Applications of Conformal Methods to Relativistic Trace-Free Matter Models

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Applications of Conformal Methods to Relativistic Trace-Free Matter Models School of Mathematical Sciences Queen Mary, University of London Applications of Conformal Methods to Relativistic Trace-free Matter Models Adem E. Hursit Supervised by Dr. Juan A. Valiente Kroon Submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy Declaration I, Adem E. Hursit, confirm that the research included within this thesis is my own work or that where it has been carried out in collaboration with, or supported by others, that this is duly acknowledged below and my contribution indicated. Previ- ously published material is also acknowledged below. I attest that I have exercised reasonable care to ensure that the work is original, and does not to the best of my knowledge break any UK law, infringe any third partys copyright or other Intellectual Property Right, or contain any confidential material. I accept that the College has the right to use plagiarism detection software to check the electronic version of the thesis. I confirm that this thesis has not been previously submitted for the award of a degree by this or any other university. The copyright of this thesis rests with the author and no quotation from it or infor- mation derived from it may be published without the prior written consent of the author. Signature: Date: Details of collaboration and publications The content of this thesis is based in the following papers: AE Hursit and JA Valiente-Kroon, Conformal wave equations for the Einstein- tracefree matter system. In preparation Adem E. Hursit 2 3 Abstract Conformal methods have proven to be very useful in the analysis global properties and stability of vacuum spacetimes in general relativity. These methods transform the physical spacetime into a different Lorentzian manifold known as the unphysical spacetime where the ideal points at infinity are located at a finite position. This the- sis makes use of conformal methods and applies them to various problems involving trace-free matter models. In particular, it makes progress towards the understanding of the evolution of unphysical spacetimes perturbed by trace-free matter as well as the behaviour of the the matter itself. To this end, evolution equations (wave equa- tions) are derived and analyzed for both the unphysical spacetime and the matter. To investigate the relation between solutions of these wave equations to the Einstein field equations, a suitable system of subsidiary evolution equations is also derived. Furthermore, this thesis looks in detail at the behaviour of an unphysical spacetime coupled to the simplest matter trace free model: the confomally invariant scalar field. Finally, the system of conformal wave equations is used to show that the deSitter spacetime is non-linearly stable under perturbations by trace-free matter. 4 Acknowledgements I would like to take the time to thank a few individuals for their help and support. First and foremost, I would like to thank my supervisor, Juan Antonio Valiente- Kroon, to whom I am greatly indebted to. His limitless patience, clear exposition and willingness to advise on a multitude of topics, even outside of the academic content of this thesis have been invaluable to me. In advance, I would like to thank my examiners: Christian Boehmer and Peter Hogan, for taking the time to read my thesis. I would also like to thank the members of the Queen Mary General Relativity research group for their friendship and support. I consider it a privilege to have worked among such a great research group and I'm proud to call them friends. Finally, an extra special thanks to my family, who have supported me in more ways than one, more that I could possibly express in words. In particular, special thanks goes out to my parents, not only for the various forms of support even during times when I didn't deserve it, but more than anything, for believing in me and never giving up on me. Contents 1 Introduction7 1.1 General Relativity and the Einstein equations . .7 1.2 The initial value problem in GR . 11 1.3 Outlook of the thesis . 15 2 Mathematical prerequisites of GR 17 2.1 Differential Geometry, Manifolds and Tensors . 17 2.2 Connections and curvature . 21 2.3 Bianchi Identities . 27 2.4 Conformal transformations of connections . 31 3 The Conformal Einstein Field Equations 33 3.1 Derivation of the Conformal Field Equations with Matter . 33 4 The Conformal Wave Equations 42 4.1 Auxiliary results . 42 4.2 Derivation of the conformal wave equations . 45 4.3 The subsidiary equations . 50 5 Analysis of Trace-Free Matter Models 64 5.1 Conformally coupled scalar field . 64 5.2 Electromagnetic field . 67 5.3 Yang-Mills fields . 71 5.4 Perfect Fluids . 