| LECTURE | SUMMARY |
|---|---|
| R1 | All pages of Relativity by Einstein |
| R2 | All pages of The Physical Foundations of General Relativity by D W Sciama |
| R3 | GRFN, pp. 1 - 6; GRFN Section 1.5 |
| R4 | GRFN, Where are the bases? |
| R5 | GRFN, Appendix B, The Chinese connection. Study Eqn. (B.1). Also read Appendix B.3 |
| R6 | GRFN, Appendix A. You may avoid the discussion on Maxwell's equations and their tensor character. |
| R7 | GRFN, Newtonian gravitational potential, geodesics, and rotating frames |
| R8 | GRFN, Chapter 4. Read carefully these pages. You should understand this material well |
| C1 | Galilean principle of relativity | Arguments for strong gravity and for General Relativity | Practical uses of STR & GR | GPS | Strong gravity & the universe & Newtonian considerations | Postulates of STR | Maxwell's Equations and the constancy of c |
| C2 | Proper & coordinate time | time dilation | length contraction | Spacetime diagrams | Mass of a particle | More about the photon | Mass-energy equivalence | Spacetime metric |
| C3 | Spacetime metric | Einstein summation convention | Metric tensor of STR | Geodesics in the gravitational field | Equivalence Principle | Time is curved and its curvature as Newtonian gravity |
| 1 | Introduction to GR; Inertial Mass = Gravitational Mass; March's Principle; Coordinates in Euclidean Space; Natural Basis and Basis at P; Examples using Spherical Coordinates |
| 2 | Dual basis; Suffix notation; Einstein summation convention; Contravariant vectors; Covariant vectors; Dot product of two vectors; Matrix notations |
| 3 | Tangents and gradients, Length of a curve; Line element; Coordinate transformations in Euclidean space; Metric tensor |
| 4 | Kronecker tensor; Jacobian matrix of coordinate transform; 2D surfaces in Euclidean space; Vector fields on curved spaces; Metric properties on 2D surfaces; Euclidean verses Curved space; Manifolds; General covariant and contravariant vectors |
| 5 | Tensor fields on manifolds; Definition of a tensor of type (r + s); Transformations of various tensors; Multiplication of tensors: Outer and Inner products; Contraction of tensors; Symmetric and skew-symmetric tensors; Associated tensors; Metric properties; Riemannian manifolds; Pseudo-Riemannian manifolds; Length of vectors; Null vectors; Angle between vectors; Length of curves; Line elements of manifolds; Spacetime of GR; Tiemlike, Spacelike, and Null vectors |
| 6 | Pseudo-Riemannian manifold of GR; Straight lines of flat space; Geodesics; Global view of spacetime; Geodesic deviation; Nature of physical laws of SR and GR; Massive bodies and curvature of spacetime; How gravity in Newtonian theory arises; Equation of geodesics; Calculation of the connection coefficients (Christoffel symbols of the second kind) |
| 7 | Affine parameters; Affinely parameterized geodesics in N-dimensional Riemannian or pseudo-Riemannian manifolds; Parallel transport of vectors along curves; Path dependence of vectors parallely transported along curves on manifolds; The meaning of connection coefficients; Transformation of Christoffel symbols of the first and the second kinds |
| 8 | Conceptual basis of parallel transport of vectors along curves on manifolds; Derivative of a tensor; Absolute derivative of a tensor; Absolute derivative of contravariant and covariant vectors; Absolute derivative for higher order tensors; Covariant derivative and its tensor character; Covariant derivative of higher order tensors |
| 9 | Covariant derivative of the metric tensor; Geodesic coordinates; Spacetimes of General Relativity and Special Relativity; Postulates of STR; Invariant intervals; Metric tensor of SR; Proper time interval |
| 10 | World line; Derivation of Lorentz transformations; Homogeneous and inhomogeneous Lorentz transformations; Covariant metric tensor of SR; Tensors in SR; Limit of SR when v << c; Time dilation; Simultaneity; Length contraction |
| 11 | Spacetime diagrams; World velocity; 4-momentum; 4-force; Spacetime of GR; Geodesic coordinates at P; Metric tensor around P; Curved spacetime of GR; Flat spacetime of STR; Generalization of 4-vectors in SR to those in GR; Motion of free particles and of photons in GR; Comparison between Newtonian mechanics and GR |
| 12 | Equation of the motion of a slowly moving particle in a quasi-static gravitational field; Derivation of the same using "t = coordinate time" as a parameter; Consideration of a reference frame that is not rotating; Deduction of the zero-zero component of the metric tensor by going to the Newtonian limit; Discussion of imaginary Coriolis, Centrifugal, and Gravitational forces; Their connection to the connection coefficients and hence to the coordinate systems |
| 13 | Field equation in Newtonian mechanics; Momentarily Comoving Reference Frame (MCRF); Perfect fluids in STR; Energy-Momentum-Stress Tensor for a perfect fluid in STR; Divergence of tensors; Divergence of Energy-Momentum Tensor; Relativistic Equation of Motion; Relativistic Continuity Equation |
| 14 | Generalization of the energy-momentum tensor to GR; Its divergence; Curvature tensor; Riemann tensor and its symmetry properties; Cyclic and Bianchi identities; Einstein's tensor and its divergence |
| 15 | Curvature and parallel transport; Geodesic deviation; Einstein's field equations and their weak behavior at the weak field limit; Field equations for the empty spacetime |
| 16 | Fundamental assumptions of the Schwarzschild solution; Derivation of the Schwarzschild metric; Solutions of the field equations in empty spacetime and the Schwarzschild spacetime; Schwarzschild radius and black holes; Nature of the radial coordinate "r" and its bounds |
| 17 | Proper distance and proper length in Schwarzschild spacetime; Clocks in this manifold; Radar sounding (4th Test of GR); ie, how radar pulses bounced off of Mercury arrive on Earth with a time delay (measured in relation to time interval in flat spacetime) |
| 18 | Gravitational redshift; Derivation of equations of motion for a particle moving in the Schwarzschild spacetime |
| 19 | Equations of motion of a particle moving in the Schwarzschild spacetime; Comparison with results of classical mechanics; Vertical free-fall; Vertical free-fall to r = 2 m; Coordinate and proper time for the free-fall; Coordinate speed of the free-fall |
| 20 | Circular motion; Proper period of circular motion measure by an observer at r; Proper period measured by an observer on the particle; Possibility of photons circling at r = 3m |
| 21 | Paths of photons in the Schwarzschild geometry; General equation of motion; Possibility of photons circling at r = 3m; Detection of light rays emanating from a particle falling to r = 2 m |
| 22 | General theory of perihelion advance; Calculation of perihelion advance for the planet Mercury; General theory of bending of light near a massive object; Light bending near the limb of the sun |
| 23 | Theory of the geodesic effect; Rotation of the spatial part of a vector in the negative phi direction; Observation of the geodesic effect via orbiting gyroscopes in spaceships orbiting Earth |
| 24 | Eddington-Finkelstein coordinates and black holes; Geodesics and null geodesics in v - r plane; Schwarzschild radius; Objects falling to black holes; Light emitted by falling objects; Proper time to fall |
| 25 | Instability of Newtonian universe; Fundamental observations of the universe; Cosmological principle; Scale factor R(t); cosmic time, t; Robertson-Walker line element; Perfect fluid approximation of the universe |
| 26 | Friedmann equation; Different models of the universe; Cosmological redshift; Epoch of the Big Bang |