Overview

The book employs oscillatory dynamical systems to represent the Universe mathematically via constructing classical and quantum theory of damped oscillators. It further discusses isotropic and homogeneous metrics in the Friedman-Robertson-Walker Universe and shows their equivalence to non-stationary oscillators. The wide class of exactly solvable damped oscillator models with variable parameters is associated with classical special functions of mathematical physics. Combining principles with observations in an easy to follow way, it inspires further thinking for mathematicians and physicists. Contents Part I: Dissipative geometry and general relativity theory Pseudo-Riemannian geometry and general relativity Dynamics of universe models Anisotropic and homogeneous universe models Metric waves in a nonstationary universe and dissipative oscillator Bosonic and fermionic models of a Friedman–Robertson–Walker universe Time dependent constants in an oscillatory universe Part II: Variational principle for time dependent oscillations and dissipations Lagrangian and Hamilton descriptions Damped oscillator: classical and quantum theory Sturm–Liouville problem as a damped oscillator with time dependent damping and frequency Riccati representation of time dependent damped oscillators Quantization of the harmonic oscillator with time dependent parameters

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9783110514957

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Table of content: Chapter 1 Introduction Part I Dissipative Geometry and General Relativity Theory Chapter 2 Pseudo-Riemannian Geometry and General Relativity 2.1. Curvature of Space Time and Einstein Field Equations 2.1.1. Einstein Field Equations 2.1.2. Energy Momentum Tensor 2.2. Universe as a Dynamical System 2.2.1. Friedman-Robertson-Walker (FRW) Metric 2.2.2. Friedman Equations 2.2.3. Adiabatic Expansion and Friedman Differential Equation Chapter 3 Dynamics of Universe Models 3.1. Friedman Models 3.1.1. Static Models 3.1.2. Empty Models 3.1.3. Three Non-Empty Models with � = 0 3.1.4. Non-Empty Models with � 6= 0 3.2. Milne Model Chapter 4 Anisotropic and Homogeneous Universe Models 4.1. General Solution 4.1.1. Constant Density and Zero Pressure 4.1.2. Constant Pressure and Zero Density 4.1.3. Absence of Pressure and Density Chapter 5 Barotropic Models of FRW Universe 5.1. Bosonic FRW Model 5.2. Fermionic FRW Barotropy 5.3. Decoupled Fermionic and Bosonic FRW Barotropies 5.4. Coupled Fermionic and Bosonic Cosmological Barotropies Chapter 6 Time Dependent Gravitational and Cosmological Constants 6.1. Model and Field Equation 6.2. Solution of the Field Equation 6.2.1. G (t) H 6.2.1.1. Inflationary Phase 6.2.1.2. Radiation Dominated Phase 6.2.2. G (t) 1/H 6.2.2.1. Inflationary Phase 6.2.2.2. Radiation Dominated Phase Chapter 7 Gravitational Waves in Non-Stationary Universe and Dissipative Oscillator 7.1. Linear Gravitational (Metric) Waves in Flat Space Time 7.2. Linear Gravitational (Metric) Waves in Non-Stationary Universe 7.2.1. Hyperbolic Geometry of Damped Oscillator and Double Universe Part II Variational Principles for Time Dependent Oscillations and Dissipations Chapter 8 Lagrangian and Hamilton Description 8.1. Generalized Co-ordinates and Velocities 8.2. Principle of Least Action 8.3. Hamilton's Equations 8.3.1. Poisson Brackets Chapter 9 Damped Oscillator: Classical Quantum 8.4. Damped Oscillator 8.5. Bateman Dual Description 8.6. Caldirola Kanai Approach for Damped Oscillator 8.7. Quantization of Caldirola-Kanai Damped Oscillator with Constant Frequency and Constant Damping Chapter 10 Sturm Liouville Problem as Damped Parametric Oscillator 10.1. Sturm Liouville Problem in Doublet Oscillator Representation and Self-Adjoint Form 10.1.1. Particular Cases for Non-Self Adjoint Equation 10.1.2. Variational Principle for Self Adjoint Operator 10.1.3. Particular Cases for Self Adjoint Equation 10.2. Oscillator Equation with Three Regular Singular Points 10.2.1. Hypergeometric Functions 10.2.2. Confluent Hypergeometric Function 10.2.3. Bessel Equation 10.2.4. Legendre Equation 10.2.5. Shifted-Legendre Equation 10.2.6. Associated-Legendre Equation 10.2.7. Hermite Equation 10.2.8. Ultra-Spherical (Gegenbauer) Equation 10.2.9. Laguerre 10.2.10. Associated Laguerre Equation 10.2.11. Chebyshev Equation I 10.2.12. Chebyshev Equation II 10.2.13. Shifted Chebyshev Equation I Chapter 11 Riccati Representation of Time Dependent Damped Oscillators 11.1. Hypergeometric Equation 11.2. Confluent Hypergeometric Equation 11.3. Legendre Equation 11.4. Associated-Legendre Equation 11.5. Hermite Equation 11.6. Laguerre Equation 11.7. Associated Laguerre Equation 11.8. Chebyshev Equation I 11.9. Chebyshev Equation II Chapter 12 Conclusion References List of Tables Appendices Appendix A Preliminaries for Tensor Calculus A.1. Tensor Calculus A.2. Calculating Christoffel Symbols from Metric A.3. Parallel Transport and Geodesics A.4. Variational Method for Geodesics A.5. Properties of Riemann Curvature Tensor A.6. Bianchi Identities; Ricci and Einstein Tensors A.6.1. Ricci Tensor A.6.2. Einstein Tensor Appendix B Riccati Differential Equation Appendix C Hermited Differential Equation C.1. Orthogonality C.2. Even/Odd Functions C.3. Recurrence Relation C.4. Special Results Appendix D Non-Stationary Oscillator Representation of FRW Universe D.1. Time Dependent Oscillator D.1.1. Delta Function Potential Appendix E Bianchi Model: An Alternative Way to Model the Present-Day Universe E.1. BI Models E.2. Isotropization of BI models into FRW Universe E3. Evolution of anisotropic deviations from FRW in decoupling E3.1 Isotropization criteria of the radiation and matter dominated BI model Indexing

