1 = 0 0 −1 0 , L−1 = 0 1 0 0 , L0 = 1 2 0 0 −12 . (3.12) 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] Jµν = xν ∂ ∂xµ − xµ ∂ ∂xν. (3.14)
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)
L0 = 1 2(Jux+ Jvy) L−1 = 1 2(Jxy − Juv− Jxv+ Jyu) L1 = 1 2(Jxy − Juv+ Jxv − Jyu) (3.16) for the generators of SL(2, R)L, and
˜ L0 = 1 2(Jux− Jvy) ˜ L−1= 1 2(Juv+ Jxy − Jxv − Jyu) ˜ L1 = 1 2(Jxv + Jyu+ Juv+ Jxy). (3.17) 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 AdS
3So 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 Ba˜nados,
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
com-bination of a Killing vector is itself a Killing vector): ξ = r+
` Jux− r−
` Jvy, (3.20)
in which the r± can be seen as parametrizing constants for now. We will attribute
must be larger than 0 to exclude closed timelike lines in the quotient space. We thus find that
ξ · ξ = r 2 + `2(u 2− x2) + r 2 − `2(v 2− y2) = r 2 +− r2− `2 (u 2− x2 ) + r−2, which means that the region − r2−`2
r2
+−r2− < u
2 − x2 < 0 is where ξ · ξ > 0. We can
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: I. u2− x2 > `2, with r2 −< ξ · ξ < ∞ II. 0 < u2− x2 < l2, with r2 − < ξ · ξ < r+2 III. − r 2 −`2 r2−+r2 − < u 2− x2 < 0, with 0 < ξ · ξ < r 2
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)
Where A(r) = `2 r 2− r2 − r2 +− r−2 , B(r) = `2 r 2− r2 + r2 +− r2− τ = 1 ` r+t ` − r−φ , χ = 1 ` −r−t ` + r+φ . On each of the three patches, this brings us to the metric
ds2 = −(r 2− r2 +)(r2− r2−) `2r2 dt 2+ `2r2dr2 (r2− r2 +)(r2− r2−) dr2+ r2dφ − r+r− `r2 dt 2 . (3.21) Here, 0 < r < ∞, −∞ < t < ∞, and −∞ < φ < ∞.
We now have a spacetime metric that takes the form
−N2(r)dt2+ N−2(r) + r2(dφ + Nφ(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: ξ = r+ ` ( √ A sinh χ∂u+ √ A cosh χ∂x) − r− ` ( √ B sinh τ ∂y+ √ B cosh τ ∂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 transfor-mations 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 invari-ants: scalars that are coordinate-independent. They show if there are places in the
spacetime that have a true singularity in the curvature. One of these invariants is the Kretschmann scalar:
K = RµνρσRµνρσ. (3.24)
For example, for the Schwarzschild black hole (the most simple four-dimensional black hole), with the line element
ds2 = 1 − rs r dt2− dr 2 1 − rs r − r 2 (dθ2+ sin2θdφ2), (3.25) the Kretschmann scalar is given by K = 48Gc42rM6 2. 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] , K = 12
`4, (3.26)
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)
We then get the Euclidean black hole metric ds2E = (r 2− r2 +)(r2+ r2−) `2r2 dt 2+ `2r2 (r2− r2 +)(r2+ r−2) dr2+ r2(dφ + ir+(r−) `r2 dt−) 2. (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 t0E = r+tE+ r−φ, φ0 = r+φ − r−tE r02= r2− r2 + r2 ++ r2− . (3.32) Which leads us to the metric
ds2E = r 02 `2dt 02 E + `2dr02 1 + r02 + (1 + r 02 )dφ02. (3.33) If we now take the near-horizon limit, r0 → 0, or r → r+, the non-angular part
looks like a plane:
r02dt02E + dr02.
It is non-singular provided we identify t0E as an angle. So, we identify t0E ∼ t0 E+ 2π.
Going back to our original coordinates, this means
∆t0E = 2π = r+∆tE + r−∆φ and ∆φ0 = 0 = r+∆φ − r−∆tE,
leading to the identification of our original coordinates: (tE, r, φ) ∼ tE+ 2π`2r + r2 ++ r2− , r, φ + 2π`r− r2 ++ r−2 . (3.34)
So upon imposing smoothness at the horizon, the Euclidean time tE has a
period-icity we will call β, and a periodperiod-icity in the φ-direction we call Φ:1 Φ = 2π` 2r − r2 ++ r−2 , β = 2π`r+ r2 ++ r2− . (3.35)
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).
