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On Cosmologies with Torsion Nikolaos Chatzarakis On Cosmologies with Torsion Nikolaos Chatzarakis Register Number 13603 B.Sc. Thesis Supervisor: Christos G. Tsagas Department of Physics Aristotle University of Thessaloniki Greece Friday 19 February 2016 On the Cosmology of the Einstein-Cartan Universe Nikolaos Chatzarakis Abstract In this thesis, we will discuss the construction of a cosmological theory in a Riemann-Cartan space-time. This theory presents similarities to the relativistic theory of cosmology, and thus we will present the two theories at the same time, comparing the results. We begin the analysis by introducing the reader to the geometry of a Riemann-Cartan space-time, where torsion is non-zero and appears as the antisymmetric part of the affine connection; this causes several interesting changes to the classical geometric features of the Riemann geometry, most notably in the curves of zero \acceleration" and the form of the curvature tensor and its components. From this geometrical background, we are introduced to the field equations of the Einstein-Cartan theory and we focus on the differences of the \matter-to-geometry" coupling between this theory and the General Relativity; one of the most noticeable results is the immediate coupling of spin to torsion. We also take a look into the conservation laws in this theory of gravity, since they appear to take different forms from their usual relativistic ones. Then, we continue with the kinematics of this theory, presenting the most interesting changes that torsion causes in the kinematic equations (e.g. the Rauchaudhuri equation). Finally, we construct a homogeneous and isotropic cosmological model that bears all the symmetries of the relativistic Friedmann model; this model has the Robertson-Walker metric, but also a scalar fields related to torsion. 1 2 Contents 1 Preface 5 2 Introduction to the Einstein-Cartan Theory 11 2.1 The Structure of a Riemann-Cartan Space-time . 11 2.1.1 The Metric Tensor . 11 2.1.2 The Affine Connection . 12 2.1.3 The Torsion and Contorsion Tensors . 13 2.2 The Effects of Torsion . 15 2.2.1 The Parallel Transport and the Unclosed Parallelogram . 15 2.2.2 The Co-moving Transport and the Ricci Identity . 16 2.2.3 Autoparallel and Geodesic Equations . 18 2.3 Curvature with Torsion . 19 2.3.1 The Curvature Tensor . 19 2.3.2 The Ricci Curvature . 21 2.3.3 The Weyl Curvature . 22 2.3.4 The Weitzenb¨ock Identities . 24 3 The Field Equations of the Einstein-Cartan Theory 27 3.1 The Einstein-Hilbert Action . 27 3.2 The Field Equations . 28 3.2.1 The Variation in respect to the Metric . 29 3.2.2 The Variation in respect to the Contorsion . 29 3.2.3 The Algebraic Form of the Field Equations . 30 3.2.4 The Curvature Tensor as a Field Equation . 31 3.2.5 The Effective Field Equations . 33 3.3 Matter Fields . 34 3.3.1 The Matter Lagrangian . 34 3.3.2 The Energy-Momentum density tensor . 37 3.3.3 The Spin density tensor . 40 3.4 Conservation Laws . 42 3.4.1 The Conservation of Energy and Momentum . 42 3.4.2 The Conservation of Spin . 45 3.4.3 The Conservation of Angular Momentum . 46 3.5 The Equations of Motion . 47 3.5.1 The Mathisson-Papapetrou-Dixon Equations . 48 3.5.2 The Dixon-Souriau Equations . 50 3.5.3 Supplementary Conditions . 51 4 The Kinematics and the Dynamics of the Einstein-Cartan Theory 52 4.1 The Kinematic Parameters . 52 4.1.1 The Raychaudhuri Equation . 53 4.1.2 The Shear Propagation Equation . 55 4.1.3 The Vorticity Propagation Equation . 58 3 4.1.4 The Constraints . 59 4.2 Electrodynamics and Magnetohydrodynamics . 61 4.2.1 Maxwell Equations and Conservation Laws . 61 4.2.2 Propagation of Electromagnetic Waves . 67 4.2.3 Kinematics of a Charged Fluid . 70 4.2.4 Kinematics of a Magnetized Fluid . 71 4.3 The \Electric" and \Magnetic" Parts of the Weyl Tensor . 72 4.3.1 The Propagation Equations . 73 4.3.2 The Constraints . 76 4.4 Comparison to Other Analyses . 78 4.4.1 The Relativistic Approach of Spinning Fluid Kinematics . 79 4.4.2 The Einstein-Cartan Approach of Spinning Fluid Kinematics . 80 5 Constructing a Cosmological Model in the Einstein-Cartan Theory 83 5.1 The Spatial Curvature . 83 5.2 The Friedmann-Lema^ıtre-Robertson-Walker Model . 85 5.2.1 The Metric, the Torsion and the Matter Fields . 86 5.2.2 The Evolution of the Universe . 87 5.2.3 The Friedmann Equations . 88 5.3 Solutions to the Friedmann Model . 90 5.3.1 The Dust Solution . 92 5.3.2 The Radiation Solution . 93 5.3.3 The Stiff Matter Solution . 94 5.3.4 The Inflation Solution . 94 5.3.5 The Curvature Solution . 95 5.3.6 The Vacuum Solution . 96 5.3.7 The Static Solution . 97 5.4 Comparison to Other Analyses . 