Oscillatory Models in General Relativity (De Gruyter Studies in Mathematical Physics)

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

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

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