University of Amsterdam 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: 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 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. 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 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, 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 introduc- tion 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 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. 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.