Table of content: Chapter 1 IntroductionPart I Dissipative Geometry and General Relativity TheoryChapter 2 Pseudo-Riemannian Geometry and General Relativity2.1. Curvature of Space Time and Einstein Field Equations2.1.1. Einstein Field Equations2.1.2. Energy Momentum Tensor2.2. Universe as a Dynamical System2.2.1. Friedman-Robertson-Walker (FRW) Metric2.2.2. Friedman Equations2.2.3. Adiabatic Expansion and Friedman Differential EquationChapter 3 Dynamics of Universe Models3.1. Friedman Models3.1.1. Static Models3.1.2. Empty Models3.1.3. Three Non-Empty Models with = 03.1.4. Non-Empty Models with 6= 03.2. Milne ModelChapter 4 Anisotropic and Homogeneous Universe Models4.1. General Solution4.1.1. Constant Density and Zero Pressure4.1.2. Constant Pressure and Zero Density4.1.3. Absence of Pressure and DensityChapter 5 Barotropic Models of FRW Universe5.1. Bosonic FRW Model5.2. Fermionic FRW Barotropy5.3. Decoupled Fermionic and Bosonic FRW Barotropies5.4. Coupled Fermionic and Bosonic Cosmological BarotropiesChapter 6 Time Dependent Gravitational and Cosmological Constants6.1. Model and Field Equation6.2. Solution of the Field Equation6.2.1. G (t) H6.2.1.1. Inflationary Phase6.2.1.2. Radiation Dominated Phase6.2.2. G (t) 1/H6.2.2.1. Inflationary Phase6.2.2.2. Radiation Dominated PhaseChapter 7 Gravitational Waves in Non-Stationary Universe and Dissipative Oscillator7.1. Linear Gravitational (Metric) Waves in Flat Space Time7.2. Linear Gravitational (Metric) Waves in Non-Stationary Universe7.2.1. Hyperbolic Geometry of Damped Oscillator and Double UniversePart II Variational Principles for Time Dependent Oscillations and DissipationsChapter 8 Lagrangian and Hamilton Description8.1. Generalized Co-ordinates and Velocities8.2. Principle of Least Action8.3. Hamilton's Equations8.3.1. Poisson BracketsChapter 9 Damped Oscillator: Classical Quantum8.4. Damped Oscillator8.5. Bateman Dual Description8.6. Caldirola Kanai Approach for Damped Oscillator8.7. Quantization of Caldirola-Kanai Damped Oscillator with Constant Frequency and Constant DampingChapter 10 Sturm Liouville Problem as Damped Parametric Oscillator10.1. Sturm Liouville Problem in Doublet Oscillator Representation and Self-Adjoint Form10.1.1. Particular Cases for Non-Self Adjoint Equation10.1.2. Variational Principle for Self Adjoint Operator10.1.3. Particular Cases for Self Adjoint Equation10.2. Oscillator Equation with Three Regular Singular Points10.2.1. Hypergeometric Functions10.2.2. Confluent Hypergeometric Function10.2.3. Bessel Equation10.2.4. Legendre Equation10.2.5. Shifted-Legendre Equation10.2.6. Associated-Legendre Equation10.2.7. Hermite Equation10.2.8. Ultra-Spherical (Gegenbauer) Equation10.2.9. Laguerre10.2.10. Associated Laguerre Equation10.2.11. Chebyshev Equation I10.2.12. Chebyshev Equation II10.2.13. Shifted Chebyshev Equation IChapter 11 Riccati Representation of Time Dependent Damped Oscillators11.1. Hypergeometric Equation11.2. Confluent Hypergeometric Equation11.3. Legendre Equation11.4. Associated-Legendre Equation11.5. Hermite Equation11.6. Laguerre Equation11.7. Associated Laguerre Equation11.8. Chebyshev Equation I11.9. Chebyshev Equation IIChapter 12 ConclusionReferencesList of TablesAppendicesAppendix A Preliminaries for Tensor CalculusA.1. Tensor CalculusA.2. Calculating Christoffel Symbols from MetricA.3. Parallel Transport and GeodesicsA.4. Variational Method for GeodesicsA.5. Properties of Riemann Curvature TensorA.6. Bianchi Identities; Ricci and Einstein TensorsA.6.1. Ricci TensorA.6.2. Einstein TensorAppendix B Riccati Differential EquationAppendix C Hermited Differential EquationC.1. OrthogonalityC.2. Even/Odd FunctionsC.3. Recurrence RelationC.4. Special ResultsAppendix D Non-Stationary Oscillator Representation of FRW UniverseD.1. Time Dependent OscillatorD.1.1. Delta Function PotentialAppendix E Bianchi Model: An Alternative Way to Model the Present-Day UniverseE.1. BI ModelsE.2. Isotropization of BI models into FRW Univer

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Esra Russel, New York University Abu Dhabi, United Arab Emirates, Oktay Pashaev, Izmir Institute of Technology, Turkey

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