Table of Contents

1. Part I – Introduction and Basic Structure
   1. Introduction
      1. Purpose of this document
      2. Approach
      3. Intended audience
      4. Final remarks
2. Part II – Derivation of General Relativity
   1. General Relativity
      1. The Equivalence Principle
      2. Curvature of Space-Time
      3. Covariant and Contravariant Vectors and Dual Vectors
      4. Covariant and Contravariant Transformations of Tensors
      5. Christoffel Symbols and the Covariant Derivative
      6. Geodesic Equation and Christoffel Symbols
      7. Christoffel Symbols Expressed in Terms of the Metric Tensor
      8. Geodesic Equation and its Newtonian Limit
      9. Generalizing the Definition of the Metric Tensor
      10. The Riemann Curvature Tensor
      11. Symmetries and Independent Components
      12. Bianchi Identity and Ricci Tensor
      13. Energy-Momentum Tensor
      14. The Einstein Tensor
      15. The Einstein Field Equations
      16. Summary of the Final Formula for General Relativity
3. Part III – Physical Interpretations
   1. Schwarzschild Metric
   1. Discussion of the Schwarzschild Metric
   2. Relation between the Schwarzschild Metric and Noether’s Theorem
   3. Physical Interpretation of the Schwarzschild Metric
   4. Experiments: Confirmation of General Relativity
4. Part IV – Experiments and Verifications
   1. Experiments Confirming Einstein’s Theory
   1. Experiment 1 – The Hafele & Keating Experiment with the Schwarzschild Equation
   2. Experiment 2 – Motion of Particles in Schwarzschild Geometry
   3. Experiment 3 – Deflection of Light
   4. Experiment 4 – Perihelion Precession (Mercury)
   5. Experiment 5 – Shapiro Time Delay
   6. Time Relation between an Observer on Earth and the Center of the Sun
   7. Alternative Derivation of the Orbital Equation
   8. Calculation of a Projectile Trajectory
5. Part V – Coordinates and Formal Analysis
   1. Coordinate Systems
   1. Rectangular (Cartesian) Coordinate System
   2. Experiment 2 – Non-Orthogonal Coordinate System
   3. Experiment 3 – Curved Coordinates
   4. General Form of a Coordinate System
   5. The Metric Tensor and Einstein Notation
   6. Transformation between Two Coordinate Systems
   7. Transformation between Cartesian and Polar (Infinitesimal) Coordinates
   8. Exercise: Applying the Metric Transformation Formula
   9. Further Considerations on Co- and Contravariant Transformations
   10. Considerations on the Minkowski and Schwarzschild Formulas
   11. Schwarzschild: “On the Gravitational Field of a Mass Point According to Einstein’s Theory”
6. Part VI – Validation of the Theory
   1. Verification that the Schwarzschild Metric Satisfies the Einstein Field Equations
   1. Verification Using the Full Field Equations
   2. Verification of and within the Schwarzschild Metric
   3. Verification of the Ricci Tensor Components in Schwarzschild Coordinates
   4. Verification of the Schwarzschild Solution Using a Simplified Form of the Field Equations
   5. t, x, y, z (Modified Polar) Coordinates
   6. Verification of Ricci Components in Spherical Coordinates
7. Part VII – Questions and Discussion
   1. Answers to Questions
   1. Derivation of the Schwarzschild Formula to Proper Time
   2. Explanation of Einstein’s Transformation Formula
   3. Answers to Questions Concerning Schwarzschild
   4. Detailed Derivation of Einstein Equation (57) from Equation (53)
   5. Question Regarding an Equation in Einstein’s Original Work (English Version)
   6. Question Regarding Einstein’s Equation (69)
8. Appendices
   1. Appendix 1 – Formulas of General Relativity
   2. Appendix 2 – Derivation of Derivatives of the Christoffel Symbols
   3. Appendix 3 – Mathematical Elaboration of Schwarzschild
   4. Appendix 4 – The Schwarzschild Formula Extended to Electric Charges
   5. Appendix 5 – Schwarzschild Solution Inside a Mass
   6. Appendix 6 – Derivation of Gauss’s Theorem
   7. Appendix 7 – Derivation of the Laplace and Poisson Equations
   8. Appendix 8 – Tidal Forces
   9. Appendix 9 – Special Relativity
   10. Appendix 10 – Specific Angular Momentum
   11. Appendix 11 – Considerations on Rotation
   12. Appendix 12 – Derivation of the Euler–Lagrange Equation
9. Bibliography & Web Resources