97 5.4.1 The Relativistic Approach of Spinning Fluid Cosmologies . 97 5.4.2 The Einstein-Cartan Approach of Spinning Fluid Cosmologies . 98 6 Conclusions 103 6.1 The Main Results . 103 6.2 Cosmological Implications . 104 4 Chapter 1 Preface In 1915, Albert Einstein proposed a theory that could, on certain circumstances, summarize all known physical laws -with the exception of the laws on the microscopic scales that were being constructed at the very time by the first quantum theories. This theory tried to perceive every physical phenomenon as taking place in a four-dimensional space-time, whose geometry was not the typical euclidean one. On the contrary this space-time was curved and its curvature was relative to the physical phenomena that took place. Even deeper, the curvature of the space-time was expressing these physical phenomena, while the physical phenomena were depending on the curvature; this space-time could not exist without any physical phenomena taking place on it -without matter, energy and momentum existing on it- and it was a form of existence of these physical phenomena -of matter, energy and momentum included. This was the General Theory of Relativity. One could say that the General Theory of Relativity was an extension of another theory Einstein himself had proposed ten years before: the Special Theory of Relativity. The latter was a theory for understanding the -microscopic- physical phenomena that developed high velocities -very close to the one of light. In this theory, the four-dimensional space-time is flat, pseudo-euclidean; this did not allow any strong phenomena to occur, neither could include macroscopic phenomena, such as gravity. Gravity until that time was perceived as a force acting between two bodies that have mass -the property of matter related to gravitational interactions. Of course, this idea that was originally posted by Isaac Newton was highly criticised, since the very concept of force was not very stable -it did not seem natural at all. Many physicists and mathematicians have tried to remove this \curse" from the physical theories, but until then this was not made possible. However, the opinion that the light speed is the upper limit for any velocity, and as a result of any transfer, blew a new breath to this old controversy. It seemed impossible that the information of the gravitational field would be transferred immediately throughout any distance via the mechanism of the gravitational force; the electromagnetic field was not acting like this at all. It is remarkable that, a few years after Einstein published his Special Theory, the French mathematician Henri Poincar`estated that the space-time of this theory -known as Minkowski space-time- should be curved in the presence of mass and that the information of the existence of this mass should travel across the space-time with the speed of light, just like a wave. Einstein -with some help from K. Karatheodori and D. Hilbert- managed to find this connection of physics and geometry Poincar`ewas trying to point out, by using the Riemannian geometry. The latter is an approach of the geometry of curved spaces; it allows us to study the curvature of these space-times and perform any kind of \measures" on it. Einstein, with this theory, united the physical phenomena to the curved space-time, which he would use in order to describe the gravitational field. The idea is simple: the space-time exists only when physical phenomena occur and is curved proportionally to these phenomena. In simpler words, the presence of mass, energy or momentum not only \creates" space-time on which the previous will \live", but it also curves this space-time according to the density of them. As a result, the physical systems -eg. a material body or a physical process- do not \feel" any gravitational force; they \live" in a curved space-time whose 5 curvature they understand as gravity. The geometry that Einstein used in order to describe these -the Riemann geometry- is based on one simple fact: the connections of the space-time are non-zero and symmetric. As a result, they can describe the curvature immediately. The metric tensor of the space-time is turned into the main variable for the description of the space-time and, at the same time, the main variable for any physical theory in this space-time. On the other hand, no other geometrical variable appears as necessary for this description; the majority of them are considered to be zero. One of those variables is the torsion of the space-time. The torsion is related to the antisymmetric part of the connection, which we have demanded to be zero, so that we would exclude its effects.
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