1It is important to note that when working in Lorentzian formulation, the periodicities Φ and
β are ∝ r21 +−r2−
Fig. 3.2: The Euclidean BTZ black hole has a solid torus topology, with a non-contractible cycle in the φ-direction, and a non-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´e coordinates, the metric of BTZ is
`2 dz2(dz
2+ dwd ¯w). (3.36)
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) M =z + w ¯w z w z ¯ w z 1 z , (3.37)
where the metric is given by ds2 = `
2
2T r(M
−1
dM M−1dM ), (3.38) which returns (3.36). Taking a quotient space in this formulation now explicitly means that
M = gM g† (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
g = e π(r++ir−) ` 0 0 e−π(r++ir−)` , ! ; g†= e π(r+−ir−) ` 0 0 e−π(r+−ir−)` ! . (3.40) The eigenvalues of these matrices are
λ = ±π(r++ ir−)
` (3.41)
for g, and for g† we find
λ(†)= ±π(r+− ir−)
` . (3.42)
The coordinates (w, z) are given explicitly by z = r 2 +− r2− r2− r2 − 12 exp r+ ` φ + ir− ` tE (3.43) w = r 2− r2 + r2− r2 − 12 exp r++ r− ` (φ + itE) , (3.44)
so that our matrix M becomes
M = r r2 −+r2+ r2+r2 − + r2−r2 + √ r2 −+r2+ er+φ+r−tE` r r2−r2 + r2 −+r+2 ei(−r−φ+r+tE )` r r2−r2 + r2 −+r2+e i(r−φ−r+tE )` r r2+r2 + r2 −+r2+e −(r+φ+r−tE )` . (3.45)
If we now do (3.39), in the (t, r, φ) coordinates, we find
gM g†= r r2 −+r+2 r2+r2 − + r2−r2 + √ r2 −+r+2 er+(φ+2π)+r−tE` r r2−r2 + r2 −+r2+e −ir−(φ+2π)+r+tE ) ` r r2−r2 + r2 −+r2+ eir−(φ+2π)−r+tE )` r r2+r2 + r2 −+r+2 e−(r+(φ+2π)+r−tE )` (3.46) so we can see that taking the quotient gM g† = 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): ξφ= r+ ` Jux− ir− ` Jvy = r+ ` (L0+ ˜L0) − ir− ` (L0− ˜L0). (3.47)
This leads to ξ = r+− ir− ` L0+ r++ ir− ` ˜ L0. (3.48)
In terms of the generators of SL(2, R), L1 = 0 0 −1 0 , L−1 = 0 1 0 0 , L0 = 1 2 0 0 −12 , (3.49) in which representation L0 = ˜L0, we can write our identification matrices as
g = exp π(r−− ir−) ` L0 ; g† = exp π(r−+ ir−) ` L0 . (3.50) 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).
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 loc-ally -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
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 spacelike hypersurfaces of constant time Σt.1 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 every-where 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
1In this formulation, three-dimensional gravity is often referred to as (2+1)-dimensional
grav-ity 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.
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
ac-tion is varied using the metric. As the metric carries more informaac-tion 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 Nµ respectively, connect these spacelike surfaces. A metric in
this decomposition takes the following general form:
ds2 = −N2dt2+ gij(dxµ+ Nµdr)(dxν + Nνdr) (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 defined by 2B and the spacelike slices t
1 and t2. The boundary metric is the
metric on2B, 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.
4.1.4 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
P |t2 =
∂Scl
∂x2
, (4.2)
and the Hamiltonian as
H|t2 = −
∂Scl
∂t2
. (4.3)
Analogously to these equations, in a foliated spacetime with a boundary metric γµν, and a hypersurface Σtmetric gij we can define the components of the angular
momentum of the spacetime as
Pij|t2 =
δScl
δgij
. (4.4)
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:
Tµν = √2 −γ
δScl
δγµν
. (4.5)
The above expression holds close similarity to the stress-energy tensor in the Ein-stein equation, TEinsteinµν = √2 −g δSm δgµν .
It is important to note that even though Tµν 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.
