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Damping Oscillatory Models in General Theory of Relativity

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Damping Oscillatory Models in General Theory of Relativity DAMPING OSCILLATORY MODELS IN GENERAL THEORY OF RELATIVITY A Thesis Submitted to the Graduate School of Engineering and Sciences of Izmir˙ Institute of Technology in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE in Mathematics by Esra TIGRAKˇ ULAS¸ June 2007 IZM˙ IR˙ We approve the thesis of Esra TIGRAKˇ ULAS¸ Date of Signature ..................................... 27 June 2007 Prof. Dr. Oktay PASHAEV Supervisor Department of Mathematics Izmir˙ Institute of Technology ..................................... 27 June 2007 Prof. Dr. Recai ERDEM Department of Physics Izmir˙ Institute of Technology ..................................... 27 June 2007 Prof. Dr. E. Rennan PEKUNL¨ U¨ Department of Astronomy and Space Sciences Ege University ..................................... 27 June 2007 Prof. Dr. Oguzˇ YILMAZ Head of Department Department of Mathematics Izmir˙ Institute of Technology .................................. Prof. Dr. M. Barıs¸ OZERDEM¨ Head of the Graduate School ACKNOWLEDGEMENTS I would like to express my gratitude to my supervisor, Prof. Dr. Oktay Pashaev, for his support, encouragement and patience throughout my thesis study. I always felt lucky to have a supervisor as him. Besides teaching me valuable lessons regarding academic research, he always found time for listening to my problems. Finally, his editorial advice and help were essential to the completion of this thesis. ABSTRACT DAMPING OSCILLATORY MODELS IN GENERAL THEORY OF RELATIVITY In my thesis we have studied the universe models as dynamical systems which can be represented by harmonic oscillators. For example, a harmonic oscillatior equation is constructed by the transformation of the Riccati differential equations for the anisotropic and homogeneous metric. The solution of the Friedman equations with the state equa- tion satisfies both bosonic expansion and fermionic contraction in Friedman Robertson Walker universe with different curvatures is studied as a conservative system with the harmonic oscillator equations. Apart from the oscillator representations mentioned above (constructed from the universe models), we showed that the linearization of the Einstein field equations produces harmonic oscillator equation with constant frequency and the linearization of the metric on the de-Sitter background produces damped harmonic os- cillator system. In addition to these, we have constructed the doublet and the Caldirola type oscillator equations with time dependent damping and frequency terms in the light of the Sturm Liouville form. The Lagrangian and Hamiltonian functions are calculated for all particular cases of the Sturm Liouville form. Finally, we have shown that zeros of the oscillator equations constructed from the particular cases can be transformed into pole singularities of the Riccati equations. iv OZET¨ GENEL GOREL¨ IL˙ IK˙ TEORIS˙ INDE˙ SON¨ UML¨ U¨ OSILASYON˙ MODELLERI˙ Tezimde harmonik salınıcı olarak betimlenebilen dinamik evren modelleri c¸alıs¸ıldı. Bu modellere ornek¨ olarak homojen ve isotropik olmayan evren ic¸in