A Course in Gravitational Theory

This represents the effort to display all of the essential tools and applications of gravitational theory. This can be thought of as an addition to How to be a Great Amateur Theoretical Physicist.

The goal here is to physically motivate everything. If I can figure out a way of writing an exciting introduction in words, then I will do it. After that, of course, the mathematical gloves will come off.

From the beginning I will use Mathematica and a free package for it called xAct to perform many calculuations. I will begin by doing everything by hand, once I feel that enough detail has been given to allow the reader to follow the ideas I will turn the calculations over to Mathematica (after demonstarting that it gives the same answer).

This requires either Mathematica 8 or later, or the free Mathematica CDF Viewer, though the viewer cannot run the programs, (you can find that here). You will also need to download the MAST Writing Style into the folder SystemFiles/Front End/Stylesheets. You can download that here. Once you load this file into the folder rename it MAST Writing Style 3. Reload Mathematica and it will be there.

First Unit: An Survey of Gravity Theory

This first unit introduces the major ideas of gravitational theory along with introducing the mathematics necessary to study gravitation. For this first unit I will assume that you have completed the equivalent of all of the material in Basic Mathematics and Basic Mechanics from How to be a Great Amateur Theoretical Physicist, noted above.

Introduction

1. Why Study Gravity?

Classical Mechanics

2. Space, Time, and the Galilean Transformation

3. The Mathematics of Space and Time

4. Mathematical Objects in Space and Time

5. Newtonian Mechanics

6. Solving the Equations of Motion

7. The Conservation Laws of Newtonian Mechanics

8. The Two-Body Problem and The Restricted Three-Body Problem

Classical Fields

9. The Nature of Fields

10. The Mathematics of Fields

11. Mathematical Objects in Fields

12. Newtonian Gravitational Theory

13. Electromagnetic Fields

14. Matter Fields

Special Relativity

15. Minkowski Spacetime and Lorentz Transformations

16. The Mathematics of Minkowski Spacetime

17. Mathematical Objects in Minkowski Spacetime

18. Relativistic Kinematics

19. Relativistic Mechanics

20. Relativistic Electrodynamics

21. Relativistic Matter Fields

General Relativity

22. The Equivalence Principle and Curved Spacetime

23. The Mathematics of Curved Spacetime

24. Mathematical Objects in Curved Spacetime

25. General Relativity

26. The Schwarzschild Solution

27. Black Holes

28. Gravitational Waves

29. Cosmology

The Quantum World

30. Why Study Quantum Mechanics?

31. Quantum Mathematics

32. Mathematical Objects in Quantum Mechanics

33. Quantum Mechanics

34. Quantum Fields

35. Quantum Gravitational Theory

Second Unit: Classical Gravity Theory and Special Relativity

Introduction

1. Why Study Classical Physics and Special Relativity?

2. The Three Pillars of Mathematical Physics

3. Some Notions of Mathematical Physics

Lagrangian and Hamiltonian Mechanics

4. Configuration Space

5. The Calculus of Variations

6. The Principle of Stationary Action

7. Lagrangian Mechanics

8. Conservation Laws and Invariance

9. Phase Space

10. Hamiltonian Mechanics

11. Thermodynamics

12. Dynamical Systems

13. Poisson Brackets and Symplectic Geometry

14. Canonical Transformations

15. The Hamilton-Jacobi Equation

16. Action-Angle Variables

Classical Thermal Physics

17. Kinetic Theory

18. Classical Statistical Mechanics

19. Statistical Thermodynamics

20. Random Processes and Diffusion

Classical Theory of Fields

21. Formulating a Field Theory

22. Rigid Bodies

23. Fluid Mechanics

24. Elastic Bodies

25. Hooke's Law

26. The Electromagnetic Field

27. The Stress Tensor

28. Polarized Materials

29. Plasmas

Special Relativity

30. The Minkowski Manifold

31. The Lorentz Transformation

32. The Stress-Energy Tensor

33. Relativistic Electromagnetic Fields

34. Relativistic Matter Fields

35. Extended Phase Space and Relativistic Fields

Third Unit: Mathematical Requirements

Introduction

1. Review of the Three Pillars of Mathematical Physics

Algebra

2. Algebraic Structures

3. Groups

4. Rings and Fields

5. Vector Spaces

6. Linear Transformations and Operators

7. Matrices

8. Inner Product Spaces

9. Algebras

10. Tensors

11. Exterior Algebra

Analysis and Topology

12. Topology

13. Measure Theory

14. Integration Theory

15. Banach Spaces

16. Distributions

17. Hilbert Spaces

Geometry and Topology

18. Curves

19. Surfaces

20. Manifolds

21. Vector Fields

22. Tensor Geometry

23. Exterior Calculus

23. Integration on Manifolds

24. The Lie Derivative

25. Connections and Curvature

26. Geodesics

27. Tensor Analysis

28. Synge's Theorem

29. Homology

30. Homotopy

31. Harmonic Forms