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 AdS
3To 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
asymptot-ically 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
ds2 = dρ2+ gij(xk, ρ)dxidxj, i, j = 1, 2; (4.6)
in which we can write gij in the Fefferman-Graham expansion [2]:
gij = e2ρ/`g (0) ij (x µ) + g(2) ij (x µ) + g(4) ij (x k) + ... (4.7)
This expansion indicates what happens to the metric asymptotically - by which we mean for large ρ; which is the radial coordinate in this case - defined dρ2 = dr2
relation to the radial coordinate r we use. So, when we talk about an asymptotic spacetime, we mean as r → ∞. The leading part of the metric is dρ2+e2ρ/`gij(0)(xµ), and the subleading parts the terms gij(2) 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 constant Λ = −`12, S = − 1 16πG Z M dd+1x√g(R − 2 `2). (4.8)
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]:
ds2 = N2dr2+ γµν(dxµ+ Nµdr)(dxν+ Nνdr). (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 to ˆnr, 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
γµν ≡ gαβeαµe β
ν, Θµν ≡ ∇βnαeαµe β
with eα µ =
∂xα
∂yµ, where xα are the coordinates on the spacetime and yµ 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 γµνΘµν = Θ.
We can, for simplicity, write
Θµν =
1
2∂rγµν. (4.11) The reason we can do the above for the extrinsic curvature is that [30] it is sym-metric: Θµν = Θνµ, (4.12) so we can rewrite Θµν = 1 2(∇νnµ+ ∇µnν) = 1 2(Lrγµν), (4.13) 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:
R =2 R + Θ2− ΘµνΘ
µν+ 2∇ρ(∇σnρnσ− nρ∇σnσ). (4.14) 2R is the Ricci scalar in two dimensions:
2
R = γµνRλµλν. (4.15) Now, using Gauss’ theorem,
Z V ∇αAα √ −gd4x = I ∂V AαdΣα (4.16)
and √−g = nµ√γ, we can write the last term in the decomposed Ricci scalar as
−2 I ∂V (∇µnνnµ− nν∇µnµdΣρ) = −2 I ∂V ∇µnµ √ γd2x = −2 I ∂V d2x√γ Θ. (4.17) This way we get the following gravitational (Einstein-Hilbert) action:
SEH = 1 16πG Z d2xdr√γ R(2)+ Θ2− ΘµνΘ µν− 2Λ − 1 8πG Z ∂M d2x√γ Θ. (4.18)
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:
SGH = 1 8πG Z ∂M d2x√γΘ. (4.19) After adding this term, we get for the action
SEH+GH = 1 16πG Z M d2xdη√γ 2R + Θ2− ΘµνΘ µν− 2Λ . (4.20)
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 δS = − 1 16πG Z ∂M d2x√γ (Θµν− Θγµν) δγ µν. (4.21)
Using (4.5), we find for the stress tensor thus far Tµν = 1
8πG(Θ
µν− Θγµν) . (4.22)
4.2.2 Finding a Finite Stress Tensor
Now, this stress tensor is undefined as r → ∞. In [5] a method is used in which Tµν is seen as a conformal field theory on the boundary of AdS
3. 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 AdS3, with Sct =
R
∂MrLct, they have found
Lct = −
1 `
√ −γ.
This means that when we add this term to the action, we get a stress tensor that looks like Tµν = 1 8πG Θµν− Θγµν− 1 `γ µν . (4.23)
ds2 = ` 2 r2dr 2 +r 2 `2(−dt 2+ dx2), (4.24)
where the metric at the boundary is given by −γtt = γxx = r
2
`2 and the radial normal
vector is ˆnµ= r`δµ,r, we get the following values for the stress tensor components: 8πGTtt = − r2 `3 − 1 `γ tt 8πGTxx = r2 `3 − 1 `γ xx 8πGTtx= − 1 `γ tx.
For AdS3, the added term in the action cancels the divergences, and actually yields
Tµν = 0. Thus, because of the added term −1`γµν, 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
de-scribed above can be used to find the quasilocal stress tensor. To do this, the BTZ metric for large r is used:
−r2+ r2 ++ r2− `2 dt2+ r 2+ r2 ++ r2− r4 `2dr2 −r+r− ` dtdφ + r2 `2dφ 2, (4.25)
with which the stress tensor components Ttt and Ttφ can be found [5]:
Ttt = r2 ++ r2− 2π`3 (4.26) Ttφ = − r+r− π`2 . (4.27)
Then we integrate setting x → `φ, over the interval 0 to 2π, for the mass and angular momentum: M = Z 2π 0 r2++ r2− 2π`3 `dφ (4.28) J = Z 2π 0 −r+r− π`2 `dφ. (4.29)
Basically, we see that using this technique, we find M = M and Pφ = J . Depending
and the Poincar´e patch 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):
ds2 = −(r 2− r2 +)(r2− r−2) `2r2 dt 2+ ` 2r2dr2 (r2− r2 +)(r2− r−2) + r2dφ − r+r− `r2 dt 2 , in which we now know M = r2++r2−
8G`2 and J =
2r+r−
4G` .