Ric- cati diferansiyel denkleminin dogrusal˘ don¨ us¸¨ um¨ unden¨ olus¸turulan harmonik salınıcı den- klemlerini elde ettik. Buna ek olarak, Friedman diferansiyel denklemlerinin ve durum denkleminin Friedman Robertson Walker evreninde ortak c¸oz¨ um¨ unden¨ bosonik olarak genis¸leyen ve fermionik olarak buz¨ ulen¨ evren modellerini farklı egrilikler˘ ic¸in korunumlu sistem olarak harmonik salınıcı denklemler ıs¸ıgında˘ inceledik. Yukarıda ki harmonik salınıcı evren modellerinden farklı olarak, Einstein denklemlerinin Minkowski uzayında dogrusallas¸tırılmasının˘ sabit frekanslı standart harmonik salınıcı denklemini ve aynı den- klemlerin de-Sitter uzayında dogrusallas¸tırılmasıyla˘ da son¨ uml¨ u¨ harmonik salınıcı den- klemlerinin uretildi¨ gini˘ gosterdik.¨ Bunlara ek olarak, zaman bagımlı˘ son¨ umleme¨ ve frekans biles¸enlerine sahip olan c¸ift ve Caldirola tipi son¨ uml¨ u¨ salınıcı denklemleri, Sturm Liouville formu yardımıyla elde edildi. Sturm Liouville formunun tum¨ ozel¨ durumları ic¸in Lagrangian ve Hamiltonian fonksiyonları elde edildi. Son olarak, ozel¨ durumlar ic¸in elde edilmis¸salınıcı denklemlerinin sıfırlarının Riccati denklemlerinin kutup noktalarına don¨ us¸ebilece¨ gini˘ gosterdik.¨ v TABLE OF CONTENTS LIST OF TABLES . xi CHAPTER 1 . INTRODUCTION . 1 I DISSIPATIVE GEOMETRY AND GENERAL RELATIVITY THEORY 7 CHAPTER 2 . PSEUDO-RIEMANNIAN GEOMETRY AND GENERAL RELATIVITY . 8 2.1. Curvature of Space Time and Einstein Field Equations . 9 2.1.1. Einstein Field Equations . 10 2.1.2. Energy Momentum Tensor . 11 2.2. Universe as a Dynamical System . 12 2.2.1. Friedman-Robertson-Walker (FRW) Metric . 12 2.2.2. Friedman Equations . 16 2.2.3. Adiabatic Expansion and Friedman Differential Equation . 19 CHAPTER 3 . DYNAMICS OF UNIVERSE MODELS . 23 3.1. Friedman Models . 23 3.1.1. Static Models . 24 3.1.2. Empty Models . 25 3.1.3. Three Non-Empty Models with ¤ = 0 . 25 3.1.4. Non-Empty Models with ¤ 6= 0 . 29 3.2. Milne Model . 29 CHAPTER 4 . ANISOTROPIC AND HOMOGENEOUS UNIVERSE MODELS 31 4.1. General Solution . 34 4.1.1. Constant Density and Zero Pressure . 36 4.1.2. Constant Pressure and Zero Density . 38 4.1.3. Absence of Pressure and Density . 41 vi CHAPTER 5 . BAROTROPIC MODELS OF FRW UNIVERSE . 42 5.1. Bosonic FRW Model . 42 5.2. Fermionic FRW Barotropy . 47 5.3. Decoupled Fermionic and Bosonic FRW Barotropies . 49 5.4. Coupled Fermionic and Bosonic Cosmological Barotropies . 52 CHAPTER 6 . TIME DEPENDENT GRAVITATIONAL AND COSMOLOGICAL CONSTANTS . 56 6.1. Model and Field Equation . 56 6.2. Solution of The Field Equation . 61 6.2.1. G(t) » H .............................. 62 6.2.1.1. Inflationary Phase . 65 6.2.1.2. Radiation Dominated Phase . 66 6.2.2. G(t) » 1=H ............................. 67 6.2.2.1. Inflationary Phase . 70 6.2.2.2. Radiation Dominated Phase . 71 CHAPTER 7 . METRIC WAVES IN NON-STATIONARY UNIVERSE AND DISSIPATIVE OSCILLATOR . 73 7.1. Linear Metric Waves in Flat Space Time . 73 7.2. Linear Metric Waves in Non-Stationary Universe . 81 7.2.1. Hyperbolic Geometry of Damped Oscillator and Double Universe . 82 II VARIATIONAL PRINCIPLES FOR TIME DEPENDENT OSCILLATIONS AND DISSIPATIONS 86 CHAPTER 8 . LAGRANGIAN AND HAMILTONIAN DESCRIPTION . 87 8.1. Generalized Co-ordinates and Velocities . 87 8.2. Principle of Least Action . 