Swooning Over Relativity: Understanding The Math Behind General Relativity

Title\nContents\nPreface\n1. Co-ordinate systems\n	1.1. Cartesian co-ordinates\n	1.2. Polar co-ordinates\n	1.3. Co-ordinate Conversion (Polar to Cartesian)\n	1.4. Co-ordinate Conversion (Cartesian to Polar)\n	1.5. Spherical co-ordinates and their Conversion\n	1.6. Oblique co-ordinates\n	1.7. Cantor co-ordinates\n	1.8. Distance between Intervals\n	1.9. Summary of co-ordinate systems\n2. Advanced Calculus\n	2.1. Partial derivatives\n	2.2. Total differential and total derivatives\n	2.3. Second order partial derivatives\n	2.4. Taylor series\n	2.5. Summary\n3. Vectors and Matrices\n	3.1. What is a matrix?\n	3.2. The Identity Matrix\n	3.3. Symmetric matrices\n	3.4. Diagonal matrix\n	3.5. Vectors\n	3.6. Vector multiplication\n	3.7. Vector fields\n	3.8. Gradient of a scalar field\n	3.9. Divergence of scalar field\n	3.10. Euclidean metric in Cartesian co-ordinates\n	3.11. Euclidean metric in Polar co-ordinates\n	3.12. Euclidean metric using Spherical co-ordinates\n	3.13. Finding the inverse of Euclidean metrics for Cartesian, Polar and Spherical co-ordinates\n	3.14. Index notation\n	3.15. Summary\n4. Newtonian Mechanics\n	4.1. Gravitational field\n	4.2. Gravitational potential energy\n	4.3. Gravitational potential field\n	4.4. Poisson’s equation\n	4.5. Summary of Newtonian Mechanics\n5. Special Relativity\n	5.1. Representation of units\n	5.2. Space-time diagrams\n	5.3. The K-factor\n	5.4. Simultaneity and casualty\n	5.5. Lorentz Transformation\n	5.6. Space-time intervals\n	5.7. Proper time\n	5.8. Proper Length\n	5.9. More about Simultaneity\n	5.10. Relative speed in General Relativity\n	5.11. Minkowski metric\n	5.12. Conservation law\n	5.13. Four-velocity\n	5.14. Relativistic Momentum and Energy\n	5.15. Four-force\n	5.16. Conclusion\n6. General Relativity\n	6.1. Manifolds\n	6.2. Differential Geometry\n	6.3. Overview of Tensors\n	6.4. Tensor representation and transformations\n	6.5. Tensor Algebra Pre-requisites\n	6.6. Connection coefficients\n	6.7. Covariant differentiation\n	6.8. Maxwell Equations\n	6.9. Parallel Transport of Vectors\n	6.10. Geodesics\n	6.11. Riemann curvature tensor\n	6.12. Ricci tensor\n	6.13. The principle of equivalence\n	6.14. Principle of general covariance\n	6.15. Principle of consistency\n	6.16. Space-time\n	6.17. Geodesics in space time\n	6.18. Geodesics on Earth\n	6.19. Geodesic deviation\n	6.20. The Energy-momentum tensor\n	6.21. Dust\n	6.22. Perfect fluids\n	6.23. Covariant divergence of Tμv\n	6.24. Einstein field equations basics\n	6.25. More about Einstein field equations\n	6.26. Derivations of the Einstein field equation\n	6.27. Conclusion\nResources used and acknowledgments\nAbout Author