80 6 The Conformal Scalar Field System 92 6.1 Warped product metrics . 92 6.2 Computation of the geometric quantities for warped product metrics . 93 6.3 Evolution equations for the scalar field system . 97 6.4 Conformal geodesics . 106 7 Initial Data for the Conformal Wave Equations 111 7.1 The 3+1 Decomposition in General Relativity . 111 7.2 Decomposition of the Conformal Field Equations . 115 7.3 Construction of initial data for the conformal evolution equations . 125 5 Contents 6 8 Stability of the de Sitter spacetime under trace-free matter per- turbations 128 8.1 Stability - basic ideas . 128 8.2 The de Sitter background solution . 131 9 Conclusions 135 Bibliography 137 1 Introduction 1.1 General Relativity and the Einstein equations The theory of General Relativity (GR), first developed in 1915 by Einstein, is one of the greatest achievements in modern physics. In essence, it postulates that gravity is not a force, but is the result of spacetime curvature [5]. A spacetime is a pair (M; gab), where gab is the metric and M is a 4-dimensional manifold. A manifold is an object which on a small, local scale looks different than on a global scale. An example of this is a sphere; on a small enough scale of distance a sphere looks like a flat plane, but if you look at it from a large enough scale of distance you find that this is not the case. It is for this reason that the Earth appears flat from the point of view of anyone standing on its surface. The metric is an object that describes how things are measured on the manifold; together these two objects form a spacetime. The universe that we live in is a 4-dimensional spacetime, since we need four components to describe the behaviour of objects living in the spacetime (three spatial coordinates and one time coordinate). This 4-dimensional spacetime becomes curved in the presence of matter. This curvature is the very source of what we perceive as the force of gravity. The metric is the main object of interest in GR since, as mentioned beforehand, it describes the measurement of distances on a manifold, however the measurement of distances on the manifold can be used to quantify curvature. This means that the metric can be thought of as the relativistic analogue of the gravitational potential, as in classical mechanics one obtains a force by differentiating the necessary potential; likewise in GR, one obtains the curvature (analogous to the gravitational tidal forces) by differentiating the metric. Precisely how the curvature of spacetime is related to the presence of matter is given by the Einstein field equations (EFEs) 1 8πG R − g R + λg = T ; (1.1) ab 2 ab ab c4 ab where Rab is the Ricci curvature tensor, which describes the curvature of the space- time, R is the Ricci scalar curvature that is derived from Rab, λ is the cosmological constant and Tab is the energy-momentum tensor (also known as the stress-energy tensor), whose individual components describe the matter fields. Hence, the left hand side of the EFEs completely describes the curvature and the right hand side describes the distribution of matter. One can obtain the equations that describe the 7 1.1: General Relativity and the Einstein equations 8 vacuum of space simply by setting Tab in (1.1) to zero and then taking the trace of the resulting equation, which gives Rab = λgab: (1.2) Despite their relatively simple appearance, the EFEs are extremely complicated and very difficult to solve, so much that Einstein himself predicted that no one would ever be able to find a solution to the equations. However, less than a year after the EFEs were introduced to the scientific community, Karl Schwarzchild successfully derived the simplest solution of the EFEs [32]. Since then a large number of exact solutions have been found; much research has been devoted to the understanding of these exact solutions and how they can be applied to various problems in physics. 1.1.1 Conformal methods in GR Despite the success of GR in describing gravity and the nature of the universe, there are noticeable limitations to the theory when one wants to extract useful information. Two major problems were apparent when the theory was still considered new. The first problem was related to the uniqueness of solutions. One of the main guiding points that was used when developing the theory was that it should be invariant under coordinate transformations. Physically, this means that any law in nature should not depend on a particular choice of coordinates, since physical quantities do not change based on our point of view. GR is indeed invariant under a coordinate transformation, as a result it becomes possible to work in any system of coordinates and produce correct results. However, this strength of the theory is at the same time one of its biggest weaknesses.
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