Week 1: Review of Newton’s non-relativistic theory of gravitation: Inverse square law, the notion of gravitation field intensity vector, Newton’s scalar potential, the Poisson equation for Newton’s gravity, Uniqueness, mean value, Earnshaw's theorems, and Multipole expansion of the Scalar potential. Principles of Equivalence of Galileo and Newton, Tidal Forces on extended objects due to gravity.
Week 2: Review of Special Relativity: Inertial frames, Lorentz/Poincaré transformations, Invariant spacetime interval (Minkowski metric), Minkowski spacetime, timelike, spacelike, and lightlike separated events, lightcones, Lorentz group & its generators.
Week 3: Review of Special Relativity (Continued): Relativistic index notation (4-vector notation), Scalars, Vectors, Tensors, etc., Invariant tensors, Differential Forms and Exterior Calculus.
Week 4: Special relativistic formulations of point particles (free) and Newton’s laws of motion, Covariant equations for fluid mechanics.
Week 5: Fluid mechanics (Continued): Stress-Energy-Momentum Tensor, fluid mechanics, Maxwell theory, Alternative derivation of Maxwell’s equations (in terms of potentials) purely from Lorentz symmetry and conservation of charge.
Week 6: Lorentz covariant equations for gravity ala Maxwell’s equations: Fierz-Pauli field (symmetric tensor potential) and Fierz-Pauli equation, Gauge symmetry of Fierz-Pauli field. Coupling to matter: From Minkowski metric to curved metric, gauge invariance as general coordinate transformation symmetry.
Week 7: Mathematical Preliminaries I: Elements of point set topology: topological spaces, homeomorphisms, Topological manifolds, Differentiable manifolds, Differential geometry: Charts & Atlases, Tangent space, cotangent space, functions (tensor fields), Pseudotensors and curves on manifolds, smooth maps, pushforward and pullbacks, flows, Lie derivatives.
Week 8: Mathematical Preliminaries II: Riemannian manifolds: metric and geodesic equation, Parallel transport and affine connections, Levi-Civita connection (Christoffel symbols), Covariant derivatives, Killing vector and Conservation laws, Geodesic deviation equation & the Riemann Curvature tensor, Ricci tensor, Ricci scalar and the Einstein field equation, Properties of the Riemann tensor, Newtonian limit.
Week 9: Linear approximation of Einstein field equations: Recovering Fierz-Pauli equation, Lorenz gauge, gravitational waves: polarization and detection, and gravitational radiation.
Week 10: Non-linear vacuum solution: Deriving the Schwarzschild Solution for metric outside a spherical mass distribution, Uniqueness theorem, Thin shell, and Newton’s theorem. Coordinate and curvature singularities.
Week 11: Timelike and lightlike geodesics in the Schwarzschild spacetime: ISCO, Photon Sphere, Precession of the perihelion of planets, Deflection of light, Gravitational redshifting and time-dilation, Shapiro’s radar-echo delay.
Week 12: Homogeneous & Isotropic Cosmological Models: FLRW universes, Friedmann and Acceleration Equation, Cosmological constant, De Sitter universe, Hubble expansion and galactic redshifts. The Hot Big-Bang: From Planck era to Baryogenesis, The Cosmic microwave background, Dark Matter and Dark energy, Origin of Structure.
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