87 8.3. Hamilton’s Equations . 89 8.3.1. Poisson Brackets . 91 vii CHAPTER 9 . DAMPED OSCILLATOR: CLASSICAL AND QUANTUM . 93 9.1. Damped Oscillator . 93 9.2. Bateman Dual Description . 94 9.3. Caldirola Kanai Approach for Damped Oscillator . 95 9.4. Quantization of Caldirola-Kanai Damped Oscillator with Con- stant Frequency and Constant Damping . 96 CHAPTER 10.STURM LIOUVILLE PROBLEM AS DAMPED PARAMETRIC OSCILLATOR . 105 10.1. Sturm Liouville Problem in Doublet Oscillator Representation and Self-Adjoint Form . 105 10.1.1. Particular Cases for Non-Self Adjoint Equation . 108 10.1.2. Variational Principle for Self Adjoint Operator . 109 10.1.3. Particular Cases for Self Adjoint Equation . 111 10.2. Oscillator Equation with Three Regular Singular Points . 112 10.2.1. Hypergeometric Functions . 118 10.2.2. Confluent Hypergeometric Function . 120 10.2.3. Bessel Equation . 122 10.2.4. Legendre Equation . 124 10.2.5. Shifted-Legendre Equation . 125 10.2.6. Associated-Legendre Equation . 127 10.2.7. Hermite Equation . 128 10.2.8. Ultra-Spherical(Gegenbauer)Equation . 129 10.2.9. Laguerre . 131 10.2.10.Associated Laguerre Equation . 132 10.2.11.Chebyshev Equation I . 133 10.2.12.Chebyshev Equation II . 135 10.2.13.Shifted Chebyshev Equation I . 136 CHAPTER 11.RICCATI REPRESENTATION OF TIME DEPENDENT DAMPED OSCILLATORS . 138 11.1. Hypergeometric Equation . 140 11.2. Confluent Hypergeometric Equation . 140 viii 11.3. Legendre Equation . 142 11.4. Associated-Legendre Equation . 143 11.5. Hermite Equation . 144 11.6. Laguerre Equation . 145 11.7. Associated Laguerre Equation . 146 11.8. Chebyshev Equation I . 147 11.9. Chebyshev Equation II . 148 CHAPTER 12. CONCLUSION . 149 REFERENCES . 149 APPENDICES APPENDIX A. PRELIMINARIES FOR TENSOR CALCULUS . 153 A.1. Tensor Calculus . 153 A.2. Calculating Christoffel Symbols from Metric . 154 A.3. Parallel Transport and Geodesics . 155 A.4. Variational Method For Geodesics . 156 A.5. Properties of Riemann Curvature Tensor . 158 A.6. Bianchi Identities; Ricci and Einstein Tensors . 161 A.6.1. Ricci Tensor . 162 A.6.2. Einstein Tensor . 162 APPENDIX B. RICCATI DIFFERENTIAL EQUATION . 165 APPENDIX C. HERMITE DIFFERENTIAL EQUATION . 168 C.1. Orthogonality . 169 C.2. Even/Odd Functions . 170 C.3. Recurrence Relation . 170 C.4. Special Results . 170 APPENDIX D. NON-STATIONARY OSCILLATOR REPRESENTATION OF FRW UNIVERSE . 171 D.1. Time Dependent Oscillator . 172 ix D.1.1. Delta Function Potential . 171 x LIST OF TABLES Table Page Table 10.1 Coefficients and parameters of the special functions. 113 Table 10.2 Dampings and frequencies for the special functions. 115 Table 10.3 Intervals [a; b] of the special functions. 116 xi CHAPTER 1 INTRODUCTION Modern mathematical cosmology started in 1917 when Albert Einstein applied his gravity model in order to understand the large scale structure of the universe. This model was built using his general theory of relativity which was constructed in 1916 by using the Riemannian Geometry. The first problem faced by his model was that it is not static, while it was generally believed that the universe is static. The model predicted that the universe is expanding while observations, of that time, did not support
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