32. Lie Groups and Lie Algebras

33. Vector Bundles

34. Fiber Bundles

35. Connections and Bundles

Fourth Unit: General Relativity

Introduction

1. The Purpose of a Detailed Study of General Relativity

General Relativity

2. Principles of General Relativity

3. The Field Equations

4. The Action Principle

5. Lagrangians

6. The Stress-Energy Tensor

7. The Structure of the Field Equations

8. Weak Fields

9. Mass and Angular Momentum

10. Conservation Laws

11. Field Sources

Some Advanced Ideas

12. The Tetrad Formalism

13. Geometrical Algebra

14. Spacetime Algebra

15. Geometrical Calculus

16. Christodoulou's Memory Effect

17. The Hamiltonian Approach

18. Hyperbolic Equations

19. Symmetric Spaces

Physics in Curved Spacetime

20. Geometrical Optics

21. Electrodynamics

22. Thermodynamics

23. Kinetic Theory

24. Fluid Mechanics

25. Tests of General Relativity

Black Holes

26. The Schwarzschild Solution

27. Schwarzschld Black Holes

28. Reissner-NordstrÝm Black Holes

29. Kerr Black Holes

Approximations and Waves

30. The Post-Newtonian Approximation

31. Perturbations and Gauge Transformations

32. Gravitational Waves

Cosmology

33. Friedmann-Robertson-Walker Cosmology

34. Vacuum Energy

35. De Sitter Spacetime and Anti de Sitter Spacetime

Fifth Unit: Applications of General Relativity

Introduction

1. Applying General Relativity

2. The GPS Cluster

Astrophysics and Compact Objects

3. Stellar Interiors and Stellar Evolution

4. Stallar Pulsations and Rotations

5. Cold Equations of State

6. White Dwarf Stars

7. Magnetism of Stars

8. Quantum Mechanics

9. Quantum Field Theory

10. Nuclear Field theory

11. Neutron Degeneracy

12. Neutron Stars

13. Rotating Neutron Stars and Pulsars

Black Holes

14. Gravitational Collapse

15. Properties of Black Holes

16. Black Hole Perturbations

17. Stationary Black Holes

18. Gravitational Effects of Black Holes

19. Black Hole Electrodynamics

20. Black Hole Astrophysics

Accretion

21. Binary Systems and Accretion

22. The Fluid Dynamics of Accretion

23. Accretion onto Neutron Stars

24. Accretion onto Black Holes

25. Active Galactic Nuclei

Gravitational Waves

26. Propagation of Gravitational Waves

27. Generation of Gravitational Waves

28. Sources of Gravitational Waves

29. Detection of Gravitational Waves

Cosmology

30. Cosmological Models

31. Cosmological Singularities

32. The Early Universe and The Very Early Universe

33. Quantum Cosmology

34. The Future of the Universe

35. Structure Formation

Sixth Unit: Advanced Topics in General Relativity

Introduction

1. Review of General Relativity

2. Matter Fields

Advanced Mathematical Methods and Formulations

3. Symmetry Classes

4. Ordinary Differential Equations in General Relativity

5. Functional Analysis

6. Methods for Solving Einstein's Equations

7. Elliptic Equations

8. Hyperbolic Equations

9. The Initial-Value Formulation

10. The Cauchy Problem

11. Constraints

12. Hyperbolic-Elliptic Systems

13. The Variational Method

14. Global Methods

15. Global Hyperbolicity

16. Global Existence Theorems

17. Tetrad Methods

18. Conformal Methods

19. Generation Techniques

20. Groups of Motions

21. Spinors

22. The Newman-Penrose Formalism

23. Axisymmetric Solutions

24. Asymptotic Behavior of Solutions and Asymptotically Flat Spacetimes

Arenas of General Relaivity

25. Null Surfaces

26. Causal Structure

27. Singularities

Advanced Physics in General Relativity

28. Relativistic Fluids

29. Relativistic Kinetic Theory and The Einstein-Vlasov System

30. Gravitational Waves

31. Cosmological Global Existence Theorems

32. The Einstein-Scalar Field System

33. Regge Calculus

34. Superspace

35. Pregeometry

Seventh Unit: Quantum Gravity

Introduction

1. Why Quantum Gravity?

2. Review of The Hamiltonian Formulation of General Relativity

3. Kaluza-Klein Theory

4. The Quest for Quantum Gravity

5. Complex Geometry

6. Category Theory

7. C*-Algebras

Quantum Mechanics

8. Review of Quantum Mechanics

9. The Quantum Mechanics of Closed Systems

10. Decoherence

11. Generalized Quantum Mechanics

12. The Problem of Time

13. The Spacetime Approach to Non-Relativistic Quantum Mechanics

14. Canonical Quantization

15. Geometric Quantization

16. Relativistic Quantum Mechanics

Quantum Field Theory

17. Quantum Field Theory

18. Effective Field Theories

19. Abelian Gauge Theories

Loop Quantum Gravity

20. The Holonomy Flux Algebra

21. The Quantum *-Algebra

22. Representation of the *-Algebra

23. Kinematical Constraints

24. Hamiltonian Constraints

25. Coherent States

26. Matter

27. Kinematical Operators

28. Spin Foam

29. Quantum Black Holes

Other Ideas

30. Euclidean Path Integrals

31. Discrete Quantum Gravity

32. Noncommutative Geometry and Spacetime

33. Asymptotic Safety

34. Causal Histories

35. Cellular Automata

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