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.
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.
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
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
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
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
30. Why Study Quantum Mechanics?
31. Quantum Mathematics
32. Mathematical Objects in Quantum Mechanics
33. Quantum Mechanics
34. Quantum Fields
35. Quantum Gravitational Theory
1. Why Study Classical Physics and Special Relativity?
2. The Three Pillars of Mathematical Physics
3. Some Notions of Mathematical Physics
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
17. Kinetic Theory
18. Classical Statistical Mechanics
19. Statistical Thermodynamics
20. Random Processes and Diffusion
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
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
1. Review of the Three Pillars of Mathematical Physics
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
12. Topology
13. Measure Theory
14. Integration Theory
15. Banach Spaces
16. Distributions
17. Hilbert Spaces
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
1. The Purpose of a Detailed Study of 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
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
20. Geometrical Optics
21. Electrodynamics
22. Thermodynamics
23. Kinetic Theory
24. Fluid Mechanics
25. Tests of General Relativity
26. The Schwarzschild Solution
27. Schwarzschld Black Holes
28. Reissner-Nordstrøm Black Holes
29. Kerr Black Holes
30. The Post-Newtonian Approximation
31. Perturbations and Gauge Transformations
32. Gravitational Waves
33. Friedmann-Robertson-Walker Cosmology
34. Vacuum Energy
35. De Sitter Spacetime and Anti de Sitter Spacetime
1. Applying General Relativity
2. The GPS Cluster
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
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
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
26. Propagation of Gravitational Waves
27. Generation of Gravitational Waves
28. Sources of Gravitational Waves
29. Detection of Gravitational Waves
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
1. Review of General Relativity
2. Matter Fields
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
25. Null Surfaces
26. Causal Structure
27. Singularities
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
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
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
17. Quantum Field Theory
18. Effective Field Theories
19. Abelian Gauge Theories
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
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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