We can write our metric in terms of the mass and angular momentum specifically [8]: 8GM − r 2 `2 − 16G2J2 r2 dt2+ dr 2 −8GM + r2 `2 + 16G2J2 r2 + r 2 dφ − 4GJ r2 dt 2 . (4.30) With this metric, we can calculate the curvature tensors and Christoffel connec-tions using the definiconnec-tions (2.3), (2.4), (2.5), and (2.6).
Rtt = 2(−M `2+ r2) `4 , Rrr = − 8r2 4r4+ `2(J2− 4M r2), Rφt = J `2, Rφφ = − 2r2 `2 , (4.31) out of which we can find the Ricci scalar by contracting:
R = −6
`2. (4.32)
As the Ricci scalar is dependent only on the cosmological constant (Λ = −`12) we
4.3.2 Event Horizons Using Killing Vectors
To calculate certain properties of the black hole, we need to know what its sym-metries, 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 in this way [7, 21]. We write their components as χµ= (∂α)µ= δµα, 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 field: χµ = 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
χµχµ= gµνχνχµ= 0. (4.33)
We know that the Killing vector is timelike before it hits the horizon; so we have the extra condition that
χµχµ < 0 for r → ∞. (4.34)
Using the form (4.30) of the metric, we get for the norm χµχµ = −(−8GM +
r2 `2)A
2− 8GJAB + r2
B2. (4.35) To satisfy the condition (4.34) we must demand that A`
> |B|. We are free to set A = 1 for simplicity as there are no further restrictions on it, and thus we end up with
χµχµ= −(−8GM +
r2
`2) − 8GJ B + r
2B2. (4.36)
The value for B is then found by setting the norm to zero as above. In terms of metric coefficients, we then get
B± = −gtφ± q g2 tφ− gφφgtt gφφ r± . (4.37)
To find the event horizons, we look for which values of r grr has a pole. This is when −8GM + r 2 `2 + 16G2J2 r2 = 0,
so the horizons in terms of M and J are
r± = ` v u u t8GM ± 8G q M2− J2 `2 2 . (4.38)
Now we can plug in these values in the expression for B we found earlier, which gives B+ = 4GJ r2 + = r− `r+ , B− = 4GJ r2 − . (4.39)
When plugging in either of the values for r±, the part
q g2
tφ− gφφgtt disappears,
so that we end up with the equivalence of B±= − gtφ gφφ r± .
By analogy, the angular momentum associated to the Kerr metric is Ω = dφdt = −gtφ
gφφ. We rename B± = Ω±. We end up with the Killing vector field
χµ= ∂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 section (3.5) but in Lorentzian signature so r2
− → −r2−; β = 2π` 2r + r+2 − r2 − and Φ = 2π`r− r+2 − r2 − ,
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
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 con-stant 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]:
1. (−χβχβ);α = 2κχα r+, 2. χ α ;βχβ = κχα r+, 3. κ 2 = −1 2χ α;βχ α;β r+. (4.41)
Using any of these gives the same value for the BTZ black hole’s surface gravity: κ = 8G q −J2 l2 + M2 r+ = r 2 +− r2− `2r + (4.42) The relationship between a black hole’s surface gravity and its temperature (known as the Hawking temperature), is
TH = κ 2π = r+2 − r2 − 2π`2r + (4.43) 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 dM = κ
8πdA + Ω+dJ. (4.44) A is the area of the event horizon r+. To find it, we use a covariant surface integral:
A = Z
p|g|dx. (4.45)
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.
A = Z 2π 0 q r2 +dφ = 2πr+. (4.46)
Now we can fill in for (4.44): A = 2πr+ = 2πl
r 8GM +8G q M2−J 2 l2 2 into dA = dA dMdM + dA
dJdJ . After some manipulations, we end up with
1 π G q M2− J2 l2 r+ dA = dM − J 2r2 + dJ ; 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:
SBH =
A
4G (4.47)
Specifically for the BTZ black hole, this gives for its entropy SBT Z =
r+π
2G. (4.48)
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
of a field, we see [22] that the amplitude of moving from configuration φ1(t1) to φ2(t2) is given by hφ2, t2|φ1, t1i = Z 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
Tr exp(−βH) = Z
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
β−1 = κ 2π = TH = r2 +− r2− 2π`2r + . (4.52)
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 SCS[A] = k 4π Z Tr(A ∧ dA + 2 3A ∧ A ∧ A), (5.1)
